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Mathematics and Modernism: Jorge Luis Borges, Samuel Beckett

and J.M. Coetzee

Baylee Brits

Submitted in fulfilment of the requirements of the degree of Doctor of Philosophy.

Faculty of Arts, University of New South Wales 31st March, 2015

Word Count: 88,023

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THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: BRITS First name: BAYLEE Other name/s: JAYNE Abbreviation for degree as given in the University calendar: PhD School: SCHOOL OF ARTS AND MEDIA Title: Mathematics and modernism: Jorge Luis Borges, Samuel Beckett and J.M. Coetzee Faculty: ARTS AND SOCIAL SCIENCES Abstract 350 words maximum: (PLEASE TYPE) This thesis contributes to the understanding of the intricate relation between number, narrative and modernity by analysing the connection between the late nineteenth-century revolution in the mathematics of the infinite and selected texts of twentieth century modernist fiction. This ‘revolution’ in the infinite refers to Georg Cantor’s discovery of the ‘transfinite’ numbers, which measure multiple infinities of different ‘sizes.’ I will focus on the significance of number in key works of three twentieth-century writers: Argentine miniaturist Jorge Luis Borges, Irish novelist and playwright Samuel Beckett, and South African novelist J.M. Coetzee. Each of these writers participate in a different instantiation of literary modernism in vastly different regional and linguistic contexts. What unites them is the centrality of number to the literary innovations that shape and define their work. Through an analysis of number and the transfinite in fiction, I will forge a formal and conceptual connection between two domains regularly understood to be antithetical, without allowing one domain to ‘cannibalise’ the other. Studies of mathematics in literature frequently render numbers as metaphors or symbols, which I argue removes from mathematics that disciplinary and representational difference that makes it appealing to modernist project of literary supervenience. Additionally, this thesis makes a contribution to the relation between literary form, most importantly allegorical form, and the conceptual, textual and epistemological status of the numeral. In these texts, at the end of the allegorical ‘rainbow’, we find not another text or transcendental meaning, but a number. This understanding of numerical allegory allows for an account of the way that numerical form undergirds narrative perspective, description and the undescribable, and the relationship between the efficacy of the symbol and the fictional world. Formally, this thesis maps the reciprocity between mathematics and literature along the lines of the lemniscate. This strategy maintains the autonomy of the tropological and semiotic domain of literature, and the presentational domain of mathematics, but allows the two domains meet at a certain point. This point is the ‘transfinite.’ Where this point touches the mathematical domain, it takes the form of Cantor’s transfinite numbers, where it faces the literary domain, it constitutes a type of formal allegory, which I will call ‘transfinite allegory.’ Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). .................................................................. ........................................................... .......................... ... ... Date Signature Witness The University recognises that there may be exceptional circumstances requiring restrictions on copying or conditions on use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of restriction may be considered in exceptional circumstances and require the approval of the Dean of Graduate Research.

FOR OFFICE USE ONLY

Date of completion of requirements for Award: THIS SHEET IS TO BE GLUED TO THE INSIDE FRONT COVER OF THE THESIS

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ORIGINALITY STATEMENT

‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’

Signed ............................................................. Date .................................................................

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Since Copernicus, man has been rolling from the center towards X. Friedrich Nietzsche

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TABLE OF CONTENTS

0. Introduction 6 0.1 Critical Precedents 18 0.2 Contemporary Methods 26 0.3 A Generic Literature 34

1. Transfinite Allegory 39 1.1 The Missed Encounter 40 1.2 Cantor’s Transfinite 43 1.3 Mallarmé and Meillassoux: Fixing the Infinite 57 1.4 Allegory and Enigma 71

2. Jorge Luis Borges and the Measure of Prose 76 2.1 Supplanting the Symbol for the ‘Thing Itself’: Borges’ Ultraist Beginnings 82 2.2 ‘Funes, His Memory’: A Transfinite Technogenesis of Perception 89 2.3 ‘The Library of Babel’ and ‘The Book of Sand’ 103 2.4 ‘The Lottery of Babylon’ 119 2.5 What is a Transfinite Allegory? 125

3.1 Continuous Deformation in Samuel Beckett’s Novels 133 3.11 A Mania for Symmetry: Molloy and the Deformation of Language 138 3.12 Permutation and Division in Watt 156 3.2 Mathematics Contra Empiricism: The Radicalisation of Naturalism in Samuel Beckett’s Late Work 178 3.21 All Strange Away and Imagination Dead Imagine: Imagination by Numbers 181 3.22 Quad: Counting Back and Forwards in Time 199 3.23 Worstward Ho and the Deictic Count 213

4. J.M. Coetzee and the Name of the Number 220 4.1 ‘Literature in the Lap of Mathematics:’ The Quantification of Style 223 4.2 In the Heart of the Country: Freedom and Equality 233 4.3 The Childhood of Jesus and Mathematical Nominalism 252 4.4 Counting as One, Rather than Counting to One 270

Conclusion: Actual Literary Infinities 276 Works Cited 283 Acknowledgements 297

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0 INTRODUCTION

This dissertation contributes to the understanding of the intricate relation between

number, narrative and modernity. It analyses the connection between the late

nineteenth-century revolution in the mathematics of the infinite and selected modernist

narratives. This revolution in mathematics came about through Georg Cantor’s

development of ‘transfinite’ numbers, which measure multiple ‘actual’ infinities of

different sizes. Hitherto, it seemed ludicrous to measure or speak of the relative size of

different infinities. Cantor brought infinity, the antithesis of human knowledge, into the

rational sphere through the transfinites. The notation for the transfinite is the Hebrew

letter aleph (א). The transfinite numbers measure the cardinality of infinite sets,1 that is,

the number of elements in the infinite set of, for instance, natural numbers.2 The

discovery of a number that measures the infinity of the numbers constitutes a transgressive

reformulation of our ideas about the ‘actual,’ the infinite, and what is termed the

‘generic’: a word used to describe something that may have measure but not determination.3

The very notion of a measure of the infinite that has certain finite capacities (one can

add or multiply transfinite numbers) constitutes a radical refutation of the laws of

1 Cardinality is the measure of objects in a set, whereas ordinality is the order type of the set. 2 A set is a collection of distinguishable objects. Set theory arose from Cantor’s work on cardinality: the

number of objects in an infinite set. Cantor managed to distinguish the cardinality of two infinite sets: the

set of real numbers, and the set of natural numbers. Cantor proved that the set of real numbers could not

be counted, but that the set of natural numbers could (and, as such, that one could establish a cardinality

for the set of natural numbers). Whereas the natural numbers have a cardinality of 0א the set of real

numbers is greater than this set. Here lies the radical proposition of infinite sets of different ‘sizes.’ 3 Here I am glossing Peter Hallward’s excellent definition. See: Peter Hallward, ‘Generic Sovereignty:

The Philosophy of Alain Badiou,’ Angelaki: Journal of the Theoretical Humanities 3, no. 3 (2008), 87.

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number and infinity, which demands an interrogation of the very foundations of

mathematics. Cantor’s ‘transfinite’ revolution is a remarkable moment of

epistemological reaching, one that I hope to show held signal attractions for modernist

writers who were grappling with the limits of expression and striving, to use Samuel

Beckett’s words, to ‘eff the ineffable.’4

Joseph Warren Dauben, the preeminent Anglophone scholar of Cantor’s legacy,

compares the advent of transfinite set theory to the Renaissance ‘transition in advancing

from the closed world of Aristotle’s universe to the infinite world of the post-Copernican

era’ which was ‘in many respects a painful and traumatic one, but profound in its

implications for the subsequent history of Western thought.’ 5 Cantor’s transfinite

numbers would provoke just such a shift in mathematics: a reorientation of the ground

upon which mathematics is founded. The connection between this shift in the

foundations of mathematics and the tremors that reverberated through innovative

literary work in the subsequent decades has not hitherto been widely acknowledged by

scholars of literary modernism.

In this study I will focus on the significance of number, the transfinite and the

infinite in key works of three twentieth-century writers: Argentine miniaturist Jorge Luis

Borges, Irish novelist and playwright Samuel Beckett, and South African novelist J.M.

Coetzee. Each of these writers participates in a different instantiation of literary

modernism in vastly different regional and linguistic contexts. They represent various

tendencies of modernism, including post-colonial modernism and ‘late’ modernism,

4 The line from Watt is as follows: ‘[...] perhaps also because of what we know partakes in no small

measure of the nature of what has so happily been called the unutterable or ineffable, so that any attempt

to utter or eff it is doomed to fail, doomed, doomed to fail.’ See: Samuel Beckett, Watt, ed. C.J. Ackerley

(London: Faber and Faber, 2009), 52-53. 5 Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, New Jersey:

Princeton University Press, 1990), 4.

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their lives spanning the nineteenth to the twenty first centuries, their careers spanning

Spanish modernismo, the Parisian avant-garde of Jolas and Joyce, and North American

post-structuralism.6 What unites these three writers is the centrality of number and, in

particular, the idea of the transfinite to the literary innovations that shape and define

their work. It is number that will mediate, for each of these writers, one of the defining

constraints of literary modernism. This constraint is the imbrication in a formal

paradox. Modernist art veers between the hubristic ambition for the representative

capacities of art and a lack of faith in precisely the elements and techniques from which

that art is formed. The ambition for total expression coupled with a critical lack of faith

in the substances available for expression is perhaps even the fundamental artistic

contradiction in modernism. Into this gap comes the truth affording capacity of number.

Each of these writers comes to literature wrestling with the limits and limitations

of language. Jorge Luis Borges, in 1921, produced an ‘ultraist manifesto’ with a Spanish

avant-garde group that called for a liberation of the symbol from ornamental language

in favour of ‘primordial metaphor.’7 Samuel Beckett famously elaborates, in a 1937

letter to Axel Kaun, his exasperation with his medium: ‘Grammar and Style!’ Beckett

writes, ‘To me they seem to have become as irrelevant as a Biedemeier bathing suit or

6 I am referring, here, to Tyrus Miller’s concept of ‘late modernism.’ For Miller late modernism is ‘a

distinctly self-conscious manifestation of the aging and decline of modernism, in both its institutional and

ideological dimensions. More surprising, however, such writing also strongly anticipates future

developments, so that without forcing, it might easily fit into a narrative of emergent postmodernism.’

See: Tyrus Miller, Late Modernism: Politics, Fiction, and the Arts Between the World Wars (Berkeley and Los

Angeles: University of California Press, 1999), 7. 7 See Beatriz Sarlo’s discussion and translation (the most authoritative translation that I have come across)

of the ultraist manifesto in her book: Beatriz Sarlo, Jorge Luis Borges: A Writer on the Edge, ed. John King, 2nd

ed. (London and New York: Verso, 2006), 113.

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the imperturbability of a gentleman.’ 8 J.M. Coetzee condemned the effects of

conventional literary voice for creating stultifying identities for readers and writers in a

1984 essay entitled ‘A Note on Writing’: ‘A is written by X’ comes to function as ‘a

linguistic metaphor for a particular kind of writing [...] where the machine runs the

operator.’9 In each case, these authors find the received structures of language and

expression to be hindrances to their artistic task. The argument that I develop over the

following chapters claims that these three writers took number to be an instance of

presentation rather than representation, valuable precisely for its mathematical identity

and difference to natural language. It is the status of the numeral as a ‘presentational

mark’ that makes it key to resolving limitations inherited from traditional literary modes

of style and address. These three writers worked with number in fiction without co-

opting it into a maelstrom of literary fantasies of the ‘meaning’ of the numeral (be it

alienating, unlucky, perpetual, and so on). It is this formal engagement with number

that signals an advent of a novel relationship between prose work and mathematics.

As the choice of authors for this dissertation indicates, this is not a periodising

study. I am not seeking to theorise or describe the approach to number of a broad

historical movement (using, say, one of the usual historical boundaries of modernism,

1914-1939). Rather, my use of the terms ‘modernism’ and ‘modernist’ tend towards an

inductive rather than deductive method. While I begin with some general remarks, these

are in order to situate my selected authors rather than to establish a theory of

modernism and number of which they are instances. This is not to say that fruitful

comparisons and conclusions are not available, but they are established from the

8Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929-1940, ed. George Craig and Dan Gunn

(Cambridge and New York: Cambridge University Press, 2009), 518. 9 J.M. Coetzee, ‘A Note on Writing,’ in Doubling the Point: Essays and Interviews, ed. David Attwell,

(Cambridge: Harvard University Press, 1992), 95.

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‘ground up,’ as it were. This is a selective foray as opposed to a comprehensive

cartography. It is also a prolegomena to a different sort of literary criticism, one that

tries to trace the contact points between two discrete spheres of knowing: narrative and

number.

In the last fifteen years, there has been a shift in the way that the term

‘modernism’ is used. The term is no longer bound to describing a period or delimited to

a few decades at the beginning of the twentieth century, and it can encompass various

instantiations.10 My three exemplars of modernism bear family resemblances: they

demonstrate, in different ways, a shared suspicion of the potency of expressive language;

a consequent desire to expand or explode the linguistic capacity for expression; a self-

conscious refutation or reworking of dominant Anglophone or continental poetic forms

or prose genres, most importantly poetic symbolism and the tradition of the naturalist

novel; a scepticism towards perception and its purchase on a ‘reality’ or a ‘real’; and an

overt engagement with technological innovation, particularly new technologies of

perception such as photography, film, and the personal computer, insofar as these

10 This shift is well summed up in the editorial introduction to a recent special issue of Filozofski Vestnik on

‘modernism revisited.’ Tyrus Miller and Aleš Erjavec not that ‘Since the 1970s and 1980s when the

concept of postmodernism was advanced and hotly debated, the concept of ‘modernism’ was not simply

superseded, but also itself became a major object of criticism, questioning, negation and reinscription’ (9).

The implication, here, is that modernism has breached its historical bounds and constitutes an aesthetic

endeavour that has maintained its rigour and potency at various moments over the twentieth-century, and

now again holds relevance. This is ratified by Julian Murphet. In his introduction to the inaugural issue of

Affirmations: Of the Modern Murphet writes that ‘Modernism and modernity have now completed the second

decade of a remarkable, some would have said unpredictable, revival of intellectual fortunes. [...] In any

event, this ‘Modernism’ is different in kind from the variable emphases on organic form, Verfremdungseffecte,

abstraction, and existentiality of an earlier moment’ (1-2). Modernism is protean without being relative,

attached to a certain historical movement and locale but not bounded by it, diverse without being

unrecognisable. See Tyrus Miller and Aleš Erjavec, ‘Editorial,’ Filozofski Vestnik 35, no. 2 (2014), and

Julian Murphet, ‘Introduction: On the Market and Uneven Development,’ Affirmations: Of the Modern 1,

no. 1 (2013).

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technologies dissolve the ‘aura’ that once demarcated art from other aesthetic forms.

The difference of their historical contexts, cultural legacies and participation in national

or cosmopolitan literatures provides, here, a diversity that is counterpoint to a shared

formal procedure that revolves around number and the transfinite.

The literary context for this operation must be situated within a broader

centrality of number for biological and political life. There are two overarching

narratives that focalise such a shift: the narrative of the Enlightenment dissolution of

quasi-ecclesiastical hierarchies of being, and the ascendency of a ‘regime’ of number

through the biopolitical organisation of life in the twentieth century. Enlightenment

science and philosophy was a process that severed the ‘great chain of being’ by virtue of

eliminating a stable hierarchy of existence, posited in some absolute. The concept of the

‘great chain of being’ or scala naturae was originally developed by Aristotle (though

analogous concepts were developed by other Ancient Greek philosophers, notably Plato)

as a mode of hierarchic empirical classification of animals and humans. In Ray

Brassier’s words, it was the Enlightenment ‘disenchantment of the world’ and the

‘coruscating potency of reason’ that ‘shattered the ‘great chain of being.’11 However,

rather than a deformation of this sequence of existence, we see instead a replacement of

this version of order and hierarchy with another hegemonic narrative: that of

quantification.

The Enlightenment ‘rupture’ of a sequence or narrative of existence such as the

‘great chain of being’ did not give rise to new non-linear or non-totalising modes of

thought and organisation. Instead, the Enlightenment ushered in an era of

quantification. Divine or transcendental orders of being were replaced by a new, secular

11 Ray Brassier, Nihil Unbound: Enlightenment and Extinction (Hampshire and New York: Palgrave MacMillan,

2007), xi.

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absolute: the number line. In the words of J.L. Heilbron, editor of The Quantifying Spirit in

the Eighteenth Century, ‘the later 18th century saw a rapid increase in the range and

intensity of application of mathematical methods.’12 The proliferation of mathematics

across the sciences of the eighteenth century saw the establishment of a single marker of

intellectual vigour: the numerical measure of the object, and, bound up with this, the

quantification of those objects offering themselves for scientific measure.13 This would

hold for population as much as for plants, animals, and the heavenly bodies. It was this

same predominance of quantification that would produce ‘reification,’ the concept

developed by Marx to capture the process by which human attributes or acts become

identified and evaluated on the same basis by which things and commodities are

identified and evaluated. Reification is the extension of a quantificatory method and

spirit into human life. It is the process by which the ‘spirit of quantification’ comes to be

12J.L. Heilbron, ‘Introductory Essay,’ in The Quantifying Spirit in the Eighteenth Century, ed. Tore Frängsmyr,

J.L. Heilbron, and Robin E. Rider (Berkeley and Los Angeles: University of California Press, 1990), 2. 13 Gunnar Broberg’s study of the significance of quantification to the natural science and history of the

eighteenth-century clearly establishes the dual nature of quantification for science: the potential objects of

study proliferate with the new mechanisms of measure (what Broberg calls ‘the challenge of plenitude’) at

the same time that the method of study is itself an act of quantification. Broberg writes: ‘The word

‘quantitative’ applies to natural history during the second half of the eighteenth-century in two distinct but

related ways: as characteristic of the object and of the method of study. As for objects, the sheer number

of known and estimated forms forced new approaches to the storage and retrieval of information; as for

method, these new approaches were instrumentalist and, in the dominant system of Linnæus,

mathematical. These features—the overwhelming flow of information and the determination to inventory

and survey it for useful purposes—characterize much of the learned activity of the late Enlightenment’

(45). The clearest example of this is found in the quantification of animal species: the number of

discoveries of new species in the eighteenth-century triggered many estimates of the total number of

species on the planet. As the century wore on, the estimates increased exponentially. See Gunnar

Broberg, ‘The Broken Circle,’ in The Quantifying Spirit in the Eighteenth Century, ed. J.L. Heilbron, Tore

Frängsmyr, and Robin E. Rider (Berkeley and Los Angeles: University of California Press, 1990).

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radicalised in the twentieth century, producing what French philosopher Alain Badiou

calls the ‘despotism of number.’14

For Badiou, the ‘singular condition’ of thought in the twentieth century will be

number:

Having interred the pathologies of an unbridled will, the happy correlation of a

Market without restrictions and a Democracy without shores would finally have

established that the meaning of the century lies in pacification or in the wisdom

of mediocrity. The century would thereby express the victory of the economy, in

all senses of the term: the victory of Capital...15

The victory of the economy is consolidated by the biopolitical quantification of life and

norms constructed through numerical parameters.16 The ascendency of number in

social and political life translates, for Badiou, into a century that sustains a regime of

number whilst also, crucially, forgetting the mathematical proofs of the continuum, the

transfinite, and the very basis for launching what Badiou would refer to as ‘the count.’

In Badiou’s summation, number now ‘governs our conception of the political, with the

14 Alain Badiou, Number and Numbers, trans. Robin Mackay (Cambridge: Polity Press, 2008), 1. 15 Alain Badiou, The Century, trans. Alberto Toscano (Cambridge: Polity Press, 2007), 2-3. 16 Although Badiou does not directly situate his claims in terms of biopolitical organisation, though in this

particular tirade we can find echoes of Michel Foucault’s reading of the braiding of quantification, capital

and control in the twentieth century. See, for instance, Foucault’s comments on quantification in The

Birth of Biopolitics: ‘This means that it must be possible to analyse the simple time parents spend feeding

their children, or giving them affection as investment which can form human capital. [...] This means that

we thus arrive at a whole environmental analysis, as the Americans say, of the child’s life which it will be

possible to calculate, and to a certain extent quantify, or at any rate measure, in terms of the possibilities

of investment in human capital.’ See: Michel Foucault, The Birth of Biopolitics: Lectures at the Collége de France

1978-1979, ed. Michel Senellart, trans. Graham Burchell (Hampshire and New York: Palgrave

MacMillan, 2008), 229-230.

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currency [...] of the majority’; it ‘governs the quasi-totality of the ‘human sciences’’ as

well as the economy.17 Above all, for Badiou ‘Number informs our souls. What is it to

exist, if not to give a favourable account of oneself?’18 What Badiou registers, here, is a

complete ‘naturalisation’ of the numeric identities of life and social organisation. For

Badiou, this century relies upon an obfuscation of the relation of number to ontology

(or, if we accept Badiou’s terms, mathematics as ontology) that at once creates a tyranny

of number without retaining anything of the truth of number, the truth of mathematics.

In Badiou’s thought, the twentieth century will see a substitution of politics with

management, love with sexuality, and mathematics with technologies that abet

bureaucratization. This is the age where the other life of number, in pure mathematics

and in particular in set theory, is forgotten, and number exists only to produce the

statistic.

What is the relationship, then, between Cantor’s ‘actual infinities’ and the

Enlightenment and twentieth century passions for quantification and the reification,

indeed, the numerical ‘despotism’ that Badiou refers to? Is Cantor’s ‘actualisation’ of

that which is deemed beyond human science and thought another instance of

quantification, and, if we follow Badiou, a ‘forgetting’ of the foundations of

mathematics? Cantor’s transfinite numbers occupy a liminal position as regards

quantification, seeming to embrace it on one level, and utterly reject it on another.

Cantor’s generation of a numerical infinite that is not necessarily tied to a withdrawn

ideal or divinity (which I will explain at length in the following chapter) at once renders

an idea that was previously ‘transcendental’ or simply elusive ‘actual,’ a task that may

seem, on surface level, to be radically positivist or naively quantificatory. On the one

17 Badiou, Number and Numbers, 2-3. 18 Ibid. 2-3.

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hand, Cantor’s ‘actual infinities’ ratify the Lockean privileging of experience as a source

of knowledge. It is precisely this unity of the actual (what we can see and work with: a

concrete number form) with a revolution in noetic and metaphysical sequence that we

find echoed in Cantor’s transfinites. And yet, simultaneously, Cantor’s work would

disturb the central Enlightenment connection between the actual and a consistent,

continuous number line that has the capacity to measure any of science’s objects. It is

this affirmation of ‘actual’, material measures for that which exceeds and confounds the

finitude of experience that presents a novel mathematical discovery pertinent to the

form and possibilities entailed in modernist literary infinities.

This dissertation investigates the formal, theoretical and artistic possibilities of an

actual infinity as it is manifest in Cantor’s transfinite numbers. Through an analysis of

number and the transfinite in fiction, I will forge a formal and conceptual connection

between two domains regularly understood to be antithetical, without succumbing to

the pitfall peculiar to this type of analysis: allowing one domain to co-opt the other. The

mathematical or numerical literary criticism that does exist has often been characterised

either by a reduction of mathematical form to simply a technique of abstraction, or a co-

opting of mathematics as concept or analogy, negating the unmitigated difference

between writing as presentation and writing as representation.

For many literary critics, mathematical writing and signs occupy a domain that

is irreconcilable with the subjective and phenomenal world of art, except when

commuted to metaphor. This circumvents both the textual existence of the numeral, as

well as its distance from the identity of signs in natural language. Number is an instance

of pure presentation distinct from representation, which nonetheless remains a mark or

a ‘symbol.’ It is this dual status of the mathematical mark (existing as a ‘presentation’ yet

occupying a textual domain) that is comprehended and exploited in art, and must be

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realised by a criticism capable of retaining the same autonomy for mathematics. The

near-ideogrammatic form of ∞ (the lemniscate; the sign for infinity) captures this

inconsistent and contingent bridge between visual-semiotic implication and numerical

referent, conjoining both quantity and quality in the signifier for the absolute infinite.

The numeral remains a mark and susceptible to symbolic attribution yet properly –

mathematically – must exist outside signification. There is a similar bridge between the

experience of the numeral and its capacity for abstraction: numerical identity is at once

both quotidian – governing the minutiae of everyday life � and transcendental. We use

numbers all the time, without there being any clear signified origin for the numeral. We

encounter number every day, define our actions and identities by number, and yet it

remains ‘transcendental’ in the sense that it occupies precisely the definition of the

‘presupposed’: it exceeds the limits of language because it has no referent other than

itself, and is fully and immanently available at all times, and for this very reason is rarely

interrogated itself. What is a number? How do we establish the validity of the numerical

stakes in continuity, consistency, and the role of the number as the ultimate ‘unit’? Such

questions pinpoint a problem that will in the twentieth century become literary as well

as mathematical, social as well as scientific.

Formally, this thesis maps the reciprocity between mathematics and literature

along the lines of the lemniscate. This mark provides an image of the reciprocal

structure that I map here, without properly conforming either to a presentational or a

representational scheme. The lemniscate maintains the autonomy of the tropological

and semiotic domain of literature on one of its loops, and the presentational domain of

mathematics on another, but it also allows the two domains meet at a certain point. This

point is the ‘transfinite.’ Where this point touches the mathematical domain, it takes the

form of Cantor’s transfinite numbers; where it faces the literary domain, it constitutes a

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type of formal allegory, which I will call ‘transfinite allegory.’ But this visual model

should not be interpreted to imply a literature of the ‘infinite,’ here. Cantorian set

theory and proofs of the actual infinite do not lead to a literature of the infinite, just as

Cantor’s ‘actual infinite’ is absolutely not collapsible into a general ‘infinite’ or ‘infinity.’

We might find a literature of the infinite precisely where Andrew Gibson locates the

most vehement rejection of actual infinity: in Romanticism, where there are many

analogies for an Absolute or an infinite, but where the infinite is ultimately immaterial,

absent, or intangible referent.19 In his theorisation of ‘the pathos of intermittency’ in

Beckett and Badiou, Gibson summarises the distinction between a Romantic and a

modern infinity. The key discrepancy between the two essentially resides in a rejection

of mathematics. In Romanticism, ‘We are separated from infinity by an uncrossable

frontier’ and that frontier is the ‘boundless exteriority, an openness without end.’20

Indeed, in Romanticism the infinite is not just intangible but too often it stands in for or is

the intangible. I will argue that in the instances modernist fiction at issue here we come

to see a transfinite rather than infinite operation, an actual literary infinity rather than

simply a literary infinity. This is what allows an embrace again of the link between

mathematics and literature, and what distinguishes a modernist reciprocity between

prose and mathematics from a romantic rejection of a mathematical infinite.21

19 Andrew Gibson, Beckett and Badiou: The Pathos of Intermittency (Oxford and New York: Oxford University

Press, 2006), 6-7, 51-2. 20 Ibid. 6. 21 It may seem that the charge of a ‘rejection of the mathematical infinite’ is unfair, here, in particular

because Cantor’s proofs most certainly come after the bulk of what can be considered literary

romanticism (and the proofs were hardly accessible to a Anglophone literary community), but I do not

think that this charge has to imply a failure to anticipate Cantor. The issue of the truth or falsity of infinity

is one that is hardly new and, indeed, has persisted along the lines of an Aristotelian versus Platonic view

of the world: a divergence that I will articulate, in my chapter on Samuel Beckett, as a distinction between

viewing multiplicity as genera or as generic. So this avoidance of the actual infinite in favour of the ideal

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*

0.1 CRITICAL PRECEDENTS

There is a rich and extensive engagement in nineteenth-century literature with the

empirical sciences, despite the fact that this century also saw the consolidation and

specialisation of scholarly disciplines. Yet pure mathematics, as a speculative or formal

rather than empirical science did not have the same presence in literature or literary

studies as biology, for instance.22 This is because literature and mathematics are at once

infinite is not anachronistic, and the embrace of an infinity that is ideal, and ephemeral, can be seen as an

opposition to Cantorian thought without any direct engagement between these philosophies and Cantor’s

mathematics. 22 The novels written by the ‘realists’ and the ‘naturalists’ of the nineteenth-century (as well as into the

twentieth-century) are the most evident testimony to this. Laura Otis, in her preface to her anthology,

Literature and Science in the nineteenth century, explains that it was only in the 1830s that science took on a

definition that separated it from general skill: ‘At an 1833 meeting of the British Association, William

Whewall proposed the term ‘scientist’ for investigators who until then had been known as natural

philosophers. In the nineteenth century, ‘science’ came to signify the study of the natural and physical

world. Until that time it had denoted any sort of knowledge or skill, including the ‘science’ of boxing. In

contrast, the notion of a ‘split’ between literature and science, or a ‘gap’ to be ‘bridged’ between the two,

was never a nineteenth century phenomenon’ (xxv). Despite elaborating a distinction between science and

skill, or science and literature, Otis’s history of the interaction between literature and science does not find

a great separation between the two but rather suggests that science and literature overlapped in terms of

subject matter. Otis justifies the separation of mathematics from science in her anthology on the basis of

the fact that nineteenth-century mathematics, along with technology, would have been considered an ‘art’

not a ‘science’ ‘...indicating the use of practical skill rather than systematic inquiry’ (xxvii). Where

mathematics is included in the anthology, it is done so in the legacy of August Comte and the positivist

circles around Comte, who ‘understood mathematics and physics as the source of rigorous laws and

consequently the foundation on which other disciplines might be based’ (xxvii). Otis thus places

mathematics at the start of her anthology to represent the way in which writers turned from these rigid

confines to richer, non-positivistic sciences dealing with evolution and other empirical findings. See: Laura

Otis, ‘Introduction,’ in Literature and Science in the nineteenth century (Oxford and New York: Oxford

University Press, 2002).

19

inextricably intimate and implacably oppositional: poetry needs its metrics, but the

values we ascribe to literature – humane, affective, figurative, narratological – are far

from the perceived abstractions and bloodless symmetries of mathematics. Unlike, say,

biology, mathematics has traditionally been considered an art, aligned with the poetic or

expressive, and yet the form of writing peculiar to mathematics – pure presentation – is

excluded from what we understand to be relevant to literature or, even, to the exercise

of language. In the work of Borges, Beckett and Coetzee we will see a shift in this

polarity, harnessing rather than negating that distance and embracing those paradoxes

of the number: textual and not semantic, quotidian and radically abstract, the tool of

empiricism and objectivity yet fundamentally grounded in a void or zero, or the

irrational, or the infinite. In other words, in literary modernism we see mathematics

become intimate with prose precisely on the basis of its irreducible difference from

prose, not its relationality.23

It is perhaps this ‘irreducible difference’ between natural language and

mathematical notation that has made mathematical elements or influences a rare topic

for literary critics and theorists. The examples of engagements with mathematics in

fiction or poetry tend to be singular studies that are not interventions in a wider

disciplinary or theoretical constellation. This is, in part, a result of the modern isolation

23 Another juncture at which mathematics resembles or echoes artistic practice is in the controversy

surrounding whether mathematical innovations constitute ‘creations’ or ‘discoveries.’ This debate is

particularly pertinent to Cantor, and Mary Tiles sums up the issue of creation (or in Tiles’ terms,

‘invention’) versus discovery succinctly: ‘Did Cantor discover the rich and strange world of transfinite sets

(which Hilbert was to call Cantor’s Paradise) or did he (with a little help from his friends) create it? Are set

theorists now discovering more about the universe to which Cantor showed them the way, or are they

continuing the creative process?’ Mary Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s

Paradise (Mineola, New York: Dover, 2004), 1.

20

of mathematics from the humanities. In his foreword to Morris Kline’s Mathematics in

Western Culture, Richard Courant notes that

After an unbroken tradition of many centuries, mathematics has ceased to be

generally considered as an integral part of culture in our era of mass education.

The isolation of research scientists, the pitiful scarcity of inspiring teachers, the

host of dull and empty commercial textbooks and the general educational trend

away from intellectual discipline have contributed to the anti-mathematical

fashion in education.24

He may as well have written ‘the anti-mathematical fashion in literary criticism.’ Work

on literature and mathematics is relatively scant in comparison to other disciplinary

interests. The critical studies of literature and mathematics that do exist fall into several

categories essentially based on the way in which they conceive of and elaborate the link

between ‘literature and mathematics.’ There are three broad categories that divide the

existing work: single author studies, studies of ‘compossibility,’ and, finally and most

importantly, studies of ‘literary numbers.’

In the first category, there is a wide ranging, substantial collection of mostly

single author studies that pursue a mathematical influence, conundrum or episode in a

work of literature.25 Each of the chapters that constitute this dissertation benefit from

24 Richard Courant, ‘Foreword,’ in Mathematics in Western Culture, (Oxford and New York: Oxford

University Press, 1953), v. 25 There are few exceptions to the ‘single author’ tendency in critical work bringing literature and

mathematics together. Robert Tubbs’s recent book, Mathematics in Twentieth Century Literature and Art,

examines ‘artists and writers who explicitly employed mathematical ideas to express their aesthetic ideals

or create their works’ as well as ‘artists’ and writers’ use of mathematical images or forms or methods in

the creative processes’ and ‘the use by theoreticians of mathematical concepts to examine those

21

such single author studies. In the first chapter, ‘Transfinite Allegory,’ I will discuss at

some length Quentin Meillassoux’s recent study, The Number and the Siren, which provides

a deciphering – an unorthodox and speculative mathematical reading – of Mallarmé’s

revolutionary poem Un Coup de Dés Jamais N’Abolira Le Hasard.26 In the second chapter, I

refer to mathematician William Goldbloom Bloch’s monograph, The Unimaginable

innovations’ (ix). Tubbs’s work does not investigate any particular artistic goal in employing mathematical

techniques and concepts. Rather, Tubbs see it as ‘only natural that artists, writers and others would

incorporate mathematical thinking into their attempts to express their artistic ideals’ because

‘mathematical thinking was no longer the private domain of mathematicians’ and ‘mathematical ideals’

had ‘entered into the daily discourse of artists and intellectuals’ (xi). This fascinating survey of

mathematics in an eclectic range of art practices thus accepts mathematics as one of many possible

sources of reference and stimulation for artists, without theorising any necessity or link that may coexist

between these. Such an assumption thus naturalises number and avoids any critical reflection on

number’s constitutive relation to literature, identity or society. This study is thus interesting in its

nomination of examples, but its wider literary impact is neutralised by its uncritical relation to the

thematic it approaches. See Robert Tubbs, Mathematics in Twentieth-Century Literature and Art (Baltimore:

Johns Hopkins University Press, 2014).

The other significant multi-author study is Miranda B. Hickman’s The Geometry of Modernism: Vorticist Idiom

in Lewis, Pound, H.D. and Yeats. Hickman’s work provides an analysis of the significance of geometry to

those working in the legacy of Vorticism. Hickman’s focus is not, however, on formal innovation based on

geometries, but on a geometric idiom used to ‘imagine and figure ideal cultural conditions, bodily states,

philosophical attitudes and epistemological methods’ (2). As such, this fascinating study of geometry in

several modernist authors is restricted to the ‘instructive and reformative’ modes of these writers work (2).

Given that this work seeks to study geometric language as opposed to the introduction of geometry into

poetry, it remains of little direct use for the present study. See Miranda B. Hickman, The Geometry of

Modernism: The Vorticist Idiom in Lewis, Pound, H.D., and Yeats (Austin: University of Texas Press, 2005).

Another exception to this is the work of N. Katherine Hayles. Hayles has written wide- ranging,

definitive works on literature and technology, in particular electronic literature and cybernetic theory.

Although not directly associated with mathematics, cybernetic theory is derived from mathematical

innovations in the twentieth-century, most famously Turing’s work on computability. See N. Katherine

Hayles, How We Became Posthuman: Virtual Bodies in Cybernetics, Literature, and Informatics (Chicago and

London: University of Chicago Press, 1999), and N. Katherine Hayles, Electronic Literature: New Horizons for

the Literary (Notre Dame, Indiana: University of Notre Dame Press, 2008).

26 Quentin Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés,’ trans. Robin

Mackay (Falmouth: Urbanomic, 2012).

22

Mathematics of the Library of Babel, which is the most important and wide ranging extended

study of mathematics in Borges’ oeuvre.27 Bloch’s work on mathematics in Borges’

stories pursues not a coherent narrative of mathematics and literary form but rather

embraces and describes the eclectic mathematical forms relevant to Borges’ stories. The

second full-length study of mathematics in Borges’ work takes a similar approach.

Guillermo Martínez’ Borges and Mathematics provides a full index of the range of

mathematical themes in Borges work and, like Bloch’s study, investigates eclectic

manifestations of mathematics in the short stories. The themes that Martinez

investigates include infinity, abstraction and concretion, and artificial intelligence.

Martinez concludes by developing a direct connection between literary form and

mathematics: the short story, for Martinez, is a ‘logical system’; the link between the

fiction and mathematics resides in the function of the author as a ‘manipulator of

systems.’28

There are multiple short studies that have collectively established the centrality

of mathematics in Samuel Beckett’s work. Together, these studies provide the most

important precedent for work on mathematics in Beckett’s oeuvre. Chris Ackerley,

Hugh Culik and Brian Macaskill have written articles that illuminate varied

mathematical elements in Beckett’s work and catalogue mathematical episodes in his

oeuvre; elaborate the literary significance of mathematical modes of negation and

incompleteness; and theorise the connection between literary and mathematical

27 William Goldbloom Bloch, The Unimaginable Mathematics of Borges’ Library of Babel (Oxford: Oxford

University Press, 2008). 28 Guillermo Martinez, Borges and Mathematics, trans. Andrea Labinger (West Lafayette, Indiana: Purdue

University Press, 2012), 69-70.

23

uncertainty, respectively.29 On the work of J.M. Coetzee, Peter Johnston’s important

thesis, ‘Presences of the Infinite: J.M. Coetzee and Mathematics’ is indispensible both

for its archival detail and theoretical innovation. Perhaps the most important aspect of

this work is Johnston’s consideration of Coetzee’s interest in Isaac Newton’s conviction

that the gap between natural language and mathematical language could be bridged.

This sets an important precedent for methodologies for literary work on mathematics

that I hope to contribute to: the most important intersection between mathematics, and

the point at which such an intersection becomes of vital importance to literary studies

more generally, is found in the bearing that mathematical language has on the

expressive, representational and cognitive capacities of natural language and vice versa.

29Macaskill’s essay, ‘The Logic of Coprophilia: Mathematics and Beckett’s Molloy’ analyses ‘an affinity

between Beckett’s style and numerical practice’ but also rejects a ‘naïve belief in mathematical certainty’

(14). Indeed, it is mathematical uncertainty that Macaskill focuses on, looking at Molloy’s ‘logic of [...] being

and the logic of literature which delivers his being’ through Aristotelian logic, Gödel’s incompleteness

theorem and Russell’s paradoxes. Mackaskill produces a masterful analysis of Molloy’s relation to zero, in

particular the ‘zero’ of the anus, the ungenerative orifice that so preoccupies and disturbs Molloy.

Hugh Culik’s ‘Mathematics as Metaphor: Samuel Beckett and the Esthetic of Incompleteness’ has a

similar focus to Macaskill, looking at the necessary incompleteness of ‘descriptive sufficiency’ (132). Culik

works with the ‘Pythagorean struggle with irrational numbers’ (132) and Beckett’s development of

Pythagorean metaphors. Culik focuses largely on the novel Murphy, the mind-body problem, and the

‘Pythagorean ambition to mathematize the world’ (143), in order to elaborate the significance of

mathematical models for the very possibility of Beckett’s art. Culik’s essay is particularly significant, here,

because he connects Beckett’s formal solutions to representational issues with mathematics and – briefly –

his modernist context.

Ackerley’s entry on ‘mathematics’ in the Grove Companion to Samuel Beckett details instances and

engagements with mathematics across the entire oeuvre of Beckett. This brief article is encyclopaedic in

nature and does not posses the scope or intention for detailed analysis. See Hugh Culik, ‘Mathematics as

Metaphor: Samuel Beckett and the Esthetics of Incompleteness,’ Papers on Language and Literature 29, no. 2

(1993): 131–51; Brian Macaskill, ‘The Logic of Coprophilia: Mathematics and Beckett’s ‘Molloy,’’

SubStance 17, no. 3 (1988) ; C.J. Ackerley and S.E. Gontarski, The Grove Companion to Samuel Beckett (New

York: Grove Press, 2004), 347-358.

24

The second category of literary critical work on mathematics and literature is

named ‘studies of compossibility.’ I borrow the term ‘compossible’ from Alain Badiou,

who uses it to articulate a shared space of possibility between different procedures,

including art, science, love and politics. These four domains are not collapsible for

Badiou, and thus philosophy creates a shared, mediating space in which these particular

possibilities become comparable. The studies oriented towards a ‘compossibility’ of

literature and mathematics thus seek to find a shared space whereby forms and

potentialities of each domain come to overlap. These works are diverse philosophical

and theoretical studies of mathematics and culture, and are indispensible to any

consideration of how mathematics may influence and operate in literature. Preeminent

amongst these studies are the work of Wilhelm Worringer, Arkady Plotnitsky, Morris

Kline and Jeremy Gray. These studies find compossibility between mathematics and

literature in the shared formal commitment to abstraction30 or in a shared conceptual

trajectory.31

30 Two works consider the relevance of mathematics to culture in terms of abstraction. The first of these is

Wilhelm Worringer’s 1914 study Abstraction and Empathy. Worringer’s extended essay is a study of

geometric abstraction in art, which theorises a form of art that does not stake its success on soliciting

empathy. Worringer engages with Theodore Lipps’ theory of art, which formalises the notion that

aesthetic experience is conditioned by an affect: empathy. Worringer adopts Lipps’ psychologising form of

analysis but argues that there is a second essential psychic dimension that forms the necessary counterpart

to empathy. This counterpart is abstraction. For Worringer, abstraction becomes expression where

abstraction is ‘emancipation from all the contingency and temporality of the world-picture’ (44). See

Wilhelm Worringer, Abstraction and Empathy (Chicago: Ivan R. Dee, 1997).

The second study that establishes abstraction as a conduit between mathematics and culture is

Jeremy Gray’s Plato’s Ghost: The Modernist Transformation of Mathematics. Plato’s Ghost is a history of

mathematics that articulates broad changes in the discipline – including the changing status of

mathematics relative to the other sciences and the transformative effects of several path-breaking proofs.

Gray collects these varied events in mathematics to a broader intellectual shift that he calls modernism

and compares this ‘mathematical’ modernism to ‘cultural’ modernism that occurs roughly during the

same period (1890-1930). In Gray’s analysis it is the increasing levels of ‘autonomy,’ abstraction and

formalisation in mathematical work that provides the major points of comparison between mathematical

25

The third and final category comes under the umbrella of ‘literary numbers.’

‘Literary numbers’ signifies a substantial departure from the other two modes of

enquiring into literature and mathematics. In these works, the authors consider a

mathematics internal to literature. Two works that exemplify this perspective are Sigi

Jöttkandt’s First Love: A Phenomenology of the One and Mathew Wickman’s ‘Robert Burns

and Big Data; Or, Pests of Quantity and Visualisation.’ In First Love, Jöttkandt considers

psychoanalytic and philosophical conceptions of the ‘one’ and analyses the ways in

which literature homogenises a ‘one’ or establishes a ‘count.’32 Matthew Wickman’s

studies and wider socio-cultural shifts. See Jeremy Gray, Plato’s Ghost: The Modernist Transformation of

Mathematics (Princeton and Woodstock: Princeton University Press, 2008). 31 Arkady Plotnitsky’s work on physics and philosophy links the two domains on the basis of shared

concepts, most significantly ‘complementarity.’ Whilst Plotnitsky’s work is only indirectly related to

mathematics, the rigour and detail of his studies provides an important precedent. Plotnitsky has paved

the way for work that connects literature and mathematics in two books that draw connections between

physics and philosophy. In The Knowable and the Unknowable Plotnitsky draws comparisons between non-

classical physics, which is associated with a scientific method that accepts a degree of ‘unknowability’ in its

object, Lacan’s use of mathematics for his psychoanalytic work and the connections between Derrida’s

philosophical work and relativity. In Complementarity Plotnisky develops his work connecting Derrida’s

‘general economy’ to Bohr’s ‘complementarity.’ Plotnitsky analyses the ‘affinities, differences, and

interactions’ between ‘complementarity and Derridean deconstruction’ producing an elegant account of

the way that deconstruction (and in particular the development of a theory of ‘general economy’) allows

philosophers to address the theory of complementarity developed in physics, which would not be

permitted in other philosophical matrixes (191). See Arkady Plotnitsky, The Knowable and the Unknowable:

Modern Science, Nonclassical Thought, and the ‘Two Cultures’ (Ann Arbor: University of Michigan Press, 2002),

and Arkady Plotnitsky, Complementarity: Anti-Epistemology After Bohr and Derrida (Durham: Duke University

Press, 1994). 32 Jöttkandt’s book takes up the question of Number (or, in her Badiouian vocabulary, the ‘count- as-

One’) in literature, specifically the significance of the count in the ‘literary tradition of First Love’ (7).

While focused on interrogating a philosophical and psychoanalytic concept of the ‘One,’ Jöttkandt

deploys both a Badiouian (and Lacanian) understanding of what it means to count to one, and looks at

how this takes place in a psychoanalytic constitution of the first love as a ‘one.’ Whilst Jöttkandt’s readings

differ from mine in their overarching theme and theoretical reference points, this book is a vital precedent

to understanding how counting, as the constitution of number, occurs in and through fiction. See Sigi

Jöttkandt, First Love: A Phenomenology of the One (Melbourne: Re.Press, 2010).

26

important study of poetry and quantity focuses on the work of Robert Burns. This

article looks not at numbers in Burns’ work, but focuses on Burns’ status as a national

poet, and hence his status as a ‘unity amid multiplicity.’33 Wickman considers the

interpretation of Burns’ status in the ‘era of big data’ against the backdrop of the literary

and reception history of Burns. This essay is one of the handful of examples of criticism

that turns the text loose on our quotidian assumptions about ‘quantity and magnitude,’

showing that the text possesses a numerical configuration that refutes naïve notions that

numbers can be applied to a literary text in analysis whereas a literary text cannot, in

reverse, be a tool by which to illuminate the tyranny of the number. Although in content

and focus this work is starkly different to Jöttkandt’s book, the two share a similar

perspective on the relationship between number and literature. They share the

conviction that a proper literary study of numbers addresses not only how literature may

refer to or use mathematics, but how this reference or use relates intimately to a literary

mathematics: the enumerative acts of prose, for example. It is this critical perspective,

this understanding of the conjunction ‘literature and mathematics,’ that I seek to

contribute to in the present study.

*

0.2 CONTEMPORARY METHODS

Three literary theorists represent the possibilities of confronting the mathematical stakes

of literature as well as the methodological pitfalls of integrating mathematics into literary

theory. It is worth dwelling on the divergence of these methods for the study of number

33 Matthew Wickman, ‘Robert Burns and Big Data; Or, Pests of Quantity and Visualisation.’ Modern

Language Quarterly 75, no. 1 (2014), 7.

27

and literature in order to illuminate the essential element determining such studies: the

definition of- and philosophical stakes in number. Each of these theorists occupies a

vastly different position in literary theory: J. Hillis Miller, Franco Moretti and Steven

Connor have little in common other than a preoccupation with number. The vast

methodological and critical difference between their studies can be traced back to

irreconcilable concepts of number. J. Hillis Miller, for instance, does not develop a

numerical methodology for literary studies but rather looks at a philosophical extension

of number, which transforms number into a concept. Miller’s work on ‘zero’ is

contained in a volume entitled Zero Plus One. In Zero Plus One Miller engages with

number as a concept, as an expressive ground that literature operates through and

enquires into. In Miller’s gloss, infinity is ‘zero’s reciprocal’ and it ‘is a feature of zero

that it is incommensurate with all other numbers, or rather, it is a Number and not a

Number.’34 Crucially, this work on zero as a concept is part of a wider reflection on

comparative literature. Miller’s work on zero is contextualized by the necessity to go

beyond national literatures and their comparison to a genuine world literature that

exceeds national peculiarities. His contentions about zero and comparative literature are

contained as two parts in a single volume (Zero Plus One) linked because they are both

bound up with the ‘untranslatable, idiomatic and obscure.’35 Indeed, Miller claims that

the ‘zero dimension’ of literature corresponds to ‘the language crisis in comparative

literature.’36 Miller uses zero to collect and describe acts or literary productions or

34 J. Hillis Miller, Zero Plus One (Valénzia: Biblioteca Javier Coy d’estudis nord-americans, 2003), 11. 35 Ibid. 36 Ibid.

Number has been persistently related to the idea of universal communication and Miller seems to be

operating in something of a tradition of thought on number and forms of ‘Esperanto.’ In the work of the

writer Zalkind Hourwitz, for instance, we find a similar claim to number functioning as the antidote to the

specificity and national identity and diversity of languages. Robin E. Rider summarises Hourwitz’s project

28

modes of expression, and in this sense zero becomes simply a descriptor, a cipher of a

cipher rather than a mathematical entity operating within a work. Here, zero is the

partially semiotic sign for that which exceeds linguistic capture. In this study, number is

in danger of being used to express something, functioning as a concept, metaphor or

allegory of precisely that which is divergent from literary form. In other words, number

is forced into participating in a connotative or representational scheme. The problem

here is that in a mathematical sense zero is simply not idiomatic or obscure, and is only

untranslatable insofar as it is not within a linguistic economy (and perhaps, then,

nontranslatable rather than untranslatable). In order to understand precisely what

mathematical difference, and the writing of ‘presentation,’ may afford literature one

needs a revised notion of zero: one acutely aware of the shifts between zero as

conceptual metaphor and zero as number; even whilst the identity of the latter allows

the creation of the former.

On the other hand, Franco Moretti’s work on number and literature avoids the

metaphoric traps of Miller’s analyses but collapses the significance of number for

literature into quantitative data production: an empirical tool to pursue distributive

tendencies in literary works.37 For the last decade or so Moretti’s critical work on the

as follows: ‘The idea that mathematical concepts, especially the concepts of number and their

representation by Arabic numerals, were universally comprehensible, held persistent appeal to those who

aimed at universal, rational languages. In 1801, for example, Zalkind Hourwitz proposed a Polygraphie that

relied on the assignment of a number to each word in the basic polyglot dictionary. The same number

thus served to designate words with the same meaning in several languages.’ For a discussion of this see

Robin E. Rider, ‘Measure of Ideas, Rule of Language: Mathematics and Language in the eighteenth-

century,’ in The Quantifying Spirit in the Eighteenth Century, ed. Tore Frängsmyer, J.L. Heilbron, and Robin E.

Rider (Berkeley and Los Angeles: University of California Press, 1990), 125. 37 This is part of a wider shift towards quantitative methodologies in literary studies. Matthew Wickman

sums up this turn to the quantitative particularly well:

The incorporation of ‘big data’ into the humanities is largely characterized by a spirit of

pragmatism—a surprising turn, perhaps, after so many decades of ‘big theory’ about the impact

29

novel has revolved around developing a quantitative methodology to study novels,

driven by the polemical claim that:

literary history will quickly become very different from what it is now: it will

become ‘second hand’: a patchwork of other people’s research, without a single

direct textual reading. Still ambitious, and actually even more so than before (world

literature!); but the ambition is now directly proportional to the distance from the text:

the more ambitious the project, the greater must the distance be.38

As in Miller’s work, this ‘numerical’ literary study is essentially about world literature:

about reading and approaches to texts that emerge from different times and national

contexts. Like Miller, Moretti uses number in the context of a study of world literature,

undertaken through a methodology appropriate to planetary scope: quantitative studies,

which deal only in quantifiable data, usually but not necessarily numerical, formed

through counts, measures or pinpoints. Once coordinates are established a diagram can

of new technologies. In literary studies electronic tools, processing massive amounts of

information, make it possible to identify previously undetectable relationships between disparate

corners of the canon or within the seams of individual texts, and the interest and anxiety inspired

by this new wizardry seem evident in growing numbers of digital humanities symposia, data

visualization websites, and conference panels on new media. In this environment familiar

arguments concerning the impact of technology on everyday life and on our conception of the

human (think of popular refrains involving the cyborg, the posthuman, and so on) are receding

behind the growing conviction that we simply need to get on with this new work of inputting,

encoding, and uploading. (1)

In the same article Wickman provides a critique of Moretti that is also based on an accusation of naiveté regarding number. See Wickman, ‘Robert Burns and Big Data; Or, Pests of Quantity and Visualisation.’ Modern Language Quarterly 75, no. 1 (2014). 38 Franco Moretti, ‘Conjectures on World Literature,’ New Left Review 1 (2000), n.pag.

30

be produced, and this secondary representation then analysed to establish fluctuations,

topographies or evolutions. ‘Distance’ is the key predicate of valid world literature

studies, and the form of reading that that these studies should take. Moretti posits an

untapped numerical resource that exists only between texts, and not only that, but

between very many texts: the truth of texts being based on a resonance that occurs

cumulatively. His distance from the texts is achieved through a naturalisation of

numeracy and an absence of any critical relation to the social regime of numeracy that

he participates in. In his direct connection between abstraction and ambition, Moretti

misplaces the actual distance of his readings. In fact, this work is not distant at all. The

supposed ‘distance’ is achieved by the ubiquity of digital technologies and computation:

the work is only ‘distant’ to the texts by virtue of embracing an absolute intimacy with

data modelling. This is an example of a study that ratifies number as the appropriate

means by which to achieve scope, perspective and generality in order to produce greater

‘awareness’ of broad trends in literary history. Such an affirmation, however, loses all

critical relation to the number, indeed, we can say that number in Moretti’s work is

substantively ‘forgotten.’

The third, important instance of recent ‘numerical’ criticism comes much closer

to a specificity of mathematics. Steven Connor’s important work on numbers in Carroll,

Dickens, Huxley and Beckett also focuses on the crucial but problematic status that

number occupies in mathematics and works to reclaim the shared project of literature

and mathematics on the basis of a ‘differentiated indifference’ at the heart of numerical

existence.39 This dissertation shares many aspects of the task that Connor undertakes,

but differs in perspective on the relation between modernism and a mathematical 39 Steven Connor, ‘‘What’s One and One and One and One and One and One and One and One and

One and One?’ Literature, Number and Death,’ Steven Connor, n.d., (accessed 12/12/2014),

http://stevenconnor.com/oneandone.html, n.pag.

31

‘actual’ infinity. Many writers and literary theorists or critics maintain a Romantic

conception of number even whilst participating in modernism in every other sense, yet

what I am concerned with here is an intersection between modernist work and a

modern concept of number and the transfinite. Connor argues that ‘modernist writers

[...] mount a sustained assault against the real of number, determined to assert quality

over quantity, determined to assert the hazy, nebular, indefinite or indistinct’ and whilst

this holds true for many writers, it is also indicative of a modernism that retains a

Romantic conception of the infinite.40 What is compelling about Connor’s investigation

into mathematics and modernism is his recognition of number not only as a tool of the

critic, but as something deployed by writers and poets towards formal ends, and as such

his work details an array of literary ‘attractions’ to number, including the graphic,

material presence of number in print culture and in the cityscape. Connor makes much

of the fact that definitions of number as well as processes of counting can inspire both

‘giddiness’ and ‘horror.’41 In this, Connor’s work is groundbreaking and approaches

mathematics in literature from the perspective that undergirds Badiou’s criticism of the

‘tyranny’ of number in the twentieth century. Key to his work is an understanding of the

difference between quotidian number and the foundations of those quotidian numbers,

which lie in the further reaches of pure mathematics and, as he suggests, simply don’t

have the habitual surety that we associate with numbers. He makes a crucial distinction

between ‘monotonous numbering’ and what is properly numerical, in what he loosely

terms a ‘Bad Infinity’ and a ‘Good Infinity’ of numbers, co-opting number as a hero in

an imagined war against the quantitative. However, pitting ‘Good Infinity’ as the hero

in the war against the quantitative, neglects the ‘actuality’ of modern infinities in favour 40 Ibid. 41 Steven Connor, ‘Hilarious Arithmetic: Annual Churchill Lecture at the University of Bristol,’ Steven

Connor, n.d., (accessed 12/12/2014), http://stevenconnor.com/hilarious.html, n.pag.

32

of a desired ‘relationality’ and ‘inexactitude.’ This actual infinity sits in start contrast to

precisely this numerical sublimity that preoccupies Connor. In his haste to enlist

mathematics for a pseudo-ethical position of an ever receding horizon of thought and

possibility, and to expound the heroic possibilities of number as a refutation of all

exactitude and independent identity, Connor neglects what many take to be the most

important advance in pure mathematics in the last century and a half: the founding of a

new form of modern mathematics based precisely on a concretion of numerical identity

rather than a reiteration of any ‘nebulousness.’ Instead of a ‘Good Infinity’ or a ‘Bad

Infinity’ in modern mathematics we are dealing with an ‘actual infinity.’ If mathematics

is the signifier of a regime of number whose foundations are remote or absent, then

mathematics cannot also be the locus of an ethical remedy for this: mathematics has no

space for the ‘ethical’ set; only the empty set.

The demand, now, is to understand literary experimentation with a form of

notation that is other to the word; that is more and other than a submissive, suggestible,

evocative symbol that conforms to the metaphysics of the literary. Number has always

been present as the ‘logos alogos’ 42 of language and poetics but – to echo Dan

Mellamphy’s Beckettian rendering of this – comes to function as the essential point of

non-being, the negated twin self, of modernist prose through its difference from

linguistic signification. There are two instances of literary work on number that stand

out, here, as realizing precisely this goal. In an important essay on ‘the pursuit of

number,’ Marjorie Perloff looks at mathematics in the work of Russian futurist Velimir

42 For a reading of Beckett’s ‘etre manqué assasine’ in relation to the ‘logos alogos’ of fiction, see: Dan

Mellamphy, ‘Alchemical Endgame: ‘Checkmate’ in Beckett and Eliot,’ in Alchemical Traditions from Antiquity

to the Avant-Garde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 552.

33

Khlebnikov (1885-1922) and Irish poet W.B. Yeats (1865-1939).43 Both writers were

interested in numerical models of history and being: in Yeats’ case the ‘Great Wheel of

History’ or the ‘Four Faculties,’ and in Khlebnikov’s case the geometric models that he

took to underlie the progression of history. Perloff argues that the numerical

preoccupations in the work of both of these poets is not a preservation of Romanticism

but is thoroughly modernist, drawing out the distinction between the ‘ultimate

abstraction’ of number (using Alfred North Whitehead’s term) and the contrasting

specificity, and indeed materiality, of numerical value. Perloff focuses on Khlebnikov’s

vision of being and the world as a great vibrating string, which connects abstracted

order that governs events to a resonant materiality and logical system. By demonstrating

the material consequence of Khlebnikov’s mathematics, Perloff shows that the

association of abstraction, determinism and materiality with resonance if not reason

enacts the transition from a romantic idealism to a modernist vision of the world or

universe.

This vision of a modernist mathematics, divorced from either a romantic or

positivist relation to number and the infinite, is captured in another seminal work on

literature and mathematics. In Beckett and Badiou: The Pathos of Intermittency, Andrew

Gibson argues that romantic philosophy broke the integral link between mathematics

and philosophy. Gibson notes that ‘romantic philosophy more or less completely

separates philosophy from mathematics, with Hegel playing the decisive role. The anti-

philosophical stance of positivism does no more than mirror the anti-mathematical

43 Marjorie Perloff, ‘The Pursuit of Number: Yeats, Khlebnikov, and the Mathematics of Modernism,’ in

Poetic License (Chicago: Northwestern University Press, 1990).

34

stance of romanticism.’44 The romantic conception of infinity directly renders it as a

metaphor for

a boundless exteriority, an openness without end. [...] By the same token, infinity

becomes an object of insatiable yearning. It is placed within a structure that

opposes it to transience, historical mortality, the birth and death of ideas.45

This is not a problem in itself, but it evacuates infinity of its peculiarly mathematical

import and prevents a genuine literary relation to mathematics. Gibson identifies this

romantic conception of the infinite as an ‘auratic conception of the infinite,’ (which

closely resembles the Burkean sublime) whereas the positivist notion of the infinite lies in

antithesis to the generation of the aura; the other side, we might say, of the same coin

and at an equal distance from any ‘actual infinite.’46

It is, then, unsurprising that we see a mathematics of the infinite return to art - if

not philosophy - at the precise moment in which culture must confront the loss of aura:

the moment at which art must adapt to ‘an age of technological reproducibility.’47 This

moment occurs at the start of the twentieth century, with European and Anglophone

modernism. For each of the writers that I am considering, here – Borges, Beckett and

Coetzee – the relation to number is bound up with a relation to the mechanisation of

44 Andrew Gibson, Beckett and Badiou: The Pathos of Intermittency (Oxford and New York: Oxford University

Press, 2006), 6-7. 45 Ibid. 46 Ibid. 7. 47 Walter Benjamin, ‘The Work of Art in the Age of Its Technological Reproducibility,’ in The Work of Art

in the Age of Its Technological Reproducibility and Other Writings on Media, ed. Michal W. Jennings, Brigid

Doherty, and Thomas Y. Levin (Cambridge and London: The Belknap Press of Harvard University

Press, 2008), 19.

35

either the image or the poem: for Borges, the filmstrip; for Beckett, the photograph,

sound recording, and television screen; for Coetzee, computer programming. In these

cases, the problem of an art without aura parallels the problem of an infinite without

sublimity. Perloff’s understanding of the necessity of materiality to a modernist

mathematics, and Gibson’s rendering of the romantic conception of the infinite as

‘auratic’ together articulate the necessity and the specificity of a vision of the infinite in

an age of rapid technological innovation.

*

0.3 A GENERIC LITERATURE

Perloff and Gibson offer a means by which to analyse the centrality of number to

modernist work without delimiting number to either measurement or metaphor. This

dissertation makes a contribution to the relation between literary form, most

importantly allegorical form, and the conceptual, textual and epistemological status of

the numeral by attempting to traverse just such a conceptual and methodological line.

In these texts by Borges, Beckett and Coetzee, we find not another text or

transcendental meaning at the end of the allegorical ‘rainbow,’ but a number. The

understanding of ‘numerical’ allegory developed here allows for an account of the way

that numerical form undergirds narrative perspective, description and the

undescribable, and the relationship between the efficacy of the symbol and the fictional

world.

In the following chapters I will argue that number facilitates a particular

aesthetic end for Borges, Beckett and Coetzee. I will argue that number introduces a

generic aspect to the medium of writing in two ways. The generic that I will present here

36

emerges from set theory and is most recently elaborated as a philosophical-

mathematical concept by Badiou. The concept of the generic, used metaphorically in

Badiou’s magnum opus Being and Event, expresses the existential correlate of the

mathematical concept developed by the famed set theoretician Paul Cohen.48 For

Badiou, following the language of Cohen, the null set or empty set in mathematics is

also called the ‘generic set,’ in the sense that it is ‘indiscernable.’49 As such, mathematics

affords us a particular sense by which to use the term generic that is distinct from other

common uses of the terms ‘genre’ or ‘general.’ The mathematical generic indicates not a

‘purity’ or an ‘origin’ but, instead, a set without attribute, or a set that is unspecified. A

mode of ‘generic’ writing doubles back on the fictional task to focus not on the

representation of a world (attributes, specificities) but on the conditions of that

representation.

Indeed, a generic work doubles back on itself to focus on what description,

definition and representation necessarily exclude. I will show here that this generic

literature is a literature that demands of its audience not reading so much as writing. This

is a literature that relies less on the creation of a perspective (a vision or familiarity with

48 Zachary Luke Fraser claims that Badiou’s use of this phrase is ‘essentially metaphorical.’ Z.L. Fraser,

‘Badiou Dictionary: Generic,’ Form and Formalism, 2011, (accessed 12/12/2014)

http://formandformalism.blogspot.com.au/2011/03/generic-entry.html. 49 Of peripheral interest to the wider question of mathematics and literature is the fact that several of the

most important mathematicians of the twentieth century also harboured unrealised literary ambitions.

Paul Cohen himself envisaged, at times, putting aside his mathematical work to write novels: ‘He spoke

several languages. He played the piano. His ambitions were seemingly unlimited and he spoke, from one

moment to the next, of becoming a physicist, a composer, even a novelist.’ [Sylvia Nasar, A Beautiful Mind

(New York: Simon and Schuster, 1998), 237.] Cohen is, in this respect, perhaps close to the Fields

Medalist Alexander Grothendiek, who, after his truncated career as a mathematician penned a thousand

page creative memoir, or Fernando Zalamea, the Colombian mathematician who has authored several

novels. Cantor, of course, considered himself as much a philosopher as a mathematician, and made

several attempts to be transferred from the mathematics department to the philosophy department at his

university.

37

a world, with a person or a life) so much as a pre-scriptive creativity. The word prescriptive

superficially indicates an attempt to enforce a rule, but this sense of enforcement

emerges from the portmanteau of ‘before writing.’ ‘Pre-scriptive’ literature harnesses

that which exists before writing – the number – rather than that which is ‘in’ or ‘of’

writing in order to create fictions of precisely that which is indiscernable; to evoke not a

discernable world but, rather, its unseen, unheard and above all unwritten twin.

The three writers I select here for in-depth treatment are heterogenous. I make

no claim that they are representative or exemplary of a broad literary movement.

Rather, the analyses that follow are comparable to drilling explanatory holes in an

untapped well – the links between number and narrative established here offer analytic

forays into three key oeuvres in what is no doubt a wider field of ‘mathematics and

modernism.’ The first chapter addresses the theoretical and mathematical context for

this dissertation. In this chapter, I situate Cantor’s transfinite numbers, and his

diagonalisation proofs, in terms of the wider mathematical context of the nineteenth

century and in terms of the impact of these proofs in the twentieth century. In this

chapter I consider the possibility of Cantor’s proofs as a triumph of modernity and even

as ‘modernist’ and present two engagements with literature and mathematics that set the

scene for the following analyses. The first of these is a short story: Paul Valéry’s short

text on the poet Verlaine and the mathematician Poincaré passing each other on the

street. The second of these is critical: French philosopher Quentin Meillassoux recently

discovered a ‘code’ at the heart of Stéphane Mallarmé’s Un coup de dés jamais n’abolira le

hasard. Meillassoux argues that this code involves a poetic ‘concretion’ of the infinite.

Following this, I will outline a model for comparison between Cantor’s ‘actual infinite’

and Mallarmé’s ‘actual finite’: transfinite allegory. I will theorise ‘transfinite allegory’ as

the model shared by Cantor’s proofs and literary infinities of Mallarmé and of those

38

who work in the legacy of the actual infinite. ‘Transfinite allegory’ will also provide the

model by which literature can approach mathematical infinities without subsuming

mathematics to a representational economy.

The second chapter deals with a writer that exists at the crossroads between the

nineteenth century and the twentieth century, European modernism and the South

American periphery, indeed Symbolisme and modernism. In this chapter I will analyse

the relation between number and total worlds, and number and the capacities and limits

of prose representation in the short stories of Jorge Luis Borges. I will show how Borges

inverts the traditions of prose description to create fictional worlds that defy the limits of

representation.

In the following two-part chapter (Chapter 3.1 and Chapter 3.2) I analyse two of

Beckett’s novels, three short prose works and a play for television. Chapter 3.1 focuses

on the transformation of naturalism in Molloy and Watt through different numerical

models, most significantly permutation. Beckett’s prose connects mathematical and

literary crises of foundation by attempting what Beckett would call a literature of the

‘unword’ that is decidedly mathematical. The repeated negation of semantic reference

and the radicalisation of naturalist description both rely on a reworking of the numeracy

of prose. In Chapter 3.2 I analyse the numeral as a mark of representation that cannot

be deemed with ‘condition’ or ‘content’ in Beckett’s later work, focusing on All Strange

Away, Imagination Dead Imagine, Quad and Worstward Ho. This status of the numeral allows

Beckett to invert the naturalist ambition for fictional worlds to achieve a ‘generic

literature.’

The fourth chapter is titled ‘J.M. Coetzee and the Name of the Number’ and

considers Coetzee’s investment in structuralism and the ‘quantification’ of style, and the

role of number in two of Coetzee’s novels. Here I consider, in particular, Coetzee’s

39

engagement with the uniqueness of the symbol, what counts as ‘I’ and, most

importantly, what allows for the consistency of the count. I will argue that the site of

formal experimentation in these novels is the generic operation of – to use Badiou’s

terms – what is considered to ‘count-as-one.’

1 TRANSFINITE ALLEGORY

There is no doubt that we cannot do without variable quantities in the sense of the

potential infinite. But from this very fact the necessity of the actual infinite can be

demonstrated.

George Cantor, 1885

The unique number / which cannot be another

Stéphane Mallarmé, 1897

No one will expel us from the paradise that Cantor has created.

David Hilbert, 1925

Is there a fruitful connection between literary infinities and mathematical ones? How

might we conceptualise a relation that is mutually illuminating without the need to insist

on similarity? How might such a connection be drawn without eroding the specificity of

each of these domains? By and large, literature and mathematics have been represented

as two separate worlds that bypass each other without any interaction. This planetary

metaphor is not without its own value: the relationship between these two worlds does

not involve direct communication; these worlds are not co-constituted and do not

resemble each other. What they share is relation to a third element: the source of gravity

around which they both revolve. If we take literary and mathematical innovation to be

something more than contingent cultural flux, we might claim that these two worlds

41

inevitably fall into a relationship with each other via the conduit of a common

orientation: the infinite. In this short, contextualising chapter I will address what can be

called two ‘transfinite’ endeavours in literature: Mallarmé’s coded poetry, and Cantor’s

transfinite numbers, and opportunities for situating these endeavours in terms of an

allegorical structure. But first, an excursus on distance.

*

1.1 THE MISSED ENCOUNTER

The short text ‘Passage de Verlaine’ by Paul Valéry (1871-1945) is one of the rare works

that address the distance between mathematics and literature. Valéry, friend and

champion of Stéphane Mallarmé, is the inimitable poet who is also a polymath. Valéry

read modern mathematics and was versed in the perennial problems posed by the

numerical infinite, especially as it is figured in Ancient Greek thought.1 He is distinctive

for privileging mathematics as the discipline in which the imagination is as its most

adventurous:

In comparison to the dream, the real is a convention. And the same applies for the

pure imagination. [...] The imagination is always curiously timid. It rarely risks

combinations far from all probable use and reality. It is the mathematicians who

are led furthest from this by the necessity to interpret or demonstrate their

1 Valéry’s poem ‘The Graveyard by the Sea’ is one of the best examples of Valéry’s practice of drawing

connections between states of being and mathematical problems. See Paul Valéry, ‘The Graveyard by the

Sea,’ in Selected Writings of Paul Valéry, trans. Malcolm Cowley, C. Day Lewis, and Jackson Mathews (New

York: New Directions Publishing, 1950), 41.

42

equations when they want to generalise or study the whole domain. They write

with a greater generality than they can see. And afterwards, they attempt to see.2

Valéry’s place as a hinge between worlds of science and poetry is best exemplified in

‘Verlaine and Poincaré.’ Here, Valéry describes a scene at the Jardin du Luxembourg,

where two profound intellects – the symbolist poet Paul Verlaine (1844-1896), and the

mathematician, Henri Poincaré (1854-1912) – very nearly cross paths on a daily basis,

without encounter or even awareness of each other. Poincaré frequently passes along

the same street as Valéry, alone, hunched, and with a gaze that is ‘empty and fixed.’3

Minutes later, Verlaine heads down the same street to a tavern ‘[...] Flanked by his

friends, leaning on a woman’s arm... [he] would speak, pounding on the pavement, to

his small devoted retinue.’ 4 At that time, Verlaine was France’s most significant

symbolist poet, whose goal, as put most famously by Jean Moréas in the Le Symbolisme

was to ‘clothe the Ideal in a perceptible form.’5 The only link between Poincaré and

Verlaine is Valéry, the silent observer who takes note of the scene from a distance.

Valéry’s anecdote portrays two gods traversing the streets of Paris, both divine but

whose ‘spiritual distance’ is ‘immense.’6

What is particularly striking in Valéry’s anecdote is the fact that the poet speaks

to his entourage, whereas the mathematician, solitary, does not. This allegorises the

means by which mathematics is distinguished from the empirical sciences as well as 2 Valéry quoted in Karin Krauthausen, ‘Paul Valéry and Geometry: Instrument, Writing Model,

Practice,’ Configurations 18 (2010), 243. 3 Paul Valéry, ‘Verlaine and Poincaré,’ in Mathematical Lives: Protagonists of the Twentieth Century, ed. Claudio

Bartocci, Renato Betti, Angelo Geurraggio, Robert Lucchetti and Kim Williams, (Berlin and Heidelberg:

Springer-Verlag, 2011), 25. 4 Ibid. 5 Jean Moreas, ‘The Symbolist Manifesto,’ in Manifesto: A Century of Isms, ed. Mary Ann Caws (University of Nebraska Press, 2000), 50. 6 Valéry, ‘Verlaine and Poincaré,’ 26.

43

literature and the arts, a distinction that can be articulated in a variety of ways but

which in essence pertains to the distinction between presentation and representation.

Pure mathematics only became distinguished as a discipline distinct from other sciences

in the eighteenth century, on the basis that it does not measure or record empirical data

but produces proofs that are pursued not for the sake of utility but the truth that these

proofs would formalise.7 In Valéry’s anecdote, it is the divergence between expression

and silence, sociality and isolation that is the initial deciding factor in the mutual

exclusivity between the figure of the mathematician and the figure of the poet. And yet

they are united by the same pursuit of truth rather than measurement.

Valéry’s story describes a missed encounter that is perhaps synecdochic of the

separation between mathematics and poetry. At the same time, through an emphasis on

purity, abstraction and divinity, Valéry manages to gesture towards a shared sphere:

What incomparably different effects the sight of even the same street could

produce in those two systems that followed so quickly upon one another. In order to

conceive of it, I had to choose between two admirable orders of things that were

mutually exclusive in appearance, but that resembled on another in the purity and

depth of their purpose...8

Valéry retained a lifelong obsession with the relation between the artistic

imagination and geometry.9 His topological notions of the imagination find form within

his poems, which echoes his own deformation of mathematical orthodoxy: for Valéry, as

Karin Krauthausen summarises it: ‘Riemann planes, [...] are states.’10 In ‘Passage de

Verlaine’ it is the figure of Valéry, the author, who can see the two magisteria passing

7 This definition was popularized by the Bourbaki group. See Nicolas Bourbaki, Elements of Mathematics: Theory of Sets (Berlin and Heidelberg: Springer-Verlag, 1968), 6-7. 8 Valéry, ‘Verlaine and Poincaré,’ 26. 9 Valéry quoted in Krauthausen, ‘Paul Valéry and Geometry: Instrument, Writing Model, Practice,’ 236. 10 Ibid. 238.

44

each other. Valéry manages to form his own account of a relation between the two –

finding analogues between mathematical entities and the phenomenal world. Yet what

is crucial to emphasise here, and what this short story shows us, is the difference between

two worlds that both privilege ‘purity’ and ‘depth of purpose.’ In the following two

sections I will present another ‘passage’ of two minds, albeit one that does not have the

convenient scene of the Parisian street at which they intersect. Stéphane Mallarmé and

Georg Cantor both developed forms of ‘actual infinity’ in the nineteenth-century,

though they would have been unaware of each other’s work and certainly the poetry of

Mallarmé and the mathematics of Cantor have rarely been read in terms of each other,

save in the work of Badiou and, indirectly, Quentin Meillassoux. The link that I will

hazard between Cantor’s transfinite numbers and the literary ‘actual’ infinities (for

which Mallarmé’s work is the representative, here) takes the form of allegory, which I

will discuss in the concluding section of this chapter.

*

1.2 CANTOR’S TRANSFINITE

What is the ‘paradise’ that Cantor created? This phrase comes from David Hilbert’s

famous call for mathematicians to resolve issues in ‘set theory’ with the rallying

declaration that no one could ‘expel’ the mathematicians from the ‘paradise’ that

Cantor had created.11 ‘Paradise,’ here, describes the radical new potential offered to

mathematics by the ‘set theory’ that follows from Cantor’s proofs, as well as a radical

formalisation of the foundations of mathematics. It is also, of course, a biblical reference.

11 This famous and now oft-quoted reference to paradise originates in ‘Über Das Unendliche’, a lecture delivered in 1925. For a discussion of Hilbert’s phrase and its implication for notions of mathematical creativity see Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise, 1.

45

Hilbert here has replaced the Garden of Eden with set theory: a secular paradise that

only the fecklessness of mathematicians can ruin. There is no deity to expel the

mathematicians from the paradise: once the roof is in existence, its offer of paradise is

eternal, and the challenge merely has to be met by future generations. There are two

crucial co-implicated issues that are resolved in Cantor’s proofs: the first is the numerical

instantiation of the actual infinite. The second is the stabilisation of the number line, and

most importantly and controversially the ‘proof’ of the continuum. Whilst both of these

profound steps in mathematics resolved many deeply troubling conundrums in

mathematics, these steps also generated a whole new domain of paradoxes: paradise had

its own problems. In this section I will summarise Cantor’s discoveries and the

interpretation of these discoveries as ‘modern’ or ‘modernist,’ beginning with Cantor’s

proofs of the ‘uncountability’ of the real numbers. Proving the discrete existence of

numbers is a key metamathematical problem that has two significant implications. All

mathematics can be built once you have a coherent theory of the whole number. But

the problems that prevented such a theory proved insoluble until the development of

Cantorian set theory. The need for a theory of the discrete existence of numbers and

their placement in a continuum arose from the existence of irrational numbers.

Irrationals pose a problem for number theory as a whole because they never reach a

limit; these are numbers with decimal places that never stop. The irrational is in other

words a crevasse without floor between two whole numbers that thwarts attempts to

prove the stability and consistency of the numerical continuum. If these numbers get

ever more refined and ever smaller, the idea of counting upwards from one whole

number through the decimals to the next number is an impossibility: a destination that

keeps receding (a problem most famously represented in Zeno’s Paradoxes). Potentially

infinite quantities disrupt arithmetic because they follow different rules: this other and

46

uncountable universe of irrational numbers (and their association with infinitesimal

quantities) would be demarcated from the countable infinity of whole numbers by

Cantor.

The most important theory that attempted to stave off the problems that

irrationals posed for arithmetic, prior to Cantor’s work, was written by the French

mathematician Augustin-Louis Cauchy (1789-1857), who posited that the endless

exactitude of the infinitesimal could be ignored by virtue of a more important criteria

for whole numbers: the limit. Cauchy demonstrated that whole numbers can be

differentiated by virtue of the fact that they constitute a limit for a series: ‘When the

successive values attributed to a variable approach a fixed value indefinitely so as to end

by differing from it as little as is wished, this fixed value is called the ‘limit’ of all the

others’ but one which was not, crucially, a limit that affected whole numbers.12 Cauchy

would stipulate the problem with irrationals in terms of a definition of numerical limit:

‘thus an irrational number is the limit of the various fractions which furnish more and

more approximate values of it.’13 The whole number would then be a limit of a

convergent series. However, this theory would not overcome the impasse of the

irrational, as Cauchy’s most important interlocutor in number theory, Karl Weierstrass

(1815-1897) showed. This deeply attractive theory of numerical identity requires a

definition of a real number; as such it does not prove the existence of such numbers in a

continuum. Cantor’s stabilisation of the number line comes from, on one level, actually

affirming the distinct identity of the irrationals from the natural numbers. Cantor

proved that irrational numbers (along with all the category of all the real numbers) are

uncountable: it is this fact that was proved in his famous ‘diagonal argument.’ An

12 Cauchy is quoted in: Julian Havil, The Irrationals: A Story ofthe Numbers You Can’t Count On (Princeton and Woodstock: Princeton University Press, 2012), 238. 13 Ibid.

47

irrational is simply any number that cannot be expressed as a fraction, or, more simply,

a number with a decimal string that does not end; it is one of the forms of ‘real’ number.

π is the best known example of an irrational number. Douglas Hofstadter aptly describes

Cantor’s diagonal argument as a ‘twist.’14 Hofstadter’s classic text Gödel, Escher, Bach

contains one of the clearest descriptions of the uncountability proofs: ‘What Cantor

wanted to show was that a ‘directory’ of real numbers were made, it would inevitably

leave some real numbers out – so that actually the notion of a complete directory of real

numbers is a contradiction in terms.’15 The transfinite numbers are ‘directories’ of

infinite sizes. Cantor proved this by demonstrating that there is always one number not

included in a register of all real numbers between zero and one. He did this by

constructing a hypothetical table that assigns a number to all the real numbers (placed in

rows) as such:

1) 1.1438934578935096783758349654374632944343228938472389 ... (row 1)

2) 1.422895347543589345748935734895748395834756348975847 ... (row 2)

3) 1. n ... (row n)

Cantor proceeds by identifying a number ‘d’ (here I am following, for convenience,

Hofstadter’s notation) from a diagonal extraction from the decimal extensions (here, d

could be constructed from the numbers that appear in bold and underlined). Here

things get strange, and this is where the proof is perhaps best approached as a thought

experiment. Imagine that 1 is subtracted from each of the decimals in the number ‘d.’

This would produce a totally new number – and surely this ‘new’ number should be

14 Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, 20th anniversary edition, (London and New York: Penguin Books, 2000), 420. 15 Ibid. 421.

48

somewhere in that complete directory of real numbers. The astounding answer is that it

is not. Why? Because we have subtracted one digit from ‘d’ it is now not the same as the

original ‘d’ and it is different from the number in row one, row two, indeed all the rows

of the table. It is a real number and cannot possibly be on the list of real numbers. More

importantly, to use Hofstadter’s words, ‘the set of integers is just not big enough to index

the set of real numbers.’16 In other words, to again make use of Mary Tiles’ wonderful

line, the question ‘how many points in a line’ is answerable for the natural numbers, but

not the reals: the number of real numbers is, and will by definition always be,

uncountable.17 This point in the history of mathematics identifies what will come to

characterise a new sequence in mathematics, a break from both classicist and logicist

praxis, with significant implications for the philosophy of novelty and contingency.18

It is worth taking note at this juncture of the form of Cantor’s intuitive proof.

The claim for uncountability here rests on a number being used in two ‘senses’: as the

index and as that which is indexed. This diagonal ‘method’ of Cantor’s would be

replicated in at least two of the most significant mathematical proofs produced in the

twentieth century: Kurt Gödel’s ‘Incompleteness Theorems’ and Alan Turing’s ‘On

Computable Numbers.’ Both Gödel and Turing relied on the problems that arise from

16 Ibid. 17 Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise, 10. 18 Most significantly, this uprooted stable Kantian critique as the predominant philosophical point of

departure (idealism or another weaker correlationism as the only possible engagement with the ‘real’) and

the Cartesian order of the world (the proof of an unintelligible but existent God, the cogito as the site of

novelty and thought). Quentin Meillassoux summarises this very well: ‘...it is precisely this totalisation of

the thinkable which can no longer be guaranteed a priori. For we now know – indeed, we have known it at

least since Cantor’s revolutionary set-theory – that we have no grounds for maintaining that the

conceivable is necessarily totalisable. For one of the fundamental components of this revolution was the

detotalisation of number, a detotalisation also known as “the transfinite.”’ See: Quentin Meillassoux, After

Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London and New York: Continuum,

2008), 103.

49

numbers being used as both an index and as that which is indexed, or, in other words,

self-reference. Essentially diagonalisation is a process of applying a principle to that very

principle, or, in the words of mathematician Haim Gaifman, ‘diagonalization then

consists in applying to an object the function it represents.’19 The essential importance of

self-reference in Cantor’s proofs, and in subsequent developments in set theory will

become clear in the final section of this sub-chapter, where I address a literary form of

self-reference: allegory, and its possible relation to a transfinite poetics or prose.

There are two profound consequences of the uncountability proofs. On the one

hand, this proof enables mathematics to be a properly deductive rather than inductive

enterprise. Secondly, this enables arithmetic to overcome the problems presented by the

infinite: most acutely the problems that emerge from irrational numbers. There are,

then, two inflexions of the infinite, as Cantor in his Contributions to the Founding of the Theory

of Transfinite Numbers (two texts published in 1895 and 1897) recognises: the destination of

the Number line and the infinite quality of the irrational number.

Cantor’s ‘infinite sets’ are represented by transfinite numbers. A transfinite

number is an actual infinity, without being the absolute infinity. Cantor represented the

transfinite cardinal numbers with aleph letters, which linked the finite numbers that we

use to count with their infinite potentialities, ‘fix[ing] the infinite’ as Quentin

Meillassoux would say.20 A transfinite number is particularly difficult to understand,

because, whilst it is an infinite number (in the sense that there is a transfinite number for

any infinite set), Cantor wanted to reserve the absolute infinite for God. In Morris

Kline’s words, ‘Cantor’s greatness lies in his perception of the importance of the one-to-

one correspondence principle and in his courage to pursue its consequences. If two 19 Haim Gaifman, Chapter 0: The Easy Way to Gödel’s Proof and Related Topics, 2007, (accessed 30/11/2014)

http://www.columbia.edu/~hg17/Inc07-chap0.pdf, 2. 20 Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés,’ 99.

50

infinite classes can be put into one-to-one correspondence then, according to Cantor, they have the

same Number of objects in them.’21 By establishing a one to one correspondence between two

sets (matching the objects of one set to the objects of another), one can tell the Number

of objects in the set (or whether one is greater or lesser). For instance (in an example

used by Kline) the set:

a) 1 2 3 4 5 6 7 ...

Can be said to have the same number of objects as the set:

b) 6 7 8 9 10 11 12 ...

simply because one can show that there is a 1-1 correspondence between these sets. By

showing that some sets of numbers do not have a 1-1 correspondence Cantor can prove

that some sets are ‘larger’ than others. For instance, completely counter-intuitively, the

number of squares and the number of natural numbers is the same, because for each

natural number there is a square and hence both sets have a cardinality of 0א, but

ordinal numbers, which cannot be put in a one-to-one correspondence with natural

numbers or squares become 1א. Thus Cantor demonstrates the existence of two

different sizes of infinities, thereafter extrapolating the possibility of not one infinite, but

infinite countable infinities, of different ‘sizes.’ Cantor numbered these different sized

infinities with what he initially called ‘symbols of infinity.’22. Of the transfinite numbers,

Cantor writes:

21 Morris Kline, Mathematical Thought From Ancient to Modern Times, vol. 1 (Oxford and New York: Oxford

University Press, 1990), 398. 22 Akihiro Kanamori, ‘Cohen and Set Theory,’ The Bulletin of Symbolic Logic 12, no. 3 (2008), 353. ‘1-1

correspondence’ has been known as ‘bijection’ since the Bourbaki account of set theory. Bijection is

simply the existence of corresponding pairs in two sets. There is no bijection if two sets are of different

sizes. Bijections can only apply to finite sets; for infinite sets one replaces such a bijection with a cardinal

Number. In the case of finite sets, a cardinal would just be a number counting the elements of a set, but in

51

Every aggregate of distinct things can be regarded as a unitary thing in which

the things first mentioned are constitutive elements. If we abstract both from the

nature of the things given and the order in which they are given, we get the

‘cardinal Number’ or ‘power’ of the aggregate, a general concept in which the

elements, as so-called units, have so grown organically into one another to make

a unitary whole that none of them ranks above the others.23

Transfinite numbers are exempt from the problem that infinitesimals cause for whole

numbers. Take, for example, ω (omega – the order type of natural numbers), which is

the lowest transfinite ordinal number. Where ω is ‘the limit to which the variable finite

whole Number v tends’ then ‘ω is the least transfinite ordinal Number which is greater

than all finite numbers; exactly in the same way that √2 is the limit of certain variable,

increasing, rational numbers, with this difference: the difference between √2 and these

approximating fractions becomes as small as we wish, whereas ω – v is always equal to

ω.’24 This is thus not a matter of saying that whole numbers approach ω but rather that

these numbers are all an equal distance from ω. The attribution of a transfinite number

to the set of natural numbers (but not to the set of reals) stabilises a continuum (a

countable infinity) and solves the arithmetical problems associated with infinitesimals

and numerical identity generally. In this, Cantor has introduced Platonism, and a

infinite sets, a transfinite number is used. The use of transfinite cardinals to describe infinite sets measure

different ‘sizes’ of infinity, and hence implies different infinities. 23 Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Mineola, New York: Dover,

1955), 74. 24 Cantor, Contributions to the Founding of Transfinite Numbers, 77.

52

positive account of the infinite - a numerical value of the infinite - into mathematics.25 In

David Foster Wallace’s words, after Cantor, the world ‘spins, now, in a new kind of all-

formal Void.’26 This void cuts right to the heart of what can be considered true about

mathematics, and the connection between the mathematical proof and the real.

Cantor occupied a liminal position between romanticism and an emergent

modernism: he was convinced, on the one hand, that his proofs were delivered to him

by God, and went to significant lengths to involve the church in his work. For the

Catholic clergy of the nineteenth century Cantor’s proofs appeared heretical by virtue of

what can be described as their ‘Platonism’: they posit actual infinities accessible to

abstract mathematics and, theologically, the ‘actual infinite’ is supposed to be reserved

for God. In this, Cantor appears as an Adam figure, grasping at forbidden knowledge in

the Garden of Eden. And yet it is precisely this echo of the Platonic that divided the

mathematical response to his theories. For Gauss, this Platonism was heretical in itself:

infinity was only a manner of speaking and should not be formalised as an entity beyond

this. Whereas it was precisely this Platonism that would make Cantor a hero in the eyes

of mathematicians like Hilbert and others who sought, in the early decades of the 21st

century, to axiomatise mathematics in a way consistent with Cantor’s proofs.

The mathematician Jeremy Gray directly associates Cantor with what he calls a

‘mathematical modernism.’ In Gray’s work Cantor is only one of the actors in a wider

transformation of mathematics that occurs at the end of the 19th and start of the 20th

25 A Platonist would take all numbers to be actually existent, and this would necessarily include the

transfinite numbers. As such, for a Platonist Cantor discovers rather than creates the transfinites. A

formalist, on the other hand would take Cantor to have created the transfinite numbers. Cantor believed

squarely in the former: the numbers were actually existent, and this fact was made indisputable by his

proofs. 26 David Foster Wallace, Everything and More: A Compact History of Infinity, 2nd edition (New York: W. W.

Norton and Company, 2010), 305.

53

centuries. In Plato’s Ghost: The Modernist Transformation of Mathematics, Gray argues that

‘the period from 1890 to 1930 saw mathematics go through a modernist

transformation.’27 Gray’s book charts a shift in mathematics, arguing that the year 1900

saw a substantial shift in the way that mathematics was carried out: both in terms of its

institutional status (the ‘autonomy’ of mathematicians from scientists, in particular) as

well as intellectual ambition (new visions of what constitutes a proof, for instance),

nominating Hilbert’s Grundlagen der Geometrie as perhaps the ‘single exemplary work that

ushered in modernism.’28 For Gray, this applies to the whole of mathematics, or at least

its constituent sub- disciplines, rather than one particular event or proof:

taken together all the changes in mathematics here described and the

connections to other intellectual disciplines that were then animated constitute a

development that cannot be described adequately as progress in this or that

branch of mathematics (logic and philosophy) but must be seen as a single

cultural shift, which I call mathematical modernism.29

Crucially, Gray ties this mathematical ‘modernism’ to a new Platonist perspective in

mathematics. Whilst Gray acknowledges that there is ‘little direct influence’ of cultural

modernism upon his notion of mathematical modernism, he does construct a definite

basis for the comparison.30 Most significantly for my work, Gray situates the link

between these two modernisms that does not construct a similarity between the two, but

27 Jeremy Gray, Plato’s Ghost: The Modernist Transformation of Mathematics (Princeton and Woodstock:

Princeton University Press, 2008), 1. 28 Ibid. 5. 29 Ibid. 4. 30 Ibid. 7.

54

rather uses a third, material metaphor to place the two domains in an allegorical

relation to each other:

Some features of bats and birds, or ichthyosaurs and dolphins, are alike because

the requirements of efficient flying or swimming promote them, but one species

does not inherit them from the other. The common features in the present case

are hard to discern, but the sheer size of society, its extensive diversification, the

existence of cultural activities remote from immediate practical needs, and their

high degree of cultural hegemony are certainly present in each.31

This is, in a certain sense, a complicated means by which to articulate the fact that

Cantor and the modernists emerged from the same zeitgeist: Gray points, here, at an

affinity without necessarily establishing direct influence. This dissertation cannot hope to

address whether, broadly understood, the discipline of mathematics can be understood

to have practiced a form of modernism analogous, say, to what we mean by modernism

in music. This would require an assessment of the wider late nineteenth century and

early twentieth century developments in mathematics. Rather, I am considering quite

the inverse of Gray’s topic: the presence of mathematics in literary modernism. By

articulating the points at which mathematics becomes relevant to literature, or to any of

the arts, we can start to reflect on the possibility of a modernism in mathematics from

the ‘other side,’ finding literary affinities with a ‘modernist’ mathematics.

Alain Badiou has made the strongest case for Cantor’s breakthrough constituting

both a triumph of modernity and a reintroduction of Platonism into thought. Unlike

Gray, Badiou does not argue for a modernism internal to mathematics, so much as a

31 Ibid. 8.

55

broad achievement of a ‘modernity’ in mathematics that echoes similar achievements in

art and politics (the most notably example of comparison that Badiou makes is between

Cantor and Mallarmé). As Andrew Gibson points out, although Badiou’s work revolves

around a philosophy of truth and event, he implicitly also provides a theory of

modernity in the domains of art, love, science and politics. In Gibson’s words, for

Badiou ‘the aesthetic domain catches up with the political domain with the first great

modern writer, Mallarmé, with the emergence of Cantor, the scientific domain catches

up too...’32 Such a comparative modernity provides the most important precedent for a

comparison between mathematics and literature. For Badiou,

a new definition of a number follows from the approach of Dedekind and

Cantor, and then of Zermelo, von Neumann and Gödel (which we shall call the

set-theoretical or ‘platonising’ approach) determines Number as a particular case

of the hierarchy of sets. In this context, ‘being a Number’ is thus referred back to

an ontology of the pure multiple, whose great Ideas are the classical axioms of set

theory.33

Badiou goes so far as to claim that the radical implications of Cantor’s theory were

played out in literature independently of any direct engagement with mathematics. For

Badiou, Mallarmé is the example par excellence here.34 With the advent of set theory

truth could no longer proceed from a concept of the essential, or the whole. However,

Badiou argues that the modern mathematicians responded not by creating a new

understanding of number, a new philosophy that could accommodate the multiplicity of 32 Gibson, Beckett and Badiou: The Pathos of Intermittency, 257. 33 Badiou, Number and Numbers, 8. 34 See, for instance, Ibid. 13.

56

number but rather they evacuated the concept. For Badiou, the ‘modern’ infinite was

realised by Cantor, who refused to let the definition of the infinite lapse into being ‘only

the secret strengths of the finite.’35 He argues that every age holds a certain conception

of infinity and its place regarding divinity, destiny, materiality and desire and Cantor’s

‘pure interruption’ in numerical succession constituted a radical break with those

conceptions of his age.36 The infinite is no longer a ‘law without limit’ or a completed

totality, but a measure outside of the limits of numeric repetition.37 In light of the

refutation of wholeness, the mathematicians of the early twentieth century developed an

understanding of types of numbers, each with their own apparatuses by which one

works with these numbers or arrives at these numbers. As such, Badiou claims that it

was not mathematics but literature that first assumed the challenge of the modern:

although mathematicians were bestowed with Cantor’s revolutionary truths, they

suppressed the radical consequences of these truths in order to develop types of

numbers.

For Badiou, then, the transfinite numbers are a direct model of the achievement

of the modern. But does the transfinite directly offer anything to literary innovation? Is

there any bridge of connection between this mathematics and experimental language?

Cantor was convinced that the transfinite numbers were one form of the infinite: a

mathematical form. He thus left room for a divine form of the infinite separate from this

mathematical form. Dauben summarises Cantor’s position on this succinctly:

35 Alain Badiou, ‘Infinity and Set Theory: How to Begin with the Void,’ The European Graduate School

Lectures, 2011, accessed 30/11/2014, http://www.egs.edu/faculty/alain- badiou/articles/infinity-and-set-

theory/, n.pag. 36 Ibid. 37 Ibid.

57

For example, there were a number of different ways in which Cantor felt the

concept of infinity could be regarded. One of these was the form it assumed ‘in

Deo extramundano alterno omnipotenti sive natura naturante,’ or the Absolute.

When the infinite served in this capacity he regarded it as capable of no change

or increase, and he said that it was therefore to be thought of as mathematically

indeterminable. Were it determinable, then it would have been limited in some

manner.38

Mathematics is simply one version of the logos alogos: one means by which to regard

infinities.39 This is a kind of allegorical structure: the ‘in abstracto’ is the mathematical

other of a theological presence. We have an ‘actualisation’ of that which – by traditional

terms – is supposed to formalise the antithesis of actualisation. Cantor’s proofs thus offer

perhaps the most important model from which to explore other means by which to

‘regard’ infinities; inscribed in his very own proof is the notion of a finite supplement of

an infinite in Deo.

38 Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, 245. 39 The term logos alogos seems to have been created by the Pythagoreans, according to Simóne Weil. In

Weil’s words, ‘One of the first problems they [the Pythagoreans] ran up against was that the diagonal of a

square could not be expressed as a rational number. The proof of this is to be found in an appendix to

Book X of Euclid’s Elements. The Pythagoreans used the Greek word logos to refer to a number, and seem

to have coined the phrase logos alogos (a logos that is not a logos) to refer to the incommensurability of the

diagonal of a square, which we now refer to as an irrational number.’ [Simóne Weil, Lectures on Philosophy,

trans. Hugh Price (Cambridge and New York: Cambridge University Press, 1978), 116]. The term logos

alogos has been used outside of this direct mathematical reference to the irrational numbers, however. In

Dan Mellamphy’s work, it is used to refer to a ‘wordless language’ and to the ‘word unheard’ in Beckett’s

work [Dan Mellamphy, ‘Alchemical Endgame: ‘Checkmate’ in Beckett and Eliot,’ in Alchemical Traditions

from Antiquity to the Avant-Garde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 552]. This extended

meaning allows the term to refer to something coexisting in or as its antithesis.

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There are two forms of ‘doubling’ and paradox in Cantor’s work that I take to

be pertinent to other modes of regarding infinities, particularly the literary mode. The

first instance of a doubling is this creation of an actual but not absolute infinity: just one

mode in which infinities are manifest, according to Cantor. The mathematical

transfinite doubles absolute infinity in an actual and countable realm, formalising a

paradox that, as I have just claimed, constitutes a logos alogos. The second form of

doubling and paradox occurs within the proof itself: Cantor uses numbers to index

numbers, producing an impasse in the process of indexing and demonstrating that the

real numbers are not countable. This mode of self-reference is the same form that

produces antinomies of all sorts, including semantic antinomies (most famously the

Liar’s Paradox: ‘all Cretans are liars’). The transfinite thus shows us a connection

between certain forms of doubling (which occur by pairing an actual instantiation with a

‘uncountable’ or ‘transcendental’ pair; and the formalisation of paradoxes within a

proof by virtue of the ‘doubling’ of sense or self-reference) and the discovery of ‘actual’

infinities. I will begin to show, below, that literary ‘actual infinities’ share this model with

Cantor’s mathematical transfinites: in this formal echo lies the most important link

between the literary and the mathematical infinite.

*

1.3 MALLARMÉ AND MEILLASSOUX: FIXING THE INFINITE

So what might a peculiarly literary transfinite look like? And what are the precedents for

a theory of the literary transfinite? The most important articulation of a literary

transfinite, which I will assess here, emerges from a ‘discovery’ of a code in French poet

Stéphane Mallarmé’s masterpiece Un Coup de Dés Jamais N’Abolira le Hasard (hereafter

59

Coup de dés) by French philosopher Quentin Meillassoux. Meillassoux’s reading is both

shocking and controversial, but also extends a substantial record of reading Mallarmé in

terms of the infinite, of which Badiou’s interpretation is perhaps the most important

contemporary instance. Meillassoux’s ‘deciphering’ of the poem is unique by virtue of

the actual number he finds present in the poem, but also his explicit connection between

infinity and chance. The Coup de dés has traditionally been read in terms of making the

‘invisible’ visible. Valéry, one of the poem’s first readers, recalled the experience of

reading in terms of a concretion of the ephemeral:

It seemed to me that I was looking at the form and pattern of a thought, placed

for the first time in finite space. Here space itself truly spoke, dreamed, and gave

birth to temporal forms. Expectancy, doubt, concentration, all were visible

things.40

Meillassoux’s reading secures precisely this oscillation between what can and cannot be

apprehended in terms of the poetic infinite. Un Coup de dés has always been recognised as

a profound break in poetry, but this new reading of the poem in terms of number

suggests that the proto-modernist break with traditional poetic graphology and metrics,

and the poem’s focus on the generic (the ‘only the place will have taken place’) is

achieved via a poetic use of number. Meillassoux finds an endogenous code (located within

the text) in the Coup de dés, rather than a kind of exogenous code (located in some external

meaning), the kind that he claims criticism is far more comfortable with.41 There is thus

40 Paul Valéry, ‘Mallarme’s ‘Coup de Dés,’’ in Selected Writings of Paul Valéry, trans. Malcolm Cowley, C.

Day Lewis, and Jackson Mathews (New York: New Directions Publishing Corporation, 1950), 218. 41 Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés,’ 5.

60

a simplistic, cryptographical ‘solution’ that gathers together every formal aspect of the

poem under a single number.

Mallarmé wrote the Coup de dés in response to a crisis in French verse: the

introduction of vers libre that presented a threat to the traditional alexandrine. The

alexandrine (or dodecasyllable) is a twelve syllable line frequently split by a caesura after

six syllables. The significance of the alexandrine in French poetry is roughly

commensurate with the significance of pentameter in English verse.42 Mallarmé is

perhaps most famous for transforming the poetic capacities of the line by severing the

alexandrine and constructing non-linear, non-sequential metre through resonance

across words and sounds exploded from lines on the page; making the poem generate

meaning and resonance across both linear, diagonal and graphological axes. However,

this is not to say that Mallarmé was straightforwardly opposed to the alexandrine. He

had no naïve notion of vers libre necessarily surpassing the systematic strictures of the

past. Indeed, certain significant lines of the Coup de dés are rendered in alexandrine.

Mallarmé understood vers libre to be central for poetic individuation and the alexandrine

to be central to ceremonial purpose.43 There was, nonetheless, a crisis that Mallarmé

thought poetry faced. ‘Literature here is undergoing an exquisite crisis, a fundamental

crisis’, wrote Mallarmé in 1897,

The faithful supporters of the alexandrine, our hexameter, are loosening from

within the rigid and puerile mechanism of its beat; the ear, set free from an

artificial counter, discovers delight in discerning on its own all the possible

42 See: Clive Scott, French Verse-Art: A Study (Cambridge and New York: Cambridge University Press,

1980), xii. 43 Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés,’ 22.

61

combinations that twelve timbres can make amongst themselves. It’s a taste we

should consider very modern.44

The Coup de dés manifests this freeing of the line from an ‘artificial counter’ most clearly.

The poem consists of various syntactic threads but is equally dependent on a meticulous

graphology, which involves the spacing of sentences so that they work not as traditional

lines but rather, as Heather Williams notes, meaning becomes structured by the blank

spaces as much as the content of the poem.45

The Coup de dés itself is governed by number: it begins and ends with dice being

thrown. The captain of the ship, holding the dice, senses and anticipates the appearance

of a ‘unique number unlike every other number’ as he oscillates over whether to throw

the dice as the waters surge around him.46 The poem ends with the master engulfed by

the whirlpool, with just the feather of his cap sitting above the water. A siren arises from

the waves and beats her tail against the rock that destroyed the master’s ship, destroying

it. We never find out whether the master throws the dice or not: the numerical image

that we are left with, at the close of the poem, is of the Septentrion constellation, which

is rendered in terms of a count:

A CONSTELLATION

cold with neglect and desuetude not so much

44 Stéphane Mallarmé, ‘Crise de Vers,’ in Mallarmé: The Poet and His Circle, ed. and trans. Rosemary Lloyd

(Ithaca and London: Cornell University Press, 1999), 227-228. 45 Heather Williams, ‘Mallarmé and the Language of Ideas,’ Nineteenth-Century French Studies 29, no. 3

(2001), 89. 46 Stéphane Mallarmé, ‘A Throw of the Dice Will Never Abolish Chance,’ in The Number and the Siren: A

Decipherment of Mallarmé’s Coup de Dés, by Quentin Meillassoux, trans. Robin Mackay (Falmouth:

Urbanomic, 2012).

62

that it fails to number

on some vacant and higher surface the successive impact

starrily

of a total count in the making47

The poem concludes with the line: ‘Every Thought Emits a Throw of Dice.’ Despite the

appearance of a radical form of liberated verse, the Coup de dés (as befits the contingency of

numeracy in the title) is in fact structured by a single number that is not the number

twelve, as in the alexandrine, but a ‘unique number’ that Meillassoux claims is 7. The

poem revolves around number in a more fundamental way than the simple use of a

metrical arrangement. Meillassoux has, through a process of laborious, multidirectional

counting, extracted a number that essentially structures the Coup de dés. Meillassoux

reads the poem numerically on a number of different levels: he counts the words in the

poem (707), the last line (7), and analyses the symbolism of the number seven in (the

Septentrion) and the significance of ‘si,’ the seventh musical note in the poem (amongst

other significant suggestions). Meillassoux enumerates or deciphers several hidden

references to the number, reading backwards and diagonally across the poem.

Simultaneously, Meillassoux reads the poem as a reflection upon the task of poetry itself:

the shipwreck is an allegory for the crisis of verse in France, the feather from the

master’s cap is simultaneously the symbol of the writer’s quill, and the unique number

that the poem produces is one that can replace the alexandrine metric: it is a poem

about and for the future of poetry.

47 Stéphane Mallarmé, ‘A Throw of the Dice Will Never Abolish Chance,’ in The Number and the Siren: A

Decipherment of Mallarmé’s Coup de Dés, by Quentin Meillassoux, trans. Robin Mackay (Falmouth:

Urbanomic, 2012), 273.

63

Meillassoux’s claim for the Coup de dés possessing a secret numerical key is both

ratified and complicated by the fact that – as Clive Scott in French Verse Art: A Study puts

it —‘The French line is syllabic before it is accentual. That is to say, its unity is

mathematical, and the rhythmic units or measures resulting from the accents which occur

in the line are significant as fractions of the total number of syllables the line has.’48 This

‘mathesis’ of verse peculiar to French poetry goes a long way to establishing a precedent

for precisely Mallarmé’s mathematical revision of the alexandrine. Mallarmé’s insistence

on a crisis, in the above quote, is not so much a departure from implicit French metrical

tradition so much as an insistence upon its very fundamentals: a radicalisation from the

inside rather than the outside.49 Whilst traditionally, in Scott’s words, it is important to

remember that ‘accents are, of course, created by language’ not number, the

mathematical organisation of the poem still provides ‘customary sizes of measure’ and

so ‘to know that one is reading alexandrines or octosyllables is already to locate oneself

in a certain area of expectations, and to exclude certain possibilities.’50

There are two parts to Meillassoux’s discovery, however: the deciphering of the

code, and then the potential undoing of that deciphering. Meillassoux’s most important

discovery is two part: he locates a potential code in the poem and then finds a string of

ambiguities that call that code into question. In Meillassoux’s final analysis the code is

‘undecideable’ or, more accurately, it is both posited and effaced in the construction of

the poem. The number, in Meillassoux’s terms, ‘quavers.’51 And it is this quavering that

is the most significant trait of Mallarmé’s code; indeed, it is this element of contingency

in the code itself that renders it part of an infinite poetic process. For Meillassoux, this is

48 Scott, French Verse-Art: A Study, 17. 49 Ibid. 17. 50 Ibid. 51 Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés,’ 138.

64

a realization of Mallarmé’s own convictions that any insertion of the divine into poetry

must be through diffusion not representation. A throw that properly diffuses chance into

the poem would ‘produce a Number that presents the hesitation,’ that ‘would be at one

and the same time this Number premeditated by the count of the Poem – 707 – and not this

Number at all, becoming [...] a noncoded total’ or a ‘negative proof.’52 And indeed the

poem ratifies this: ‘the signs or the metaphors of Chance [...] are not Chance

itself.’ 53 And the code itself ‘quavers’ – for Meillassoux – precisely by virtue of

Mallarmé’s choice to include it. It is the trace of the finite Mallarmé, making the

decision to include the code, the ‘bequest he made of his memory’ being ultimately,

irreducibly, a finite act.54

Meillassoux argues that this is precisely the nature of a ‘unique number’: ‘If the

code was entirely unambiguous, the imperfect subjectives through which the Number is

characterised (‘were it to exist,’ ‘were it to be ciphered’, ‘were it to be illuminated,’ and

so on) would cease to be pertinent once the code was decrypted.’55 In other words, a

code that can properly instantiate a literature that makes the ‘invisible visible’ would fail

at its task if, once decrypted, it simply made its secret visible once and for all. The code

has to be perpetually hypothetical, subject to speculation without ever allowing itself to

be defined once and for all. This ‘quavering’ of the code allows it to perpetually remain

52 Ibid. 139. 53 Ibid. 141. 54 As such, ‘the Number would indeed bring together all three properties of Chance: (a) it would contain

two equally determinate opposites (707) and another number close to it, but without any relation to the

code); (b) it would be eternal (the uncertainty is forever inscribed in the meaning of the Poem, since we

can never determine the ‘correct’ solution); (c) it would be real (since it would refer to the act – perhaps

effectuated, but undecidably so – of the man Mallamé coding the Coup de dés).’ Ibid. 144. 55 Ibid. 159.

65

an instantiation of chance, just as, in the poem we never know whether the master

throws the dice.

The new number that is the essence of the Coup de dés realises Kenner’s

‘Symbolist formula’ of ‘words set free in new structures’ in two senses. The words are

literally set free from each other, and interact across the page visually, semiotically,

aurally. The poem is made manifest by an underlying structure liberated from linearity

and rhythm, a process doubled in the desecration of any pure idealisation that such an

innate and spontaneous arrangement might aspire to through the hidden (and now

revealed) code.56 The ‘unique’ number remains ‘unique’ by virtue of its status as a

perpetual hypothesis that extends to the dice throw: perpetually hypothetical, we never

know if the captain threw the dice or not. The ‘unique number’ thus remains suspended

and can only be reached through speculation of the sort that Meillassoux produces.

Mallarmé’s poetry is distinctive within symbolism in that his preoccupation with

irreducibility was not wedded to the Ideal as a whole pure form that poetry was the

conduit of. Instead, for Mallarmé, the ideal is subtly replaced with a concept of

universalism devoid of attributes or particularities, or to use Badiou’s words, a

recognition of the truth that the One is ‘Not.’ In other words, the ideal is not manifest as

a totality accessible or inaccessible but that manifests a material order never fully

determined or delimitable by language. This is well summed up by Peter Broome and

Graham Chesters, who note Mallarmé’s extension of Baudelaire’s idea of

correspondences: ‘[Mallarmé’s] work takes to their most absolute and purified

conclusion [...] the belief that behind the phenomena of the material world lies an

anterior, immutable Reality; that material forms, imprisoned in the plane of the

contingent and accidental, testify nevertheless to a pattern of meaning and a mysterious

56 Hugh Kenner, The Pound Era (Berkeley and Los Angeles: University of California Press, 1971), 187.

66

order beyond themselves...’ 57 What is crucial in this description is recourse to

contingency, which distinguishes Mallarmé’s preoccupation with the relation between

the poetic surface and more fundamental orders. In order to fully understand

Mallarmé’s relation to contingency, however, this quote needs to paired with the famous

statement that forms the title of the poem we are concerned with, here: Un Coup de dés

Jamais N’Abolira Le Hasard: a throw of the dice will never abolish chance.

Symbolist literature, in Arthur Symons’ words, was a task of rendering the

invisible world real: ‘[...] after the world has starved its soul long enough in the

contemplation and the re-arrangement of material things, comes the turn of the soul;

and with it comes the literature in which the visible world is no longer a reality, and the

unseen world no longer a dream.’58 In Mallarmé’s poem emphasis on the unseen

quickly finds its place in the silence inherent to verse, and the vagaries of style and the

fixities of language suited to material world become a hindrance to the work of the poet.

Hugh Kenner calls silence ‘the arch- Symbolist’s best-known preoccupation,’ figuring

language as a wrench preventing intimacy with the object: ‘To name is to destroy,

thought Mallarmé.’59 This vision of contingency and ‘genericity’ must supersede that

arbitrariness of naming for the contingency of a singular future. But how can a poem

inscribe the contingency of the real that is necessarily beyond it? Simply through a

suggested, but not fully secure, code posited at the heart of the poem?

The formal reciprocity between number and poem, here, is not entirely divorced

from superstitious relations between art and number. Such a superstitious relation is

57 Peter Broome and Graham Chesters, An Anthology of Modern French Poetry 1850-1950 (Cambridge and

New York: Cambridge University Press, 1976), 35. 58 Arthur Symons, The Symbolist Movement in Literature (New York: E.P. Dutton and Company, 1958), 4. 59 Kenner, The Pound Era, 136.

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exemplified best, in the early twentieth century, by Schoenberg, whose fixation with the

number thirteen is described by Jean-Michel Rabaté:

[Schoenberg] abhorred the number 13 and routinely skipped over measure 13 in

his scores. He would often renumber page 13 as well. He even went to the

extreme of deleting the extra ‘a’ in Aaron’s name from the title of his opera

Moses und Aron, in order to have only twelve letters in it. There must be a link

between this superstitious numerology and the decision made after the war to

systematize atonality by creating the twelve-tone technique.60

But while Mallarmé loads his poem with the symbolism of number he does not – and

here is the crucial difference and the literary significance – ground it in a belief

structure. Instead, the number is the perfect generative kernel for the poem. Meillassoux

understands the significance of the number to be not so much symbolic (in that it does

not imply bad luck, some affective or transcendental import, and does not signify an

essence or a possibility other than the poem itself, which revolves around the number)

but radically polysemic and, as such, standing in for chance itself.

How can one produce a literature of the ‘perhaps’ – of contingency – rather

than a literature of what ‘is’? This is one of the fundamental aesthetic questions of the

nineteenth but above all the twentieth centuries, and Mallarmé’s ‘quavering’ code is one

poetic attempt to answer it. A literature of the ‘perhaps’ offers a formal approach to an

infinite art: one that can actualise the infinite ‘perfection beyond perfection’ rather than

alluding to a temporal experience of eternity. In other words, can there be an art that is

infinite or thinks infinity rather than describing infinity or thinking about the infinite?

60 Jean-Michel Rabaté, 1913: The Cradle of Modernism (Malden and Oxford: Wiley-Blackwell, 2007), 3.

68

For Mallarmé, any art capable of thinking infinity would have to also be an art that was

entirely based on chance. Here, it is the non-affective chance rather than the experience

of endlessness that is the formal cipher for the infinite: chance, as perpetually

indeterminate, offers a kind of poetic or textual ‘actualisation’ of the infinite without

having recourse to (finite) sentiment. A literary infinity would have to realised through a

combination of the absurd, and the hypothetical, perpetually insecure idea:

In short, in an act where chance is in play, it is always chance that accomplishes

its proper Idea in affirming or denying itself. Before its existence, negation and

affirmation are exhausted. It contains the Absurd – implies it, but in the latent

state, preventing it from existing: which permits Infinity to be.61

Meillassoux would claim, thus, that the ‘unique number’ is at once literally manifest in

the poem and, simultaneously, that it can never be directly ‘extracted’ – that it will

always be suspended in chance. It is this double logic of the code that allows it to

straddle a contradiction: the code will perpetually instantiate chance. Key to this is the

relation between reader and the poem – a relation echoed in the formulations by

Symons and Kenner above. Although Meillassoux does not put it in these terms, we

might claim that the code manages to present an instance of ‘infinite chance’ and

‘unique number’ because the reader’s apprehension of the number will always exceed

poetic evidence, and the poem’s dimensions will always exceed a single reading. It is the

instability and disjunction between the reader and poem that Meillassoux is preoccupied

with: the infinite chance is rendered only through a speculative reading of the poem.

61 Mallarmé quoted in: Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés.’ 29-

30.

69

If we were to return Meillassoux’s reading of the Coup de dés to literary

interpretation, rather than ‘deciphering,’ we could recuperate this ‘ambiguity’ of the

code as not a necessary feature of the text but rather the necessary dynamic (dialectic,

even) between text and reader that is traditionally necessary to the definition of allegory.

This is not to say that the ‘unique’ number is purely a product of reading, but that

Mallarmé understood that the only way to render unique number is via allegorical form.

The definition of allegory that I refer to here is not the hackneyed, one-dimensional

‘resolution’ of a text via a single authoritative process but rather something quite

different, a perpetual dialectic of reader and writer. Frederic Jameson sums up this

vision of allegory best:

our traditional concept of allegory [...] is that of an elaborate set of figures and

personifications to be read against some one-to-one table of equivalence: this is,

so to speak, a one-dimensional view of this signifying process, which might only

be set in motion and complexified were we willing to entertain the more

alarming notion that such equivalencies are themselves in constant change and

transformation at each perpetual present of the text.62

This aspect of allegorical interpretation is relevant to Meillassoux’s insistence on the

unique number necessarily being ambiguous: the reading of the text oscillates between

the clear presence of a number and the abolishment of the code (just as the Siren

abolishes the rock upon which the Master’s ship crashed). The ‘oscillation’ resembles

precisely the polysemy and ‘constant change and transformation’ that Jameson

62 Frederic Jameson, ‘Third World Literature in the Era of Multinational Capitalism,’ Social Text, no. 15

(1986), 73.

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designates as allegorical. The most pertinent allegorical element, here, is a formal

allegory: a situation where the text allegorises its own formal processes. It is clear that

the poem directly addresses the crisis in French verse through the metaphor of the

sinking ship, and suggests a solution through the figures of the siren and the Septentrion.

In this sense it is an allegory is a ‘formal’ one in that it allegorises the very processes of its

own composition. Formal allegory was most famously theorised by Paul de Man. For de

Man interpretation is always an act of reading another meaning into the text and ‘any

narrative is primarily the allegory of its own reading.’63 This rather complex circuitry of

innate textual allegory happens via what de Man calls the ‘rhetorical model of the

trope.’64 In its structure of deferred or displaced reference, the model of trope mirrors

the model of reading itself, in de Man’s argument. In de Man’s theory of literature and

meaning all language involves a displacement between referent and significance. A

symbol of an olive branch, for instance, refers to ‘reconciliation’, but this meaning is

entirely independent of the olive branch itself. The referent – the olive branch – is

independent of its significance, which occurs by virtue of interpretation. This

displacement is essential to all language: its medium always skirts what it purports to

capture. In de Man’s powerful rereading of this essential feature of language, all figural

form in fact allegorises its own reading: the very construction of the trope, which

involves a divergence and path between referent and significance, is that same as the

process that occurs between text and reader, words on a page and the instability of their

interpretation. This we can name ‘formal allegory’: an instance where the text

allegorises its own formal processes. On one level this occurs through Mallarmé’s direct

engagement in this poem with a crisis in French verse: the entire ‘plot’ of the Coup de dés, 63 Paul de Man, Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust (New

Haven and London: Yale University Press, 1979), 76. 64 Ibid. 15.

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if we can call it a plot, revolves around a complex allegory of the destruction of a poetic

limitation (the rock) and the ascension to a celestial rather than earthly alternative: the

Septentrion. On another level, however, a formal allegory occurs precisely in

Meillassoux’s reading of the code in the text. Meillassoux discovers this anew when he

shows that the presence of a code can never be ‘finalised’ – that the suggestion of the

code will oscillate between certainty and uncertainty. The code inscribes, above all, the

poem’s own limitations in establishing semantic meaning: through reference to a

number, to composition, that ultimately undergirds any ‘pure’ symbolism. Equally, the

poem stages its own limitation: it cannot produce a ‘unique number that cannot be

another’ but can allegorise this, through the ‘quavering’ placement of the number 707

in the text. Likewise, the figure of the throw of the dice so central to Meillassoux’s

reading allegorises the process of simultaneous chance and determination found in the

very production of a poem: the inevitably unstable dialectic between writer and reader,

intention and reception, composition and distribution. The code is, then, is ultimately

allegorical in the same sense that de Man speaks of allegory.

On one level Meillassoux does recognise this in Mallarmé’s text (claiming it for

eternity, and infinity, by virtue of its self-referential delineation of its own poetic task),

but he falls into the trap of understanding this to be peculiar to a text with a code, rather

than to poetic form that embraces a certain form of allegorical doubling. My reading of

Meillassoux’s Number and Siren as an allegorical act (more properly than a deciphering)

does not refute his reading necessarily, but rather places it in a literary context that I

take to be necessary to understanding a peculiarly poetic relation to the ‘unique

number.’ Understanding the unique number – 707 – in the Coup de dés as allegorical on

the levels of both composition and interpretation allows us to recognise a form that is

also central to Cantor’s proofs: semantic antinomy.

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So, to ask the question again, then: What might a peculiarly literary transfinite

look like? The hypothesis that I will test over the following chapters is this: a literary

transfinite, a properly modern concept of the infinite in literature, will operate according

to that ‘doubling’ process crucial to Cantor’s uncountability proofs; the application of a

system to itself. Of course, this same doubling process is found in de Man’s theory of

allegory: when one applies linguistic bifurcation to itself on a different level (the

phenomenology of reading rather than the formal processes of the text) we find an

allegorical structure between the creation of meaning intra-text and extra-text. This

doubling, here, will involve the allegorical structure that we see in Mallarmé’s work, and

which Meillassoux elucidates through what he calls a ‘deciphering.’ This formal

realisation of a literary transfinite will occur through what can be named ‘transfinite

allegory.’

*

1.4 ALLEGORY AND ENIGMA

What unites the analyses by Gray, Meillassoux and Badiou is a conviction that the

innovations and implications of Cantor’s work are not exclusive to mathematics. This is

not to say, however, that the mathematical theory of multiplicity, and the grasp of

infinite extension is the same as forms or concepts of this in other domains. These three

thinkers do not even make the claim that it is analogous: this would involve a ‘similarity’

between literature and mathematics that is not visible other than with extensive

speculation. For Badiou, art and mathematics are irreconcilable but share a truth, and

become modern at different times and through the procedures that follow from different

events. In Gray’s work, it is the radical transformation of the discipline and the shift

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towards formalisation that links mathematics and literature. And for Meillassoux, there

is no direct link, only Mallarmé’s singular conviction that a ‘unique number’ could

contribute to a poetic infinite. I will argue here that from a literary perspective the right

form of comparison or linkage between these two domains is allegorical.

Rita Copeland and Paul Struck provide a basic definition of allegory that

indicates its origins in displaced speech: ‘The term itself has Greek origins: allos (other)

and agoreuein (to speak in public), produce a sense of ‘other speaking.’ In its most

common usage it refers to two related procedures, a manner of composing and a

method of interpreting.’65 The descriptor ‘allegorical’ can refer to both a form of

reading, which seeks the meaning of (or in) an ‘other speech’ as well as a mode of

composition. But indeed, this double implication of the word is perhaps not double at

all: the method of composing an allegory originates in a kind of creative reading of one

‘speech’ into another. Key to the entire process of both allegorical reading and

allegorical composition is the consistency and autonomy of each narrative. Allegorical

form rests on certain vision of composition, which anticipates a reader who can locate a

meaning, a tale or a significance that is not properly manifest in the content of the

allegory. It is this aspect of allegory that was so important to Paul de Man, and which he

‘doubles’ by reading allegorical structure into the very form of language itself.

Allegory has not always been privileged as a literary form: for Coleridge,

famously, allegory was associated with ‘translation’ whereas symbolism was a form of

‘translucence.’ It was a simple and stodgy commutation of meaning. Indeed, if we follow

the narrative of Copeland and Struck, the nineteenth century privileged symbolism over

allegory based on the perceived aesthetic immediacy of the former and the deductive,

65 Rita Copeland and Peter Struck, The Cambridge Companion to Allegory (Cambridge and New York:

Cambridge University Press, 2010), 2.

74

conceptual requirements of the latter. Even in the criticism of the last quarter century

allegory has remained a suspect category because, in Imre Szeman’s words, it is in

danger of creating ‘a presumed passage from text to context that is epistemologically

and politically suspect’ because of ‘the naive mode of one-to-one mapping that it seems

to imply.’66

And yet the definitions of allegory produced by Jameson and de Man discussed

above suggest a different potential in allegorical form and reading. In these definitions,

what becomes interesting about allegory is precisely the absence of an immanent

meaning contained within one form: it is the mutually exclusive coexistence of two

‘speeches’, two ‘narratives’ that are co-implicated precisely by being the ‘other’ of the

pair. This ‘other’ is not a measure of difference (this story claims one ending, the other

claims a divergent one) between the two texts, but a different existence of the text,

perhaps on a different plane (the transcendent values, for instance) or a different book

(the Bible). This structure befits the relationship between literature and mathematics:

two utterly autonomous worlds that are bound together in their form and implication.

At the same time, this structure befits the way that mathematics is used in literature: in

order to introduce another logos into the representational scheme of fiction. Fiction

requires this allegorical ‘doubling’ in order to properly produce its own infinity.

Allegory has been related to both revelation and to the pursuit of enigma. Here,

number functions as the enigma of language for these three writers working within and

after an era of modernism. A literary transfinite relies, as the mathematical one does, on

a ‘doubling’ of its measures. Cantor’s diagonal proofs relied on a problem created by

using numbers in ‘two senses’: on the one hand as the index and on the other hand as

66 Imre Szeman, ‘Who’s Afraid of National Allegory? Jameson, Literary Criticism, Globalization,’ The

South Atlantic Quarterly 100, no. 3 (2001), 806.

75

that which is indexed. My analyses over the following three chapters will show that

where we find a literary ‘transfinite’ (or, ‘actual infinity’), we also find just such a

‘doubling’ process. This doubling takes the form of an allegory, most importantly the

peculiarly literary form whereby a text allegorises its own formal processes.

Just as allegory involves both a method of reading and a method of interpreting,

so too does allegory occur in both directions in this study. This allegorical process is not

only a theoretical model for thinking the interconnectedness between mathematics and

literature. I’d like to suggest, in this dissertation, that it is also the mode by which

literature approaches mathematics. Here we have the Boromean knot between the

tranfinite in literature, mathematics and the model that unites the two in literary theory:

allegory. Allegory, and in particular formal allegory, is key to each of the writers that I

analyse in the following chapters. Most importantly, this allegorical process enables

them to produce prose that includes or approaches precisely that which exceeds it:

totality, being outside of language, and above all the processes of counting and

quantification that structure their prose (or, in the case of Beckett, a television play as

well). Cantor relies on a doubling of number in his uncountability proofs in order to

prove the uncountability of real numbers and to show that the infinity of real numbers is

larger than the infinite of natural numbers. It is the conjunction (and disjunction) of

these two forms of counting that produces something radically new. In Mallarmé, the

poet ‘doubles’ the metrical essence of the poem to produce a ‘quavering’ between

chance and determination, finitude and eternity, a literary crisis and its potential but

unfulfilled solution. In Borges, Beckett and Coetzee we will see a doubling of the task of

prose literature through ‘transfinite’ allegory, which indexes the enumeration of the

prose as it proceeds. This formal doubling produces a literature that revolves around its

own enigma: number, and the revelatory capacities of this.

76

77

2 CHAPTER TWO:

JORGE LUIS BORGES AND THE MEASURE OF PROSE

There is a concept which corrupts and upsets all others. I refer not to Evil, whose

limited realm is that of ethics; I refer to the infinite.

Borges, ‘Avatars of the Tortoise’

The great Argentine miniaturist, Jorge Luis Borges, held numbers to be essential to the

reconstitution of both poetry and prose, after what he saw as a saturation and loss of

vitality in each form at outset of the twentieth century. Here I will contend that this

connection between numeracy and artistic form shapes his later short prose, in

particular the allegorical form that would carry Borges’ artistic innovation past what he

took to be stagnant forms. Borges’ perspective on late nineteenth-century poetry, and

the failings of this form, are apparent in the manifestos and experiments that came out

of his participation in avant-garde groups in Spain the early 1920’s. Whilst he and his

family were stranded in Europe during World War One, the young Borges became

involved with the ultraists in Seville and Madrid, returning to Argentina in the early

1920’s convinced that primordial metaphor would rescue verse from poetic

ornamentation. Borges eventually disowned this ‘ultraist’ enterprise, but it would

provide the foundations for his persistent preoccupation with fictional worlds created

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without rigid or naïve symbolism, and his commitment to the construction of fictional

worlds through systems of abstraction, suggestion and – most importantly – a

mathematised order or imagination. Borges’ perspective on the saturation of the novel

form is made explicit in an essay on the novel published in 1954, entitled ‘A Defense of

Bouvard and Pécuchet.’ In this essay, Borges argues that Flaubert, creator of the realist

novel, is also the first to ‘shatter’ it.1 Within this discussion of Flaubert Borges makes a

significant aside about numerals in Ulysses and the death of the novel:

Flaubert instinctively sensed that death [of the novel], which is indeed taking

place (is not Ulysses, with its maps and timetables and exactitudes the magnificent

death throes of a genre?), and in the fifth chapter of the work, he condemned the

‘statistical or ethnographic’ novels of Balzac and, by extension, Zola.2

Numbers, here, are the harbingers of the death of a genre. The statistical quality of

Balzac and Zola’s narratives (an enumeration or account of a reality) perhaps give a sense of

the death of the genre, but it is in Joyce’s self conscious inclusion of these statistics in

Ulysses that we see this most clearly. James Ramey, in a study of Borges and Joyce,

agrees with Cesar Augusto Sagado that Borges’ relationship to the novel is

‘eschatological’: Borges writes in (and of) an end-time of this prose form.3 And, above all

in this short essay, Borges associates these death throes with number. Death throes, of

course, are the performance and drama of the death of the genre, which he takes to be

1 Jorge Luis Borges, ‘A Defense of Bouvard and Pécuchet,’ in Selected Non-Fictions, ed. Eliot Weinberger,

trans. Esther Allen (New York: Penguin, 1999), 389. 2 Ibid. 3 James Ramey, ‘Synecdoche and Literary Parasitism in Borges and Joyce,’ Comparative Literature 61, no. 2

(2009), 142.

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staged in Ulysses, a book that Borges had, elsewhere, called a ‘miracle.’4 In Borges’ eyes

Joyce is attuned to the failure of that novelistic relation to the ‘actual’ instituted by

Defoe, Fielding and others, whose prose numeracy grounded characters as subjects of

their world, facilitating the drama and immediacy that would make the novel such a

cultural force over the following centuries. Borges needed a new relation between prose

and number: one that does not enumerate the reality it sets out to describe or reduce

realism to the staid presence of what Flaubert recognises, in Bouvard and Pécuchet, as the

‘statistical’ or ‘ethnographic’ novels of Balzac. This critical understanding of the relation

between number and literature would be resolved in his short stories through an

unprecedented, transfinite mathesis of fiction.5

Borges had an explicit and sustained engagement with Cantor’s

transfinite numbers. He was a reader of mathematical texts, including Russell and

Whitehead’s Principia Mathematica and Kasner and Newman’s Mathematics and the

Imagination, and in his essays and stories he engaged directly with the issue of an actual

infinity. This engagement would shape many of his short stories, not only those that he

collected in the volume named for Cantor’s choice of symbol for his transfinite numbers,

The Aleph.6 Indeed, Borges took a broad interest and enjoyment in mathematics. In his

1940 book review of Mathematics and the Imagination, he predicts that, in addition to

4 Borges reviewed Ulysses in superlative terms in 1925. See: Jorge Luis Borges, ‘Joyce’s ‘Ulysses,’’ in Selected

Non-Fictions, ed. Eliot Weinberger, trans. Suzanne Jill Levine (New York: Penguin, 1999), 13. 5 For Michel Foucault, a ‘mathesis’ was the ‘general science of order; a theory of signs analyzing

representation; the arrangement of identities and differences into ordered tables.’ A ‘universal method’ of

matheis is, for Foucault, ‘algebra.’ See: Michel Foucault, The Order of Things: An Archeology of the Human

Sciences (London and New York: Routledge Classics, 2002), 79. 6 Floyd Merrell suggests that there is a crucial difference between Borges and Cantor: Cantor hoped that

the paradoxes of his set theory would be resolved, making a total system. Borges, on the other hand, had

no such hope, and reveled in the paradoxes. Floyd Merrell, Unthinking Thinking: Jorge Luis Borges,

Mathematics and the New Physics (West Lafayette, Indiana: Purdue University Press, 1991), 61.

80

‘Mauthner’s Dictionary of Philosophy, Lewes’ Biographical History of Philosophy, Liddell Hart’s

History of the War of 1914-1918, Boswell’s Life of Samuel Johnson, and Gustav Spiller’s

psychological study The Mind of Man’ this volume was likely to become one of the works

that he has ‘most reread and scribbled with notes.’7 Borges was particularly well versed

in the consequences of Cantor’s theory of the infinite, and its differentiation from

vaguer, non-mathematical visions like Nietzsche’s ‘Eternal Return.’ In an essay entitled

‘The Doctrine of Cycles,’ Borges engages with Cantor’s work on infinity in order to

disprove the ideas of Eternal Recurrence of Nietzsche’s Zarathustra. ‘[Cantor] asserts

the perfect infinity of the number of points in the universe, and even in one meter of the

universe, or a fraction of that meter. The operation of counting is, for him, nothing else

than that of comparing two series,’8 Borges explains. His illustration, as ever, is vivid:

For example, if the first-born sons of all the houses of Egypt were killed by the

Angel, except for those who lived in a house that had a red mark on the door, it

is clear that as many sons were saved as there were red marks, and an

enumeration of precisely how many of these there were does not matter. [...]

The set of natural numbers is infinite, but it is possible to demonstrate that,

within it, there are as many off numbers as even.9

In Borges’ argument, Nietzsche’s eternal return (or ‘eternal recurrence’) is impoverished

because it falls into the trap of assuming that an infinite number of particles cannot be

7 In this review Borges also refers to Bertrand Russell’s classic Principia Mathematica, which he took copious

notes from. See: Jorge Luis Borges, ‘From Allegories to Novels,’ in Selected Non-Fictions, ed. Eliot

Weinberger, trans. Esther Allen (New York: Penguin Books, 1999), 337. 8 Jorge Luis Borges, ‘The Doctrine of the Cycles,’ in Selected Non-Fictions, ed. Eliot Weinberger, trans.

Esther Allen (New York: Penguin, 1999), 116. 9 Ibid. 116-117.

81

presented an infinite number of ways by virtue of finite force. What is attractive for

Borges in Cantor’s theories of the infinite is the profusion of forms of counting that

negate Nietzsche’s single count. In Cantor’s vision, the infinity of natural numbers and

the infinity of points in space belong to different sets and are incommensurable. These

counts give way, for Borges, to an infinite generation; an intellectual prospect that

Nietzsche’s theory is devoid of by virtue of the fact that there is only one infinity, not

multiple infinities of different sizes. This modern vision of infinite generation is more

than simply an intellectual intrigue for Borges. Here, I will argue that it is the Cantorian

proofs of ‘actual infinites’ and generic difference that will provide Borges with a prose

form that possesses the vitality that he felt the novel and poetry had lost. This form I will

call a transfinite allegory.

Anthony Cascardi claims that Borges’ modernism relies on the rejection

of literature as a repetition of the world. Instead, Borges’ approach would be to create

worlds. Implicitly, Cascardi’s remark suggests a numerical task: the avoidance of

doubling and the production of a singularity, or a genuinely new ‘number.’ ‘The work of

art,’ Cascardi writes, can ‘reassert its claim to be something more or other than a

mimesis of the world, in part by reflecting on the impossibility of it ever being a full and

complete mimesis of the world.’10 In this vein, Cascardi claims that the ‘imperfections’

worked in to Borges’ worlds ‘suggest how art remembers what it was like to be a world,

not just to be like the world.’11 This is a kind of refusal to ‘clone’ the world through

description, one that entails a refusal of broader representational completion. These

imperfections, in Cascardi’s argument, speak to art’s ‘memory’ of the time in which it

10 Anthony Cascardi, ‘Mimesis and Modernism: The Case of Jorge Luis Borges,’ in Literary Philosophers:

Borges, Calvino, Eco, ed. Jorge J.E. Garcia, Carolyn Korsmeyer, and Rodolphe Gasché (New York and

London: Routledge, 2002), 116. 11 Ibid. 116.

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did create a world: the time of myth and epic that preceded the Romance and novel.

This mimesis becomes the product of another mimesis: a world contingently created

from another. This is an act not of linear progression but ‘permutations and

combinations.’12 These worlds do rely on permutations, combinations, axioms and,

most importantly, transfinite numbers. Floyd Merrell links ‘the demise of totalising

narratives’ that Cascardi is speaking of directly to Cantor’s infinities, citing Cantor’s

remark that ‘the least particle contains a world full of an infinity of creations.’13 There

are two problems, here: a literary one and a mathematical one. The first is the problem

of the literary creation of novelty and the state of Barthesian ‘exhaustion’ that Cascardi

cites: the state of cultural saturation whereby literature comes to take other literature as

its subject matter. The second is the mathematical paradox of completing the ‘set of all

sets.’ Merell puts this perfectly in his gloss of Cantor: ‘everything is contained in

everything else.’14 Looking at this through the combined perspective of literature and

mathematics, this appears, then, not as recollection of a lost mimesis (as Cascardi feels it

does) so much as a mathesis of literature; this is not a restitution of a descriptive totality,

so much as a modern awareness of the mathematical presence of multiple totalities, each

of different measure, and the representational demands made by this new concept of

infinite. A problem of literary creation and a problem of modernism come together

here. In each case what is required is the transfinite operation upon the totality of

writing, or, to follow Cantor, the totality of sets.

Number is announced, in Borges’ worlds, as presentation and not

representation, as the alogos removed from conceptual, rational and narrative

integration. In this sense, a reading of number in Borges’ work is also a means of 12 Ibid. 113. 13 Merrell, Unthinking Thinking: Jorge Luis Borges, Mathematics and the New Physics, 61. 14 Ibid.

83

probing the fictional form that, as Jaime Alazraki puts it, leaves his readers to ‘search

and strive for some unturned stone, for an undiscovered pearl waiting, iridescent – that

is, full of new insights – at the bottom of some recondite line, on the reverse of an

overlooked ambiguity, in the elusive meaning subtly intimated between the lines. Borges

has turned us all into inquisitive Kabbalists.’15 This often hidden ‘presence’ of number is

essential to the return to the Platonism of Borges’ fiction, and allows for an aesthetic

fidelity to the numerical exception from representation. It is not, however, a simple

occlusion of semantic meaning that qualifies number as an instance of textual presentation

rather than representation. In ‘Funes, His Memory,’ for instance, we have the subjective

apprehension of pure irrational number locked in the paralysed body of the character

Funes. This is typical of the new mode of allegory that Borges produces, where the

referent is not to some other world, or to some other tale (as the literature of exhaustion

might require) but to another infinity contained within but inaccessible to the

denumerable world that he sets out.

*

2.1 SUPPLANTING THE SYMBOL FOR THE ‘THING ITSELF’: BORGES’

ULTRAIST BEGINNINGS

Criticism on number in Borges’ work tends to follow grooves of either ahistorical

or adaptive interpretation. There is a plethora of scholarly work on mathematics in

Borges’ fiction that has developed explicit connections between the content of these

stories and mystical mathematics (the midrash and the kabbalah), the ‘new’ physics, set

15 Jamie Alazraki, Borges and the Kabbalah: And Other Essays on His Fiction and Poetry (Cambridge:

Cambridge University Press, 1998), ix.

84

theory, and information theory.16 In addition to this, Borges’ work has been read in

terms of discoveries in genetics and technological advances in computing and quantum

physics. By and large, these analytic connections between Borges’ stories and scientific

or technological events are adaptive: they show how elements of the story resonate with,

anticipate or echo these events, without necessarily being tied to a narrative of literary

form and development. These analyses demonstrate connectivity and the ongoing

cultural significance of Borges’ work in contexts that he perhaps would never have

anticipated. There are two full monographs devoted to mathematics in Borges’ work

both of which are ahistorical. William Goldbloom Bloch’s classic, exhaustive account of

mathematics and Borges, focuses specifically on ‘The Library of Babel,’ with a

secondary consideration of ‘The Book of Sand.’ Guillermo Martinez’s full length study

looks at a range of Borges’ stories, extending his analysis to include discussions of

artificial intelligence, logical systems, reasoning and the form of the proof. Bloch – a

mathematician – reads Borges’ imaginary library in terms of combinatorics, information

theory, real analysis, topology, cosmology, geometry, graph theory and homomorphics.

This exhaustive account of the various modes of mathematical analysis in ‘The Library

16 See for instance: Ibid.; Merrell, Unthinking Thinking: Jorge Luis Borges, Mathematics and the New Physics.

Broader critical engagement with the mathematical, scientific or abstract worlds in Borges’ stories tends to

draw out the affinity between various technological or scientific events and Borges stories. Daniel Dennett

has argued that ‘The Library of Babel’ illustrates principles of genetics [see Daniel Dennett, Darwin’s

Dangerous Idea: Evolution and the Meanings of Life (New York: Touchstone, 1995)]; Quine has noted that the

entire Library of Babel could be written from binary code [see Willard van Ormen Quine, ‘Universal

Library,’ in Quiddities: An Intermittently Philosophical Dictionary (Cambridge: Harvard University Press, 1987)];

N. Katherine Hayles has envisioned a ‘digital equivalent’ of ‘The Library of Babel,’ [see N. Katherine

Hayles, ‘Subversion: Infinite Series and Transfinite Numbers in Borges Fiction,’ in The Cosmic Web:

Scientific Field Models and Literary Strategies in the twentieth-century (Ithaca: Cornell University Press, 1984)]; and

Thomas P. Weissert has considered Borges’ stories in terms of quantum theory, [see Thomas P. Weissert,

‘Representation and Bifucation: Borges’ Garden of Chaos Dynamics,’ in Chaos and Order (Chicago:

Chicago University Press, 1991).]

85

of Babel’ is vital to any reading of mathematics in Borges’ work. The unifying strand in

Bloch’s work is the notion of ‘elegance’ in mathematics, which is simultaneously

illustrated in and resonates with the elegance of Borges’ stories. Bloch explains and

experiments with the mathematical elements of ‘The Library,’ without necessarily

associating this with a wider literary or cultural shift: he reads Borges as a

mathematician, in other words. This work is thus (deliberately) ahistorical: it separates

the mathematics in Borges’ work from its contribution to the literary impasses that

Borges work faced. In order to fully understand the emergence of such numerical

fiction, and its function in terms of symbolism and reference, it is necessary to situate the

stories within and against Borges’ participation in the Spanish and later Argentine

avant-garde, and the issues that animated these groups, in addition to his preoccupation

with both ancient and contemporary mathematics.17

Borges occupies an enigmatic position on the fringes of modernism. He was an

eclectic and highly idiosyncratic reader and his influences and allusions render his work

anomalous as regards larger literary movements of the early twentieth century.

Nonetheless, he was also an active and formative member of several literary collectives

and journals in Spain and Argentina, and his early years as a student were formed by

collective endeavours to move away from the limitations of nineteenth-century

symbolism and initiate new form of poetry focussed on a generative rather than

17 Borges’ mathematical readings are evident in the citations in both his non-fiction. In 1942 Borges made

notes on Bertrand Russell’s Principia Mathematica. Here, Borges draws connections between Russell and

Wordsworth, and Russell and Tristram Shandy. This is the most direct archival linking of mathematics

and literature in Borges’ thinking and reading. In his notes for page 358 of the Principia Mathematica,

Borges draws the comparison between the premise of Sterne’s novel and the paradoxes that Russell

discovers in the Principia Mathematica. See Jorge Luis Borges, ‘Cuaderno Avon / Green Notebook,’ (Jorge

Luis Borges Papers, Harry Ransom Humanities Research Centre, The University of Texas at Austin,

Container 1.14, 1954), n.pag.

86

evocative practice of symbolism that would undergird his approach to literature and

number, even after he abandoned poetry. The early career of Borges is defined by his

participation in the ultraismo circles, a literary collective decisive in the development of

Borges’ mathematical fiction. Despite the fact that his poems and manifestos of this time

related an extreme form of idealism, and were eventually disowned by Borges, they

provide the scaffolding for the questions if not the answers that would contribute to the

role of number in his fiction. Borges was associated, in 1921, with ultraist writers of

Madrid and Seville that opposed modernismo, the aesthetic ideological cluster that

Spanish poetry had orbited around since the 1890’s, whose most famous practitioner

was the Nicaraguan poet Rubén Darío (1867-1916), who was responsible for inspiring a

wave of modernismo writing in Latin American writers including Jose Marti, Julian del

Casal, Salvador Díaz Mirón, José Asunción Silva, and Manuel Gutiérrez Nájera.18

The term modernismo is not the Spanish version of ‘modernism’; in fact,

modernismo, although obviously connected to European modernism, describes an earlier

phenomenon in Spanish literature which is characterised by reaction against the brute

materiality of quotidian modern life, and hence an attraction to the transcendental

possibilities that the Symbolists placed in poetry and the Parnassian refutation of

positivism.19 The Ultraists were a self-styled avant-garde group whose manifesto for

poetry also harked back to the Parnassians, revolving around the poet’s work as creation

rather than representation, and a poetry that had the capacity to generate a certain

image, experience or state, above and beyond any mimetic endeavour. They gathered at

the Café Colonial in Madrid, their conversations and publications revolved around

18 Kelly Washbourne, ‘Introduction,’ in After-Dinner Conversation: The Diary of a Decadent by José Asunción

Silva (Austin: University of Texas Press, 2005), 8. 19 See Gwen Kirkpatrick’s discussion of this in: Gwen Kirkpatrick, The Dissonant Legacy of Modernismo

(Berkeley and Los Angeles: The University of California Press, 1998), 38-43.

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renewing a symbolism that they took to have become ornate and rigid within modernismo.

In Borges’s own words,

Ultraism in Seville and Madrid was a desire for renewal; it was a desire to define

a new cycle in the arts; it was a poetry written as if with big red letters on the

leaves of a calendar and whose proudest emblems – airplanes, antennae, and

propellers – plainly state a chronological newness.20

This renewal involved a shift to ‘modern’ images and content, in part, but also involved

a direct reformation of the symbol. The ultraists sought to wage a literary war against

the excesses of sentiment that they saw committed in modernismo, and did this through an

attempt to generate a primordial sort of metaphor through poetic conjunctions: ‘The

Ultraist movement’s [...] programme affirmed the image as the fundamental element in

poetry, the abolition of logical and syntactical links, and the brevity of the poem as a

formal proof of the condensation of meaning.’21 For the ultraists, primordial metaphor

was to replace stock symbols to generate a new, revitalised poetic idealism. Each

instance of primordial metaphor would produce a moment of transcendental unity, very

much in line with symbolist visions of the infinite, or the ideal ‘One.’

Ultraism was a French import. Spanish writers returning from France brought

with them texts and influences, helping to expose Borges to the French Symbolists, most

20 This quote was published in Inquisitions in 1925, and the translation cited is from Emir Rodiguez

Monegal See: Emir Rodiguez Monegal, Jorge Luis Borges: A Literary Biography (New York: Paragon House

Publishers, 1988), 173. 21 Sarlo, Jorge Luis Borges: A Writer on the Edge, 113.

88

significantly the great early experiments in generative poetry of Mallarmé and Remy de

Gourmont:22

Through a local master, Rafael Cansinos-Asséns – who had already moved to

Madrid – [the young poets of Seville] discovered the modern movement that in

only two decades had profoundly changed European literature. In little

magazines published in Madrid and Seville they had come across some of the

key names of the period: Mallarmé and Apollinaire, Marinetti and Tzara,

Cendrars and Max Jacob.23

The influence of the Symbolists and the Parnassians upon modernismo and ultraism is

complicated: there is no broad base for the influence of either of these groups. Ultraism

was a reaction against modernismo, which was deeply influenced by the French Symbolists

and Parnassians, and yet ultraism would develop its own relation to the symbol and the

ideal that also profits from the legacy of those same groups. Borges took the Spanish

avant-garde movement back to Argentina with him, publishing an ultraist manifesto in a

literary journal that he initiated. This journal was called Prisma: Revista Mural and was

printed on big poster sized cards of brown paper, the poems flanked by woodcuts by his

sister, Norah. There were only two issues of Prisma: one from December 1921 and the

other a few months later, in March 1922. Upon seeing the posters Alfredo Bianchi,

editor of the established literary magazine Nosotros, invited Borges and his collective to

publish a special issue of Nosotros on ultraism. Borges published another manifesto in this

commissioned issue of Nosotros entitled ‘Ultraismo,’ in which he outlined the tenets of the 22 For details on Borges’ readings of the French symbolists, see: Monegal, Jorge Luis Borges: A Literary

Biography, 120-121. 23 Ibid. 158.

89

movement. Sarlo paraphrases the three key elements of this manifesto: ‘Its programme

affirmed the image as the fundamental element in poetry, the abolition of logical and

syntactical links, and the brevity of the poem as a formal proof of the condensation of

meaning.’24

Ultraism involved an approach to poetry that focused on the internal coherence

of the poem (rather than external evocation) and the use of archetypes which solved the

problem of tropological language in terms of hierarchy. The archetype was a primordial

signifier, and thus not an efflorescent or ornamental use of language, but one

purportedly connected to the very origins of subjectivity. This at once aligns Borges with

certain more successful strands of European literary modernism and at the same time

differentiates him from these. For instance, Ezra Pound, in the essay Imagisme, notes that

the first rule of the imagistes is the ‘direct treatment of the ‘thing,’ whether subjective or

objective’: a rule that went along with the embrace of free verse under the banner of

poetic necessity, as opposed to poetic ‘dilettantism.’25 It is this simultaneity of treatment

of the thing itself and the emphasis on exactitude that resonates with and predates the

ultraist manifestos. And yet ultraism premised this form of direct verse upon an idealism

antithetical to most modernism: the idealism of ‘primordial signification’ and ‘pure

style.’ It is in this ambiguous status of ultraism that we see an early Platonism in Borges,

albeit one still concerned with a sublime absolute. Ultraism, seeped in a still Romantic

idealism, would be preoccupied with only one number: One.

In what follows I will show that Borges does not entirely reject the fantasy of

pure style after ultraism, though he holds it to be no longer exclusive of erudition,

history and a certain baroque effect. Rather, Borges uses number to buttress a version of

24 Sarlo, Jorge Luis Borges: A Writer on the Edge, 113. 25 F.S. Flint, ‘Imagisme,’ Poetry: A Magazine of Verse 1, no. 6 (1913), 199; 200.

90

pure style that is not puritan, anti-intellectual or, indeed, totalitarian. His move away

from ultraist ‘symbolism’ will allow him to retain a preoccupation with the internal

coherence of the text, and the inclusion of the generative principles of fiction within the

text itself. The drive for textual autonomy in ultraism constitutes Borges’ first steps

towards an art that – to again use Cascardi’s phrase - asks ‘What it is like to be a world.’

In his later work, this project comes to involve a fictional incorporation of both an origin

and the world it produces instead of being couched in a still-Romantic idealism that

seeks a link to a transcendental real in which metaphor or allegory is secured. Allegory

here becomes relevant not because there is a connection to a primordial meaning or

realm of narrative but because it instead performs a formal doubling that incorporates a

reflection of the processes of composition in the text: its own condition of meaning.

*

2.2 ‘FUNES, HIS MEMORY’:

A TRANSFINITE TECHNOGENESIS OF PERCEPTION

‘Funes, His Memory,’ (hereafter ‘Funes’) first published in La Nacion on the 7th of May

1942 and subsequently in the collection Artifices in 1944,26 presents a brief, exquisitely

crafted fictional exploration of the phenomenal and representational significance of

26 This story has also been translated by James Irby as ‘Funes, the Memorius’ which bears a more

immediate resonance with the Spanish title ‘Funes el memorioso’ (spelt as either ‘memorius’ or

‘memorious’ depending on the translator’). Although Hurley’s translation is a superior translation, his title

loses the evocation and mystery of Irby’s ‘memorius.’ Hurley decided to depart from the Spanish title in

order not to introduce a strange or idiosyncratic title that might suggest more or otherwise than Borges’

had intended with his Spanish title. In Spanish, memorious, as Hurley notes, is not an uncommon word

and signifies a ‘powerful memory.’ Whilst this is true, I feel it is imperative to include the appellation for

Funes in the title (even if the English word is perhaps rather too suggestive).

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number.27 The story takes the form of a memorial to a dead Uruguayan who possessed

an extraordinary mental transformation as a result of a horse riding accident. The story

is narrated in the first person by an unnamed narrator who is never entirely dissociated

from Borges himself. This narrator describes his two trips to the Uruguayan town of

Fray Bentos and his three encounters there with Ireneo Funes, an adolescent at their

first meeting, and later a young man. The last meeting with Funes takes place in 1887.

Funes is a gaucho, who has ‘the delicate fingers of the plainsman who can braid leather’

and who is subject to two extraordinary cognitive conditions.28

The first encounter between Funes and our narrator is brief but haunting. In

1884 the narrator visits Fray Bentos and goes out riding with his cousin, Bernardo.

Storm clouds are gathering, ‘The wind flailed the trees wildly, and I was filled with the

fear (the hope) that we would be surprised in the open countryside by the elemental

water. We ran a kind of race against the storm.’29 Against this backdrop, they encounter

a boy running along a wall. Borges’ cousin knows the child, and asks him the time;

‘Unexpectedly, Bernardo shouted out to [the boy] – Whats the time, Ireneo? Without

consulting the sky, without a second’s pause, the boy replied, Four minutes till eight, young

Bernardo Juan Francisco.’30 The boy is described as incredibly haughty, his voice ‘shrill and

mocking’, and through this brief exchange he is marked – like the gathering storm – as

an uncanny figure with extraordinary capabilities for the abstract measure of time.31

When the narrator, years later, returns to the town, it is this strange and brief encounter

27 Here I use the version in the Collected Fictions. Jorge Luis Borges, ‘Funes, His Memory,’ in Collected

Fictions, trans. Andrew Hurley (New York: Penguin Books, 1998). 28 Monegal, Jorge Luis Borges: A Literary Biography, 56; 137. 29 Borges, ‘Funes, His Memory,’ 132. 30 Ibid. 31 Ibid.

92

that prompts him to ask after ‘chronometric Funes.’32 The phrasing – chronometric

Funes – registers Funes’ capacity to count time as an attribute, a qualifier, and not yet a

proper epithet as in ‘Funes, the Memorius.’ The appellation ‘chronometic’ implies a

function, the measure of time, where metric is: ‘a binary function of topological space

that gives, for any two points of the space, a value equal to the distance between them,

or to a value treated as analogous to the distance for the purpose of analysis.’33 This

definition is helpful in emphasising the way in which Funes operates in Borges’ and the

readers’ first encounter with him: he traverses an abstract logic of space through a

binary function. A ‘binary function’ is derived from the distance between a two part

structure comprising of two points separated on a topological map, here hours, minutes

or seconds, (3.58 am to 3.59am, or x to y). In this case, the function is the analogue of

the progression of time according to a clock. The young Funes manages – without

visible effort or transition – to abstract from the situation he is in to tell the time. Here,

he is effectively able to tie continuous phenomenal existence to inorganic measure in a

profound and extraordinary way, forcing a first, and literal connection between

phenomenality and its measure, or, in other words, presence and presentation. This

analogy is the pure, isomorphic sort not common to the analogies usually found in

fiction and poetics: this metric for temporal experience is the purest of symbolism,

formalising a perfect coherence between a mode of measure and temporal existence.

At the time of the narrator’s second visit to Fray Bentos, Funes is crippled and

bedridden after a horse riding accident. But this physical debility pales in light of the

profound cognitive transformation that the accident has triggered in him. After being

knocked unconscious in the accident, Funes awakes to find ‘the present was so rich, so

32 Ibid.

33 Oxford English Dictionary, 2nd ed., s.v. ‘metric.’

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clear, that it was almost unbearable, as were his oldest and even his most trivial

memories.’34 Our narrator informs us that Funes remains incredibly proud, yet also

experiences a fundamental transformation of self. If Funes is still proud and haughty, it

could only conceivably be as a defence mechanism, or a relic of his former self: Funes’

memories are so acute that regardless of any physical impairment he is immobilized by

the perfection and acuity of each of these memories.

His transformation is a sort of ‘second birth,’ but a curious one that does not

fully inaugurate a new life:

He had lived, he said, for nineteen years as though in a dream: he looked

without seeing, heard without listening, forgot everything, or virtually

everything. When he fell, he’d been knocked unconscious: when he came to

again, the present was so rich, so clear, that it was almost unbearable, as were his

oldest and even his most trivial memories.35

This characterisation of Funes’ earlier life is less an explanation of a peculiarly amnesic

youth than a comparison with his new state: in contrast to this new total and unrelenting

awareness of detail, Funes did forget virtually everything. Now, he struggles to bear the

acuity of the present. Indeed, we might summarise this simply by observing the

catechrestic movement between the minute (as in the measure of time) and the minute

(as in the profoundly intricate or tiny) that goes on between the two encounters with

Funes. The capacity for counting that Funes possessed before his accident is an

achievement in abstract and continuous thought: the boy running along the wall seemed

34 Ibid. 135. 35 Ibid. 134-135.

94

to be able to maintain a remarkably exact track of time (a form of counting) that was not

dependent on any external factors other than a constant measure. In Funes’ post-

accident existence he is subject to an extraordinary eidetic memory, to a constant

awareness of the minute:

With one quick look, you and I perceive three wineglasses on a table; Funes

perceived every grape that had been pressed into the wine and all the stalks and

tendrils of its vineyard. He knew the forms of the clouds in the southern sky on

the morning of April 30, 1882, and he could compare them in his memory with

the veins in the marbled binding of a book he had seen only once, or with the

feathers of spray lifted by an oar on the Rio Negro on the eve of the Battle of

Quebracho.36

Fascinatingly, the narrator delivers the sense of randomness through specificity: the

inclusion of a date in the above quote emphasises Funes’ all-encompassing memory,

where even the most arbitrary day or scene is subject to an extraordinary, or apparently

infinite, retentional exactitude. This exactitude is not only retentional, though: Funes

also has an exquisite sense of detail that exceeds memory, he is aware of the nuances of

time and space that memory passes over: ‘his perception and his memory were perfect.’37

Of course, this sense of arbitrariness is conveyed through number; the use of digits in the

text emphasises both the apparently arbitrary and precise nature of Funes’ memory.

This strategic detail in the memorial brings into sharp relief the extremity of Funes’

experience: no object bears an identity but rather is a veritable barrage of other events,

36 Ibid. 135. 37 Ibid., italics mine.

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objects and forms; it cannot be determined by a single idea or name. Where number

does not suffice, the narrator clinches this through the rhetorical use of ‘every’ and ‘all,’

capturing multitude without finite measure. It is not only the memory of a book that he

encounters but all the veins in the binding: detail too inconsequential, miniscule and

undifferentiated for normal human perception. The narrator compares this state of

perception to our own geometric intuitions: ‘A circle drawn on a blackboard, a right

triangle, a rhombus – all these forms we can fully intuit; Ireneo could do the same with

the stormy mane of a young colt.’38 This ability to recall abstractions is so great in Funes

that it is inverted: this radical intuition becomes a form of phenomenal nominalism,

where perception is so acute it disqualifies abstraction. In this sense, there is also radical

equality of objects: the detail of an apparently random cloud formation on a certain

afternoon many years ago is as acute as any other memory or event in Funes’ past,

regardless of significance. This might seem to be an exacerbated awareness of singularity, in

the sense that the singularity of each object actually precludes an abstract grasp of the

quality of singularity as such. Yet Funes, immune to abstract thought, has no concept of

the singular, but purely an unrelenting perception of it. The distinction between Funes’

two conditions is at once considerable and barely distinguishable: although there is a

radical difference in the perception of detail, in both cases Funes is not so much capable

of an abstract traversal of space and time, but rather is such a traversal. In the first

instance, however, the function that accompanies Funes’ existence is tied to a

calculation system, in the second instance, it is the anarchy of pure memory. The simple

difference is the presence of a continuum in the former and a plenum in the latter.

In ‘Funes,’ we have not simply a plot structured around a mathematical ordering

system, but one that concerns the moment of correlation between mathematical order

38 Ibid.

96

and phenomenal awareness. Abstract thought is lost to Funes, and as such he is

suspended by what we might call – to modify French philosopher Bernard Stiegler’s

term somewhat – a retentional infinitude that leaves him at the mercy of his extraordinary

memory. ‘When life becomes technical,’ Stiegler writes, ‘it is also to be understood as

‘retentional finitude.’’39 In Stiegler’s theory, retentional finitude refers to the finitude of

memory, or the technical boundaries of memory; it refers to the fact that memories

cannot be contained by the wider entity of memory but are mutable and subject to

loss.40 The notion of retentional finitude is closely related to technics. For Stiegler, the

human experience of time is supplemented by technics: by inorganic measure.

Measured time is a technical division of reality into regular sections: it is the insertion of

a metric into rhythm (and in this sense, despite the fact that the metric is inserted into

rhythm, the conceptual externalisation of rhythm succeeds the metric, in Stiegler’s

terms, ‘techno-genesis structurally precedes socio-genesis’ 41 ). The younger Funes,

running along the wall, has an extraordinary technical cognitive facility: he is able to

maintain a perfect equivalence between his phenomenal existence and the ordering

abstraction of time. The older Funes is paralysed in a cognitive world whereby

overwhelming beauty or stimulation of each lived scene possesses him, rather than

Funes being able – through abstraction and categorisation – to possess the memories.

Funes does retain aspects of his former self: he can still determine correspondences

between phenomena in time, but can now ‘rewind’ or pause this: he can move back and

forwards in memory, zoom in or out and is not subject to the movement of the clock.

This is not an antinomial transformation but a developmental one. The significant

39 Bernard Stiegler, Technics and Time, 1: The Fault of Epimetheus, trans. Richard Beardsworth and George

Collins (Stanford, California: Stanford University Press, 1998), 17. 40 Ibid. 41 Ibid. 2.

97

words, here, are ‘rewind’ and ‘pause.’ Funes’ developmental arc is also the arc of the

new ‘technics of time’ and aesthetic duration found in the filmstrip. (And, notably, the

other name for an eidetic memory like that of Funes is, of course, a ‘photographic’

memory.)

It is here that we begin to see the tripartite arrangement between literature,

mathematics, and other representational technologies. ‘Just as the entire mode of

existence of human collectives changes over long historical periods, so too does their

mode of perception,’ writes Walter Benjamin in ‘The Work of Art in the Age of its

Technological Reproducibility.’42 There are two key changes in perception at work in

this scenario: the first, as I have suggested, is mathematical, the second is

cinematographic. These two are bound up together at the intersection between

perception and calculation. At one point in this story, the narrator bemoans the fact

that, in Fray Bentos, there were no phonographers, no cinematographers to capture

Funes’ abilities. Funes thus exists, literally in his two conditions, and figuratively in his

rural life, on the border between two regimes of technicity: the cinematographers were

not in Fray Bentos for Funes yet, but the future of retentional finitude would be forever

altered by the new perceptive regime of the filmstrip.

Subject to retentional infinitude, Funes is cognitively – not only physically –

paralysed. His state of exquisite memory has made him ‘mechanised.’ This may seem

counter-intuitive: is Funes’ mind not in fact reduced to some pure, not mechanised state

by virtue of its transcendence from any metric? If we pursue the impact of calculation

upon phenomenality and the existence of organic life, this is in fact not the case. For

Stiegler,

42 Benjamin, ‘The Work of Art in the Age of Its Technological Reproducibility,’ 23.

98

the technicization of science [which is also the numeration of science] constitutes

its eidetic blinding. [...] Technicization is what produces a loss of memory, as was

already the case in Plato’s Phaedrus. [...] With the advent of calculation, which

will come to determine the essence of modernity, the memory of originary

eidetic intuitions, upon which all apodictic processes and meaning are founded,

is lost.43

Here, Stiegler’s narrative of modernity falls into a tendency all too common to Borges

criticism. Gabriel Riera has diagnosed two tendencies in Borges criticism: on the one

hand those who read Borges as ‘textualist’ and postmodern, and, on the other hand,

those who see in Borges’ work ‘as stand-ins for the restitution of an auratic reality — or,

as if his staging of the ‘fables of the One’ could be equated with a mystical temptation to

which he finally succumbs.’44 In this diagnosis, Riera is referring in particular to

readings of Borges’ intertextual library and his engagement with pre-modern or pre-

Enlightenment texts, where recourse to texts that might retain a conception of the

sacred, or some divine or transcendental totality, is simply a lapse into this world-view.

Applying Stiegler’s narrative of pre-modern eidetic intuitions to Borges’ stories seems

entirely susceptible to this. Prior to modernity – this narrative tells us – consciousness

was unsullied by calculation, only to be subjected to it post-Enlightenment. And yet,

what remains fundamentally useful here – beyond any fantasy of Borges recuperating a

pre-modern unity of perception and its objects – is the connection between the ‘advent

of calculation’ and eidetic experience. Stiegler’s theory of technicisation, memory and

counting provides an important qualification for the change that Funes undergoes. The 43 Stiegler, Technics and Time, 1: The Fault of Epimetheus, 3. 44 Gabriel Riera, ‘‘The One Does Not Exist’: Borges and Modernity’s Predicament,’ Romance Studies 24,

no. 1 (2006), 56.

99

only way to modulate the acuity of his memory would be through adopting a form of

technicisation – numeration – as a means of facilitating a retentional finitude, as a

means by which to impose a mode of eidetic blinding for Funes’ mind. Using Lamarck’s

distinction between living and non-living beings, Stiegler notes that the living are

necessarily subject to organisation: biology is organisational but anti-mechanical. Here it

is the inorganic, not the organic that attends the experience of pure singularity. This

‘inorganic’ measure is precisely inorganic because not tied to finite (human) experience.

It is, rather, an avatar of an infinite filmstrip, constituted by an innumerable rate of

frames per second, each containing every detail at every time, independent of the

‘action’ or ‘point’ of the frame in a wider sequence. And, as would befit a properly

infinite filmstrip, the infinity, here, is not in the fact that the filmstrip is ‘never-ending’

but that each still has an equally denumerable infinity of points.

Funes tells the narrator that at one time after his accident he attempts to develop

a sort of number system, which he completes to over 24,000 numerals, and that he also

attempts to categorise his days. Funes ‘resolved to reduce every one of his past days to

some seventy thousand recollections, which he would then define by numbers’ but he

gives up, realising (with a quite astounding naivety) that the project is ‘interminable.’45

In this sense, the number system is redundant because it lacks a continuum: a consistent

form of both similarity and difference between numbers, as well as some regulated and

well founded system for the spacing between numbers. In other words, Funes’ numbers

have no clear relationship to a number line; there is no proper metric. Despite the fact

that Funes’ numbers appear to be independent of any numerical continuum, these

numbers are not entirely foreign or absolutely anomalous to mathematics. Funes’

numbers do not negate number: just as Funes’ memory does not constitute an

45 Borges, ‘Funes, His Memory,’ 136.

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annihiliation of memory, so too are these numbers not the antithesis of number. Rather,

this other continuum corresponds to a type of magnitude that does not require

quantification: the irrational number.

In the previous chapter I detailed Cantor’s discovery of transfinite numbers.

Transfinite numbers are always cardinal or ordinal numbers, which measure different

infinities (for instance, the infinity of natural numbers). Transfinite numbers are closely

connected to irrational numbers. Indeed, Cantor called his transfinites the ‘new

irrationals,’ because, like the irrationals, transfinite numbers are also ‘delineated forms

or modifications of the actual infinite.’46 Irrational numbers (pi is the most famous)

cannot be integrated into logorithmic sensory experience or conceptual abstract

knowledge, in much the same way that Funes’ eidetic experience (memory alogos) cannot

be retained as normal, finite memory. The Ancient Greeks would term magnitudes that

were irrational numbers ‘alogos’: without reason. Funes’ memory exceeds finitude –

including the limits of the human senses and the bounds of temporal continuity – but is

not absolute, and it constitutes a measure of a totality without indexing or containing

that totality (a transfinite number is not the end of the numerical continuum). It is thus

that any attempt at restoring a chronometric or categorical function to his memories –

Funes is said to have reconstructed whole days but this process itself actually takes whole

days – is in fact a reliving, an irreducible and incompressible rerun of the day, a number

with no cipher or concept. This overwhelming minutiae is – remarkably – an irrational

phenomenal experience. Borges here presents a form of experience excluded from prose

precisely because it operates within no metric, or is, in set theoretical terms

‘uncountable.’ The technogenesis of Funes’ later consciousness is uncountable in

46 This remark comes from a Cantor’s ‘Mitteilungen zur Lehre vom Transfinite’ written in 1887. This is

translated and cited in: Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, 128.

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contrast to, say, the metric of the watch that defined his earlier self. It does not enact a

function x ! y but, rather, phenomenal existence is tied an infinite extensity. This is not

an absolute (Funes’ memory must depart from his own existence, it is not the memory of

God) but is, rather, uncountable. And yet Funes’ uncountable memory does not make

this an irrational story, or a story that itself contains a certain uncountability. Where

Funes’ mind is associated with the irrational number, the story is more broadly

associated with a transfinite process; the type of process that can, without having to

exhaust all the digits of the irrational number, show that which is uncountable.

Upon his second trip to Fray Bentos, the narrator had been learning Latin, and

had brought several books with him: ‘In my suitcase I had brought with me Lhomond’s

De viris illustribus, Quicherat’s Thesaurus, Julius Caesar’s commentaries, and an odd-

numbered volume of Pliny’s Naturalis historia ....’47 Word gets back to Funes that he is in

possession of these books, which prompts him to send a ‘flowery sententious letter’

requesting loan of the Latin books and a dictionary ‘for a full understanding of the text,

since I must plead ignorance of Latin.’48 Days later, upon arriving at Funes’ home to

collect his two books – the Quicherat and the Pliny – Borges hears Funes reciting Latin,

specifically the ‘twenty-fourth chapter of the seventh book of Pliny’s Naturalis historia,’

whose subject is memory. 49 Given the exactitude of Funes’ memory, it is even

questionable as to whether he is properly reciting the text or whether he is remembering

the moment of reading, raising one of Borges’ favoured problems: what is the nature of

the minimal difference that constitutes repetition? Such a memory would not simply

recall the text but relive it. The narrator explains that the last words of this chapter in

47 Borges, ‘Funes, His Memory,’ 133. 48 Ibid. 49 Ibid. 133.

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the Pliny text are: ‘so that nothing is restored to the hearing of the same words.’50 The

narrator, who had sent Funes the books and dictionary spitefully, to demonstrate the

absurdity of Funes’ pretension to learn Latin with a text and the mere aid of a

dictionary, is quite stunned by Funes’ ability, and indeed the ability to speak in and

follow sentences seems to be the only kind of continuum that Funes can muster. This

last phrase from Pliny resounds as a precaution in the context of Funes’ predicament, in

that it seems to take heed of the function of memory as a bastion (so that nothing is restored)

against irrational retentional capacity – a cinematographic consciousness – that Pliny

seems to be warning against.

The true node of singularity is in Borges’ narration of Funes himself; it is Funes

who appears to us as singular and we are necessarily barred from his experience. The

text opens with a performance of the narrator’s own recollection:

I recall him (although I have no right to speak that sacred verb – only one man

on earth did, and that man is dead) holding a dark passionflower in his hand,

seeing it as it had never been seen, even had it been stared at from the first light

of dawn till the last light of evening for an entire life-time.51

Here Funes is constructed from the ‘outside’, and we see Funes as experiencing an end

of the continuum, not the end of the continuum, which is necessarily barred from

representation and phenomenality and must be embedded, through the narrator’s voice,

in the story. Funes is always at one narrative remove from the text (his voice is mediated

by the narrator) and his form of experience is never replicated, only reported. The

50 Ibid. 134. 51 Ibid. 131.

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repetition of ‘I recall him’ invokes Funes as a singularity, but a singularity that remains

fixed in time for the narrator who retains a finite existence in time and space, a finite

perception and recall. The narrator’s ‘memorial’ becomes a memory of an encounter

with Funes’ memory, a narrative testimony: ‘I recall him – his taciturn face, its Indian

features, its extraordinary remoteness – behind the cigarette.’ 52 The kernel of an

‘uncountable’ consciousness is presented without being represented, and the fidelity to

that aconceptuality is never resolved, lending an enigmatic quality to the story. What is

fascinating here in terms of the relation between mathematics and literature is that

Borges has, in the first instance, managed to embed existence- outside-of-representation

through narrative recursion, an ‘uncountable’ consciousness within a ‘countable’ one.

This intersection between number and the transition from an abstract to a

phenomenological absolute has been overwhelmingly read in the critical literature as

moral fables that are preoccupied with human finitude. Edmond Wright’s claim that the

story strips numbers of some ‘outward’ identity to reveal their ‘nakedness’ does not seem

quite right:

Remove the decimal, the binary (based on 2), the duodecimal (based on 12), the

‘undevigintal’ (based on 19, which is perfectly possible) – if truth be told, remove

a system of mnemonics based on any number whatsoever, and we are left with

the numbers in all their naked glory, each an abstract singularity worthy of the

proper name.53

52 Ibid. 131, italics mine. 53 Edmond Wright, ‘Jorge Luis Borges’s ‘Funes: The Memorious’: A Philosophical Narrative,’ Partial

Answers: Journal of LIterature and the History of Ideas 5, no. 1 (2007), 44.

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The notion of a ‘naked glory’ of numbers, whilst initially appealing, neglects the most

fundamental device that allows numerical presentation to structure a prose story here:

narrative recursion conditioning the experience of ‘singularity.’ Funes’ projects of

counting and his paralysis all contribute to a situation which is defined by the possibility of

presentation whilst never conflating mathematical writing with the writing of prose, or

uncountability with the necessarily ‘countable’ and finite consciousness of the narrator

and, even, of representation. Irrational numbers are formally revealed whilst being

simultaneously concealed; the one who utters that ‘sacred verb’ ‘remember’ has no right

to. The story incorporates its own generic, that which exists prior to (primordial to,

even, if we are to appeal back to the young Borges) the narrative.

There is an allegorical incorporation of the story’s own technicity in a dual form.

We move from a chronometric existence – the watch as the symbol of organised life – to

an irrational one; to a vision of ‘infinite’ detail held in the photograph, the film still. This

is a vision that is replicated rather than represented. Of course, then, this presentation of

‘uncountability’ renders Funes’ memory the existential equivalent of ‘ ’ in Cantor’s

notation; the cardinality of the continuum. It is this fictional ‘proof’ of uncountable

existence that brings into relief the measurable infinity of countable numbers, the world

that the narrator exists in.

*

2.3 ‘THE LIBRARY OF BABEL’ AND ‘THE BOOK OF SAND’

The volume entitled Labyrinths contains a series of stories that revolve in some way

around a labyrinthine structure. In Emir Rodiguez Monegal’s words, labyrinths are

‘according to tradition, the representation of ordered chaos, a chaos submitted to

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human intelligence, a deliberate disorder that contains its own code.’54 The ‘code,’ here,

is also the ‘clue’, a word that originates with the Greek term for the thread given to

Daedelus by Ariadne, to help him find his way out of the labyrinth that he had built to

house the Minotaur. Borges, of course, is obsessed with the labyrinth as a combination

of both order and disorder, and as a predicament that requires some mysterious key to

unlock. Borges’ most famous ‘labyrinth’ is perhaps ‘The Library of Babel,’ which is a

narrative account of a universe that takes the form of a library ordered by regular

geometric cells. ‘The Library of Babel’ is a labyrinth insofar as it is infinite and thus

contains all possible books and thus, most importantly, the ‘book of Books’ that forms

the code for all others. The existence of this book is a rumour: it has not been found by

any of the ‘librarians’ but provides the most important justification of the Library for the

librarians, as well as an impetus for their search through the books. This rumour is

derived from speculation that emerges from the logic of the library itself; the speculation

that the library contains every possible permutation of the alphabet of Babel. This book

in many ways stands in for the mythological minotaur, here: it is both the source of the

Library and that which the Library keeps hidden. The Library is thus a labyrinth in the

sense that its nature presents a conundrum: on the one hand a quest to find this essential

book, and on the other hand a metaphysical problem concerning the order and contents

of the library and the lives lived in the library. In both of these cases we are dealing with

the mystery of the thought of the library, an unspoken divine that drives the questions of

the narrator or a clue that unravels the being of the library, that shows up its pattern.

The labyrinth is thus a double bind in the sense that the appearance of the question is

also the grain, the method, of the search.

54 Monegal, Jorge Luis Borges: A Literary Biography, 42.

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The Library of Babel houses the books that the inhabitants – who are referred to

as the librarians – spend their lives combing. The narrator is an ageing inhabitant of the

Library, whose sight is failing and who seems to be reaching the end of his or her life:

‘now that my eyes can hardly decipher what I write, I am preparing to die just a few

leagues from the hexagon in which I was born.’55 The narrator’s brief and strange

account seems to be at once an outline of the only world in which he has lived his life, as

well as an explanation of the world for one who lives outside the library. In this sense,

the short story is fundamentally contradictory: the representative account of a total

universe is presented as if for the foreigner (the various details of the uniform structure of

the library are noted) or, in other words, a world that supposedly contains no outside is

presented for an outside. The story opens with the essential structures of the Library:

The universe (which others call the Library) is composed of an indefinite and

perhaps infinite number of hexagonal galleries, with vast air shafts between,

surrounded by very low railings. From any of the hexagons one can see,

interminably, the upper and lower floors. The distribution of the galleries is

invariable. Twenty shelves, five long shelves per side, cover all the sides except

two; their height, which is the distance from floor to ceiling, scarcely exceeds that

of a normal bookcase.56

How the narrator knows the height of a ‘normal bookcase’ is a mystery: he has lived his

whole life in the Library. The self-conscious estrangement that occurs here is carried

right through the story and is particularly significant given that the production of texts 55 Jorge Luis Borges, ‘The Library of Babel,’ in Collected Fictions, trans. Andrew Hurley (New York:

Penguin Books, 1998), 112. 56 Ibid.

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has such consequences for the constitution of the Library. The ‘significance’ of this is a

kind of anti-significance: this discrepancy in the form of the narrative signals the

contradiction in positing any totality at the same time that it constructs the ‘total’

Library.

The epigraph to ‘The Library’ is taken from Robert Burton’s Anatomy of

Melancholy (Pt 2., Sec. II, Mem. IV) and foregrounds the issue of textual production as a

process of combinatorics: ‘By this art you may contemplate the variation of the 23

letters....’ 57 Though Borges only provides a snippet of this section from Burton’s

masterpiece, the passage that it falls within is fascinating in terms of its use of numbers.

Burton sets out, here, to make the point that through the simple variant combinations of

twenty-three letters of the alphabet (during Burton’s time the Classical Latin alphabet

contained twenty-three letters, not yet including j, u and w) one is afforded an

exceptional imaginative range. Within a wider section on the value of literature and

discourse as a solution to melancholy, Burton cites numerical combinations of the letters

of the alphabet to give a sense of the vast possibility that emerges from language alone.

What is intriguing is Burton’s exact delineation of this realm of possibility:

ten words may be varied 40,320 several ways: by this art you may examine how

many men may stand one by another in the whole superficies of the earth, some

say 148,456,800,000,000, assignando singulis passum quadratum (assigning a square

foot to each), how many men, supposing all the world as habitable as France, as

fruitful and so long-lived, may be born in 60,000 years, and so may you

demonstrate with Archimedes how many sands the mass of the whole world

57 Ibid.

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might contain if all sandy, if you did but first know how much a small cube as big

as a mustard-seed might hold, with infinite such.58

Here, in order to give a sense of the full range afforded by the humble alphabet, Burton

in fact defaults away from the alphabet and starts listing very large numerical figures.

What ascribes the further reaches of the expressive capacity of the alphabet, here, are

numbers. What is notable about the epigraph to ‘The Library of Babel’ is the wider

numerical import of the passage from which that sentence is extracted: the

contemplation of the variation of the twenty-three letters leads to the contemplation of a

condensation and extensity of these letters, and of course Burton expresses this

condensation through numbers rather than letters. Of the very large numbers in the

theory of eternal recurrence, Borges writes: ‘This chaste, painless squandering of

enormous numbers undoubtedly yields the peculiar pleasure of all excesses.’59 Burton

seems to be applying this very technique, here. There would, conceivably, be a number

to describe the extent of variation of the twenty-three letters: a number that sits outside

of the alphabetic universe that Burton opens for his reader. The task of the librarians,

the contemplation of the books, the limits of their universe, and the chances of their

finding a ‘book of books,’ happens in the context of a just such numbers as Burton

deploys. One might say, in other words, that the capacities of language to describe and

imagine vast possibilities – even the simple, random combinatorial possibilities of the

alphabet – are undergirded by a number. Indeed, here, the ‘truth’ of the alphabet, and

so of literature, appears numerical.

58 Robert Burton, The Anatomy of Melancholy (New York: The New York Review of Books, 2001), 95. 59 Borges, ‘The Doctrine of the Cycles,’ 116.

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The narrator is careful in systematically describing the Library and the means by

which the librarians understand its potential totality. The only instance in which the

structure of the library is oriented towards anything more than housing books is in the

literally narrow provisions – two small spaces in the cells – in which the librarians may

sleep or relieve themselves. The books in the Library do not necessarily make sense:

rather, they are arrangements of letters that do not necessarily produce words, sentences

or meaning of any kind and, in fact, statistically will not do so. The ‘meaning’ is instead

produced in instantiation or permutation: each book another possible arrangement of

the letters or characters of the twenty-five letter alphabet, even if it is only the most

minimal change, of just one character. Thus there must necessarily exist a ‘written’

account of every life, and of all future events in the Library, and in this sense the Library

is infinite to its inhabitants, in the sense that it expresses, in its very form, all possibility.60

The random arrangement of letters means that ‘for every rational line or forthright

statement, there are leagues of senseless cacophony, verbal nonsense, and

incoherency.’61

60 The number for all possible combinations of the 25 orthographic symbols in the library is of course not

infinite. It is merely, as William Goldbloom Bloch points out, ‘unimaginably vast.’ The number that Bloch

comes up with is 251,312,000. This of course, pertains only to a sense of the infinite rather than the infinite

as such. It also pertains, however, to some certainty that the library never repeats a volume: a fact which is

necessarily unverifiable because any full index of the library would be an entire replication of the library.

As such, the safest ‘size’ to posit for the Library of Babel is somewhere in between 251,312,000 and ∞.

This number, however, does not include the possible arrangements of letters on the spines of the books.

Current research, as Bloch notes, posits the size of our observable universe as 1.5 × 1026 meters across.

Goldbloom calculates this in terms of cubic meters, and demonstrates – astonishingly – that our universe

‘doesn’t make the slightest dent in the Library’ in terms of cubic size. See: Bloch, The Unimaginable

Mathematics of Borges’ Library of Babel, 18-19. 61 Borges,’The Library of Babel,’114.

110

Upon introducing the Library, our ageing librarian relates the basic axioms of

his world: the first principle being that ‘the Library exists has existed ab aeternitate.’62

The narrator repeats what he or she calls ‘the classic dictum,’ supposedly a common

belief about the nature of the library, which is in fact a rephrasing of Giordano Bruno’s

description of God: ‘The Library is a sphere whose exact centre is any hexagon and

whose circumference is unattainable.’63 This is, of course, also a version of infinity

severed from the number line, the version reflected in Cantor’s uncountability proofs

(where, as Borges himself notes, all you need is a 1-1 correspondence to construct a

measure of infinity). This ‘classic dictum’ is less a statement about a cosmic distribution

impossible to imagine than a claim about the Library as instantiation. Reminiscent of

Asimov’s Nine Billion Names of God, the ‘meaning’ of the Library is the inclusion of every

possible arrangement of letters, and in this inclusion the Library is complete and is

manifest as a labyrinth. The Library presents the form and limits of everything the

librarians can know, and it is precisely these limits that form the basis for theories of the

being of the library. The second principle of the Library stipulates the number of letters:

There are twenty-five orthographic symbols. That discovery enabled mankind, three

hundred years ago, to formulate a general theory of the Library and thereby

satisfactorily solve the riddle that no conjecture had been able to divine – the

formless and chaotic nature of virtually all books.64

The librarian relates an instance where a book his father ‘saw in a hexagon in circuit 15-

94, consisted of the letters M C V perversely repeated from the first line to the last. 62 Ibid. 113. 63 Ibid. 64 Ibid. 113.

111

Another (much consulted in this zone) is a mere labyrinth of letters whose penultimate

page contains the phrase O Time thy pyramids.’65 The M C V’s demonstrate this ‘chaotic

nature of all books,’ and the significance of this text is precisely the absence rather than

presence of any semiotic important.66 The MCV’s demonstrate that this alphabet and

the dialects, languages and various forms of meaning that it produces also contains its

antithesis: the capacity for meaninglessness. What is deeply unsettling, regarding this

story, is the presence of a footnote that implies that the work we are reading is a copy:

‘the original manuscript has neither numbers nor capital letters.’67 The version of ‘The

Library of Babel’ that we are reading does include numbers – for instance, it refers to a

hexagon in circuit ‘15-94’ – and so there must be an ‘original’ copy, then, where this

would have been spelt out: fifteen-ninety four. Once again, what is behind the

distribution of the hexagons are numbers, and what signals the status of the text as a

copy and its fictionality, is the supervenience of numeracy in a universe totalised by an

alphabet.

Regardless of the limit of the orthographic symbols, we know that if the number

line is potentially infinite, then so too are the books of the Library. The conditions for

being and the conditions for knowing both revolve around the number of letters and the

possible combinations of these letters. The twenty-five orthographic symbols are also a

notation for a unimaginably large number of combinations. It is in the number twenty-

five that we see the simultaneity of the ontic and the epistemic (the being of the universe,

and the limits of what can be known in that universe), but, curiously, without the

implication of a totality. This is not ‘god’ or some other philosophical absolute

65 Ibid. 113-114. 66 Ibid. 67 Ibid. 113.

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(consciousness, perhaps) but simply the number twenty-five.68 But, in so far as these

letters can be combined to form number words, any calculation of combinations

becomes redundant, precisely insofar as numbers can continue infinitely. All one could

produce is a key for an endless creative process: those simple twenty-five orthographic

symbols that are vastly more generative of permutational combinations than the decimal

system.

The narrator details the various superstitions that have prevailed, from time to

time, in regions of the library: the Vindications, the Crimson Hexagon, the Man of the

Book. The notion that the Library contains all possible narratives, including those that

will occur in the future and those that have occurred in the past (‘the archangel’s

autobiographies’), inspires religious fervour:

When it was announced that the Library contained all books, the first reaction

was unbounded joy. [...] There was no personal, no world problem, whose

eloquent solution did not exist – somewhere in some hexagon. [...] At that

period there was much talk of The Vindications – books of apologiæ and

prophecies that would vindicate for all time the actions of every person in the

universe and that held wondrous arcana for men’s futures. Thousands of the

greedy individuals abandoned their sweet native hexagons and rushed

downstairs, upstairs, spurred by the vain desire to find their Vindication.69

This fervour comes from the most interesting (and horrifying) possibility that the Library

(literally) materialises, which is validation simply through existence: the realm of

68 Ibid. 69 Ibid. 115.

113

possibility is already delimited through the record of all possibility. Each vindication is

an individual manifestation of the necessity and reason for existence. This fervour is

reversed in another superstition held at one time by the inhabitants of the Library (or a

certain susceptible set of them): ‘the Book-Man. On some shelf in some hexagon, it was

argued, there must exist a book that is the cipher and perfect compendium of all other

books, and some librarian must have examined that book; this librarian is analogous to a

god.’70 Another form of this speculation is the presence of a total book with a circular

spine: ‘Mystics claim that their ecstasies reveal to them a circular chamber containing an

enormous circular book with a continuous spine that goes completely around the walls.

But their testimony is suspect, their words obscure. That cyclical book is God.’71 The

Library is the measure of time and life because it contains all possibility at once, already

rolled out and specified, and the Vindications or the great circular book are merely the

most explicit, extended version of this. The simple possibility of an essential book that

provides the key to the unravelling of the rest of the Library, the codex for the universe

that proceeds from it, is the most condensed version of the same impulse. This

labyrinthine morphology is distinctive because it posits a conflation of ontology and

epistemology at its core: the key that lets Daedalus escape his labyrinth. The possibilities

for the ‘art’ of twenty five orthographic symbols is at once the scope of all knowledge of

the library (epistemology), insofar as it is mediated by language, but equally the

conditions for being in the library, insofar as the world exists for housing the books and,

more minimally, the librarians who attend to the books (ontology).

Alfred North Whitehead theorises the distinction between the modern and the

pre- or non-modern on the basis of a conflation, or severing, of the link between

70 Ibid. 116. 71 Ibid. 113.

114

ontology and epistemology. For Whitehead, the modern severing between ontology and

epistemology contradicts the foundations from which modern science developed, which

are rooted (in his account in Science and the Modern World) in medieval Christianity as

much as Ancient Greek science.72 For Whitehead, Greek ‘science’ was never truly a

science, but a necessary extension of metaphysics. Rather, Whitehead contends that it is

‘scholastic divinity’ that forms the necessary ground for the Enlightenment flowering of

empiricism.73 Crucially, innovations within science emerging from scholastic divinity

relied on what Whitehead calls the ‘instinctive belief’ in the ‘secret’ and the

metaphysical consistency behind natural order: ‘I mean the inexpugnable belief that

every detailed occurrence can be correlated with its antecedents in a perfectly definite

manner, exemplifying general principles.’74 The forgotten foundation of modern science

is in the presumption of completion (the presumption of a total rationality in the

universe) and, by virtue of this completion, the presumption of consistency. In other

words, the flowering of naturalism emerged from the medieval Christian requirement

for an invisible order (a secret), that underlies and generalises the most minute natural

phenomena, creating a necessary link between ontology and epistemology.

In Whitehead’s history and theory of the Enlightenment the connection between

representation and presentation are severed by the secularisation of modern science, the 72 Murray Code provides an excellent summary of Whitehead’s perspective on the conflation of ontology

and epistemology. Even more significantly, he frames this modern ‘denial’ of its foundations in terms of a

failure to be properly modern: ‘[...] it would be better to describe Whitehead as attempting to frame a

thoroughly nonmodern naturalism in the sense outlined by Bruno Latour, who charges the moderns with

never having been truly modern since they never tried to bring all the explanatory resources of nature,

culture and discourse under one roof. They instead opened up a chasm between epistemology and

ontology by trying the former to a sensationalist theory of perception and the latter to the doctrine of

mechanistic materialism.’ See: Murray Code, Process, Reality and the Power of Symbols: Thinking with A.N.

Whitehead (Hampshire and New York: Palgrave MacMillan, 2007), 62. 73 Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1925), 12. 74 Ibid. 15.

115

disavowal of the foundational connection (the Christian connection that is also the

Parmenidean connection) between ontology and epistemology that produces the

conditions for modern science in the first place. There are two points regarding ‘The

Library of Babel’ that are significant here: the first concerns a mode of formal allegory

that emerges from this conflation of ontology and epistemology, and the second regards

the consequence of this, which is the production of an ‘in-significant totality.’ The

library revolves around a book that provides a foundational node where ontology and

epistemology are conflated: it is the key for the labyrinth, providing the source and

extent of possibility of the material universe of the library. In other words, the Library

revolves around a moment of pure presentation, not yet an instantiation or permutation,

and not a re-presentation of the 25 orthographic symbols. This ‘book’ would be entirely

future-oriented: not referring to or representing anything, but providing the molecular

form of the world that will surround it. This book provides the same function for the

librarians as what Whitehead calls the ‘secret’ that animated the Christian scholastics,

and conflates the ontic and the epistemic in the same way. However, there are two levels

to this observation: the first regarding the nature of the Library and the lives of the

librarians, the second regarding The Library as a story, and how it relates to a version of

the ‘total book’ envisaged by Mallarmé as a part of his development of an infinite art.

Just as the Library revolves around some necessary microcosm that is the total book, The

Library is reciprocally constructed around a total book: the universe of Babel, and all of

its contents. The Library is only an allegory of a total world, the extensity of language and

the function of presentation in our world insofar as the Library occupies a similar

distance to its potential ‘total book.’ The short story here revolves around an imagined

total book that justifies and provides the reason for the universe that the narrator lives

in, just as the world that this narrator tells us about orbits around the same structure

116

(which provides a rationale for the estranging contradictions of the story, whereby the

narrator explains his world as if to someone from another world, as if there were an

outside). In the case of The Library (the story) the total book is of course ‘The Library of

Babel,’ in The Library (the world) the total book is of course The Golden Book, the

Vindications, the circular book or whatever other figure of an essential or total book the

librarians might formulate, whatever they choose to name the secret that they pursue.

This reciprocity allows this story to become a curious formal allegory, whereby the

contents of the story reflect the narrative’s own creation and explanation of a total

universe.

To make this effect of ‘The Library of Babel’ clearer, it is useful to turn to a story

that also presents a kind of total book, albeit one that retains a ‘significance’ and has

vastly different effect as a result. This story is entitled ‘The Book of Sand’. This title

refers to the name of a book that is sold to the narrator of the tale, a book that seems to

be of infinite length. ‘The Book of Sand’ starts with a reflection upon the problem of

infinity and verifiability:

The line consists of an infinite number of points; the plane, of an infinite number

of lines; the volume, of an infinite number of planes; the hypervolume, of an

infinite number of volumes... No – this, more geometrico, is decidedly not the best

way to begin my tale. To say that the story is true is by now a convention of

every fantastic tale; mine, nevertheless is true.75

In this story the (again unnamed) narrator recounts how he came to own ‘The Book of

Sand’ and how he would eventually dispose of the menacing volume. The narrator

75 Jorge Luis Borges, ‘The Book of Sand,’ in Collected Fictions (New York: Penguin, 1998), 480.

117

acquires the book from a man who arrives at his home claiming that he sells Bibles but

presenting not a Bible but another ‘sacred book’ that he found in ‘northern India, in

Bikaner.’76 This book is unusually heavy and possesses other strange traits:

At the upper corner of each page were Arabic numerals. I was struck by an odd

fact: the even-numbered page would carry the number 40, 514, let us say, while

the odd-numbered page that followed it would be 999. I turned the page; the

next page bore an eight-digit number. It also bore a small illustration, like those

one sees in dictionaries: an anchor drawn in pen and ink, as though by the

unskilled hand of a child.77

The book was named the ‘Book of Sand,’ by the former owner, an illiterate man of the

lowest class in India, who named it thus because ‘neither sand nor this book has a

beginning or an end,’ a suspicion that is affirmed by the narrator’s attempts to locate the

first or last pages of the book.78 Eventually, our narrator’s life becomes twisted by his

possession of the book, he is – like Funes – subject to a form of paralysis: ‘I began noting

[the illustrations] down in an alphabetised notebook, which was very soon filled. They

never repeated themselves. At night, during the rare intervals spared me by insomnia, I

dreamed of the book.’ 79 In order to free himself he must lose the book, and,

appropriately, he chooses what he considers to be a innumerable place: ‘I remembered

reading once that the best place to hide a leaf is in the forest. Before my retirement I had

worked in the National Library, which contained nine hundred thousand books... I took

76 Ibid. 77 Ibid. 481. 78 Ibid. 79 Ibid. 483.

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advantage of the librarians’ distraction to hide the Book of Sand on one of the library’s

damp shelves.’80

The Book of Sand is a total book, but one that has no relation to the world or a

world: it is not generative in the least. The book here is pure substance (in the sense in

which substance is described above, as the material substrate of the ‘secret’ that

scholastic divinity sought in the world). It is horrific because it contains everything, rather

than the code or key for the potentiality for everything. The difference between the Golden

Book or total book in ‘The Library of Babel’ and ‘The Book of Sand’ can be

summarised in the discrepancy between a ‘significant totality’ (a totality that describes or

contains all possibilities), which describes the latter, and an in-significant totality (one

that produces the conditions for all potential materiality), which describes the former.

The idea of an insignificant totality emerges from the formulation of an

insignificant or ‘ontic’ mark, recently theorised by French philosopher Quentin

Meillassoux. Each of the superstitions in the Library centre around the notion that one

of the books holds a code from which the world of the library proceeds. As I argued

above, this is formalised as a conflation between the ontic and the epistemic, albeit an

conflation that never leads to a totality; that never reveals the ‘secret’ that Whitehead

understands the medieval Christian scientists to be pursuing. There is no complete

science behind the Library. Regardless of their semiotic meaning, regardless of whether

the marks form words or sentences, the librarians pore over the works which may or

may not have semiotic meaning: the cause for attention is not only this type of meaning

but rather the fact that the marks constitute portions of the sequence of possibility, and

indeed the book could form the codex to wider sequences, even, ultimately, the codex to

all the Library. This book that the Library revolves around is thus an instance of ‘ontic’

80 Ibid.

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writing: a mark, or a text, or some form of writing in those books that inscribes the

possible infinite destiny of the inhabitants. ‘Ontic signification’ refers to the mark used in

mathematics, and differentiates it from representational writing. Meillassoux arrives at

the term through investigating the status of mathematics as a science of ontology:

In order to resolve the problem of the absolute scope of mathematics, I began by

trying to identify a minimum requirement of any formal writing— logical or

mathematical—that distinguishes it from natural languages [langues naturelles].

I tried to reach a precise and determinative point of difference, capable of

distinguishing a symbolic, or formal, language from a natural language.81

Here, Meillassoux theorises the way that mathematics uses signs in an exceptional form,

articulating the difference between the writing of mathematics and of ‘natural’ language

in terms of a minimal difference: ‘This minimum requirement actually seems to me to

consist in a remarkable usage— a systematic and precise usage—of the sign devoid of

significance [dépourvu de sense (sic)].’ 82 This curious description of mathematical

presentation indicates a form of mathematical signification that is separated from

semiotic signification, a mark is not ‘significant’ because it is self-contained. These

criteria lead Meillassoux to assert ontic signification as a distinct form of signification:

the placeholder or mark that mathematics uses to demarcate the real. Meillassoux’s

contention that the ‘writing of mathematics’ is the use of signifiers without significance

81 This excerpt comes from a conference paper given by Quentin Meillassoux: Quentin Meillassoux,

‘Contingence et Absolutisation de l’Un,’ in Conférence Donnée À La Sorbonne, Lors D’un Colloque Organisé Par

Paris-I Sur ‘Métaphysique, Ontologie, Hénologie’ (Paris, 2008). This translation is by Fabio Gironi, and is

currently only published at: http://afterxnature.blogspot.com.au/2011/10/translation-of-meillassouxs-

contingence.html#more. All parenthetical notes are Gironi’s. 82 Ibid. (Gironi trans.)

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describes a form of sign that technically could be substituted for any other sign, and

which obeys operational protocols rather than hermeneutic ones.

What is distinctive in the total book of the Library of Babel is its status as an

ontic mark: it obeys operational rather than hermeneutic protocols, a self-contained text

that provides the ‘key’ to the Babelian universe. If indeed the Library is a total book this

is a totality without signification. The total book does not inscribe the world, but

inscribes the potentiality of the world: it is a totality that is generative without being

prescriptive. There are two possibilities regarding the total book: on the one hand, it is a

book hidden somewhere in the Library, or is an unusual form of book, like the imagined

circular book or golden book. The other ‘total book’ is the world that the story presents:

it is world within ‘The Library of Babel’ as opposed to the world within The Library of

Babel, which provides the vision for the Library without describing or representing all its

contents. Here the formal allegory of the work and the ontic signification come together:

the story itself mirrors the problem faced by the librarians of Babel, with the reciprocity

of the two worlds functioning to posit a generative but unseen ontic ‘book.’

*

2.4 ‘THE LOTTERY OF BABYLON’

The possibility of a kind of ‘ontic book’ is again explored in the story ‘The Lottery of

Babylon.’ This story – as the title makes obvious – is closely connected to the ‘Library of

Babel.’ The word ‘Babel’ can mean ‘confusion’ but also – when lengthened to Babylon

– becomes the word for the ‘gateway to God.’ The ancient city of Babylon, whose ruins

are in contemporary Iraq, is the location for the biblical myth of the ‘Tower of Babel.’

The inhabitants of the Tower were said to have one language, a story that became the

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source for the myth of ‘Babel’ as the universally understood language, the universally

shared language: here complete comprehension is also the ‘gateway’ to an absolute, to

God. In ‘The Lottery in Babylon’ we have a twisted form of the universal language.

This ‘language’ is not something universally comprehended by the inhabitants, but

rather concerns what they are universally subjected to: the radical contingency of fate,

and the absence (even if it is a potentiality of absence) of free will. In Borges’ Babylon,

the contingencies of fate are controlled by a lottery that – much like a municipal

organisation – evolves over the years to ‘better serve’ the people of Babylon. The

‘atrocious variety’ of the narrator’s identities and roles in his society is a direct result of

outcomes of this lottery: ‘Like all men in Babylon, I have been proconsul; like all, a

slave...’ he reports, ‘I have also known omnipotence, opprobrium, imprisonment.’83

Babylon, the universal language, is but an ‘infinite game of chance,’ where life happens

according to combinatorics, just as in ‘The Library of Babel.’ Unlike The Library,

however, there are only three letters of consequence in Babylon: ‘Look here, through

this gash in my cape you can see on my stomach a crimson tattoo – it is the second

letter, Beth. On nights when the moon is full, this symbol gives me power over men with

the mark of Gimel, but it subjects me to those with the Aleph, who on nights when there

is no moon owe obedience to those marked with the Gimel.’84 Aleph, Beth and Gimel,

the first three letters of the Hebrew alphabet, assign classes in Babylon and enforce the

necessity of power and submission.

The initial, by implication primitive, lotteries consisted of draws ‘in broad

daylight’ and resembled traditional lotteries in that the winners received coins.85 The

83 Jorge Luis Borges, ‘The Lottery in Babylon,’ in Collected Fictions (New York: Penguin, 1998), 101. 84 Ibid. 85 Ibid. 102.

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problem with this system was that the ‘moral force’ of the lottery was lacking.86 In this

sense, the story becomes testimony to a certain attachment to a moral necessity in

chance, and some inherent virtue in life modulated by numbers or arbitrary signs. In its

second variation, the lottery involves lucky and unlucky draws, and one could either end

up winning or owing a sum of money. The implication became that those who did not

buy tickets were cowardly, and in addition to this those who drew ‘unlucky’ numbers

and were fined were equally subjects of contempt, and hence there was a successful

moral scourge to be had. The final version of the lottery perfects this. After a revolution

inspired by the unfair drawing of multiple lots by the upper classes, the lottery is

transformed, and every citizen of Babylon is automatically a participant, creating a true

democracy of the lottery. The Company (the body that administers the lottery) becomes

the all-powerful force in Babylon and fates are decided – many times during each

citizen’s life – based on lots drawn. Radical contingency – the capacity for freedom and

wealth one day, and slavery or indeed death the next – comes to govern and normalise

the lives of the inhabitants of Babel: ‘Mine is a dizzying country in which the Lottery is a

major element of reality; until this day, I have thought as little about it as about the

conduct of indecipherable gods or of my heart.’87 The narrator goes on to recount

elements of his life and how the lotteries of Babylon have come to eventually modulate

the social order of the city:

Once initiated in the mysteries of Baal, every free man automatically

participated in the sacred drawings, which took place in the labyrinths of the god

every sixty nights and which determined his destiny until the next drawing. The

86 Ibid. 87 Ibid. 101.

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consequences were incalculable. A fortunate play could bring about his

promotion to the council of wise men or the imprisonment of an enemy (public

or private)... A bad play: mutilation, different kinds of infamy, death.88

The lottery is administered by ‘the Company’ who ensure – and here we see the key

paradox in this story – the social efficacy of the lottery. There is frequently

dissatisfaction with the form of the lottery, usually of an existential nature, which the

Company then seeks to remedy. Strangely, the very source of the lottery – the

exhilaration and submission of chance – will make it seem inadequate:

In many cases the knowledge that certain happinesses were the simple product of

chance would have diminished their virtue. To avoid that obstacle, the agents of

the Company made use of the power of suggestion and magic. Their steps, their

manoeuvrings, were secret. To find out about the intimate hopes and terrors of

each individual, they had astrologists and spies.89

The solution, then, will be to obscure the sources and ‘reasons’ for chance. This reality

of Babel is peculiar because it bears a strange relation to the real. This is not quite as

paradoxical as it sounds: everyday life is structured here by what is precisely antithetical

to the continuation of that life: utter contingency. Instead of a theory, however, this is a

surrender to what the Babylonians take to be the real:

88 Ibid. 103. 89 Ibid. 104.

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However unlikely it might seem, no one had tried out before then a general

theory of chance. Babylonians are not very speculative. They revere the

judgments of fate, they deliver to them their lives, their hopes, their panic, but it

does not occur to them to investigate fate’s labyrinthine laws nor the gyratory

spheres which reveal it.90

What Babylonians are after here is the sensation of submission to fate, which is

existentially comforting, rather than any revealed truth regarding fate. Here we find

another transformation of presentation: brute happening without reason, matter without

causality, and an ontology that seems to ‘appear’ by virtue of the constant exposure of

the citizens to the vagaries of fate, the return to generic being in the moment of

suspension in which the lots are drawn. Here, ontology appears as an affect wrought by

subjective exposure and the perpetual ‘year zeros’ that the lottery repeatedly

inaugurates. The numbers of the lottery are thus meant to provoke the ‘sensation’ of

number.91 In this story we again see an instance where the kernel of presentation is

again suggested (the ‘appearance’ of determinism in the casting of lots) but ultimately

hidden (the replacement of municipal governance with a lottery).

90 Ibid. 104. 91 One of Cantor’s precursor’s, the Italian mathematician Bolzano, developed a distinction between two

types of infinities that would be crucial to the development of the transfinite several decades later. Bolzano

defined syncategorematic and categorematic infinities, the first pertaining to potential infinities, and the

second pertaining to actual infinities. It is this distinction, primitive to the full realisation of the transfinite

as a measure of actual infinity, which is operative in Babylon: the use of a categorematic infinity (the rules

of the game and chance) to obscure or render impotent a syncategorematic infinity. It is the ever

increasing branches of chance in the lottery that shows the increasing desperation of The Company to

stave off the syncategorematic, imitating its omnipotence in ever greater degrees of detail yet still retaining

control.

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The lottery in Babylon is thus, curiously, not a speculative enterprise, but rather

a normative one. The lottery attempts to shift the fundamentally contingent process of

cause and effect to a randomised process that both originates in and dictates human

agency. The reintegration of fate into the workings of the ‘Company’ perversely

subsumes the human perception of causes behind effects into a well founded and

predictable system. The lottery creates a norm of radical contingency and

unpredictability in the lives of the citizens: unpredictability becomes what is expected,

but the form of this unpredictability must remain a mystery. When the simple roll of the

dice becomes demure, there are rumours that the Company works not through

randomness but through intimacy, compiling dossiers on its subjects and working

through favouritism, magic or deviousness. Here, we again find Borgesian narrative wit:

the narrative is finely constructed in order to conceal what it claims to present. Just as

the encyclopedia of Tlon lies, and the Library of Babel conceals, and what Funes only

appears to suffer from, to too does the world of Babylon revolve around what it necessarily

cannot avow: the kernel of the Real, the nature of presentation, and the true possibility

of a unique number. The lottery, as our narrator describes it, is the ‘insertion of chaos

into cosmos.’92 This is the annihilation of novelty through numeracy and the exertion of

a finite randomness that can show a definite link between cause and effect, assuring the

citizens of Babylon of the presence of a continuum at the same time allowing them to be

seized by sublime contingency. Just as the Company becomes more and more complex

so do the branches of contingency, and each prospect is never quite assured or quite

what it seems. Once again, the lottery here realises a kind of total book staked in

numerical chance (the buying of lots), whereby all the prospects for being and the

92 Ibid. 104.

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appearance of change and novelty are rendered by one, secretive authority, and the

shady rules by which that authority proceeds.

*

2.5 WHAT IS A TRANSFINITE ALLEGORY?

Borges begins his 1949 article, ‘From Allegories to Novels,’ with a polemical claim about

the redundancy of allegorical form: ‘For all of us, allegory is an aesthetic mistake. (I first

wrote, ‘is nothing but an error of aesthetics,’ but then I noticed that my sentence

involved an allegory.)’93 In this article, Borges traces the literary development out of

allegory, which he associates with the pre-novelistic epic and Romance, to the novel

form, and articulates why allegory in its traditional sense is a literary impossibility in his

time. The entire problem with allegory, in this essay, revolves around a loss of belief, a

loss of some ephemerality which renders the symbolic inscription ‘inseparable from

artistic intuition.’94 These are Benedetto Croce’s words, and Borges quotes him at length

to illustrate the poverty of allegory:

If the symbol is conceived as separable [from artistic intuition], if the symbol can

be expressed on the one hand, and the thing symbolised can be expressed on the

other, we fall back into the intellectualist error; the supposed symbol is the

exposition of an abstract concept; it is an allegory; it is science, or an art that

apes science.95

93 Borges, ‘From Allegories to Novels.’ 337. 94 Ibid. 95 Ibid.

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Whilst Croce appends a number of qualifications to this, the essence is clear for Borges:

if allegory is not secure on an intuitive and spiritual level it is merely, to use Croce’s

word again, a kind of ‘cryptography.’96 On the other hand, allegory is, for G.K.

Chesterton, a supplement to language, which is necessarily crude and inadequate in its

representation of the world. In this case, any perspective on allegory as a ‘justifiable’ art

form depends on a prior perspective on the representational capacities of language.

Borges’ own conclusion is that allegory is ‘a fable of abstractions:’ of Gods and destinies,

and in this sense Borges implies that that allegory is associated with pre-modern forms:

epic and pre-novelistic Romance (his example, notably, is Roman de la Rose). The novel

on the other hand is a ‘fable of individuals.’97 Only insofar as ‘the individuals that

novelists present aspire to be generic’ is there an element of allegory in the novel.

Although Borges does not claim this explicitly, the latent conclusion appears to be that

that allegorical form becomes ‘unjustifiable’ if there is no longer a belief in or sense of

abstraction. However, as we saw in the essay on Flaubert, ‘A Defense of Bouvard and

Pécuchet’ (written five years later), Borges does not see the novelistic form as defensible

either.

What form, then, must be forged from the death of the novel and the loss of a

‘justification’ for allegory? I would like suggest that it is a modified form of allegory that

Borges finds as formal replacement: one that does not require a belief in abstractions,

and one whose generic is no longer has a name, or a divinity. Many of Borges’ most

successful stories revolve around the capacities of fiction to suggest worlds beyond itself,

and the source and meaning of the extensity of fiction. But to look at this only at the

level of content misses precisely the mode in which the ‘other world’ comes into being,

96 This quote, from Benedetto Croce, is reproduced for Borges’ purposes in: Ibid. 338 97 Ibid. 339.

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which is through number. Like Mallarmé, Borges is interested in a ‘total book’: an

essential structure or principle that underlies the possibilities available to fiction, a

simultaneously genetic and generative kernel. I have argued above that Borges creates a

kind of ‘in-significant totality’ to create a genuinely generative fiction: one that produces

other worlds without the necessity of their completion or coherence. This relies not on

any spiritual foundation for allegory, but nor is it limited to that ‘intellectualist error’ or

inadequate form, an awkward science of the symbol (though it replicates this form

closely). This form is not rooted in any genericity of individual experience, either, and

takes no coherent subject-sensorium as its point of origin. Rather, this new form rests

not on divinity but on 0א and can be called ‘transfinite allegory.’

In each of the stories analysed above, we see the following mechanism at work:

the stories revolve around either an actual infinite (as in the ‘Lottery in Babylon’ or the

‘Book of Sand’) or two irreconcilable forms of numerical continuum (in ‘Funes, His

Memory’) that are never represented – determined – within the story. In each case,

number is embedded in the fiction but unseen: we find the ontic mark in the ‘Library of

Babel’ in the rumour of the Golden Book, and in ‘Funes’ we have the subjective

apprehension of transfinite number locked in the paralysed body of the character. In

‘The Book of Sand’ we see the fully realised version of this enclosure of the ontic mark,

in that the story recounts the discovery of an infinite book and hence the possibility of a

return to a unity of the ontic and the epistemic that seems to literalise the pre-

Enlightenment ‘book of the world.’ Of course, this is in turn defied by the fact that

Borges’ has ultimately produced a speculative account of the book; the existence of the

infinite book is separated twice over from the reader, just as our relation to the character

Funes is mediated by a narrator (and the memory of that narrator). In this sense, each of

these stories contain and allude to their own origins or the origins and structures of the

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lives of their characters: actual infinites that – like the Company in Babylon – bring

worlds into being without providing the rationale for these worlds or patently

contradicting the credibility of these worlds. In other words, these infinities remain an

alogos – a point of non-rationality – at the heart of the stories.

Within each of the stories there is thus a double inclusion of number, which

ultimately produces a structural allegory internal to each story. In Funes, for example,

we see number twice over: firstly in terms of Funes’ capacity to track time, and thus his

ability to attach number to a chronos, to an independent, abstract and arbitrary

movement, and secondly in the numbers deployed by the narrator to convince us of the

extensity of Funes’ perception. Here, Funes has a consciousness that is brought into

being through mechanical number; later, this will be reversed. Funes’ second ‘birth’

constructs a different epistemology, and he acquires an apparently infinite perception:

exceeding finitude without positing an absolute perception. Crucially, there is a

structural gap between the double presence of number in these texts, which brings into

relief the distinction between a transfinite number and ordinary counting in the

arrangement of presentation and its necessary effacement by representation. This

structural gap is in fact another instantiation of what Paul de Man argues is the internal

allegory essential to fictional writing. De Man’s work in Allegories of Reading and Blindness

and Insight looks at the ways that literary form constructs linkages between objects and

events and creates images through tropological uses of language, simultaneously

collapsing that very content to the abstract machinations of language itself. This is

exactly the form that Borges gives literary historical and evaluative weight to in the essay

‘From Allegories to Novels,’ where the operations of the symbol subtracted from

intuition and spiritual guarantee inevitably must, in Croce’s words, ‘ape science.’ De

Man constructs this argument through a method of reading that straddles both literature

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and criticism, which brings out the process of linguistic doubling essential to fiction. In

his critique of formalist criticism de Man notes that:

On the one hand literature cannot merely be received as a definite unit of

referential meaning that can be decoded without leaving a residue. The code is

unusually conspicuous, complex, and enigmatic; it attracts an inordinate amount

of attention to itself, and this attention has to acquire the rigour of a method.

The structural movement of concentration on the code for its own sake cannot

be avoided, and literature necessarily breeds its own formalism.98

Despite the fact that the critics ‘cry out for the fresh air of referential meaning’ in their

mappings of poetic language, the creative work of tropology undermines the possibility

of precisely the unified work that the formalist critic seeks, always leaving a residue in its

outward and endless referentiality. Writing on Rilke, de Man shows how the celebration

and content of the poems is achieved through language devices. This is a traditional

analytic reading until de Man doubles his analysis over to show how the very content of

the poems is subsumed to a dexterity with language, indeed, what is expressed pertains

not to God, the topic of the poems, but rather to the movement of language. De Man

shows through his analysis of the poetry of Rilke and others that the axis of content –

the story, the gesture or the sentiment of the poems – is inextricable from form, but also

that form, curiously, has a life of its own. De Man starts from the basic assertion that

themes – when they are put under close analysis – are seen to be indelibly linked to

modes of figuration. The most forceful of images or claims do not rest on their own

constitution but, rather, some other story, other logos beyond that theme or referent,

98 de Man, Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust, 4.

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that accompany the act of image creation or staking a claim. This autonomy of form

means that the content of the poems also functions as an allegory of what happens in the

language of the poem. This claim for internal allegory operates according to the same

principle whereby we see the two forms of number in Borges’ texts, which are placed in

an irresolvable contradiction: the formal rendering of the transfinite number points

outside of the text, to an autonomous existence, which is embedded but not contained

by the boundaries of the stories. It is this discovery of the singularity of fictional language

that is anticipated and worked with in Borges fiction: not only do we have de Man’s

thesis ratified in the ‘doubling’ of language and number in Borges’ stories, but we in fact

have a self-conscious engagement with this gap, made explicit in the use of number in

these stories.

Here I am pointing to a simultaneity between the ‘gap’ that characterises the

unique discourse of fiction in de Man’s understanding – in that all fictional texts

necessarily allegorise their own formal properties – and the ‘gap’ between the two forms

of number in Borges. Borges’ stories allegorise their own processes of composition by

bringing into relief their own ‘enigma’: an infinite presentation rather than a finite

representation. This is not, however, a clear parallel. It would seem that the significance

of the double forms of number for the story allegorises the organisation of expression

and tropological movement of the very fictional work. However, there is something

additional that disrupts this: the presence of number in the texts. The autonomous and

material aspects of language that necessarily defy both the formalist critics and any

pretension to unified poetic works are rendered in Borges’ work as the essential collapse

of ontology and epistemology in substance: ‘The Library of Babel’ must posit the

Library in order to posit, within that world, the superstition of a ‘Golden Book,’ for

instance. In other words, what we have with Borges is the replacement of a stable

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relationship between language and the world with number. Number here comes to

stand in for the link between sign and referent, and fascinatingly the restitution of a

foundation for language in Borges work is not found in natural language at all, but the

other form of language: mathematical writing or the ontic mark.

What this modified version of de Man’s theory of formal allegory affords us is a

concept of allegory that is not a reduction to content, or an addition upon content, nor a

mythological or humanist ‘resolution’ to the stories. This allows us to approach the

numerical aspects of the text that generate the tendency to allegorise and narrativise ‘out

of’ the stories, and allow us to start to approach and theorise the formal components of a

modernist mathesis of fiction. The internal allegory necessary to fiction is re-staged in

Borges’ work by the double presence of number, producing an allegory that does not

demonstrate the sheer autonomy of language from content, and the separation of the

sign from the signifier, but posits a generic and genetic device at the heart of the

machinations of language: number.

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3.1 CHAPTER THREE, PART ONE:

CONTINUOUS DEFORMATION IN SAMUEL BECKETT�S NOVELS

The art of Samuel Beckett has become an art of zero, as we all know.

J.M. Coetzee, Doubling the Point

It is a commonplace to identify ‘bareness,’ or ‘minimalism,’ or ‘lessening’ as the

governing aesthetic principle of the prose, theatre and television works of Samuel

Beckett. This attribution of ‘bareness’ rends Beckett’s artistic achievement in the terms

of subtraction: creativity is valued here for what it takes away from ‘reality,’ ‘scene’ or

‘thought.’1 The barren stage sets of Endgame and Waiting for Godot, and logorrheic

narrators of Molloy, Malone Dies and The Unnamable exemplify this visual or linguistic

‘minimum.’ This aesthetic is linked to mathematics in two senses. Firstly, it relies on

forms of stylistic subtraction to get ‘behind’ language, to remove the obfuscation from

language, or, rather, the obfuscation that is language. Secondly, this ‘minimal’

undertaking is implicated in a renovation of the novel form in the wake of the shift in 1 French philosopher Alain Badiou has written the most substantial account of ‘subtraction’ in Samuel

Beckett’s work. See Alain Badiou, On Beckett, trans. Nina Power and Alberto Toscano (Manchester:

Clinamen Press, 2003), Power and Toscano provide a succinct summary of Badiou’s notion of subtraction

in Beckett: ‘Beckett’s writing draws its force and urgency precisely from the way that it subtracts itself

from our impressions and intuitions; in other words, from the manner in which it evacuates our muddled

and spontaneous phenomenologies to reveal a sparse but essential set of invariant functions that

determine our ‘generic humanity’’ (xiii-xiv).

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the foundations of counting, the most basic system of order underpinning reality.

Beckett’s texts share with mathematics, in Ingo Berensmeyer’s words, ‘the formal

principle of reducing complexity, both linguistic and experiential, in order to gain a

higher degree of (structured) complexity.’2 Molloy claims that he has become ‘a little

less’ than ‘the creature [he was] in the beginning, and the middle [of his tale],’ a process

he exults as the most he can hope for.3 In this reflection, Molloy unites life, chronology

and narrative in an arc of beginning, middle and end, yet reverses the traditional

numerical assumption behind the arc of the novel: maturation, complication,

consequence. These stages are defined, in each case, by becoming a little more or a little

different than what was before. Beckett’s ‘lessening’ or ‘subtraction’ rather emerges from

an artistic endeavour preoccupied with foundations; preoccupied with the absence of a

solid linguistic, semiotic and symbolic ground upon which the fictional world is

constructed. 4 In Beckett’s words, this was a general ‘rupture in the lines of

communication.’5 Even this sense of a ground upon which fiction must rest indicates the

erosion of a linguistic system.

2 Ingo Berensmeyer, ‘‘Twofold Vibration’: Samuel Beckett’s Laws of Form,’ Poetics Today 25, no. 3 (2004),

466. 3 Samuel Beckett, Molloy, ed. Shane Weller (London: Faber and Faber, 2009), 28. 4 This is the antithesis, as Ruby Cohn points out, of Joyce’s ‘apotheosis of the word.’ See: Ruby Cohn, A

Beckett Canon (Ann Arbor: University of Michigan Press, 2001), 89. 5 Beckett opens his 1934 review, entitled ‘Recent Irish Poetry’, with the polemical paragraph that contains

this phrase: ‘I propose, as rough principle of individuation in this essay, the degree in which the younger

Irish poets evince awareness of the new thing that has happened, or the old thing that has happened

again, namely the breakdown of the object, whether current, historical, mythical or spook. The

thermolaters – and the pullulate in Ireland – adoring the stuff of song as incorruptible, uninjurable and

unchangeable, never at a loss to know when they are in the Presence, would no doubt like this amended

to breakdown of the subject. It comes to the same thing – rupture in the lines of communication.’ Samuel

Beckett, ‘Recent Irish Poetry,’ in Disjecta: Miscellaneous Writings and a Dramatic Fragment, ed. Ruby Cohn

(London: John Calder, 1983), 79.

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What guarantees the reliability of representation when the connection between

word and reality is itself at best a fantasy after almost a century of humanistic and

scientific enquiry that has eroded the association between empiricism and literary

realism and the individual and coherent voice? Beckett’s minimum is a search for a new

foundation in the wake of the disintegration of various atomic elements of literature: the

individual, the word, the progress of time. J.M. Coetzee’s statement – that ‘Beckett’s art

is an art of zero’ – both incorporates the critical fatigue that this necessary but now

clichéd aesthetic diagnosis involves (‘as we all know’) but also signals a crucial, and less

frequently remarked upon aspect of this aesthetic. This problem is the same problem

that shaped mathematical history of the last century and a half. In Coetzee’s

formulation, zero is the equivalent of the empty set: the set theoretical placeholder for

the mathematical generic that founds arithmetical order: the modern mathematical

division of the world into classes and objects. One must have a secure foundation for

units of language in order to install a ‘great chain of being’ that realism might

recuperate. Coetzee elaborates his initial remark on zero precisely in terms of the

mathematical quest for foundations:

If we can justify an initial segmentation of a set into classes X and not-X, said the

mathematician Richard Dedekind, the whole structure of mathematics will

follow as a gigantic footnote. Beckett is mathematician enough to appreciate this

lesson: make a single sure affirmation, and from it the whole contingent world of

bicycles and greatcoats can, with a little patience, a little diligence, be deduced.6

6 J.M. Coetzee, ‘Samuel Beckett and the Temptations of Style,’ Doubling the Point (Cambridge, MA:

Harvard University Press, 1992), p. 43.

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Here, Coetzee echoes Beckett’s famous claim about the ‘unword.’ In the oft-quoted

‘Letter to Axel Kaun’ Beckett articulates his own criticism of the ‘real of language,’

claiming that words obscure more than they reveal. In this letter, Beckett imagines an

alternative literature that circumvents such limitations. This is the literature of the

‘unword.’7 The Beckett of 1939 cannot find ‘any reason why that terrible materiality of

the word surface should not be capable of being dissolved’8 and thus saw the task for

literature as one which worked to reveal what is prior to or behind language, a ground

of genuine pre-possessive and pre-experiential communicability. From this, the capacity

to secure a single adequate affirmation – the security of the first word – should follow.9

This starting point for literary creation results in an attempt to induce literature rather

that create it: it is this aspect of Beckett’s work in which he already seems to be a late

modernist.

It is Beckett’s mathesis of prose that will fulfil the modernist promise of taking

fiction beyond or outside itself. One of the most important means by which Beckett

achieves this is through importing mathematical form into fiction, or quite literally

mapping prose form onto mathematical sign. Not only do the journeys made by

Beckett’s characters take place most frequently in circular form but Beckett produced

seven manuscript versions (never published) of a text provisionally entitled ‘The Way’ 7 Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929-1940 Edited by George Craig and Dan

Gunn. (Cambridge and New York: Cambridge University Press, 2009), 518. 8 Ibid. 518. 9 Coetzee’s ‘single, sure affirmation’ is echoed by Hugh Culik is the idea of a ‘descriptive sufficiency’: ‘In

his aesthetic agenda we see [...] a reformulation [of the modern] through a strategy that relies upon the

metaphoric power of non-literary fields such as neurology, aphasiology, and mathematics to represent two

related ideas: first, the descriptive (in)sufficiency of language, and second, the (in)ability of a formal system

to comprehend itself.’ More than an analogy between one formal system and another: mathematics is the

alogos of literature, essential, rather than supplementary, to Beckett’s modernism. Hugh Culik,

‘Mathematics as Metaphor: Samuel Beckett and the Esthetics of Incompleteness,’ Papers on Language and

Literature 29, no. 2 (1993), 132.

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and later split into two sections entitled ‘8’ and ‘∞’ that record a journey which charts

the shape of the lemniscate. This diagram is the first draft of a short, two-part text that

traces movement without locale and figure without subjectivity. In the subsequent drafts

this diagram will be rendered in prose: ‘Forth and back across a barren same winding

one-way way. [...] Through emptiness the beaten ways as fixed as if enclosed. Were the

eye to look unending void.’10 This is the most literal realisation of the ‘transfinite’

exchange that Beckett’s work maps. In this diagram, and in the prose texts, Beckett

constructs a reciprocal relation between matheme and word, soul and extension, the

mute mark and its semiotic other. He does this with a substantial dose of humour: the

text traces the path of a mathematical figure, describing a journey around a figure eight

and then a lemniscate. The development, affective import and ultimately conclusion of

this short and beautiful prose work are all based on a predetermined structure: the shape

of a mathematical sign. The prose creates the mathematical mark, just as the

mathematical mark creates the prose: this reciprocity between the matheme and the text

presents an obscure allegorical relation between the two domains: one ‘language’ (either

the prose or the matheme) is the implicit ‘other’, structuring our access to the other

‘language’ (whether prose or matheme). In Part One of this chapter I will argue that

Beckett’s work involves circular form that – by virtue of geometric inevitability of the

curve that will eventually meet its origin – negates what is usually considered to be

narrative change or transformation that must be, necessarily, linear or helical. Instead,

the multiple significance of the circle produces different narrative preoccupations and

outcomes (different ‘walks,’ if we are to follow the terms of The Way). This is not,

however, an issue of simple substitution of one narrative geometric model with another.

10 Samuel Beckett, ‘The Way’ (Carlton Lake Collection, Harry Ransom Humanities Research Center,

The University of Texas at Austin, Box 17, Folder 3, 1981).

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Beckett replaces linear narrative not only with circular form but with something at once

more subtle and more disruptive to a literary naturalism: circular, continuous

deformation.

It would be wrong to say that there is an analogy between mathematics and

literature as if the former were a coherent referent ripe for tropological manoeuvre.

Instead, I will argue that the model of relation between the two is allegorical. The

relation between the mathematics and the literature is not created by the matheme

possessing symbolic ‘import,’ so to speak. The presence of mathematics in Beckett’s

prose is not a matter of proportion (Greek: analogia) but rather a matter of ‘another’

speech (Greek: allegoria). 11 This allegory finds its equivalent in Dan Mellamphy’s

formulation, where the veil of language is undone only at an ever receding, ever elusive,

projected point: ‘It is toward this point that Beckett’s texts proceed: the aporetic point at

which diction and composition, the sayable and the said (langue and parole, in French)

criss-cross so as to weave the text as such and in so doing form the knot of linguistic

expression.’12 The ‘aporetic point,’ here, is the ‘other’ to geometric form: continuous

deformation or what would be called, in mathematical terms, topology.

I will argue here that the radicalisation of naturalist prose form in Beckett’s

novels occurs through a replacement of geometry with topology, and a replacement of

the requirement for ‘typicality’ with the requirement for ‘genericity.’ The replacement

of one with the other is less a refutation of novelistic form so much as a fuller realisation

11 The alternative perspective is found in: Culik, ‘Mathematics as Metaphor: Samuel Beckett and the

Esthetics of Incompleteness.’ Culik argues that the ‘problematic’ post-modernism of Beckett is

characterised by an analogy to mathematics, or, in other words, that Beckett uses mathematics as a

metaphor. Here, I present an alternative viewpoint: rather than a mathematics of metaphor we have

mathematics as allegory. 12 Dan Mellamphy, ‘Alchemical Endgame: ‘Checkmate’ in Beckett and Eliot,’ in Alchemical Traditions from

Antiquity to the Avant-Garde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 495.

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of it. The transition is, of course, a mathematical one. Typicality emerges from the

Greek root for ‘type’ (typikos). In the nineteenth century ‘type’ became a word close to

‘characterise’ and as such is associated with a naturalist descriptive capacity. ‘Typicality’

and ‘types’ also echo the Aristotelian category of genera: a unified identity of an animal,

plant of object that differentiates it from others. The generic, on the other hand, is the

antithesis of the ‘typical’: it is a placeholder for a multiplicity that may have measure but

not determination. I will argue that this ‘generic literature’ involves a form of counting

that does not proceed in a cumulative and linear fashion, but rather through a

continuous topological process. This installs a new naturalist numeracy that befits

generic rather than typical prose fiction.

In the following chapter, also on Beckett’s work, I will consider this

‘radicalisation of naturalism’ that occurs via the use of numbers and numerical

organisation in terms of Beckett’s later work. Here I will develop this argument in terms

of Beckett’s work outside of the novel form, looking at short prose fiction and a play for

television.

*

3.11 A MANIA FOR SYMMETRY:

MOLLOY AND THE DEFORMATION OF LANGUAGE

The two novels that Beckett is perhaps best known for, Molloy and Watt, stage various

processes of enumeration that divert rather than facilitate narrative progress. In Molloy

the famous ‘sucking stones sequence’ is merely one instance of the eponymous

character’s attempts to resolve a stable relation between body and world, pleasure and

identity through a numerical system. In Watt, too, various characters engage in forms of

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enumeration as compulsions towards order, practices that defy their purpose and negate

any ‘account’ rather than stabilise it. The events that carry the plot of Watt are

distinctive for their absence of meaningful outcome. The end of activities Watt

undertakes is, often, not so much productive as establishing an order for the sake of

order itself:

It fell to Watt to weigh, to measure and to count, with the utmost exactness, the

ingredients that composed this dish [...] and to mix them thoroughly together

without loss, so that not one could be distinguished from another, and to put

them on to boil, and when boiling to keep them on the boil.... This was a task

that taxed Watt’s powers, both of mind and of body, to the utmost, it was so

delicate, and rude.13

This activity does not create substantial difference but, rather, chaos both material and

psychological. The urgency of exactitude weighs on Watt just as it weighs on the reader

waiting for telos or substance to be attributed to the process of ordering. In other words,

what is missing from these endeavours of precision is, precisely, what or, the equivalent

in this tale, Watt: a character bearing goal, intention and disposition. This absence of

signification of value is one instance where form is more important that content, end or

significance. In this section I will look at the numerical stakes in Beckett’s novelistic

realisation of the ‘unword.’ I will argue here that the deformation of the word in Molloy

occurs through the manipulation of consistent practices of counting and geometry. This

novel thus also demonstrates the inadequacy of geometric metaphors of the use and

function of language. In Molloy, language is topological; it operates as a process of

13 Beckett, Watt, 73.

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continuous deformation rather than geometric regularity. This is another way of saying

that in Molloy Beckett achieves a form of fictional measure without definition.

Molloy is one of three novels that are now grouped as a ‘trilogy,’ which consists of

Molloy and Malone Dies, both published in French in 1951 and then translated in 1956,

and The Unnamable, published in French in 1953. Molloy lives in what he thinks is his

mother’s room – although this is not verifiable for either Molloy or the reader. Molloy

does not know how he came to reside in this room, but occupies himself with writing,

and tolerates the daily intrusion of a woman who opens the door and places his food

beside it and takes out the chamber pot, as well as the weekly intrusion of a man who

comes every Sunday – Molloy thinks it is a Sunday – to collect his papers (Molloy seems

to spend some of his time writing), return last week’s papers (which appear with a series

of editorial marks on them) and give him money. Molloy had previously taken a

rambling journey – mostly on a bicycle but eventually walking – find his mother, but

presumably has not succeeded, and has somehow ended up incapacitated in her room.

It is the tale of this journey that Molloy will narrate for his readers.

But this novel is not only about Molloy. It is also about his shadow; a detective

figure named Jacques Moran who is assigned to track Molloy down. Whilst a circular

narrative and a resemblance or echo between detective and criminal is a staple of

detective fiction this occurs in Molloy to a far more dramatic extent than is typical. The

second part of this novel is narrated from Moran’s perspective. Moran’s pursuit of

Molloy will not be structured by progress, however, but circularity: Moran comes to be

Molloy rather than finding him, or, in other words, finds him only by transforming into

him. The key discrepancy in the two narratives is that Molloy starts his narrative

bedridden, and tells his tale in the past tense, whereas Moran’s narrative is present tense

and follows him until ultimately he is bedridden too. The two narratives thus form a

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diptych in the sense that they converge in the middle; the end of Molloy’s narrative of

his journey gives way to the start of Moran’s, each narrative concluding with bedridden

character. Just like Molloy, Moran writes, recording the progression of his pursuit of

Molloy in notebooks. Both end up unable to walk: Molloy has a stiff and swollen leg in

the first place, a symptom which eventually appears in his other leg, and Moran

develops pain in his knee, and becomes, like Molloy, reliant on a bicycle. The killing

scene is similarly replicated: Molloy kills a charcoal burner with his walking stick, and

Moran assaults (and perhaps kills) an elderly man using that man’s walking stick. When

Moran closes in on Molloy he becomes Molloy, and the two journeys mirror each other:

this suggests that the narrative is neither linear nor circular, but takes the form of a

lemniscate, Moran’s journey seamlessly becoming Molloy’s.

In Beckett’s early novels there is an overwhelming preoccupation with the pair, a

repetition that establishes enough difference to allow for the narrative trajectory in the

first place, but will eventually thwart any possibility of narrative linearity. This two part

structure of Molloy, where two figures follow each other, Moran chasing and collapsing

into Molloy, is common to many of Beckett’s works, including Waiting for Godot, Mercier

and Camier, and Endgame – and constitutes a kind of structural preoccupation for Beckett.

Like Molloy, Mercier and Camier (1946) resembles detective fiction – with a detective but

without the trajectory of a crime. In Mercier and Camier novelistic development, climax

and resolution, as well as the progress of a detective story (with the accumulation of

clues leading to the solution of the crime) is replaced by a circular and – crucially –

pointless alternative trajectory. Molloy broadly does not know why he writes or why he

is in the place he is in. His narrative constitutes the limits of testimony in the sense that

there is no necessity to his extended autobiography: he is a point on a narrative map

without coordinates rather than a character with place and history. There is, of course,

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a reciprocal relation between the ‘point’ and zero. The point is the precursor sign of

zero, just as zero properly functions as a placeholder (there are ‘zero’ hundreds in 7056)

in large numbers. This reciprocity is profound in Molloy. Molloy appears less as a

character and more as a placeholder, his narrative is pointless, his journey and – as we

will see – even his use of language is circular.

This substitution of character with placeholder is exemplified early in the novel,

where Molloy relates his observation of two figures on the road. Molloy names them ‘A’

and ‘C’ and observes that they were ‘going slowly towards each other, unconscious of

what they were doing. [...] At first a wide space lay between them. They couldn’t have

seen each other, even had they raised their heads and looked about.’14 Like Molloy and

Moran, A and C walk to meet each other, until they ‘stop breast to breast’ and then

‘turned towards the sea which, far in the east, beyond the fields, loomed high in the

waning sky, and exchanged a few words. Then each went on his way.’15 A and C are not

characters: they are figures, and hence they can be without proper names, but are

instead indicated by algebraic placeholders. Mellamphy construes this same

arbitrariness found in Endgame in terms of figuration. Mother Pegg is, for Mellamphy, a

‘non-persona [who] functions as the focal point not only of that play in particular

(Endgame), but figuratively – or rather, figurelessly: that is, as a function (functionally)

rather than a figure per se – more generally in Beckett’s work.’16 In this sense A and C

do not even properly achieve the status of the figure: they retain the figure whilst

evacuating it of its substance, thereby ‘functioning figurelessly’ in Mellamphy’s

formulation.

14 Beckett, Molloy, 4-5. 15 Ibid. 5. 16 Mellamphy, ‘Alchemical Endgame: ‘Checkmate’ in Beckett and Eliot,’ 491.

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What Beckett presents, here, is not entirely alien to the novel form, so much as a

radicalisation of a crucial element of the novel. Novelists have long relied upon the

arbitrariness of the numerical figure to ground their narratives in fact: indeed facts are

required to support the apparatus of fictionality. The requirement for facts or a sense of

‘facticity’ for a fictional apparatus is, of course, one part of the paradox of fictionality.

The novel has traditionally been defined against any straightforward depiction of ‘real

life,’ society and the world, in order to embrace the narrative freedom that comes

through avoiding literal description of one’s surroundings (and avoid the charge of libel).

In Catherine Gallagher’s words, ‘Because the novel defined itself against the scandalous

libel, it used fiction as the diacritical mark of its differentiation’ and yet simultaneously,

‘its fictionality also differentiated itself from previous incredible forms.’17 This is the

double bind of fictionality: the novel must at once assert its fictional status in order to

have purchase on the world, and yet simultaneously disavow the outer reaches of fiction

– the ‘incredible.’ In Gallagher’s terms, this requires both an embrace and a disavowal

of fictionality: ‘the novel slowly opens the conceptual space of fictionality in the process

of seeming to narrow its practice.’ 18 This novelistic reliance on fact is thus not

straightforward: facts must be neither incredible nor true. They must occupy this

simultaneous paradox of the fictional. As such, characters must embody a paradox that

Beckett would have enjoyed: they must be typically singular or individual, far enough

from the world to be distinct personas yet close enough to evoke sympathy and

empathy. This paradox is embraced too closely in Molloy: rather than character being

typical through empathy and recognition, they are typical by virtue of mathesis. The

implication of a ‘mathematical typicality’ is that the object is not adequately described 17 Catherine Gallagher, ‘The Rise of Fictionality,’ in The Novel, Volume 1, ed. Franco Moretti (New Haven

and London: Princeton University Press, 2006), 340. 18 Ibid. 340.

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but rather evacuated of content: made generic – a placeholder for any content, thus in

effect exceeding and undoing any typicality whatsoever. This is novelistic naturalism

taken to an extreme, where typicality is realised through an algebraic sign (replacing

character) or subordination to seriality (replacing verisimilitude).

These algebraic figures and the circularity of the narrative are accompanied by a

narrative exhaustion. Molloy is virtually immobilised and apparently is waiting for his

death, although he doesn’t quite have the energy to perform any action that might hurry

it along. Molloy is – aside from his narrative activities – the exemplar of extreme

passivity:

All grows dim. A little more and you’ll go blind. It’s in the head. It doesn’t work

any more, it says, I don’t work any more. You go dumb as well and sounds fade.

The threshold scarcely crossed that’s how it is.19

Molloy’s reflections on his world are contorted by his awareness of the contingency of

perception, and as such alienated analytic relation to his own journey:

But now he knows these hills, that is to say he knows them better, and if ever

again he sees them from afar it will be I think with other eyes, and not only that

but the within, all that inner space one never sees, the brain and heart and other

caverns where thought and feeling dance their sabbath, all that too quite

differently disposed.20

19 Beckett, Molloy, 4. 20 Ibid. 6.

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Molloy’s sense of his own stagnating journey rends his perception at once radically

without purchase and wedded to a cold logic of probability. This strange ‘scholastic

autism’21 is exemplified when Molloy kills a man in a completely bizarre fashion: after

hitting him on the head with one of his crutches he kicks him to death by swinging over

him using his crutches as supports (it is perhaps this crime that leads to Moran’s pursuit

of Molloy in the second monologue). Slightly stranger is the geometric impulse that

operates behind this: ‘I rested a moment, then got up, picked up my crutches, took up

my position on the other side of the body and applied myself with method to the same

exercise. I always had a mania for symmetry.’22 This ‘mania for symmetry’ constitutes a

displacement of form onto formalism, the wider aesthetic inversion that swaps the

structure of passion for the passion for structure in the same way that Molloy swaps the

figurative (a character, or a potential one) for the figure (A or C).

A reading of symmetry and journey, or symmetry and act in Molloy necessarily

becomes a reading of identity and symmetry as well, which is exemplified in the unusual

naming. The fact that Molloy and Moran share an initial – indeed, a first syllable – is

perhaps unremarkable (although less so when the prevalence of the letter ‘M’ – the

perfectly symmetrical letter hallway through the alphabet – across Beckett’s oeuvre is

considered, including ‘Martha,’ Moran’s housekeeper, ‘Murphy,’ ‘Malone’ and

‘Mercier’). However, when considered in terms of Molloy’s personal theory of language,

and indeed personal application of language, this coincidence does become significant as

a symmetry. Unlike the symbol, this symmetry exists without any ‘ground’ for

significance, forcing the reader into replicating Molloy’s ‘mania for symmetry.’ Molloy’s

mother has a particular name for him, although he claims her choice is misguided, and 21 Anthony Cordingley, ‘Samuel Beckett’s Debt to Aristotle: Cosmology, Syllogism, Space, Time,’ Samuel

Beckett Today / Aujourd’hui 22 (2010), 184. 22 Beckett, Molloy, 79.

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he in turn has his name for her, which seems to be equally idiosyncratic:

She never called me son, fortunately, but Dan, I don’t know why, my name is

not Dan. Dan was my father’s name perhaps, yes, perhaps she took me for my

father [...] I called her Mag, when I had to call her something. And I called her

Mag because for me, without my knowing why, the letter g abolished the syllable

Ma, and as it were spat on it, better than any other letter would have done.23

Molloy’s rejection of his mother is symbolised here for the reader through a satirical

compliance with the appellation he is meant to use for her. Molloy’s half-resistance

presents the uniquely Beckettian structure of the ‘unword,’ a word which retains its

identity as a lexeme but is evacuated of content, whereby he must retain the word ‘Ma’

but manages to negate it by adding a velar plosive ‘g.’24 This addition of a letter makes

Molloy’s idiosyncratic language synthetic: a synthetic language adds or clusters

23 Ibid. 13 24 Moran has the same synthetic approach to language as Molloy, referring to Molloy’s mother as ‘Mother

Molloy, or Mollose’ and musing over the reasons he might add the suffix ‘ose’ to her name: Of these two

names, Molloy and Mollose, the second seemed to me perhaps the more correct. But barely. What I

heard, in my soul I suppose, where the acoustics are so bad, was a first syllable, Mol, very clear, followed

almost at once by a second, very thick, as though gobbled by the first, and which might have been oy as it

might have been ose, or ne, or even oc. (Ibid. 107)

Here, Moran splits the Molloy that exists in his mind versus the one presented to him by Gaber through a

suffix, to make ‘Molloy’ and ‘Mollose.’ He nonetheless must relinquish ‘Mollose’ ‘since Gaber had said

Molloy, not once but several times.... I was compelled to admit that I too should have said Molloy and

that in saying Mollose I was at fault’ (Ibid. 107). Jeanne-Sarah de Larquier pushes this even further, using

Molloy’s own ‘mania for symmetry’ to read into his name a graphic and alphabetic symmetry, one can

‘think of ‘M-ollo-Y,’ where the ‘ollo’ not only is symmetric, but also graphically looks like a bicycle,

Molloy’s main means of transportation,’ an exemplary instance of speculative symbolism. Jeanne-Sarah

de Larquier, ‘Beckett’s Molloy: Inscribing Molloy in a Metalanguage Story,’ French Forum 29, no. 4 (2004),

52.

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individual morphemes to create more complex ones.25 Molloy’s construction of ‘Mag’ is

derivational: out of a word and a letter a new word is produced. Derivational synthetic

constructions, however, are usually produced by combining two words to create a new

word, in opposition to relational construction whereby a new word is created to express

a grammatical construction. Here we see a kind of negative synthetic creation: Molloy

does not properly create a new word, but generates a version of it in order to negate that

word. There is, as such, never synthesis of two words, or a proper replacement of the

word, so much as a continuous negation: presenting the appropriate appellation whilst

also, each time, denaturing it.

This semi-negation of the maternal name is circular. In regular linguistic

formulations, the sign ‘Mag’ would be represented under the signifier, as such:

[Image of Mag]

______

MAG

However, in Molloy’s construction, the sign serves to partially negate rather than

underwrite the signifier, without, obviously, negating the woman herself or his relation

to her. This produces a constant but not final negation: a kind of circle of existences and

erasure. This circularity both reveals and disrupts the geometric form that structures the

efficacy of the sign, just as narrative circularity reveals the necessity of forms of

mensuration to the wider plot. This is a first instance of grammatical topology replacing

typology, or deformation replacing shape. Where linguistic typology is concerned with 25 Where isolated languages have one morpheme per word (or, with a less rigorous definition, few

morphemes per word), whereas synthetic languages will have multiple morphemes per word (the hyper

version of this is called polysynthetic language; synthesis upon synthesis).

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structure of language and construction of words through morpheme, topology – a

branch of geometry – studies continuous form, uniting the fields of set theory and

geometry to focus on formation and deformation in place. Content is replaced with

structure in this reformulation (or, better, deformation) of the juvenile idiom for mother.

Saussure’s image of language as a chain of signification does not apply here. Rather, we

have a circular process of negating whilst preserving: a linguistic replacement of

geometry with continuous deformation.

This instance of a textual topology replacing geometry echoes the form that

Jean-Michel Rabaté finds in Joyce’s Ulysses. In ‘Joyce, Husserl, Derrida: Calculating the

Literary Infinite’ Rabaté looks at the presence of infinitisation in Joyce’s work, focusing

in particular on the textual deformations of the ‘science of observable shapes.’26 Joyce’s

‘dream of an universal culture’ required a ‘textual infinite’ which, in Rabaté’s words, he

achieves by producing a conjunction between prose and geometry.27 Rabaté argues that

‘by opening itself to geometry, Joyce’s text lays the first trap for the reader (and, of

course, his characters): the endless aporias of textual infinitisation.’28 The key transition

that allows numbers to produce a literary infinite is, in Rabaté’s account, found in

Joyce’s capacity (which Rabaté argues holds for Husserl, Broch and Derrida as well) to

‘think topologically.’29 This literary definition of the topological involves the textual

equivalent of the mathematic science of continuous deformation and in this instance it

involves ‘disclos[ing] the formulae underpinning the creation of their literary space.’30

Although Rabaté does not name it as such, this process involves the combination of the

26 Jean-Michel Rabaté, ‘Joyce, Husserl, Derrida: Calculating the Literary Infinite,’ Journal of Romance

Studies 7, no. 3 (2007), 29. 27 Ibid. 44 28 Ibid. 30. 29 Ibid. 44. 30 Ibid. 44.

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geometrics of prose with formal allegory: the ‘formulae’ that ‘underpins’ a literary

domain are, in Joyce just as in Beckett, fundamentally geometrical. The process by

which to ‘infinitise’ the text involves the revelation of precisely its mathematical

underpinnings: turning textual finitude into textual infinitude by turning geometry into

the continuous measure of topology.

The same process by which Molloy disrupts linguistic geometry (through a

partial negation) holds for Moran’s use of pronouns. Moran has the same synthetic

approach to language as Molloy, referring to Molloy’s mother as ‘Mother Molloy, or

Mollose’ and musing over the reasons he might add the final syllable ‘ose’ to her name:

Of these two names, Molloy and Mollose, the second seemed to me perhaps the

more correct. But barely. What I heard, in my soul I suppose, where the

acoustics are so bad, was a first syllable, Mol, very clear, followed almost at once

by a second, very thick, as though gobbled by the first, and which might have

been oy as it might have been ose, or ne, or even oc.31

Here, Moran splits the figure of Molloy’s mother that exists in his mind versus the one

presented to him by Gaber through a suffix, to make ‘Molloy’ and ‘Mollose.’ He

nonetheless must relinquish ‘Mollose’ ‘since Gaber had said Molloy, not once but

several times.... I was compelled to admit that I too should have said Molloy and that in

saying Mollose I was at fault.’32 This tension between idiosyncratic suffix and Gaber’s

suffix is also the first sign of Moran’s confusion between Moran’s knowledge of Molloy

and Molloy actually residing inside him.

31 Beckett, Molloy, 107. 32 Ibid. 107.

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This formal circularity recalls Beckett’s participation in Eugene Jolas’ manifesto

entitled Poetry is Vertical (co-authored with Hans Arp, Thomas McGreevy and others).

This famed 1932 modernist manifesto was signed by nine modernist writers and artists

involved with the journal transition, including Beckett. The modernist impulses towards

essential depths and near mystic truths in art recorded in Jolas’ manifesto reflect

Beckett’s investigation into the distance between self and language: ‘Poetry builds a

nexus between the ‘I’ and the ‘you’ by leading the emotions of the sunken, telluric

depths upward toward the illumination of a collective reality and a totalistic universe.’33

In addition to his participation in Jolas’ vertical manifesto, Beckett gave a ‘parodic

lecture [...] in November 1930 titled ‘Le Concentrisme,’’’ in which he invents a French

literary avant-garde movement that he calls ‘‘les concentristes’’ (homophonous with ‘‘les

cons sont tristes’’). A concentriste is a ‘‘biconvex Buddha,’’ a ‘‘prism on the staircase,’’ at

once ‘‘excluant et exclu.’34 ‘Concentrisme’ realises in style what the vertical manifesto

does not: the pointlessness, the satirical tone of the manifesto and the chaotic polysemy

of the title contributes significantly to a genuine artistic call for formal divestment from

distinct and unproblematic categories of self and other, linear rather than vertical

intention. ‘Concentrisme’ hints at the radical possibilities of a topological rather than

geometric approach to fiction, one which involves negation, satire and insincerity in

order to achive a multiplication of the geometric form inherent to fiction, rather than a

rigid ‘verticisation’ of processes of composition. ‘Concentrisme’ is Beckett’s first

reorganisation of the circle of signification, a reverberating series of circles that will, in

these novels, be replaced with another more subtle mode of continuity: topology.

33 Hans Arp et al., ‘Poetry Is Vertical,’ in Manifesto: A Century of Isms, ed. Mary Ann Caws (Lincoln and

London: University of Nebraska Press, 2001), 529. 34 Beckett is quoted in: Berensmeyer, ‘‘Twofold Vibration’: Samuel Beckett’s Laws of Form,’ 471.

152

This geometric structure beneath the unusual names is complemented by events

in the narrative that can be called ‘manias for geometry.’ The now famous episode in

Molloy, the ‘sucking stones sequence,’ revolves around a circular obsession of Molloy’s.

Molloy has collected sixteen pebbles, which he seems to suck for either comfort or to

alleviate hunger:

I had say sixteen stones, four in each of my four pockets these being the two

pockets of my trousers and the two pockets of my greatcoat. Taking a stone from

the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the

right pocket of my greatcoat by a stone from the right pocket of my trousers,

which I replaced by a stone from the left pocket of my trousers, which I replaced

by a stone from the left pocket of my greatcoat, which I replaced by the stone

which was in my mouth, as soon as I had finished sucking it.35

Molloy’s stones thus circulate in a clockwise direction from pocket to pocket, Molloy

sucking each stone in turn. However, unless the stones manage to form a line in

Molloy’s pockets, which would only happen by fantastic coincidence, Molloy cannot

really know how long any of these stones has been in his pocket. The stones more or less

– or at least often – move in a clockwise direction though they never really complete a

full rotation, or, more pertinently, Molloy never knows if a complete rotation is

performed. Eventually, Molloy must come to terms with this:

I had to change my mind and confess that the circulation of the stones four by

four came to exactly the same thing as their circulation one by one [...] the

35 Beckett, Molloy, 64.

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possibility nevertheless remained of my always chancing on the same stone,

within each group of four, and consequently of my sucking, not the sixteen turn

and turn about as I wished, but in fact four only, always the same, turn and turn

about. So I had to seek elsewhere than in the mode of circulation.36

This model of circulation leaks constantly by virtue of – like every other fact in Molloy’s

life – an absence of verifiability. The rationality is idiosyncratic because partial and

based on comfort: it is only a creeping sense of the inadequacy of his system, an

experiential doubt that prompts Molloy to ‘confess’ that he must revise his system. The

physical comfort here is only in part the alleviation of hunger: it is equally the ‘bodily

need’ to ‘suck the stones in the way that I have described, not haphazard, but with

method.’ 37 The ‘reason’ behind the sucking stones is both psychological and

physiological and models, for Molloy, a coherent relation between mind and object,

language and organisation of the world.

Together with Watt, Molloy has been called an ‘attack upon Cartesian rationality’

by C.J. Ackerley, who claims that the book ‘probe[s] the soft centres of the rationalist

enterprise.’38 For Ackerley, the ‘soft centre’ of reason implies a criticism of the Cartesian

project: Cartesian rationality relies on a dualism of body and mind and takes scientific

method to be the privileged route to the attainment of truth, all of which Ackerley holds

to be deeply suspect. Molloy’s sequence certainly models some of the most urgent issues

at stake in Cartesian philosophy. Most importantly, the sucking stones illustrate Molloy’s

psychic investment in what is called ‘numerical materialism,’ a term that refers to a

formalisation of bodily extension that originated with Cartesian geometry. In Robin 36 Ibid. 65. 37 Ibid. 68. 38 C.J. Ackerley, ‘Preface,’ in Watt, by Samuel Beckett (London: Faber and Faber, 2009), i.

154

Mackay’s somewhat broader conception, numerical materialism is ‘an inquiry into the

extent and nature of number’s dominion over any philosophy calling itself a

materialism; but also an inquiry into the materiality of number and numerical

practices.’39 Yet the Cartesian stakes in the sucking stones sequence by no means

produce some direct criticism of Cartesian rationality, so much as an amendment to it.

The French mathematician and philosopher, Descartes (1596-1650) is most

famous for the ‘Cartesian coordinate system’ that forms the basic graph with four axes.

Undergirding this coordinate system are two theoretical positions: on the one hand,

Descartes allows arithmetic to be mapped onto a geometric system, and on the other

hand Descartes – in antithesis to Kant – formalises a concept of identity that makes

mind and body essentially separate entities that are nonetheless also sutured to each

other. Descartes distinguished numerical and qualitative identity: numerical identity is

the persistence of identity despite some material modification. In Cartesian terms, if I

have a leg amputated, I still retain the same ‘numerical identity’ despite the loss of a

limb. Here Descartes’ famous res cogitans and res extensa demarcate two concepts of

number that exempt the human from the law of numerical materialism. It is the soul –

the point of intersection between vertical and horizontal axes, the 0, 0 coordinate –

which remains consistent despite any greater or lesser material measure. This 0,0 point

is the source for numerical extension, and the principle that maintains identity despite

changes to this extension. This is also the point in the graph that has no horizontal

coordinates and no vertical coordinates.

Molloy’s ‘sucking stones’ is in one way a perfect material model for Descartes’

theory: a numerical and material system (even though it takes Molloy a couple of tries to

make this system ‘count’) that instantiates a geometric extension of Molloy’s soul, which

39 Robin Mackay, ‘Editorial Introduction,’ in Collapse, vol. 1 (Oxford: Urbanomic, 2007), 6.

155

is given consistency by Molloy’s ‘system’ and which, in turn, alleviates both a

psychological and a physical hunger. Here we have a geometric mapping of Molloy qua

rational being, and the sucking stones are thus an image of extension: of the consistent,

coherent ‘numbers’ of Molloy’s identity. As in Descartes own theory, this system

perpetuates the ‘sensory contradiction,’ whereby the integrity of identity – here the

individuality of each of the stones, the uniqueness and consistency of their numbers that

would ensure a proper circulation – is based in the recognition of each of the stones.

Descartes did not derive this profound origin of Western geometry in metaphysics but

rather grounded it in sense experience. Vlad Alexandrescu reminds us that ‘Descartes

relativizes knowledge of the world through the senses,’ simultaneously establishing a

‘modus operandi that is geometrical.’40 Algebraic geometry originates, here, in a model of

proprioception rather than some conceptual system divorced from human embodiment.

Proprioception is the generic origin of perception itself, referring not to perception of

the world around us so much as our sense of the extensity of our bodies. Like the

Cartesian coordinate system, Molloy’s geometrical modus operandi is reliant upon his

senses, and in order to complete his system he must enable some form of recognition

that the stones move from pocket to pocket as they should. This is less a critique of

rationality than a recognition of the proprioception – the ‘soft’ centre – essential to

reason.

Just as Molloy transforms the circulation system of language through his lexical

construction ‘Mag,’ so too does he transform the circulation of this proxy for bodily

extension. This transformation is facilitated by the word ‘trim,’ which acts as a sort of

conceptual hinge between habit and a better way of doing things: ‘One day suddenly it

40 Vlad Alexandrescu, ‘Y a-T-Il Un Critère d’Individuation Des Corps Physiques Chez Descartes.,’

ARCHES: Revue Internationale Des Sciences Humaines 5 (2003), 436.

156

dawned on me, dimly, that I might perhaps achieve my purpose without increasing the

number of my pockets, or reducing the number of my stones, but simply by sacrificing

the principle of trim,’ Molloy announces.41 Both the origin and the substance of ‘trim’

are a mystery to Molloy:

The meaning of this illumination, which suddenly began to sing within me, like a

verse of Isaiah, or of Jeremiah, I did not penetrate at once... Finally I seemed to

grasp that this word trim could not here mean anything else, anything better,

than the distribution of the sixteen stones in four groups of four, one group in

each pocket, and that it was my refusal to consider any distribution other than

this that had vitiated my calculations until then and rendered the problem

literally insoluble.42

The word trim is radically vague, here, because it is overdetermined. Trim can mean

‘the condition of being properly balanced,’ ‘in good condition or order,’ ‘to strengthen,’

‘to comfort’ as well as ‘to become pregnant.’43 The polysemy of ‘trim’ is, finally, an

ironic refutation of precisely that which it was supposed to refer to: a proper balance, a

fortification, some resolution. It is, in this sense, the perfect ‘unword’; a word that

undoes meaning in a fashion that inverts de Man’s theory of rhetorical allegory (or

formal allegory). Trim cannot itself enact that which it refers to, showing up the

impossibility of language to fortify a perfect circular system (the sucking stones). Like

Molloy’s ‘partial negation’ of the work ‘Ma,’ ‘trim’ allegorises the very process of

41 Beckett, Molloy, 66. 42 Ibid.

43 Oxford English Dictionary, 2nd ed., s.v. ‘trim.’

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linguistic expression and its incapacity for true performative equality with acts or

systems.

Molloy’s stones are for sucking, and thus circulate from his pockets to his mouth

and back, a kind of reversal of the way that words are supposed emanate from the

mouth in a semiotic rather than abstract circulation. This brings into relief the flipside of

natural language, its unacknowledged mathematical inverse: the patterns that resemble

counting beneath and within language. Here there is an indistinction between a phrase

and an actuality that belies the synthetic form of language. This structural and synthetic

minimal existence is astounding simply for the fact that structure and not semiotics, and

reason and not the subject comes first: it is the pattern of points that precedes the

content. Likewise, it is the ironic contradiction sustained in ‘trim’ that demonstrates the

impossibility of a descriptive adequacy (a circular reciprocity between word and

referent), resulting in, perhaps, the kind of broken totality whereby number, numerical

extension and quality are confused. Mathematics in Molloy is, then, not an allegory of

some intended meaning or a diagnostic tool: it is the expression of the non-semiotic

jump between reason and material (partial, recurrent negation), mind and extension

(proprioception), language and referent. This is where the organisation of sign and

referent defaults from a geometric analogy to a topological one.

*

3.12 PERMUTATION AND DIVISION IN WATT

Watt was written between 1942 and 1944 whilst Beckett was in hiding in France during

the Second World War, but it would only be published in English in 1953, and only

much later translated into French by Beckett himself and published in 1968. Watt

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features two key figures named ‘Watt’ (What) and ‘Knott’ (Not). In this novel issues of

matter, identity and negation are explored through processes characteristic of

mathematical enquiry: demonstration, refutation, compatibility and completion. Much

has been written characterising this text as an attempt to assert rationality in the

irrational world of France controlled by the Vichy regime and the violence of World

War II. John J. Mood has called Watt ‘[Beckett’s] most devastating depiction of the cul-

de-sac of modern Western rationalistic philosophy.’ 44 The connection between

rationalism and mathematics here revolves around the possibility of the stability of the

category, and in the symbolic markers that mathematics provides to mark the gait of

reason. This subversion is enacted, on one level, through replacing novelistic

conventions with logical or mathematical claims that stand in contradiction to their

context, a process that allows Beckett to bring out the contradictions entailed in

exercises of objective rationality. These conventions, and the process of collapse between

identity and entity, are the preeminent devices by which Beckett achieves his

‘devastating depiction’ of Western rationalism.

The plot of Watt follows an unusual count:

As Watt told the beginning of his story, not first, but second, so not fourth, but

third, now he told its end. Two, one, four, three, that was the order in which

Watt told his story. Heroic quatrains are not otherwise elaborated.45

Aside from the a-b-a-b correspondence between odd and even numbers, there is no

necessary relation to a heroic quatrain in Watt. This emphasis on (arbitrary) structure is

44 John J. Mood, ‘‘The Personal System’ - Samuel Beckett’s Watt,’ PMLA 86, no. 2 (1971), 255. 45 Beckett, Watt, 186.

159

coupled with an emphasis on – as J.M. Coetzee puts it – the ‘fictiveness of the fiction,’46

through the inclusion of remarks on the plausibility of the narrative. (Coetzee’s example

pinpoints this: ‘Haemophilia is [...] an enlargement of the prostate. But not in this

work.’47) This stress on plausibility serves to foreclose both the development of character

and any representation of change through the constant reference to its own fictionality.

Here I will look at how this rupture in novelistic bounds or discourse is inextricably

bound up with the processes of counting in this novel. Describing these two narrative

markers as a negation or undoing of the discourse of the novel is an attempt to recognise

the dual affirmation and negation that repeatedly surfaces in Beckett’s texts. Beckett’s

subversion of the novel form is thus produced through retaining certain of the

conventions of narration, whilst subsequently exposing and undermining such

conventions, whereby ‘the aesthetic goal is one of an immanent presentation of states of

affairs that, in not changing, nevertheless expose their voids.’48 In Watt we see an

attempt to renounce the conventions of the novel to reveal the generic capacities of

language, or, in other words, to evacuate the predicates of the novel that rely on the

evocation of transformation and verisimilitude in order to expose precisely the ‘gaps’ or

perforations (to use Beckett’s term) of language. By migrating mathematical forms –

notably forms of permutation and divisibility – into the novel, Beckett simultaneously

reveals and undermines the enumeration of naturalist and realist forms of narrative,

presenting a third way that again disperses coherent character and situates both voice

and identity outside of the speaking subject. This ‘third way’ takes the form of narrative 46 J.M. Coetzee, ‘The Comedy of Point of View in ‘Murphy,’’ in Doubling the Point: Essays and Interviews, ed.

David Attwell (Cambridge: Harvard University Press, 1992), 37. 47 Beckett, Watt, 102. 48 Julian Murphet, ‘The Mortification of Novelistic Discourse in Beckett’s ‘Trilogy,’’ Paper presented at

Beyond Historicism: Resituating Samuel Beckett, University of New South Wales, Sydney, Australia, 7-8th

December 2012.

160

permutation, a numerical term which migrates to the literary in order to describe

enumeration without progression, or an attempt at completion or totalisation of some

system that, by its very nature, cannot be either completed or contained.

The short poem in the Addenda is part reflection on Watt’s narrative, part

parody of the whole category of addenda, footnotes or other textual miscellany, and part

poetic comedy. The poem seems to enquire after the entitlement to tell the story of the

‘old man,’ presumably Watt, here, which, through reiterations of the question, creates

an equivalence between this act and ‘nothingness’:

who may tell the tale

of the old man?

weigh absence in a scale? mete want with a span? the sum assess

of the world’s woes? nothingness

in words enclose?49

Here, the asyndeton creates a sense of accumulation, and the questions that (graphically

and literally) pile on top of one another produce iterations rather than poetic or

syntactic progression or culmination. Each of these questions is oriented towards the

limits of linguistic and aesthetic possibility. As Mood rightly points out, the mathematics

of this novel demonstrates a ‘meticulously flawed form’ of Watt: ‘Not just in the content

(the helplessness of Watt as developed in characterisations and plot, and the use of

philosophical and literary allusions) but also in the form is this impotence and ignorance

portrayed.’50 These repetitions, and the negation of meaning that they effect, occur

49 Beckett, Watt, 247. 50 Mood, ‘‘The Personal System’ - Samuel Beckett’s Watt,’ 263.

161

frequently throughout the text. The most significant instances of this, outside of this

exemplary poem in the Addenda, are the permutative lists that Watt and others

undertake.

There is a circulation of servants at Mr Knott’s house. One servant works

upstairs, and the other downstairs. When a new servant arrives at the house – always

out of the blue, it seems – then the servant on the upper floor must leave immediately.

The servant on the lower floor subsequently moves upstairs, and the new arrival begins

work downstairs. We learn of this process through the events that follow Watt’s arrival

at the house – namely, the departure of Arsene, the servant on the upper floor – and,

later, through the arrival of Arthur and the departure of Erskine. After Watt’s arrival at

the house, Arsene gathers his belongings together and embarks upon a bizarre

monologue before leaving. This monologue is an exemplary surge of ‘weigh[ing]

absence in a scale,’ ‘met[ing] want with a span,’ and ‘the sum assess[ment] of the worlds

woes.’ Arsene compares the shifts in his capacities for reason to the barely perceptible,

sudden shift in the particles of reality:

The change. In what did it consist? It is hard to say. Something slipped. There I

was, warm and bright, smoking my tobacco-pipe, watching the warm bright

wall, when suddenly somewhere some little thing slipped, some little tiny thing.

Gliss – iss – iss – STOP!51

Here, there is an allusion to the ground shifting; all the component parts of reality are

maintained, but something, inexplicably, and without event, had changed. The short

sentences indicate the confounding of Arsene’s narrative voice: the sentences are

51 Beckett, Watt, 43.

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scattato until – for a moment – he relates his state of being prior to the change and

temporarily takes up a conventional narrative voice: ‘There I was, warm and bright,

smoking my tobacco-pipe....’ The immediate departure from this confident narration is

a transition away from imagery and into sound; this ‘Gliss-iss-iss’ is the culmination of

those short sentences in a final stammer between sound and word. In Arsene’s

experience of a shift in reality, which he compares to a shift in sand on a dune, we see a

kind of inverse or forbidden anagnorisis: a discovery is indeed made, a variation in the

world apprehended, but the content of this is subtracted from the experience.

Everything is different and the absolute hold of reality undermined, but there is no

positive content to this: there is an involuntary and unspecific sense of transition. Watt

has a very similar impression as regards the story of the Galls. The Galls are a father-son

outfit who arrive at Mr Knott’s to tune the piano. What perplexes Watt about the event

is precisely that a palpable experience of ‘nothing’ seems to occur, persist, or grate at a

normal sense of reality:

What distressed Watt in this incident of the Galls father and son, and in

subsequent similar incidents, was not so much that he did not know what had

happened, for he did not care what had happened, as that nothing had

happened, that a thing that was nothing had happened, with the utmost formal

distinctness, and that it continued to happen, in his mind, he supposed, though

he did not know exactly what that meant.52

52 Ibid. 76.

163

This sense of the ‘utmost formal distinctness’ of nothing is in fact formally realised in the

cataloguing, or enumeration, that Arsene undertakes, and which will be replicated by

Watt. In a tirade seemingly on the general state of things, Arsene bemoans:

The Tuesday scowls, the Wednesday growls, the Thursday curses, the Friday

howls, the Saturday snotes, the Sunday yams, the Monday morns, the Monday

morns. The whacks, the moans, the cracks, the groans, the welts, the squeaks,

the belts, the shrieks, the pricks, the prayers, the kicks, the tears, the skelps, and

the yelps.53

Arsene’s lists are virtually unreadable due to their excessive accumulation and a sense of

the madness of permutation: a form of counting that at once enumerates new and

distinctive elements yet equally seems not to progress or possess forward motion. Whole

pages in Watt are given to this form of permutative list, and, as such, reading happens at

the level of page as opposed to the level of the line. Although in Arsene’s monologue we

have an array of different events, sounds, or gestures, each different from the next, the

grammatical repetition and the parataxis homogenises the various elements of the list,

collapsing into a group of ‘one thing after the next.’ Yet another list is genealogical. This

list painfully enumerates familial lines: ‘And the poor old lousy old earth, my earth and

my father’s and my mother’s and my father’s father’s and my mother’s mother’s and my

father’s mothers and my mother’s fathers and my father’s mother’s father’s’ and so on.54

The list concludes with ‘An excrement,’ a feature that becomes a refrain when Arsene

53 Ibid. 46. 54 Ibid.

164

starts another list and ends it with ‘a turd.’55 In other words, these permutations

constitute a means towards no end, or towards an end that is only waste or filth. This

permutation ends, ultimately, with a corporealisation of the count: the non-progressive

enumerative cycle (a partial- but not full negation of any other enumerative act) runs

until its stops, or is stopped, by a return to the body.

Watt himself will enumerate lists very similar to those of Arsene, for instance in

regards to the affair with the Lynch family, where he meticulously processes all of the

Lynch generations: here, Watt attempts to ‘complete’ the Lynch family. This is made

even more explicit when these lists become outlines of various possible states of affairs;

the solution to the problem of Mr Knott’s leftovers, for instance, is dealt with by

enumerating possible situations with the number of objections listed next to them:

1. Mr Knott was responsible for this arrangement, and knew that such an

arrangement existed, and was content.

[...]

12. Mr Knott was not responsible for the arrangement, but knew that he was

responsible for the arrangement, but did not know that any arrangement existed,

and was content.56

This enumeration is preoccupied with potential situations and their extensions, rather

than events that might directly affect Watt. As such, for Watt, enumeration exorcises a

logical problem without solving it, precisely because of the absence of any finality to that

enumerative procedure. These iterations become meaningless excrement both for

55 Ibid. 47. 56 Ibid. 74-75.

165

Arsene, for Watt, and for the reader, because the evocation of meaningful difference is

replaced by the assault of permutation.

Beckett’s ‘permutation’ constitutes a distinct narrative ‘count’ separate from the

numerical stakes of other modes of representation, including, to use Georg Lukács’

scheme, ‘narration’ and ‘description.’ Lukács’ essay ‘Narrate or Describe,’ published in

1936, holds two different forms of writing, or ‘creative methods’ in opposition.57 The

two forms – narration and description – both occur in literary naturalism, and Lukács

uses examples from Tolstoy and Zola to illustrate the respective paradigms. Naturalism

is the literary movement with conviction in the capacities of realism to accurately and

adequately represent the everyday. For Lukács naturalism constitutes an aesthetic

procedure that exceeds the historical bounds of late nineteenth and early twentieth-

century French tendency in prose fiction (as exemplified by Zola) and, as such, Lukács

considers Beckett a naturalist because he is above all concerned with the ‘triviality of

everyday existence’ and his plots do ‘not articulate the essential relevance in an event

and in the reaction to event.’58 Lukács explains the separation between realism and

naturalism using the analogy of visual art: ‘it is ultimately the writer’s approach to reality

that determines whether he produces a painting or a photograph, an articulate

statement or a mute babbling.’59 The photograph, here, is a mute babbling – the new

technics of image subordinated to ‘painting’ because it relies on ‘technical virtuosity’

rather than the rich gestural ambiguity of realism.

57 Georg Lukács, ‘Narrate or Describe?,’ in Writer and Critic and Other Essays, ed. Arthur Kahn (London:

The Merlin Press, 1978), 110. 58 Georg Lukács, ‘Preface,’ in Writer and Critis and Other Essays, ed. Arthur Kahn (London: The Merlin

Press, 1978), 14. 59 Ibid.

166

‘Narrate or Describe’ extends this opposition between realism and naturalism by

discussing two naturalist forms: description and narration. In Lukács’s examples there is

an external viewpoint in the case of description, but in narration events are related from

the perspective of a character. The opposition between the two aesthetic regimes is thus

a division between stasis and dynamism. Narration manages to integrate an event into

the work thereby allowing the reader to experience the event whereas description

objectifies an event, facilitating a much more limited reading experience. The key

difference between these two modes, for Lukács, is the narrative temporality that they

facilitate, and the sense (or lack) of vitalism that arises from this temporality. Description

renders an event outside of temporality, privileging elements over experience, and

creating a kind of narrative stasis: a scene or event total in its constitution and suspended

outside of time.60 In each form we have a totality: in description, we have a total world,

with as many elements as possible included, whereas in narration, we have a total

experience. Each form has a different unit of measure: description attempts totality by

including as many elements of a scene as possible (each element contributing to

quantity) whereas narration achieves totality by subsuming all these elements to the

privileged literary end: readerly experience. In other words, Lukács could frame these

two modes of fiction in terms of two different definitions of count: description relies on

counting as determining a number or reciting the order of numerals whereas narration

relies on ‘counting’ as bearing significance or particular import. More importantly,

every literary work, despite its final attribution, will inevitably contain elements of both

narration and description: whilst for Lukács the former should take precedence,

moments of description are nonetheless necessary. Here, we see the unique numerical

60 Lukács, ‘Narrate or Describe?’ 111. Gérard Genette theorized this in terms of duration, naming it

‘descriptive pause.’

167

literary purview, whereby a quantitative count and qualitative count are

interchangeable and co-implicated.

These two literary forms – naturalism and realism, description and narration –

are also ‘two basically divergent approaches to language,’ both of which rely on a

certain cataloguing of plausible objects to achieve their literary, and ideological, effect.61

Just as words, rather than events, can be the sources of causality in Molloy, so too in Watt

logic and enumeration impose themselves on the speakers producing the spinning out of

a voice that confoundingly combines both of Lukács’s aesthetic regimes: both the stasis

of description (the content of the lists) but equally the dynamism of narration (we witness

these episodes of permutation happen to Watt) are present in a third form of narrative:

permutation. What is the significance of permutation, then, for narrative vitality, and for

the relation between narration and object, and narration and experience? In Watt,

narration and description are given over to the duration of the fit of enumeration –

literally, in the stasis experienced by both character and reader. These permutative

passages are ‘fixations’: rather than conveying the dynamism of experience, or creating

a world, these passages instead – and paradoxically in Lukács� terms – create an

experience of stasis. In other words, ‘permutation’ combines the antithetical elements of

narration and description and confounds them both.

In Watt, the principle of novelistic description is forced into temporary

suspension when there is no economy, when the act of enumeration cannot stop or

become accumulation. For Watt, these enumerations are not purely exercises of

rationality, to attain an objective scrutiny upon his world, but are instead compulsive

and cathartic episodes. Arsene comments on his own ‘personal system,’ and the desire to

keep it contained and complete: 61 Ibid. 120.

168

my personal system was so distended at the period of which I speak that the

distinction between what was inside it and what was outside it was not at all easy

to draw. Everything that happened happened inside it, and at the same time

everything that happened happened outside it.62

This description, which encompasses a combination of compulsion and catharsis, is also

a departure from either narration or description to a third mechanism that presents a

very different view of the human and language. This mechanism is ‘permutation.’

Watt’s enumerative activity formalises and exorcises the varied suspicions and

confusions that haunt him: ‘For Watt considered, with reason, that he was successful, in

this enterprise, when he could evolve, from the meticulous phantoms that beset him, a

hypothesis proper to disperse them, as often as this might be found necessary.’63 Here,

rationality follows a cathartic function: ‘for to explain had always been to exorcise, for

Watt.’64 In this experience of stasis, facilitated by a numerical regime that integrates both

‘senses’ of the verb ‘to count’ Beckett produces not only the numerical stakes inherent to

literature but, equally, the humour inherent to mathematics. This ‘humour’ is well

summed up in Gilles Châtelet’s reflection on the mathematical figure: ‘Why do figures

fascinate so many simple souls, and the impatient, always so fond of references and

certainties? Almost by definition, a figure is not open to discussion; there is indeed an

imbecilic virility to the number, stubborn and always ready to hide behind a kind of

scientific immunity.’65 This ‘imbecilic virility’ of the number fittingly describes the

62 Beckett, Watt, 43. 63 Ibid. 77. 64 Ibid. 78. 65 Gilles Châtelet, To Live and Think Like Pigs, trans. Robin Mackay (New York: Sequence Press, 2014), 51.

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paradoxical status of Beckett’s formal permutation. This permutative organisation

achieves an experience of narrative stasis. The lists effect, for Watt, a catharsis by virtue

of an exhaustion of reason, the experience of having let something that should, properly,

be antithetical to experience, run its course. In this, the text suspends literary duration.

Unlike words, the ‘universe of number’ is unproblematically coextensive with itself, each

and all integers immediately present, without dependence on the temporal process of

articulation. It is through this default from description to permutation that Beckett

achieves, in Watt, the evocation of a timeless realm.

The episodes of counting, and their distinctly permutative nature, can be read,

together with Watt’s theory about the croaking frogs, in terms of a triptych of

unorthodox systems of counting. What is astounding about Watt is precisely the

eclecticism of different counting forms, each, however, included by virtue of its capacity

to contradict the naturalist count, the stoppages that these counts put on the possibility

of flowing, conventional narrative. This final form of prohibitive count harks back to the

initial, onomatopoeic stumbling that replaced Arsene’s monologue, the: ‘Gliss-iss-iss,’

where we saw a first turn away from semiotics to signal a content-less experience in the

only way possible: through structure, here by the reproduction of a rhythm through

sound. Later in the novel, the arrivals and departures of servants at Mr Knott’s house

are represented as the three different voices in a chorus of croaking frogs. Watt had

earlier, on his way to Mr Knott’s house, encountered another kind of numerical singing,

quite distinct for its utter randomness in the text, whilst also conforming to the other

modes of permutative counting, which end with excrement or void. The song runs as

follows: ‘Fifty-two point one / four two eight five seven one / four two eight five seven

one / oh a bun a big fat bun...’ concluding with ‘and everyone is gone home to

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oblivion.’66 The singing of the frogs presents a development from this. Each of the

counting systems is reduced to a different sound, non-human and non-linguistic and,

emphatically, non-symbolic. After leaving Mr Knott’s, Watt considers how much time

he spent there, and attempts to work out a logic behind the different times spent at the

house by different servants, and whether these correspond to each other (each servant’s

period in the house being dependent on the arrival of new servants, which triggers the

departure of the servant on the top floor). Whilst preoccupied with this, Watt recalls an

experience from a long time ago, when he was young, sober, and also lying in a ditch,

and heard the chorus of three frogs:

Krak! – – – – – – –

Krek! – – – – Krek! – – Krik! – – Krik! – – Krik! – 67

Each enumeration here takes on a non-linguistic form of numbering (Krik, Krak, or

Krek) that differs from the other two. Here, algebraic marks are replaced by sonic

variations. In the case of Watt’s theory about the croaking, each numbering form has a

sonic domain, even if this is an arbitrary attribution, and thus has a continuity that is

based on rhythm and the breaking of silence (aurally permutative); this corresponds

exactly to Watt’s own musings about the abstract laws of the movement of servants –

hypothetically ‘Tom,’ ‘Dick’ and ‘Harry’ – whereby the narrator concludes that:

[...] it was not the Tomness of Tom, the Dickness of Dick, the Harryness of

Harry, however remarkable in themselves that preoccupied Watt, for the

66 Beckett, Watt, 28. 67 Ibid. 117.

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moment, but their Tomness, their Dickness, their Harryness then, their then-

Tomness, then-Dickness, then-Harryness; nor the ordaining of a being to come

[...] as in a musical composition bar a hundred say by say bar ten [...] the time

taken to have been true, the time taken to be proved true, whatever that is.68

These rhythmic sonic bursts that order possible states of affairs (regarding ‘Tom,’ ‘Dick’

and ‘Harry’) crystallise the process of permutation in an aural form. This is a final,

additional algebraic instance of permutation which is certainly a ‘worsening’ of

numeracy in that the elements of permutation are no longer logical possibilities but

amphibian cries. The hermeticism of such a number system, and an assumption of a

sonic algebra to stand in for the musical notation, parodies mathematics by revealing

‘imbecilic’ limits of numeracy. Reading the croaks as counts, and indeed as ordering

counting systems, presents counting in a non-linear and non-textual fashion, a non-

human modality, even. This number-as-animal-sound is the culmination of the ‘virile

imbecility’ described by Châtelet: each croak is complete and self-contained in its

meaning, like a number, and its power and totality rests precisely in this solipsism.69

The tabulated ‘croaking’ system, combined with other permutation systems in

Watt reads pictorially as well as textually, and entirely in keeping with Lukács’s diagnosis

of naturalism as closer to a photograph than the painting.70 Like the impression of light

on photosensitive paper, naturalist description is immune to content, or, in other words,

‘blind’ to the meaning and significance of the overall picture. The episodes of

permutation construct a type of difference that is distinct from the establishment of

68 Ibid. 127. 69 Croak, of course, is a synonym for death, and this completes, on a symbolic level, this negation of a

narrative count. 70 Ibid. 117-118.

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traditional descriptive differentiation. The type of difference and specificity created by

permutation can be read as a radicalisation of the naturalist form along the same lines

that Lukács sees a radicalised pictorial naturalism in the photograph. These lists are

logically and aesthetically unnecessary because they are reducible to sets, and reducible

to a continuing and consistent principle. The futility of permutation and the fact that it

confounds narrative occurs because it refutes both definitions of count inherent to

narration and description: the quantitative count and the qualitative count. Instead,

permutation, lacking meaning in itself, gestures towards a metacount: one native to the

cardinal and ordinal numbers. In other words, what these permutations reveal are not

their component parts (there is no meaningful or rich multiplicity) but rather a principle

of division. The shift here is from the substance of the count to the logical exhaustion of

the set; the structural laws of the fictional world start to become an issue of the integrity

of the set when the naturalist project is taken to its limit. In other words, these lists

realise, quite perfectly, answers to those contradictory questions of the poem from the

Addenda: here we have the effect of nothingness measured. No content is produced but

we wind up with a measure nonetheless: the measure that we find in the repeated form

of each notch in the permutation; a measure that makes the rest of the permutation

redundant. In these lists we have the first, and clear illustration of the limits of novelistic

prose. These lists fail aesthetically by virtue of their radicalisation of a naturalism: their

attempt to achieve a complete description divorces these passages, and by extension the

plot of the novel, from significance or importance or progressive development. This then

isolates the mode of description alien to the novel form: generic description, which

revolves around a world that spins inward or outward to the null set, as Beckett desires

his fiction to do. Generic description is the silenced accompaniment to description that

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instigates ‘genera’, the various discrepancies in the attributes of a differentiated and vivid

world.

More importantly, the anti-aesthetic generated by the mathematics of the text

relies on an abstract difference in the implicit metaphysical foundations of regimes of

description and narration. Lukács evaluates the regimes of naturalism and realism

according to a concept of difference that echoes Aristotelian difference, which is based

in ‘genera’ or types, rather than Platonic difference, which is based in division. This

theoretical discrepancy is acted out in the paradoxes that plagued early twentieth-

century set theory. In 1901 the British philosopher and logician Bertrand Russell

discovered a paradox in Cantor’s set theory; a paradox that was discovered around the

same time by Ernst Zermelo, who did not publicise his discovery. Russell would

elaborate on this paradox in the first volume of Principia Mathematica, which he co-

authored with Alfred North Whitehead and published in 1903. Russell and Whitehead

would develop ‘Type Theory’ in response to the paradox, whilst Zermelo would – with

Abraham Fraenkel – go on to develop the ultimately much more successful Zermelo-

Fraenkel Axiomatic Set Theory. Russell’s paradox follows the same lines as earlier

logical and mathematical paradoxes, most famously the ‘Richardian paradox,’ the

‘Burali-Forti paradoxes,’ and the ‘liar’s paradox.’ Some years later, in his ‘Introduction

to Mathematical Philosophy,’ Russell explains his paradox as such:

normally a class is not a member of itself. Mankind, for example, is not a man.

Form now the assemblage of all classes which are not members of themselves.

This is a class: is it a member of itself or not? If it is, it is one of those classes that

are not members of themselves, i.e. it is not a member of itself. If it is not, it is

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not one of those classes that are not members of themselves, i.e. it is member of

itself.71

The paradox arises when one asks whether this set – the set that contains all sets that are

not members of themselves – is a member of itself or not. In the same sense as the

Richardian paradox, this set appears to be necessarily both a member of itself, and not a

member of itself: hence the paradox.

The paradoxes appear as soon as one tries to apply mathematical systems to

themselves, or, in other words, to internalise the principles of set theory (or another

system like arithmetic) within set theory itself.72 One of the most important paradox that

71 Bertrand Russell, Introduction to Mathematical Philosophy (London: Spokesman Books, 2008), 136. 72 Similar to Richard’s paradox is the ‘Burali Forti’ paradoxes. The ‘Burali-Forti’ paradoxes were

discovered in 1897, and bear the name of the mathematician who uncovered them: Cesare Burali-Forti.

Burali-Forti noticed that any demarcation of a set of all ordinal numbers is another ordinal number itself.

Ordinal numbers classify sets according to their order type, and thus a set of ordinals necessarily produces

another ordinal, constantly producing ‘one extra’ object outside the set. In a similar fashion to Richard’s

paradox, what happens here is that the set can never be contained, because the very action of creating the

set produces another, extra ordinal, one that contains – and is thus larger – than all of the ordinals in the

set. Moreover, the simple arithmetical intervention of adding one to that new ordinal number creates an

even bigger ordinal number, and so the set is never stable anyway.

Richard’s paradox, or the Richardian paradox, was detailed in a paper in 1905 by the French

mathematician Jules Richard. The Richard paradox is particularly relevant given that it concerns the

distinction between mathematical entities and the language that is used to describe them, as well as

mathematics and metamathematics. Russell’s description of Richard’s paradox is as follows: ‘Consider all

decimals that can be defined by means of a finite number of words; let E be the class of such decimals.

Then E has 0א terms; hence its members can be ordered as the 1st, 2nd, 3rd, .... Let N be a number

defined as follows: If the nth figure in the nth decimal is p, let the nth figure in N be p + 1 (or 0, if p = 9.

Then N is different from all the members of E, since, whatever finite value n may have, the nth figure in N

is different from the nth figure in the nth of the decimals composing E, and therefore N is different from

the nth decimal. Nevertheless we have defined N in a finite number of words, and therefore N ought to be

a member of E. Thus N both is and is not a member of E’ in: Bertrand Russell, ‘The Theory of Types,’ in

From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1932, ed. Jean Van Heijenoort (Cambridge:

Harvard University Press, 1967), 153-154.

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predates Russell’s paradox is the liar’s paradox, of Ancient Greek origin, and simpler

perhaps by virtue of its semantic nature. The paradox is contained in the sentence: ‘this

sentence is false.’ The paradox is here found in the fact that if the sentence is indeed

false, then what it states is true. However, if the content relating to the sentence is true,

(indeed, the sentence is false) then this would contradict the assertion that the sentence

makes. As such, the seemingly simple assertion contradicts itself and formalises a

paradox. What is notable about these paradoxes is that they are related to a discrepancy

between a principle and its provability, which, as I will now explain through the set

theoretical terminology of Russell’s paradox, is a problem relating to the impossibility of

consistency between entity and identity that arises from a certain type of categorisation:

the divisive order necessary to set theory, that is generic but not of genera, not

pertaining to the constitution of differences.

The type of difference relevant in set theory requires all units to be the same and

mathematically exchangeable. Summarising the German philosopher Rudolph Carnap,

Albert Lautman writes:

The rules of syntax are intended to establish the conditions under which

properties can be attributed to a proposition that comes from the possibilities of

its admission in the deductive system being studied, such as, for example, the

Richard’s paradox is a ‘semantic antinomy,’ because it relies not on a logical paradox but rather thwarts

definitions of truth and falsehood: Richard opens up a certain class, and then proves that class impossible

on its own terms. Richard’s paradox involves the possibility of clearly and completely defining a set of

numbers, and, like the other paradoxes, proposes that one imagine the constitution of such a set. Once

one has listed all the numbers definable by a certain principle then one can claim a set x of all the

countable numbers and demarcate y as the number which is not countable by the rule of definition. What

has happened here is that y has inadvertently been defined and as such must be included in x, although

this is inevitably a paradox. As such, one can never complete the set.

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following properties: to be demonstrable, to be refutable, to be compatible

(meaning several propositions) to be complete (meaning a system of axioms).73

The continuity between elements, this syntax of equation, is a question of the integrity of

the set, and the form of difference that the set articulates, in opposition to other regimes

of difference.

This differentiation between the genera and the mathematical sets are best

described by Lautman, who construes this divergence as one that replicates the

distinction between the Aristotelian work on species versus Platonic division. Lautman

explains the ordering processes of axiomatic set theory as pertaining to division rather

than genera:

It is therefore not Aristotelian logic, that of genera and species, that plays a part

here, but the Platonic method of division, as taught in the Sophist and the

Philebus, for which the unity of Being is a unit of composition and a starting

point for the search for principles that are united in the Ideas.74

Lautman sees a fundamental Platonism reflected in the processes of arithmetic, algebra

and multiplication. Regarding the ‘philosophical importance of the activity of

dissociation in mathematics’ (essentially the activity of forming sets) Lautman notes that

73 Albert Lautman, Mathematics, Ideas and the Physical Real, trans. Simon B. Duffy (London and New York:

Continuum, 2011), 15. 74 Albert Lautman, Mathematics, Ideas and the Physical Real, trans. Simón B. Duffy (London and New York:

Continuum, 2011), 41.

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certain notions of elementary arithmetic and algebra, which seem simple and

primitive, envelop a plurality of logical or mathematical notions, delicate to

specify but in all cases clearly distinguishable from one another. It is in this way

that arithmetic equality is the only equivalence relation such that the countability

of the individuals of a set is conflated with the countability of classes of

equivalent individuals as defined by this relation.75

What this mathematical explanation is referring to is the philosophical coordinates that

exist but are perhaps not recognised when undertaking simple or seemingly self- evident

mathematical activities like arithmetic. Here, Lautman is pointing to the ‘countability’

of individuals in a set and the countability of classes, which are equivalent in arithmetic.

One can also apply arithmetic principles across classes. This is an instance of a wider

phenomenon that Lautman is trying to specify: the division of classes and individuals

does not sacrifice the generality of arithmetic. Here we have division without genera,

which, to add a potentially confusing claim to this, allows for the generic integrity of

arithmetic (genera and generic being utterly distinct terms in this discursive framework).

Axiomatic set theory relies, thus, on a conceptual scheme of division as opposed to

genre. Set theory does not rely on sets to demarcate different genres, or species, from

each other to avoid applying improper axioms or functions to those sets. Rather, the

reliance on sets exists for the purpose of sustaining division rather than ‘extended

types.’76

In Watt, permutation replaces normal narrative counting, thus substituting one

regime of difference with another: one which enacts a kind of hyper-naturalism,

75 Ibid. 40. 76 Ibid. 42.

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resulting in a halting of narration, a suspension (for both reader and character) of plot

and fictional world, as the lists run their course. But this is not only an act of taking

naturalism to its extreme, not only a matter of amplifying it, but also indicating a

fundamentally different form of difference (and the grouping of sets). This partial

negation of the dominant aesthetic enumerative procedure here reveals the distinction

between this form of difference and the meaningful difference created in description and

narration: this difference is based on set theory, and on the division of the worlds into

classes and genitive structures, whereas the latter mode of difference is based on

meaningful difference between elements. In the former, we are concerned with a

principle that elaborates its way down a certain set: that of the Lynch family. The

division that is enacted here is not a matter of meaningful difference – the individuality

of the members of the Lynch family, a recognition of their existence – so much as an

attempt to apply the genitive principle and elaborate the parameters of a set. This

‘difference in difference’ is one way that the novel is comedic: in Watt we find an

inappropriate form of difference in the place of another. Yet more fundamentally it is

this form of radicalised description and Platonic discourse that makes Watt a

‘mortification of novelistic discourse’77 or an ‘anti- novel’78: a prose event that achieves

its subversion of the novel form through elements that are mathematical in register, and

relate to the metaphysical consequences of mathematics, in other words, to the

enumerative organisation of the world.

77 Murphet, ‘The Mortification of Novelistic Discourse in Beckett’s ‘Trilogy.’’ n.pag. 78 John Bolin, Beckett and the Modern Novel (Cambridge and New York: Cambridge University Press, 2013),

5; 71; 164.

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3.2 CHAPTER THREE PART TWO

MATHEMATICS CONTRA EMPIRICISM: THE RADICALISATION OF NATURALISM IN BECKETT�S

LATE WORK

That’s how he reasons, wide of the mark, but wide of what mark, answer us that.

Samuel Beckett, Texts for Nothing

In the preface to Writer and Critic, Lukács writes that ‘With its forms of organisation, its

science, and its techniques of manipulation, modern life moves relentlessly toward

reducing the word to the mechanical simplicity of a mere sign.’79 The sign, here, is the

word without depth or warmth: it is language dissociated from its animation by human

beings. For Lukács the transmutation of word to sign implies

a radical departure from life, for the dynamism of everyday language derives

precisely from its always being either less or more in vocabulary and syntax than

mere signs: less in that in its ambiguity it skirts the essence of the object being

79 Georg Lukács, ‘Preface,’ in Writer and Critic and Other Essays, ed. Arthur Kahn (London: The Merlin

Press, 1978), 11.

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discussed, more in that in its very imprecision it articulates the concrete essence

of an entire concrete complex.80

Beckett’s late short fiction, theatrical and television works continue his task of creating

an art of the ‘unword’: a word that betrays the ‘grammar and style’ or the ‘surface’ or

language and presents itself as an opening to what is outside or beneath the phenomenal

fabric of dynamic speech and writing.81 In this chapter I will demonstrate how this work

– whether textual or televisual – realises this shift from the word to the sign by creating

prose that attempts to evacuate traditional elements of the phenomenal: narrative

experience, production or representation of a world and vivid imagery. In these late

works Beckett is concerned with the two most basic predicates of experience: space and

time, which, the Kantian tradition, are considered to be conditions of phenomenality,

rather than content of phenomenality.82

S.E. Gontarski frames Beckett’s later artistic concerns in terms of abstract

boundaries, claiming that ‘Despite his early insistence on ‘keeping our genres more or

less distinct,’ Beckett seemed in this later phase of his work to have stretched beyond

such limitations, beyond the generic boundaries to examine the diaphanous membrane

separating inside from outside, perception from imagination, self from others, narrative

from experience....’83 I take the most important of these ‘generic boundaries’ to be this

distinction between conditions of phenomena (space and time) and content of

phenomena (the predicate). The formal device that enables this transition exists between

80 Ibid. 11. 81 Beckett, The Letters of Samuel Beckett, Volume 1: 1929-1940, 518. 82 Immanuel Kant, Critique of Pure Reason, trans. Marcus Weigelt, Critique of Pure Reason (London:

Penguin Classics, 2007), 65. 83 S.E. Gontarski, ‘Introduction, From Unabandoned Works: Beckett’s Short Prose,’ in Samuel Beckett

The Complete Short Prose 1929-1989 (New York: Grove Press, 1995), xxix.

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these two pillars of experience: the numeral, the mark of presentation that is neither

condition nor content, but which exists in the paradoxical and nebulous space of

measuring conditions and potential conditions. In this chapter I will analyse the numerical

presentation of space and time – topos and duration – in these works, as well as the

transformation of syntax into a technique of measure. In the previous chapter, I looked

at the numerical stakes in Lukács’ definition of naturalism, whereby the realist achieves

an expressive narrative immersion in the scene but the naturalist, on the other hand,

enumerates the scene.84 The naturalist, in Lukács' rendering, stands outside of the

fictional scene that he or she presents and as such is associated with the mechanisation

and objectivity of the photograph, whereas the realist retains the gestural quality of the

painting. Here, I again will look at Beckett’s naturalism in numerical terms, exploring

the reformulation of spatiality in prose and television work from a domain of measure to

a domain of magnitude. It is this transition, from measure to magnitude, and from

content of phenomena to conditions for phenomena, that will enable Beckett to produce a

‘generic’ rather than a ‘general’ literature, radicalising the tenets of naturalism against

itself. This is, again, a realisation of both Beckett’s early commitment to the ‘unword’ as

well as an inversion of the naturalist task: the writer stands outside of the scene without

the illusion of the veridical, indeed, the writer stands outside of the fictional scene

precisely to divest him or herself of the task of the veridical and claim, for art, what is on

the ‘other side’ of the membrane of textual representation.

The 1950s were a transitional decade for Beckett and the production of

theatrical and prose works of this period are characterised by subordination of the word

to a particular kind of ‘sign’: the geometric coordinate. Beckett’s first work for the

theatre was En Attendant Godot, written in 1952 and translated into English in 1953. Fin de

84 See Lukács, ‘Preface,’ 14.

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Partie, Actes sans Paroles I and Actes sans Paroles II and Krapp’s Last Tape would all follow in

the extraordinarily prolific output of the next few years. The short prose works All Strange

Away and Imagination Dead Imagine are exemplary transitional pieces in that the prose is

given over to instructions for the creation or direction of a spatial scene, with narrative

attempts to both ‘deaden’ the imagination and to produce the barest generation of an

abstract, cognitive spatial product. Ruby Cohn claims that ‘Beckett came close to

painting still lives in movement,’ an idea that captures the various contradictions of

Beckett’s later short prose and theatre, in particular the concern with duration and

extension, in the form of contortion in or by time and space.85 In All Strange Away,

Imagination Dead Imagine and Worstward Ho we see a move away from a preoccupation

with the circulation system of words and the developments of idiosyncratic or subversive

logics. Instead, we see a consideration of the spatial stakes of numeracy, or an

engagement with number and image that attempts to reveal a material grain that exists

behind language, which, in these works is a topos that precedes and shapes duration.

Appropriately, Beckett is not interested in representing space and time (which would

contradict a genuine definition of conditions of phenomena) so much as generating it.

Once again, number fulfils the function of a modernist generic, a formalisation ‘behind’

form, radicalising the modes of counting within naturalism.

*

3.21 ALL STRANGE AWAY AND IMAGINATION DEAD IMAGINE:

IMAGINATION BY NUMBERS

85 Cohn, A Beckett Canon, 31.

183

All Strange Away (1964) is both a chronologically and formally transitional work,

appearing between Beckett’s prose and his theatre or television productions. The first

line of All Strange Away forms the title of Imagination Dead Imagine, which was initially

written as a shorter version of All Strange Away before being developed into a separate

piece. This latter text still retains much of the content of All Strange Away but is devoid of

the more substantial pornographic passages of the earlier text.

All Strange Away has at best a tenuous narrative thread. A protagonist, who we

know nothing of, produces a one-paragraph monologue that is set in what Cohn

identifies as a ‘hemicycle.’ A ‘hemicycle’ is, Cohn explains, the same as a semicircle, but

the latter word ‘lacks the resonances of enclosure – hem – and of repetition – cycle.’86

The hemicycle is described as ‘bonewhite’ and the ‘floor like bleached dirt’ and thus the

text, like the play Endgame, gives the impression of being set in a skull.87 All Strange Away is

divided into two sections, the first without a title and the second entitled ‘Diagram.’ The

prose describes a scene – imagined variously lit and unlit – initially populated by a man,

observed by the two onlookers Jolly and Draeger Praeger Draeger, subjected to images

and memories almost as a patient in a psychiatric or psychoanalytic clinic might occupy

a world of fantasy watched by distant, observant eyes. Subsequently these figures will

disappear and the space will be filled only with Emma; her face, ‘arse,’ knees and feet

each aligned to one of the coordinates of the space.

All Strange Away is narrated in a stream of consciousness in an instructive register

(‘take his coat off, no, naked, all right, leave it for the moment’88) and, as in Imagination

Dead Imagine, the reader is enjoined to imagine the elements of the scene literalising the

86 Ibid. 286. 87 Samuel Beckett, ‘All Strange Away,’ in Samuel Beckett The Complete Short Prose 1929-1989, ed. S.E.

Gontarski (New York: Grove Press, n.d.), 173. 88 Ibid. 170.

184

demand that the traditionally fictive work places upon its audience. The scene is

described using coordinates – ‘Five foot square, six high, no way in, none out’ –

privileging an efficacy of description and a breathless, crowding of clauses that befit the

entrapment of the closed space.89 We are told that the ‘someone’ in the ‘place’ is talking

to himself ‘in the last person,’ which proposes a simultaneity between the narrator and

the man in the space – but this locus of the voice is only hypothetical.90 The narrator

breathlessly stipulates the predicates of the scene, the contortions of the body in the

scene, and the shifts in light and colour, producing a monologue that is formally closer

to stage directions (or an author’s notebook) than to a work in its own right. As such,

these two texts constantly configure themselves as a supplement to fiction rather than an

original; fictionality is first and foremost the concern of the text rather than fiction per

se. The constant pauses and the listing structure of the sentences give the sense of an

abbreviated speech and the text comes to resemble an instruction manual, dense with

cues for the reader to abandon imagery: ‘Islands, waters, azure, verdure, one glimpse

and vanished, endlessly, omit.’91 Staples of the exotic (islands) or vivid (azure) are evoked

only for the purpose of exorcism (‘one glimpse and vanished’), a process familiar from

the cathartic lists in Watt.

Death pervades both Imagination Dead Imagine and All Strange Away. The narrator

of Imagination Dead Imagine must ‘crawl out of the frowsy deathbed and drag it to a place

to die in’ and the numbers that begin, from the first page, to structure the monologue

strike one as symptomatic of narratorial exhaustion.92 The refrain that closes so many of

89 Ibid. 169. 90 Ibid. 91 Samuel Beckett, ‘Imagination Dead Imagine,’ in Samuel Beckett The Complete Short Prose 1929-1989, ed.

S.E. Gontarski (New York: Grove Press, 1995),182. 92 Beckett, ‘All Strange Away,’ 169.

185

the narrator’s sentences – ‘that again’ – instills a weariness in the monologue

compounded only by the numbers which demarcate the closed scene in which the

narrator finds himself. Imagination Dead Imagine opens by evoking an imagination after life

or, an imagination after some sort of finitude: ‘No trace anywhere of life, you say, pah,

no difficulty there, imagination not dead yet, yes, dead, good, imagination dead

imagine.’93 There is a curious narrative logic to this. The imagination is, initially not

dead and then dies, and this is affirmed (‘yes, dead, good’). Yet this quickly becomes a

cycle: the imagination ‘kills itself’ in order to imagine itself dead again in order,

precisely, to resurrect itself. The reader must imagine a dead imagination, an

imagination that is almost but not entirely negated. This imagination exists but is

obsolete, or – like Lukács’s sign – lifeless, without breath or animation. The injunction

to imagine a dead imagination is an injunction to imagine thought outside of the human

or beyond a certain finitude and thus presents a paradox that reveals the limits of

literary representation: eternal, fossilised mental worlds. Or, in other words, the

abstraction of forms outside of human experience or existence, reminiscent of what

Quentin Meillassoux calls the ‘glacial world’ of primary qualities.94 All Strange Away is

rendered in topological coordinates that facilitate this same sense of obsolete

imagination. The body in the text contorts according to four angles of the quadrilateral,

rendered as the algebraic points A, B, C and D (and e, f, g, and h that set the bounds of

the ceiling), which contract to reduce the size of the space they demarcate.

Just as in Watt, where the descriptive, evocative and narrative conventions of the

novel are doubled back in the fiction to create a kind of anti-novel, in All Strange Away

and Imagination Dead Imagine these same conventions are rejected. The rejection, this

93 Beckett, ‘Imagination Dead Imagine,’ 182. 94 Meillassoux, After Finitude: An Essay on the Necessity of Contingency, 115.

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time, comes not in the form of permutation but in a narrative that is structured around

magnitude, rather than traditional forms of measure. This use of magnitude rather than

measure echoes the pre-Enlightenment subject-centred measure famously depicted in

Da Vinci’s ‘Vitruvian Man,’ the Renaissance exemplar of the connection between art

and mathematics. In All Strange Away and Imagination Dead Imagine, as in the ‘Vitruvian

Man,’ we see images structured by a mathematics that proceeds from the subject rather

than a countable system; a mathematics against objective measure.

All Strange Away begins with the two conceptual coordinates that will govern the

piece, imagination and place:

A place, that again. Never another question. A place, then someone in it, that

again. [...] Five foot square, six high, no way in, none out, try for him there.

Stool, bare walls when the light comes on, women’s faces on the walls when the

light comes on. [...] Light off and let him be, on the stool, talking to himself in

the last person, murmuring, no sound, Now where is he, no, Now he is here.95

One of the defining features of these two works is the use of coordinates to map the

scene and provide a substitute for images of contorting bodies. In the previous chapter I

argued that Watt and Molloy double the traits of the novel to stage, indicate, or inflate

these traits to work within and against the form at the same time. The two short texts

under consideration here replicate this structure. All Strange Away literally commands the

reader to imagine scenes whilst simultaneously evacuating their imagination of imagery.

In this text imagery is replaced by coordinates that are abstract and mathematical and

thus do not produce what we might traditionally associate with an image or an

95 Beckett, ‘All Strange Away,’ 169.

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imaginative experience. This is compounded by the question of who the narrator is

addressing. There is often little to suggest that the narrator of All Strange Away is

recounting anything other than an imaginative scene within his or her own head: a

‘fancy,’ to use the narrator’s word, and indeed the white rotunda literally resembles the

inside of a skull.96 But injunctions like ‘see how he crouches down and back to see’

suggest that his vision is also either observation or communication.97 There are barely

perceptible changes in speaker or address; initially the narrative voice seems to address

the man in the room or whoever is directing him, but then, with a self-reflexive remark

turns outwards to address the reader: ‘Light flows, eyes close, stay closed till it ebbs, no,

can’t do that, eyes stay open, all right, look at that later.’98 This oscillation between a

kind of stream of consciousness and compulsive injunction or instruction that

accompanies it creates an increasingly confounded series of positions with the

introduction of geometric coordinates:

Call floor angles deasil a, b, c and d and ceiling likewise e, f, g and say Jolly at b

and Draeger at d, lean him for rest with feet at a and head at g, in dark and

light, eyes glaring, murmuring He’s not here, no sound, Fancy is his only hope.99

The hemicycle thus contains a cube and here we have a radicalisation of naturalism as

Lukács defined it: the fictional scene is mapped utterly without the limitations of

96 Both James Knowlson and Graham Fraser note that the rotunda appears to be the inside of a skull. See:

James Knowlson, Damned to Fame: The Life of Samuel Beckett, Damned to Fame: The Life of Samuel Beckett

(London: Bloomsbury, 1996), 531 and Graham Fraser, ‘The Pornographic Imagination in All Strange

Away,’ Modern Fiction Studies 41, no. 3 (1995), 516. 97 Beckett, ‘All Strange Away,’ 171. 98 Ibid. 171. 99 Ibid. 171.

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perspective within the scene but from a thought outside the scene. This is then not the

depiction of a scene witnessed or experienced by the narrator (as would befit realism, in

Lukács’ terms) so much as a scene that extends from the narrator’s imagination, from a

cognitive or imaginative position outside of the narrative scene. If indeed these scenes

are set in a skull, and in the mind of the narrator, then there is a contradiction

formalised here: the imagination is both the subject from which these visions emerge as

well as being its object.

The positions of the figures in this scene become increasingly obscene and

visions or views of Emma capture not a whole being but – like a series of photographs

rather than a film or a painting – fragments of her: ‘First face alone, lovely beyond

words, leave it at that, then deasil breasts alone, then thighs and cunt alone, then arse

and hole alone.’100 Emma is split into eight images, each of which is focussed on

individually. The coordinates that map Emma’s contortions are increasingly obsessive.

The exactitude of the positions eliminates the sensuality that might convincingly

accompany fantasy or pornography, suspending imagery and producing an imagination

of utter abstraction: ‘For nine and nine eighteen that is four feet and more across in

which to kneel, arse on heels, hands on thighs, trunk best bowed....’101 It is notable that

these are not Cartesian coordinates, where the body might be mapped out on a two

dimensional graph in a stable representation. Instead, the positions to which body parts

are assigned are points. This form of measure requires only the notion of distance

between A, B, C and D. In other words, this is no longer Cartesian measure, but

topology, the study of place under a continuous ‘deformation’ (no longer a geo-metry,

measure of the earth, but a topo-logy, a study of place). Topology emerged from a

100 Ibid. 171. 101 Ibid. 172.

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combination of geometry and set theory and focuses on properties of spaces undergoing

a change, such as the emergence of one object from another. Emma’s ‘measures’ are

algebraic but these signs do not stand in for numbers that can be derived from an

equation. Instead, we have coordinates without stable measure that are put in flux by

the narrative progression, which – in a diminution perhaps typical of Beckett’s prose –

literally retracts the cube that Emma is contained within. In Beckett’s early publication

on Marcel Proust, he introduces the idea of a literary geometry that describes its object

without measuring it:

It will be impossible to prepare the hundreds of masks that rightly belong to the

objects of even his most disinterested scrutiny. He accepts regretfully the sacred

ruler and compass of literary geometry. But he will refuse to extend his

submission to spatial scales, he will refuse to measure the length and weight of

man in terms of his body instead of in terms of his years.102

Here, in All Strange Away, we have Beckett’s own realisation of this literary principle.

Where Proust might measure a man in terms of time, Beckett retains a spatial form and

substitutes measure for magnitude.

The second section, ‘Diagram,’ is utterly preoccupied with Emma, instead of

than the man of the earlier section, and Emma’s bizarre interaction with either a grey

rubber ball, or a ‘sprayer bulb.’ ‘Diagram’ consists of a fantasy revolving around

Emma’s contorting body although whether or not Emma and the positions we are told

to imagine her in are products of the man’s fantasies or an external narrator is unclear.

This ambiguity is maintained by virtue of the injunctions to imagine, the instructive

102 Samuel Beckett, Proust and Three Dialogues with Georges Duthuit (London: Calder and Boyars, 1987), 12.

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register and the physical impossibility of the contortions, rendering the scenes as

potential rather than actual. ‘Diagram’ quickly turns into a kind of detached

pornographic scene: ‘Imagine him kissing, caressing, licking, sucking, fucking and

buggering all this stuff, no sound.’103 This scene is again rendered in points a, b, c and d,

evoking an abstracted sexual combinatorics that can potentially be plotted or reenacted

by virtue of a graph, but, most interestingly, a graph with no necessary measure:

[...] Rotunda then as before no change for t he moment in dark and light no

visible source spread even no shadow slow on third seconds to full same off to

black two foot high at highest six and a half round good measure [...] floor

bleached dirt or similar, head wedged against wall at a with blank face on left

cheek and the rest the only way that is arse wedged against wall at c and knees

wedged against wall ab a few inches from face.... 104

Both duration and imperial measure are cited in the descriptions, but these are

contained with a space that is essentially measured by four letters: a, b, c and d. Here,

the alphabet is being used ‘numerically,’ hijacked for use in a second order that is

outside of the order of natural language. At the same time, we have a kind of

literalisation of the process of composition: the letters of the alphabet construct an

imaginary scene and delimit the movement within that scene.

The figure is placed in fantasy, in the imagination not the world, by virtue of this

move to geometry that renders the scene abstract in the same gesture that it describes

movement and position. The contortions of Emma’s body in the space recall a failed,

103 Beckett, ‘All Strange Away,’ 171. 104 Ibid. 178.

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de-eroticised peep show. The mathematical presentation of the scene recalls

Baudrillard’s theorisation of the relationship between ever more ‘real’ – exact and high

definition – images and their purchase on intelligibility and visibility. Writing on the

distance between virtual reality and the ‘real world,’ artificial reality and the artificial,

and, pornography and desire, Jean Baudrillard claims that

The highest definition of sex (in pornography) corresponds to the lowest

definition of desire. The highest definition of language (as computer coding)

corresponds to the lowest definition of sense. Everywhere high definition

corresponds to a world where referential substance is scarcely to be found any

more.105

This pairing – sex and language – echoes the dynamic in All Strange Away between

passages that enumerate sexual acts literally within the cognitive space (the skull) but (by

virtue of the topology) with reference to only a cerebral, hypothetical scene.106 Just as

105 Jean Baudrillard, Art and Artefact, ed. Nicholas Zurburgg (London: Sage, 1997), 26-27. 106 The Marquis de Sade produced exquisitely numerical erotic scenes. Gaëtan Brulotte notes ‘All this is

rendered with a strong penchant for mathematics, according to which to count and to measure have an

erotic value all of their own [...] For Sade, eroticism is the opposite of wild abandon’ (57). This association

between mathematics and the erotic emerges precisely from what would seem to be their antithesis. For

Brulotte Sade inverts the formalism in ‘monastic, military and industrial traditions’ (57), transforming the

strictures of these domains into the capacity for jouissance. For a broader discussion of this, see Gaëtan

Brulotte, ‘Sade and Erotic Discourse,’ Paragraph 23, no. 1 (2000), 57.

The inversion of rules to produce jouissance involves commuting precisely these rules to the domain not

of reason, where they purport to emanate from and influence, but the domain of the senses, the ‘logic’ of

here emerging not from reason but desire. The erotic potential of the number is, much more frequently in

Sade, simply an issue of quantification. More broadly, Sade’s ‘calculations’ emerge from desire and the

impetus for pleasure: ‘The faculty given me by Nature whereby I may dispose myself in a favorable sense

toward such-and-such an object and against some other, depending upon the amount of pleasure or pain

I derive from these objects: a calculation governed absolutely by my senses [...]’ (34). Numbers, here, are

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numbers provide ever greater exactitude for the scene, so too is there a loss of image,

exactly the task of a text that seeks to ‘imagine dead imagination.’107

In the Critique of Pure Reason Kant theorises the a priori conditions for experience

in an argument that – if read in an imperative rather than declarative mode – echoes

the tone and task of All Strange Away:

Remove from your empirical concept of a body everything that stems from

experience, one by one: the colour, the hardness or softness, the weight, and

even the impenetrability, and there still remains the space which the body (now

entirely vanished) occupied, and this you cannot remove.108

In Kant’s figuring mathematics is only a priori insofar as mathematical

statements contain their concept in the predicate, as opposed to requiring experience to

associate the concept with the predicate. For Kant, space-time is the condition of

calculations of the senses: the exact inversion of their usual function in discourse, scholarship, society. This

is a perennial artistic truth: the dry objectivity of the rule barely obscures the debauchery implicit in its

very stringency. See Marquis de Sade, Juliette, trans. Austryn Wainhouse (New York: Grove Press, 1968).

107 Such an idea is echoed by Ihab Hassan in his model of the relationship between Joyce and Beckett.

Hassan uses Elizabeth Sewell’s opposition between nightmare and number to describe the distinction

between Joyce and Beckett. While Joyce is the writer of ‘nightmare,’ Beckett is the writer of ‘number’:

‘The language of Nightmare is that of confusion and multiple reference; it creates a world in which all is

necessary, all significant; everything is there at once. But the language of Number empties the mind of

reference; it creates a world of pure and arbitrary order; nothing there is out of place’ (67). Here, Joyce’s

‘maximal’ prose stands in for proliferation in and of language, which is opposed to the ‘language of

Number’ of Beckett, whose overwhelming feature is to ‘empty’ rather than ‘fill.’ Hassan goes on to note

that ‘the structure of Beckett’s work is miraculously empty – anything can be made to fill it – as the

structure of Joyce’s is ineluctable. There is profound parody in this; the parody of archetypes of numbers’

(67-68), If numbers, indeed, serve to ‘empty the imagination,’ they would occupy the paradoxical space

that the line ‘imagine dead imagination’ evokes. See Ihab Hassan, ‘Joyce-Beckett: A Scenario in Eight

Scenes and a Voice,’ in Paracriticisms: Seven Speculations of the Times (Champaign: Illini Books, 1984). 108 Kant, Critique of Pure Reason, 40.

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phenomenality, and it is the intuitive a priori notion of space that produces one aspect of

mathematics – geometry, and the intuitive a priori notion of time that produces another

aspect of mathematics – counting and analytical mathematics. Such a priori

propositions are ‘indispensib[le] [...] for the possibility of experience itself.’109

Beckett was familiar with Kant’s work and, indeed, as Gontarski and Ackerley

point out in their companion to Beckett, the narrator of Company summarises one of

Kant’s central claims: ‘Pure reason? Beyond experience.’110 A version of this statement –

pure reason? beyond imagination? – seems to be at issue in these two texts. The

coordinates – A, B, C, D – of All Strange Away and Imagine Dead Imagination are not simply

alphabetic marks divorced from words but are, here, marks of partial a priori, what Kant

called the ‘synthetic a priori’ in his Critique of Pure Reason. The concept of the a priori is

relevant here in that it refracts the significance of topological marks in All Strange Away in

terms of a broader artistic project of a ‘generic’ literature. For Kant the structures that a

priori knowledge follows are informed not by experience (‘synthetic’ knowledge), but by

reason, hence being termed ‘analytic.’ An ‘analytic’ proposition is true by virtue of the

terms it contains (the famous example: ‘All bachelors are unmarried’) as opposed to

some reference to the world (‘all bachelors are unhappy’). However, a priori knowledge,

whilst separate from empirical evidence, is not completely separate from experience. In

Kant’s formulation the a priori has a restricted relation to experience: it may contain its

predicates and, as such, not require any additional information from experience, even if

the definition of identity of those predicates requires some experience in the world.111

From this, Kant concludes that ‘knowledge a priori is either pure or impure.

Pure knowledge a priori is that with which no empirical element is mixed up, it is 109 Ibid. 40. 110 Ackerley and Gontarski, The Grove Companion to Samuel Beckett, 295. 111 See Kant, Critique of Pure Reason, 38.

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knowledge that is the pure product of cognition. For example, the proposition, ‘Every

change has a cause,’ is a proposition a priori, but impure, because change is a

conception which can only be derived from experience.’112 It is only time and space that

are pure a priori. This does not, crucially, mean that space and time are intrusions of the

noumenal into perception, but instead that they are pure products of human cognition.

Space and time are, in other words, the limits of reason, the ‘rose-tinted glasses’ that

human cognition colours reality with. For Kant, analytic a priori statements are ‘pure’ a

priori statements (and, will constitute ‘transcendental logic’113) that relate to time and

space, whereas synthetic a priori statements require understanding of a concept or

definition but not necessarily experience. Mathematical propositions assume the status

of ‘synthetic a priori’ – which is a qualified a priori that requires a degree of experience

to understand the terms but not the truth of a proposition. Such cognitions are not

verified by observation or experience (they are not objective moments of understanding)

but emerge from the domain of transcendental logic.114

Despite the restrictions put on mathematical language by Kant, there is a crucial

revelation regarding non-semiotic language here. Language stripped of its semiotic

capacity is reduced to either propositions of possession or relation: space and time are

the ultimate subtraction from experience to pure cognition. Beckett rejects the ‘dust’ of

words and imagery in order to work with the conditions of experience, which, in these

terms, are spatial: the flux of A, B, C and D. Lukács’ sign, a word without life or breath,

is here perfectly correlated with the topological point, the terms by which spatial or

temporal possession or relation are articulated. The sign, then, is that which indexes

conditions of experience rather than content of experience. The a priori is thereby also 112 Ibid. 2. 113 Ibid. 97. 114 Kant, Critique of Pure Reason, 39.

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the form of proposition that exists behind the ‘veil’ of world experience that Beckett

refers to in his formulation of the ‘unword’: ‘more and more my language appears to me

like a veil which one has to tear apart in order to get to those things (or the nothingness)

lying behind it.’115

All Strange Away and Imagine Dead Imagine outline their fictional worlds not through

evocation or propositions that might recall the world but rather through points. Like All

Strange Away, Imagination Dead Imagine also moves quickly into passages of measurement,

which outline the points that demarcate the scene and the positions that the figures take

up. The rotunda is expressed in terms of four points: ‘Two diameters at right angles AB

CD divide the white ground into two semicircles ACB BDA.’116 Crucially, a point is – in

the Euclidean definition – ‘that which has no part.’117 Here, we have a strange instance

of the a priori: it is not the statement here that reflects its definition, but the component

parts which themselves are a priori, outside of empirical verification and both

transcendental and material. These geometric points are marks which have no content

or extension and are symbols that exist evacuated of substance or import by virtue of the

fact that they themselves occupy no space or time (the point is literally ‘that which has

no part’) and generate rather than represent time and space.

Take, for instance, one of the lines from the opening of Imagination Dead Imagine:

‘Till all white in the whiteness the rotunda.’118 What this absence of punctuation

suggests is a passage: one moves through the sentence on the premise of the ‘till,’ which

is a contraction of ‘until,’ a preposition. We start, then, in this sentence, with an axiom

115 Beckett, The Letters of Samuel Beckett, Volume 1: 1929-1940, 518. 116 Beckett, ‘Imagination Dead Imagine,’ 182. 117 Euclid, The Thirteen Books of the Elements, ed. trans. Sir Thomas L. Heath, 2nd ed. (New York: Dover

Publications, 1956), 155. 118 Beckett, ‘Imagination Dead Imagine,’ 182.

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of linkage: we are held, pre-posed or pre-posited, even, before all becomes white. The

object of the preposition, the ‘white,’ is doubled over. All becomes white in the

whiteness, a transformation happens against minimal difference. We start with a

temporal preposition and are moved from this, by the modification from white into

whiteness, into a locus: the rotunda. This sentence moves from preposition to position

through merely five words, a shift from time to space. Here the absence of conjunctions

indicates a lack of logical relation and equally a lack of possession or subjectivity. This is

abetted by the absence of ‘I’ or ‘my’ in the piece, which renders such a description

abstract. Likewise even the rotunda cannot unambiguously have any attributes. The

whiteness is connected to the rotunda only through incomplete movement but is not

clearly a quality of it. Similarly, the narrator of this piece speaks of a process of ‘fall’ and

‘rise’ based on certain variables and according to certain parameters. The shift from a

rise to a fall, or a fall to a rise, may be sparked by ‘pauses of varying length, from the

fraction of the second to what would have seemed, in other times, other places, an

eternity.’119 Such a flux of temporal experience can only arise from a process that

equally has no objective referent. The language ‘the rise now fall, the fall rise’ gives us

verbs of incomplete movement that are without object and occur only through the

deformation of one modality of change into another (the rise will turn to fall). The

grammar here is constructed, oddly, through the removal of punctuation and clauses or

words that facilitate logical relations, moving away from syntactic implication and

inference to transition purely across points in a domain or scene. The minimal content is

only of time and space, two factors that Kant identified as pure cognitive products,

independent of human experience and, as such, a priori.

119 Ibid. 183.

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Of course, Kant’s understanding of mathematics and the a priori cannot quite

account for this ‘mathematics of magnitude’ rather than measure. A precursor to this

‘measure without count’ exists: it is a pre-Enlightenment mathematics iconicised in Da

Vinci’s drawing entitled ‘Vitruvian Man.’ Da Vinci’s Vitruvian Man and Beckett’s

‘Emma’ are comparable by virtue of a shared anatomical magnitudes. The ‘Vitruvian

Man’ is a late 15th century drawing by Da Vinci which details the proportions of a man

in terms of a circle and a square, and purports to illustrate the way in which man’s

proportions echo the geometric essence of the universe: symmetrical, smaller units of the

body exist as units of measure for the rest of the body (the length of the fingers equals

that of the palm, the length of a foot is equal to four palms, and so on), the body

possesses a central point (the navel) when mapped within a circle. For Da Vinci, the

Vitruvian Man is a topological endeavour, where we take topology to be an exercise of

generating number in domain. Contrary to the simplistic assumption that the Vitruvian

man provides or represents empirical measure, and hence some logo- or phallo-centric

norm – however godly – Da Vinci’s work is in fact a radically anti-empirical piece

(albeit avant la lettre). This Renaissance ideal man is notably also a man whose ideality is

determined not by measure but by magnitude. Here the hand is the base unit that

determines proportions: the human is a microcosm of a wider and universal proportion,

and there is no unit, no mechanism or constant that would indicate or allow objective

measure. In All Strange Away, Beckett has relinquished sensible proportion but, like Da

Vinci, places the figure in a square and animates it, moving Emma into positions with

coordinates, turning a question of harmony and proportion into a exercise in

pornography:

198

For nine and nine eighteen that is four feet and more across in which to kneel,

arse on heels, hands on thighs, trunk best bowed and crown on ground. And

even sit, knees drawn up, trunk best bowed, bed between knees, arms round

knees to hold all together. And even lie, arse to knees say diagonal ac, feet say at

d, head on left cheek at b.120

Beckett’s Anti-Vitruvian figures are positioned in terms of various magnitudes, defying

harmonic mathematics for obscene mathematics, and debauching perfect existence in a

circle or square through the contradiction between eroticism and mathematics. This

might, indeed, also be a riff on the notion of ‘obscene’, a term which is thought to come

from the Latin ‘onto’ – ob – and the Latin for filth, caenum, but also, in English (and

French) becomes onto-scene or a putting in scene.

This use of the topological transition away from measure to magnitude must be

recognised in terms of a mathematical tradition that exists on either side of the

Enlightenment: in twentieth-century innovations in mathematics, as well as early-

modern mathematics. Pre-enlightenment mathematics mapped the world according to

magnitudes, as we see in the use of the hand to measure and construct a cosmic

geometry. In turn the use of ideal shapes and biological units to express geometrical

correspondence rather than measure instals a topological universality sundered from

objectivity. Topology defies objectivity in the same sense that Renaissance mathematics

does by virtue of the combination of arithmetic and geometry in this branch of

mathematics. The division between arithmetic and geometry echoes, as the French

philosopher and mathematician Albert Lautman put it, the philosophical opposition of

120 Beckett, ‘All Strange Away,’ 172.

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the relation between rule (analysis) and domain (geometry).121 Where geometry was

concerned with actual place, the practice of arithmetic was not bound to ‘reality’ in the

same way. Topology combines the study of domain with the arithmetical practice of

what would, in philosophical terms, be considered speculative rules: rules that do not

necessarily have to exist or be ratified in reality. In other words, topology does not

require an object, but, rather, a spatial possibility: a relationship between one point – x –

and another – y.

Of course, the subject of these two short texts echoes Kant’s famous

transcendental subject, which is always an x: ‘By this ‘I,’ or ‘He,’ or ‘It,’ who or which

thinks, nothing more is represented than a transcendental subject of thought = x, which

is cognized only by means of the thoughts that are its predicates.’122 Appropriately x

moves only in the skull, and talks to himself in the ‘fourth’ person, a voices that takes the

self as both subject and object at the same time (the most famous example of the rarely

used ‘fourth person’ comes from The Strange Case of Dr Jekyll and Mr Hyde). This

formulation is incredibly close to the subject Beckett has at work in All Strange Away. In

pursuit of a possible literature that can – like mathematics – write the a priori, the

unword, Beckett has replaced character with the transcendental subject, with x,

achieving a form of algebraic art with no central substance, only (much like the

Unnamable) ‘the thoughts that are its predicates.’123 This is the second incarnation of

Beckett’s replacement of character with a point. In Part One of this chapter on Beckett

we saw how in Molloy the character is ‘situated’ in terms of Cartesian coordinates: at the

0, 0 point allocated to the soul on the graph that delineates numerical extension. In this

case, the cogito is the medium for rationality, through which reason moves and extends 121 Lautman, Mathematics, Ideas and the Physical Real, 9. 122 Kant, Critique of Pure Reason, 142. 123 Ibid. 319.

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itself, making it another point, another x, on the graph for which there is no centre.

Here, we have the same process of replacing character with x but it happens in a

different way. In this situation, the subject becomes x by virtue of its existence as

thinking subject without identity. The subject’s thoughts constitute its predicates,

distributing alphabetic markers in a space that emerges from the mind. This is a

literature of the dead imagination. The points describe a scene outside of the world,

outside, even, of what we know sensually and experientially to be possible. This

‘literature without measure’ is also, then, a literature of in-finitude, existing as creation-

by-magnitude outside of experience and even imagination, insofar as imagination is

connected to the finitude of experience. This is the contradiction that Beckett manages

to inhabit in this later work: he has produced a literature of x: a literature of that which

we take to stand outside of phenomenal experience, after the death of the imagination.

This x, then, is the late modernist domain of the imagination.

*

2.23 QUAD: COUNTING BACK AND FORWARDS IN TIME

Beckett’s television play, Quad, also revolves around an x. Beckett pursued work based in

images and sound as much as text in the latter part of his career. Jim Lewis, who worked

with Beckett on the television play Quad at Suddeutscher Rundfunk ‘recounted a

conversation that he had with him in 1982, in the course of which Beckett said that

every word he used seemed to him to constitute a lie and that music (in the sense of

rhythm) and image were all that were left for him to create.’124 This suspicion of the

124 John Haynes and James Knowlson, Images of Beckett (Cambridge: Cambridge University Press, 2003),

49.

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word is echoed in a conversation between Beckett and Lawrence Harvey also reported

in Haynes and Knowlson’s A Portrait of Beckett, in which Beckett claims that: ‘‘Words’ [...]

‘are a form of complacency’; writing was, he said, as if one were ‘trying to build a

snowman with dust’; nothing holds together.’’125 What Beckett is articulating here is

essentially a problem with syntax. In these lines Beckett registers a disintegration of the

glue that binds words together just as much as the inadequacy of the solitary word itself.

In All Strange Away and Imagination Dead Imagine Beckett worked with topological points

that articulate magnitude rather than objective or standardised measure. In Quad,

Beckett produces a television play in which the work is structured by a numeracy that is

also disconnected from the number line. Here, I will consider Beckett’s achievement of

another formal paradox that echoes the paradoxical magnitudes of All Strange Away. Just

as Beckett managed to generate an imagination of no image, as well as an image of a

‘dead imagination’ in this earlier prose work, in Quad Beckett produces a series of

symbols without ‘significance’ (as well as being ‘signs without life’ in Lukács’ terms). This

nullified symbolism constitutes a mathesis of literature and, reciprocally, a literature of

that ‘glacial world’ of the mathematical imagination.

In Quad four dancers from the Stuttgart Preparatory Ballet School, picked for

their slight body shape, performed in the play in which the only sound was a

background drumming that sounds like a rapid metronome. This drumming appears

less as musical accompaniment than a rhythmic aid tapping out a brisk aural measure.

Each dancer wears a different coloured hooded robe (red, yellow, blue or white), which

obscures their individual features so that they appear only as a hunched anonymous

figure. All of the action of the play conforms to the title: the dancers in turn trace the

outlines of two squares – two quads, one square inside the other – with their footsteps.

125 Ibid.

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Given that there are no visible square lines on the ground, the dancers appear to

traverse both a memorised or externally set path as well as create that path, those quads,

through their journey.

Quad was conceived as a ‘crazy invention’ for the television in 1979 but only

completely developed and produced in 1981, the same year, notably, in which Beckett

also wrote The Way.126 The Way, the two part text discussed in the previous chapter,

traces two journeys around a figure, in the first instance the number 8 and the second

instance a flipped version of 8, the lemniscate ∞. Walking is a preoccupation in many of

Beckett’s texts and The Way presents a condensed form of this. The entire text revolves

around two walks that trace a number and the symbol for the infinite, which are, the

text rather blandly informs us, essentially the same walk: ‘Same pace and countless time.

Same ignorance of how far. Same leisure once at either end to pause or not. At either

groundless end. Before back forth or back.’127 What is significant here is the symbolic

fallacy that forms the heart of these two perambulatory follies and their conclusion. This

fallacy emerges from the conflation of natural number and infinite sign in a mode other

to both mathematics and semantics: the mark. The Way draws meaning from the shape

of the number rather than that which it signifies: violating the mathematical system of

non- or in-significant marks from which the number emerges to turn it into an

ideogram. Quad too revolves around a walk that is based in geometric form, but in this

performance the figures move around a square. There are several squares or forms

generated by the number four that structure this performance: the first is the quadrat,

referred to in the title. ‘Quadrat’ is the word for ‘square’ in German and ‘quad’ in

126 Beckett cited in: Cohn, A Beckett Canon, 370. 127 Samuel Beckett, ‘The Way’ [handwritten drafts and notes with revisions, 1981] (Carlton Lake

Collection, Harry Ransom Humanities Research Center, The University of Texas at Austin’ Box 17,

Folder 3).

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English alludes to ‘quadruple’ – any sequence of four. Sequences of four or multiples of

four structure the entire play: there are four dancers, each wearing one of four colours,

who move around the square along eight diagonal paths changing direction when they

meet one of the four corners of the exterior square or meet the domain of point ‘E’ in

the centre. There are collectively sixteen different walks (four squared, notably)

undertaken by the dancers in four series. As such, we have a ‘proliferation’ of the four-

part sequence, repeated on multiple levels of the composition and choreography. Thus

Quad commits the same symbolic fallacy that we see in The Way, where an underlying

referent, the lemniscate, functions as a kind of ‘mute allegory’ standing in for what

should be symbolised or revealed, but bearing no semiotic meaning itself. The ritual that

might allegorise a sacred principle, evoke or give deference to a god, or construct some

other transcendental symbolism is devoid of a clear referent: the colours, movement and

organisation of the figures cannot be attributed to any clear allegorical reference. Unlike

The Way, in Quad there is no ‘reference’ to a lemniscate but a proliferation of the number

four, both unseen ‘beneath’ the performance in seriality and seen, in the number of

figures, colours and the sides of the square.

The figures in Quad traverse, in small, staccato steps, the four sides of the square,

entering the square one after the other, and leaving in the same order. The first dancer

enters the scene at ‘A’ – the upper left hand corner, and completes the first walk (Course

1 is described as ‘AC, CB, BA, AD, DB, BC, CD, DA’). When this figure arrives at

point A, dancer ‘3’ enters that stage and completes that same walk. The two other

dancers follow. The play begins and ends with ‘unbroken movement.’128 Like All Strange

Away and Imagine Dead Imagination, in Quad movement is structured by letters of the

alphabet that demarcate boundaries of a scene and possibilities for traversal. There is a

128 Samuel Beckett, ‘Quad,’ in Collected Shorter Plays (London: Faber and Faber, 1984), 291.

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disruption, however, to this seemingly simple permutation: the figures traverse the sides

of the square, and cross the square diagonally, but avoid a small, central point

designated as ‘E’ (though it would be far too speculative to suggest that this signifies,

again, Emma in the centre of ABCD that we saw in All Strange Away) by swerving away

from the centre when they approach the middle, implicitly constructing a small central

void within the quad with their movements. It is precisely this insertion of points, the

rapidly advancing steps, which construct the measure themselves, each taking its

meaning as a count only from its consistency as regards the previous point, and the one

that might follow.

This play, described by Beckett as a ‘ballet for four people’ contains no spoken

language or text, some commentators going as far as claiming it signals a substantial

departure from language.129 Graley Herren reads Quad in terms of Martin Esslin’s

anecdote from ‘A Poetry of Moving Images’: ‘When I first met Beckett twenty-five years

ago, he mentioned, half jokingly, that he was trying to become ever more concise, ever

more to the point in his writing—so that perhaps at the end he would merely produce a

blank page.’130 Herren argues that Quad is a manifestation of this ‘blank page,’ a work

that properly indicates a ‘divorce’ from language.131 Quad does indeed seem to take the

form of a blank page: the play consists of the activity of four dancers walking around the

outlines of two squares, without plot, event, antagonism, or any language. Rather, time

and space are measured in Quad through permutation – the turn taking of the figures –

and the rhythm of the drumming, matched by the dancer’s staccato movements around

the squares. However, if Beckett has divested his work from the spoken word here, he

129 Graley Herren, ‘Samuel Beckett’s Quad: Pacing to Byzantium,’ Journal of Dramatic Theory and Criticism

15, no. 1 (2000), 45. 130 Ibid. 56. 131 Ibid. 45.

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has not necessarily achieved a complete ‘divorce’ from writing. The play is still

structured by five letters of the alphabet. Quad then signals less a departure from

language so much as a reduction from the word to the sign at the level of composition.

The geometric choreography is at once reliant on a certain type of

mathematically significant coding but equally seems to evoke its opposite: an occult or

superstitious ritual. This combination of number and ritual (the mundane or scientific

and the transcendental) echoes the non-theistic rituals developed by Mallarmé. Here,

this ritual is concentrated in the number four: the abstruse ‘key’ to a foreign or archaic

ritual as well as the numerical fate of the performers. Minako Okamuro has suggested

that Quad has its origins in the ‘alchemical dance.’132 Okamuro reads Quad in terms of

the symbolism of the square and the circle, which are found in mandalas, the lunar

phases and the alchemical wheel, as theorised by Carl Jung as well as W.B. Yeats, both

influences on Beckett.133 A circle embedded in a square was an ancient alchemical

symbol, one that appears again in the image of the Vitruvian man discussed in the

previous section on All Strange Away. The small central ‘point’ in the middle of the scene

appears to many commentators as a circle, however in Beckett’s own graphical

rendering of the choreography this is not at all clear: ‘E’ is, in Beckett’s initial diagram,

only a point in the middle of a cross section of a square. To again use Euclid’s definition

we have a point, ‘that which has no part,’ identified only by an accompanying letter,

rather than any shape or reference that does have substance, that does partake in a

wider symbolic system.

And yet, this ritualistic play is ‘pointless’: there is no clear referent at the end of

the numerical allegory. Where in Yeats’ work numbers might have signified an 132 Minako Okamuro, ‘Alchemical Dances in Beckett and Yeats,’ Samuel Beckett Today / Aujourd’hui 14

(2004). 87. 133 Ibid.

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arrangement of centre and extension, in Quad the ‘proliferation’ of four has no

genealogy or outcome other than, as we will see below, in a reference to its very

medium. The repetitive, geometric movements and the regularity of the drumming that

measures the dancers’ movements around the square indicate a purposiveness to the

walking that goes beyond expression: it seems that the dancers are carrying out a pre-

determined process. If this is a choreography, it is a choreography that is

indistinguishable from the playing out of a code, perhaps even just the playing out of a

single number. The aesthetic apprehension of regularity, rigidity, and geometric form

provides a ‘sense’ or impression of code prior to a cognitive verification of such a code.

The four hooded figures move around the square in various orders, the play finishing

abruptly when the dancers have completed all possible arrangements, or in other words,

the ending being coterminous with the point in which each dancer has completed each

of the possible walks (there are four, each traversal starting at a different corner). This

‘numerical fate’ of the performance is a code known as the ‘Gray Code’, which, in

homage to Quad, is also named the Beckett-Gray Code. A ‘Gray Code’ is defined by Joe

Sawada and Denis Wong as ‘an ordering of combinatorial objects such that any two

successive objects differ by some pre-specified constant amount;’134 in other words, the

gaps between objects (here dancers) are maintained, and using this regularity, a finite

number of different combinations can be produced. The graph that delineates the

choreography represents both the need for the dancers to complete the walk four times

as well as the impossibility of this permutation becoming a complete or perfect cycle (i.e.

there must be six stages rather than four):

134 Joe Sawada and Denis Wong, ‘A Fast Algorithm to Generate Beckett-Gray Codes,’ Electronic Notes in

Discrete Mathematics 29 (2007), 572.

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Series One Series Two Series Three Series Four

Stage One White - - - Yellow - - - Blue - - - Red - - -

Stage Two White Blue - - Yellow White - - Blue Yellow - - Red Blue - -

Stage Three White Blue Red - Yellow White Red - Blue Yellow White - Red Blue Yellow -

Stage Four White Blue Red

Yellow

Yellow White Red

Blue

Blue Yellow White

Red

Red Blue Yellow

White

Stage Five - Blue Red Yellow - White Red Blue - Yellow White Red - Blue Yellow White

Stage Six - - Red Yellow - - Red Blue - - White Red - - Yellow White

Sawada and Wong explain the difficulty of coding such a procedure by outlining the

strictures that Beckett required for his production: a four part code with four figures

cannot be permutated in a minimal fashion. Saweda and Wong summarise the difficulty

in producing a four bit Gray code with Beckett’s requirements for the sequence of the

dancers:

At the end of each time period, one of the four actors [...must be] either entering

or exiting the stage; he wished the play to begin and end with an empty stage

and he wished each subset of actors to appear on stage exactly once. Observe

that this problem is equivalent to finding a cyclic Gray code on length 4 binary

strings. However, Beckett wanted an additional restriction on the scripting: the

actor that leaves the stage must be the one who has currently been on stage for

the longest time.1

Given that there is no possible 4-bit Beckett-Gray code, repetitions of certain

arrangements were required in order to enumerate every possible combination of

dancers. Sawada and Wong take these restrictions as the parameters of a mathematical 1 Ibid.

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problem, and set out to develop an algorithm for finding the code. In a computational

universe what is behind the image (the ‘proto-language’ of the image) is code.

In addition to a religious or spiritual suggestion, and the suggestion of a hidden

numeric ‘key,’ there is another potential allegorical reading here. Quad appeared in the

decade where televisions had been installed in most homes, a decade in which a new

essence of the image installed itself: no longer the film still but the light nodes that

produce the images on a television screen. Julian Murphet has argued that Beckett’s

plays Breath and Not I achieve a form of ‘televisual modernism’ that he had failed to

achieve in earlier works such as Film. Rebuffing the definition of modernism as an

‘autotelism’ of medium, Murphet looks precisely at the impurity of medium as an index of

Beckett’s modernist achievement in these plays. Murphet claims that Beckett has

understood his medium as ‘no longer two separate analogue tracks on one strip of

celluloid, but of two discontinuous types of electromagnetic transmission: the television

image and the radio signal used to broadcast TV sound.’2 This point holds, equally, for

Quad, which, in Murphet’s terms ‘elevated these constitutive material facts into the

aesthetic superstructure of the works and used them to allegorise the medium from

within.’3 The material allegory central to Quad involves a replication of the four light

nodes in television – one red, one white, one blue and one yellow – travelling across the

image of the screen in the form of the hooded dancers. Here, the ‘secret’ of the

performance is of course the material rather than transcendental secret, as well as the

determinism of the performance: the four light nodes in the television, the techno-

ontology of the performance. And yet this technical allegory also appears simultaneously

as a transcendental one. Quad thus also allegorises precisely its own process of

2 Julian Murphet, ‘Beckett’s Televisual Modernism,’ Critical Quarterly 51, no. 2 (2009), 67. 3 Julian Murphet, ‘Voice, Image, Television: Beckett’s Divided Screens,’ Scan: Journal of Media Arts Culture

3, no. 3 (2006): n.pag.

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composition: the use of structure and suggestion in a ritual that simultaneously reveals

and conceals something sacred.

Here, we have the suggestion of spiritual ritual without any theological reference,

and we have the sense of a mathematical coding which is only revealed retrospectively,

with the permutations enumerated by the steps of the dancers. We also see a technical

allegory that produces an image of precisely the conditions of that image production. By

virtue of the fact that the ritual is divested of its meaning or necessity, and the audience

is deprived of a revealed mythos, the play tempts superstition. In each case, the process

of traversal constructs a superstitious link between the form of the mark and rhythm

(here, the footsteps, and the drumming) that leads nowhere, or, rather, that leads only

back to itself: the letters, the organisation of the walks, the production of the televisual

image. Superstition is the term for ‘belief in [...] supernatural influences,’ although this

definition is also misleading because superstition does not refer to a system of beliefs or

belief in itself but a certain form of belief, one that presumes a certain cause behind an

effect that is not evident or logical, but veers off to posit something else entirely.4 In both

cases we have a superstitious element: in the case of the Beckett-Gray codes – the

permutational, mathematical structure that conditions the play – we have the suspicion

of a principle that underlies the activity, a permutational cycle that will only be revealed

once the play ends, the first lap of possible combinations signalling the finitude of the

play. In the case of the mystical element, we have a suspicion of a practice of worship,

ceremony or evocation that is oriented towards a spiritualism or a deity, but one which

we do not know anything about. The figures are hunched and dressed in shrouds,

resembling monk or hermit like figures and seem to be performing a ritual of sorts. This

is a ritual without purpose, history or belief: one which has no referent or ideal and as

4 ‘Superstition.’ Def 7. Oxford English Dictionary online, Oxford English Dictionary, n.d. Web, 6th March

2015.

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such transforms any sense of the mystical and the archaic into an expectation of what

the ritual will be revealed as, rather than what it can be interpreted as. This is, then,

superstition that perhaps subverts the definition too far to be a proper instance of

superstition. One key element is missing, and this is belief. In both cases, the essence of

the performance follows rather than precedes the performance itself. The god, here,

only comes after the performance, the algorithm for the permutation can only come

through the traversal of the various arrangements of dancers around the square. And

the deity, or the transcendental implication that is revealed is only that which we see: the

material conditions of the image. This technical allegory produces a form of immanent

art that echoes the very form by which numbers or algebraic units are in their very

presentation fully available, unlike language, which relies on duration and

interpretation.

Quad binds together the occult and the computational to create an aesthetic that

unites mathematics with that which is purportedly its other: the illogical or, rather, alogical

transcendental. The link between the two can be expressed in terms of a secret: these

seemingly antithetical domains of ritual and code both express a hidden secret that is the

‘origin’ of a complex system or a world (a fact that echoes the form and effect of Borges’

‘The Library of Babel’). What the number shares with the superstitious is precisely its

hermeticism: the number is, like the Godhead, a tautology, existing as presentation and

not representation, indeed, perhaps removed from proper representation. Similarly,

numbers are devices of determination just as some occult mysticism may hazard a

determinism that is both material and transcendental at the same time. And yet number

posits no definable absolute. Its presentation remains as speechless and unflinching as

any effigy. The simultaneous ‘impression’ of the mystical and computational in Quad

suggest that more fundamental and effective than any code, or sense of code, is the

manipulation of time driven by the potential of the code, which initiates a process of

counting that is prospective: that manipulates the past retrospectively and that has a

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significant paranoid element. This manipulation of time is captured by the concept of

‘hyperstition,’ a term coined by Nick Land and Reza Negarestani. Hyperstition is a

portmanteau of the words ‘hyper’ and ‘superstition.’ Land and Negarestani use the

prefix ‘hyper’ to ‘flatten the transcendence of superstition,’ implying a superstition

without transcendental signifier, and hence, a signifier whose reference systems extends

not into an archaic past, an underlying essence, or an ideal realm, but rather into the

future.5 This is duration without measure, proceeding on the basis of a potential or a

speculation rather than an event or observation or perception: the term perfectly

captures the form or workings of superstition subtracted from belief. This captures the

speculative dimension of the code for the duration of the performance, the prospective

counting that happens prior to the retrospective counting. ‘Hyperstition’ allows us to

fully comprehend not only the strange and new ritualistic form of art that Beckett

pioneers in Quad, as well as crystallising a numerical development in Beckett’s work:

departing from the permutational schemes of Watt and other novels to produce an

alternative incantatory form of count attached to durée.

‘Hyperstition’ encapsulates this perversion of superstition and the curious

‘flipped time’ of the ritual: anticipatory rather than referential. Hyperstition is the

process of setting up a count whose meaning relies on its duration, its playing out or

traversal. The code, here, is not essential but speculative. In the case of the ‘quad’ in a

graph, which we can chase up the vertical and horizontal axes of the graph, but we will

only be able to measure once we reach the boundaries, once the quad becomes finite.

Until that moment (the end of the choreography, in Beckett’s case) the enumerating

system, the ‘value’ of the permutation, the completion of the Beckett-Gray code, is

always ahead of us. Hyperstition, then, helps to conceptualise Beckett’s attempt to get to

5 Mark Fisher, ‘The ‘Hype’ in Hyperstition,’ Abstract Dynamics, 25th June 2004,

http://hyperstition.abstractdynamics.org/archives/003428.html.

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a real that is not something that lies beneath the phenomenal world, but in front of it: a

potentiality that is tapped by aesthetic labour.

Concepts like hyperstition render the Kantian distinction between synthetic and

analytic a priori unstable, just as – as we saw in the previous section on All Strange Away –

the mathematics of topology does. The conditions of experience – space and time

according to Kant – are now rendered manipulable in the same way that the numbers

and equations of the synthetic a priori are subjects of mathematical study. For Hermann

Weyl, one of the founding figures of modern topology, mathematics concerns the a

priori. For Weyl, mathematics qualifies as a form of writing the a priori given that it

involves the manipulation and alteration of time, or in other words, takes time as a

fluctuating variable and because it deals with the possible as opposed to the actually

existent objects. Weyl, who at times was strongly intuitionist in his philosophy of

mathematics crucially departs in disposition from Kant, a departure that is presaged in

the capacities revealed by topology. Weyl describes the process of mathematical thinking

as ‘the step of abstraction where intuitive ideas are replaced by purely symbolic

construction,’6 and comes to argue that mathematics has both validated and moved

beyond Kant’s system of critique in its development of areas of study such as topology.

With an implicit reference to Kant, Weyl will conclude that ‘we have learned that none

of these features of our immediate observation, not even space and time, have a right to

survive in a pretended truly objective world, and thus have gradually and ultimately

come to adopt a purely symbolic combinatorial construction.’ 7 The exercise of

mathematical symbolic language is an act of human, non-objective speculation, which is

also, for Weyl, a form of a priori knowledge and writing. Mathematics for Weyl is

capable of a non-objective and hence speculative capacity; the capacity for existence 6 Hermann Weyl, ‘The Mathematical Way of Thinking,’ in The World of Mathematics: Volume Three

(Mineola: Dover Publications, 1956), 1844. 7 Ibid.

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beyond human thought or intuitive availability. Most famously, Albert Einstein’s theory

of special relativity instituted a historical shift in understanding space and time that has

undermined the Kantian account of the a priori, and its reliance on space and time.

Whilst special relativity comes from a domain separate from mathematical topology,

these two innovations share – in Weyl’s terms – a similar relation to the imagination.

Topological constructs that are not necessarily possible in experiential reality have the

same import in terms of our understanding of what conditions experience: mathematical

modelling has the capacity to exceed the human finitude conditioned by a certain model

of space and time. In surrendering objectivity (the external viewpoint, a genuine

heliocentrism) mathematics has become free to articulate non-intuitively possible

phenomena, multidimensional phenomena, and comprehend differentiation due to

relative time and space. Topological work thus avows Kant’s privileging of the subject

whilst simultaneously dismissing the notion that space and time as we experience them

condition the a priori, and thereby redefines the symbolic constructions of mathematics

as exceeding imaginative but not cognitive grasp. Beckett has taken the medium of

writing and rendered it outside of grammar and style, outside or words, even, using his

medium numerically. The hyper-Vitruvian Emma, a body mapped to topological points

rather than a standardised count, is the image outside of imagination: this is a literature

of imagination dead imagine, just as Quad is a play revolving around a material riddle that

gives nothing of its signification other than its medial origin: a subtraction to the ‘device’

upon which we see the play. This is the ultimate achievement of a naturalist art in the

age of modern mathematics as it is expressed by Weyl. The writer stands not only

outside of the world he depicts but outside of, even, the subject position native to a

world in which the human is bound by what is phenomenally available and in which

space and time are constants. This is a mode of naturalism in which the word has been

commuted past the ‘sign’ into a point, delineating a generic flux of a place without

locale, and a figure without subjecthood as we have known it.

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*

3.23 WORSTWARD HO AND THE DEICTIC COUNT

Ingo Berensmeyer has argued that ‘Beckett’s artificial language is significantly different

from Joyce’s, because what sets it off from ordinary language is its syntax, not its lexis.’8

For Berensmeyer, and others, Beckett’s distinct style rests in the connection between

words, not the words themselves. It is this syntactical element that is modulated by

mathematics in Beckett’s work. In Watt, for instance, Beckett denatures the descriptive

count of literary naturalism to produce permutational episodes that radicalise the

attempts at creating a whole or constitutive literary world. In Watt we see a shift to a focus

less on referentiality than on what syntax concerns: continuity, constitution, and sense

making. Franco Moretti’s definition of prose as forward movement bears repeating,

here:

Verse, versus: there is a pattern that turns around and comes back: there is a

symmetry, and symmetry always suggests permanence, that’s why monuments

are symmetrical. But prose is not symmetrical, and this immediately creates a

sense of im-permanence and irreversibility: prose, pro-vorsa: forward looking (or

front-facing, as in the Roman Dea Provorsa, goddess of easy childbirth).9

The contrast that the etymology indicates, here, depends on the most elusive aspect of

both prose and verse: pattern. The syntax of prose is oriented not towards symmetry but

rather, ‘the text has an orientation, it leans forwards, its meaning ‘depends on what lies

8 Berensmeyer, ‘‘Twofold Vibration’: Samuel Beckett’s Laws of Form,’’ 468. 9 Franco Moretti, Distant Reading (New York: Verso, 2013), 162.

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ahead (the end of a sentence; the next event in the plot).’’10 It is both the capacity for

continuity and impermanence that Beckett exploits in his later work. In All Strange Away

syntax is placed under experiment by virtue of a transformation of space and figure

through the use of points that operate according to topological magnitudes. Here, the

essence of prose continuity is numerical. The sentences trace an irreversible and

continuous change in imagination but the use of points rather than real coordinates

allows for this impermanence to properly realise the motility of prose: a forward

production without teleology, without progress up (or even down) some scale, some

measure of the world. This is syntax that moves towards nowhere numerically. Beckett

continues this artistic undertaking without proper syntax in Quad. In Quad Beckett

produces a phenomenology of mathematics, derived not from a rational subject but the

experience of a performance revolving around the number four and the transformation

of mathematical temporality to advent as opposed to essence. In Worstward Ho, divested

of possession and identity and using two different numerical forms to weave a plot and

poetics, Beckett returns to prose to create an experience of non-cumulative,

performative count. In this work there is one count overlaid upon another: a syllabic

measure, which rends the prose closer to poetry, is complemented by a semantic

measure, producing a fictional world created by the cement of syntax, without any

discernable object properly represented.

The first lines of Worstward Ho are a string of very short sentences, each of which

modify the next: ‘On. Say on. Be said on. Somehow on. Till nohow on. Said nohow

on.’11 We begin with ‘On,’ which appears as an imperative, but, with the subsequent

sentences, comes to feel more like simply an experimental utterance, producing the

word to consider it. The single word (and single syllable) sentence is then followed with a 10 Ibid. 11 Samuel Beckett, Company, Ill Seen Ill Said, Worstward Ho, Stirrings Still, ed. Dirk Van Hulle (London: Faber

and Faber, 2009), 81.

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two word (and two syllable) and three word (and three syllable) sentence, which is then

followed by another three-syllable sentence, and two four syllable sentences. This creates

a progressive rhythm, starting with a simple single syllable word and building upwards

to four syllables, each stage broken with a pause. The second paragraph – only three

sentences – varies the syllable count. Beginning with four syllables, the second sentence

drops to two and then, in what seems like a flood, the paragraph concludes with a seven

syllable sentence: ‘Say for be said. Missaid. From now say for be missaid.’12 The first

section modifies the word ‘on,’ mutating the word with the inclusion of various forms:

‘say,’ ‘said,’ ‘somehow,’ ‘till nohow,’ ‘said nohow.’ The second section, meanwhile,

revolves around ‘said’ and its negation: ‘missaid.’

By the fifth paragraph, the longest so far, we have the start of a kind of count:

‘First the body. No. First the place. No. First both. Now either’ which quickly descends

into submission – ‘now either’ – and quite quickly into exhaustion ‘sick of either try the

other.’13 Again, this is a performative count. A numerical accumulation never really

occurs and what we have here, instead, is simple ‘first’ and ‘now,’ and event and then

another. This is a phenomenal count just as it is a speculative count. The experience is

of the most basic units of time now and then. Here, Beckett has doubled the

preoccupation with permutation that we saw in Watt and other texts onto the very form

of the sentences, using syllables as a count to produce the traditional literary count

found in poetry: rhythm. This ‘count’ operates on a separate level to the conceptual

count – the evocation of a body in the third paragraph, and the problem with ‘first’ in

the fifth paragraph. There is a third form of counting which is not quite counting, and

involves no numbers at all, but only points. This is the form of prose counting that we

saw in All Strange Away. ‘Till’ and ‘nohow’, following each other directly, retrospectively

12 Ibid. 13 Ibid.

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make ‘somehow’ the insufficient climax, specifying a period of time that has no measure

but is merely between somehow and nohow, which then declines into negation: nohow.

With the lack of content – despite several attempts to introduce content, notably ‘a

body’ and ‘a place’ – there is no measure, and from preposition to proposition and

negation, we simply have movement between points without form. In this work we see

pure, topological drift from somehow to nohow that enacts a mode of counting not with

numbers but with deictics.

In All Strange Away and Imagination Dead Imagine Beckett contorts rather than

‘fleshes out’ the figures that appear in both short texts, replacing evocative languages

that might generate imagery with topological coordinates and spatial distribution. The

replacement of imagery with image-coordinates circumvents imagination (precisely what

these texts claim to take up, but rather defy) and allows the a priori construction of

image to intrude upon the fictional expectation of imagery. This provides a kind of

reverse and perverse enactment of the generic drive for a primitive signifier or for the

‘unword’ that Beckett seeks, producing a priori coordinates which at once generate a

literary symbolism and negates it with an absence of referentiality. However, this is not a

case of simple presentation and then negation. This is a hortatory negation that is

effected, repeatedly, through a textual pulsation. Beckett’s use of topological and a priori

coordinates in Imagination Dead Imagine and All Strange Away serve to foreground the stakes

in his frustration with language; the problem of communicating pure cognitive products,

unmuddied by the word, but also the human centrality to any desire or drive for such an

a priori. The internal undoing of the unword and the surrendering of objectivity in

mathematics both seek a universality that is not found within phenomena. In Quad, this

is developed as a phenomenology of the count: one oriented towards an axiom or a

code, a simultaneously material and transcendental numerical destiny intuited though

ritual form. In Worstward Ho this ‘double count’ also facilitates a phenomenal experience

of numeracy as the basic element of time. In other words, this attempt to count

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deictically hazards a form of counting that is entirely phenomenal, without being

attached to a fictional scene.

In each case, Beckett dismantles the word into a topological point and thus that

which is ‘before’ or ‘beyond’ experience. This is not so much an abandonment of the

word as the transformation of the word. Beckett’s experimental grammar severs normal

syntax and transports the word out of speech, insofar as the lilt and expression of speech

constitutes a norm, out of representation proper, and into a diagrammatic or durational

existence. Beckett’s work, here, does not provide ‘exemplary’ literature that achieves an

unprecedented level of generality by virtue of an adequate and complete descriptive

count but rather a genericity of literature whereby words come to take the place of signs

without animation, or, to invert Beckett’s words the capacity to ‘imagine dead

imagination.’

‘Words are dangerous tools,’ writes Weyl, ‘The scientist must thrust through the

fog of abstract words to reach the concrete rock of reality.’14 Weyl’s complaint becomes

strangely reminiscent of Beckett’s frustration at the porousness of words – his frustration

at the fact that language acts like a fog rather than rock. Does the writer of the ‘unword’

then become a scientist here, piercing through the fog of abstraction that is language?

Not in the way that one would expect. Weyl talks about a mathematics of the possible as

a means to get at the ‘concrete’; inverting one’s expectations regarding the nature of the

concrete. ‘Mathematical thinking’ is thinking in terms of functions and variables, and its

principles extend ‘over all possible, rather than over all actually existing, specifications.’15

Part of this mathematical thinking, for Weyl, is then the ‘step of abstraction where

intuitive ideas are replaced by purely symbolic construction.’16 He illustrates this ‘step of

14 Weyl, ‘The Mathematical Way of Thinking.’ 1836. 15 Ibid. 1834. 16 Ibid.

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abstraction’ (and its necessary link to possibility) with a mathematical example of time

and space that elides the words (which muddy perception) ‘past, present and future’:

A world point is represented by a point in this picture, the motion of a small

body by a world line, the propagation of light with its velocity c radiating from a

light signal at the world point O by a vertical straight circular cone with vertex at

O (light cone). The active future of a given world point O, here-now, contains all

those events which can still be influenced by what happens at O, while its passive

past consists of all those world points form which any influence, any message,

can reach O.17

What is interesting here is the insistence upon an evacuation of content in mathematics

whilst forgoing objective viewpoint (that of measure, of external reference) and hence

privileging a grammar of the domain. Here lies the inversion of naturalism. On the one

hand this work features a world represented by a narrator that still stands outside of this

world, the narrative that does not participate in his scene, is not possessed by his scene,

but at the same time this narrator relinquishes all objectivity. Of equal interest is the fact

that here the concretion afforded by the use of mathematical symbols pertains less to the

reference of each of the symbols so much as the grammar that they construct together.

This is not necessarily a grammar in the sense of a linguistic grammar, but rather one of

world-points, space-times, extension and signals, axes, bodies and motion. This is a

grammar that Beckett achieves in these four later works: All Strange Away and Imagine

Dead Imagination, Quad and Worsward Ho. This is grammar of topology; in other words, a

grammar of a world that is no longer composed of axes which determine the movement

of time, but one is which the propagation of light determines the ‘still to be’ and the

17 Ibid. 1836.

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darkness the ‘what has been,’ without any clear designation of day or night, without any

clear measure, only a magnitude.

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4 CHAPTER FOUR

J.M. COETZEE AND THE NAME OF THE NUMBER

All in all, this patchy imitation of Oxford English studies had proved a dull mistress from

whom I had been thankful to turn to the embrace of mathematics; but now, after four

years in the computer industry during which even my sleeping hours had been invaded

by picayune problems in logic, I was ready to have another try.

J.M. Coetzee, ‘Remembering Texas’

‘I know all the numbers. Do you want to hear them? I know 134 and I know 7 and I

know’ – he draws a deep breath – ‘4623551 and I know 888 and I know 92 and I know

-’

J.M. Coetzee, The Childhood of Jesus

The young John Maxwell Coetzee left Cape Town in 1962 for London to pursue the

promise of a literary life, as well as a respectable career associated with mathematics that

might support the bohemian existence of a writer. Coetzee had taken honours in two

undergraduate degrees from the University of Cape Town: in English (awarded in 1960)

and Mathematics (awarded in 1961). As the above reminiscence suggests, it was in fact

mathematics that most impressed the young Coetzee in the academic realm, whereas his

work in literary criticism (though not, it seems, the promise of a life as a poet) left much

to be desired. In London Coetzee would work as a computer programmer and write a

Masters thesis on Ford Madox Ford, whose novel, The Good Soldier, Coetzee described as

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‘probably the finest example of literary pure mathematics in English.’1 Here, Coetzee

proffers an intuitive isomorphism between literature and mathematics in Ford’s work.

However, this bold statement is also limited. Coetzee certainly did not, in his analysis of

Ford, carry out any extended investigation into or argument for a literary mathematics,

and the two domains are awkwardly joined here: literature is identified as a version of

mathematics and mathematics becomes a metaphor for a formal achievement. The

‘mathematics in English’ that Coetzee would achieve in his novels of subsequent

decades would achieve a far more subtle reciprocity between the two domains.

The only extended study of Coetzee and mathematics written to date is Peter

Johnston’s landmark doctoral thesis ‘‘Presences of the Infinite’: J.M. Coetzee and

Mathematics,’ which considers much of Coetzee’s early and middle work, tracing a

biographical link between Coetzee’s mathematical and literary pursuits. 2 Johnston

argues that Coetzee comes to an awareness early in his academic career of how

‘conceptual metaphors drawn from mathematics’ might ‘migrate into broader

conceptual discourses.’ 3 Johnston’s study is a signal analysis of how mathematics

influenced the early work of Coetzee and how this may shape his later work. Johnston

makes the compelling argument that Coetzee is in pursuit of a ‘mathematically literate

interpretation of the modern condition’ and that in Coetzee’s work literary issues are

given focus and specificity by mathematics.4 Here, I seek to do something slightly

different. My aim in this chapter is to draw out less the way that mathematics provides a

particular lens for the literary, than the way that Coetzee focuses on the mathematics

1 Peter Johnston, ‘‘Presences of the Infinite’: J.M. Coetzee and Mathematics’ (Royal Holloway, University

of London, 2013), 63. 2 Johnston writes: ‘this thesis seeks to prioritise documentation over argument to as great a degree as

possible: as such, where the ambitions and reliability of a given claim appear to vary in inverse proportion

with respect to each other, the less speculative claim will invariably be made.’ Ibid. 12. 3 Ibid. 14. 4 Ibid.

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internal to the art of literature, specifically the art of the novel. In this chapter, I take the

compossibility between mathematics and literature to be less a migration or import of

terms – less a matter of applicability – so much as a theory of language where a generic

art is the goal or issue. In other words, Coetzee’s literary interest in mathematics is

focused on allegorising the enumerative procedures of literature itself, or what I called

‘literary numbers’ in the Introduction.

In this chapter, I will consider Coetzee’s doctoral work and two of his novels: In

The Heart of the Country (1987), and The Childhood of Jesus (2013). I will argue here that

Coetzee engages with numeracy as a privileged place where aesthetics and epistemology

become indistinguishable, a collision activated by two cultural historical processes that

frame his work: the waning of South African apartheid (and within this Coetzee’s own

dispositional dissidence to the regime) and the transformations in European

structuralism that would lead to the emergence of philosophical and literary post-

structuralism in the work of Jacques Derrida and others. In the previous three chapters

of this dissertation I considered the role of number in modernist interventions in

symbolism and naturalism in the work of Jorge Luis Borges and Samuel Beckett

respectively. Here I will consider Coetzee’s preoccupation with the mathematical and

linguistic issue of what can be counted as ‘one’ and the communal stakes that undergird

this problem. I will look at this firstly in terms of the question of ‘style’ that is pursued

analytically in his doctoral work, secondly in the operations of counting and

communication in the novel In the Heart of the Country, and lastly in terms of the question

of nominalism in The Childhood of Jesus. In both of these novels Coetzee investigates what

comes to count as ‘one’ and the processes of measuring continuity that this counting

entails. I will, in concluding, theorise Coetzee’s preoccupation with the identity and

concretion of the one in terms of Alain Badiou’s theorisation of what he calls the ‘count-

as-one.’ This theory distils the significance of the operation of counting for artistic work

by virtue of the fact that it shows how counting is a ‘generic’ operation that connects

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representation and presentation. Mathematics will thus afford Coetzee a particular

fictional mode, one which allegorises its own processes of production via a focus on

linguistic mechanisms of making something ‘count.’ Such a form manages to allegorise

the process most essential to his fiction and rhetoric: the move from presentation to

representation, and the production of worlds out of generic processes.

*

4.1 ‘LITERATURE IN THE LAP OF MATHEMATICS’:

THE QUANTIFICATION OF STYLE

During his time at IBM, whilst also writing on Ford, Coetzee started to experiment with

an applied form of literary-mathematical exchange: computer poems. Coetzee made use

of his access to the IBM 1401 and 7090 to run experimental programs, creating

programs that would produce poems that contained a themed vocabulary. 5 The

program involved inputting a semantic field and verse form into the computer, which

would then arrange the vocabulary to form the skeleton of a poem ready for editing by

the programmer. 6 Coetzee’s fictionalised account of his work as a computer

5 See: J.M. Coetzee, Youth (London: Vintage, 2003), 81. 6 Johnston has provided the best documentation and analysis of Coetzee’s computer poetry. See, in

particular, Johnston, ‘‘Presences of the Infinite’: J.M. Coetzee and Mathematics,’ 42- 43 for an account of

the ‘composition’ of the computer poems and an example of one of the poems.

Details of the publication of these poems, and their significance can also be found in: J.C. Kannemeyer,

JM Coetzee: A Life in Writing, trans. Michiel Heyns (Melbourne and London: Scribe, 2012). The accounts of

these poems in Coetzee’s ficionalised autobiography and Kannemeyer’s biography differ greatly save on

the issue of literary significance. Kannemeyer notes that:

In Youth Coetzee refers dismissively to the poems written in London, and none of these were published in

magazines or otherwise preserved. He did, however, in The Lion and the Impala, II: 1, March–April 1963, a

publication of the Dramatic Society of the University of Cape Town, publish an interesting experiment

that suggests that his years as a computer programmer did at least yield some engrossing results. The

poem, also published in the Cape Times, bears the title ‘Computer poem’’ (123).

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programmer does not substantially depart from the biographical accounts. J.C.

Kannemeyer’s account of Coetzee’s years in London corroborates Coetzee’s own

fictionalised account of his time as a programmer in London and the computer poems

he produced whilst working at IBM.7 Coetzee was interested in the extent to which a

program – a machinic intelligence that reproduces forms rather than creates them –

could come close to the production of poetry that would pass for successful literary

work. It is significant that some of Coetzee’s earliest literary work pursued foremost the

limits of the ingenuity of the author. This earliest work was already forged from a

‘doubling’ of the task of literature or, in other words, a scepticism towards the authority

of the novel. These computer poems were precursors to the stylistic analysis that

Coetzee would develop in his PhD thesis written at the University of Texas at Austin in

the years 1965 to 1969, and this constellation of experiment and criticism would help to

shape the link between mathematics and language that he later explored in his fiction.

Much of this fiction would revolve around the extent to which languages are not spoken

but rather ‘speak us’ and the possibilities of a linguistic capacity to generate worlds

through genuine, rather than repetitive, creativity. These questions are, of course, yet

another instance of the question of art ‘in the age of its technological reproducibility.’8

In this instance, the mechanisation refers not to film, photography, copying or printing

techniques, but the capacity of computation to contribute to authorship. We thus will

see, in Coetzee’s work, a relationship to authorship that seems inverted. Coetzee is not

so much interested in authorship as an activity of wielding the aesthetic potential of

language, or as working with the medium in an expressive mode. Instead, authorship

becomes the activity of delineating precisely the forms that determine the aesthetic

potential, or expressive modality of the category of fiction itself.

7 See Ibid. 122-127. 8 Benjamin, ‘The Work of Art in the Age of Its Technological Reproducability,’ 19.

226

It was the tension between creative work and reproduction that Coetzee

encountered in his work for IBM, a tension that would eventually drive him back to

literary studies. Coetzee gives a fictionalised account of his work as a computer

programmer in the autobiographical-novel Youth: Scenes from a Provincial Life II:

The newspapers are full of advertisements for computer programmers. A degree

in science is recommended but not required. He has heard of computer

programming but has no clear idea of what it is. He has never laid eyes on a

computer, except in cartoons, where computers appear as box-like objects

spitting out scrolls of paper. There are no computers in South Africa that he

knows of.9

Coetzee came to computer science early: at a time prior to the ‘PC’ and mainstream

computation. This field pushed his proclivity for pure mathematics into an applied

domain of work and administration concerned with computational products with which

he was not always comfortable. In Youth, the fictionalised version of this qualm is framed

in terms of simple outflow processes and – overarching this – in relation to capital:

There is something about programming that flummoxes him, yet that even the

businessmen in the class have no trouble with. In his naiveté he had imagined

that computer programming would be about ways of translating symbolic logic

and set theory into digital codes. Instead the talk is all about inventories and

outflows, about Customer A and Customer B. What are inventories and

outflows, and what do they have to do with mathematics?10

9 Coetzee, Youth, 44. 10 Ibid. 45.

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The usurpation of pure logic by the simple inventories of accumulation irritate the

autobiographical John. The set theoretical foundations of computation are obscured for

retail operation. It is this distinction between what the fictional John refers to here as

‘mathematics’ (symbolic logic and set theory) and its transmutation and deformation in

inventories and outflow that will become significant decades later in his fiction.

Speaking of computer science as a still developing field in 1985, the

mathematician Gian-Carlo Rota explained that ‘a new unit in science is being formed

that remains to be named. It will include the best of theoretical computer science,

neurophysiology, molecular biology, psychology, and the mathematical theory of

information. It will be important to name it properly. As the Latin proverb says, ‘Nomen

est omen.’11 Here, Rota is speaking specifically of the possibility of network theory that

emerges as new structural template and mode of thought that has applications in the

pure sciences as well as physics, biology and chemistry. This ‘new unit in science’ will

come to mould cultural, scientific and economic existence. Coetzee’s work with

computers – both in his employment and his literary forays – was undertaken at the

advent of a new science, only a quarter century after Alan Turing delivered the paper

on computable numbers and Hilbert’s Entscheidungsproblem that played a significant role

in the development of the computer.12

11 Gian-Carlo Rota, ‘Mathematics, Philosophy and Artificial Intelligence: A Dialogue with Gian- Carlo

Rota and David Sharp,’ Los Alamos Science 12, no. Spring Summer (1985), 94. 12 The advent of computation in mathematics instigated a transformation in the entire approach to

numeracy. This is summarised by Jaron Lanier in a review of Gregory Chaitin’s follow up proofs to

Turing’s solution to the Entscheidungsproblem: ‘A century ago, math was thought to be an orderly Platonic

phenomenon, imperious in its perfection. The first prominent onset of weirdness came in 1931, when

Kurt Gödel showed that important systems of mathematical ideas could never be completed. In order to

get his result, he indexed mathematical ideas in a way that was somewhat analogous to the way the Web

is now indexed by services such as Google. That computational framework began to give mathematicians

a completely new perspective.’ See Jaron Lanier, ‘Two Philosophies of Mathematical Weirdness,’ American

Scientist, no. May-June (2006), n.pag. The particular transformation wrought by computation that I want

to consider is binary arithmetic, and the implications of computable and non-computable numbers. This

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The counterpart to this, in Coetzee’s personal and intellectual context, is the rise

of post-structuralism out of structuralism. Structuralism was characterised by the task of

tracing thought in language, whereas the minor shift within and against this position.

Post-structuralism, on the other hand, comprehended the double position inherent in

such a study of thought and structure: the simultaneous exercise of and uncovering of a

logic in the analysis of structure (nomen est omen, again). The capacity of the computer for

networked and systemic ‘reading’ of symbols resembles the machinic nature of

language: the grooves and networks of grammar, be they linguistic convention and

prediction (Chomsky, generative grammar13) or the psychic structures of the soul

(Lacan, the unconscious is structured like a language14). Indeed, Coetzee would, in his

doctoral work, draw a direct connection between the computer and language, and

computational mathematics and language, in the self-reflexive moments of his studies of

structuralism. In the review of Strange Attractors, detailed by Peter Johnston, Coetzee

notes a ‘general enthusiasm [...] for structuralism, that is, for systems that seemed to run

themselves without need for intervention.’15 Such a statement reveals the connection

between the mechanical and material aspect of Coetzee’s work as a programmer and

the literary criticism and theory that would found his academic life. This emphasis on

shift is characterised by Chaitin as a shift from mathematics to programming: ‘According to Pythagoras,

all is number — and by number he means the positive integers, 1, 2, 3, . . . — and God is a

mathematician. ‘Digital philosophy’ updates this as follows: Now everything is made out of 0/1 bits,

everything is digital software, and God is a computer programmer, not a mathematician!’ Gregory

Chaitin, ‘Epistemology as Information Theory: From Leibniz to Ω,’ in Alan Turing Lecture on

Computing and Philosophy, E-CAP’05, European Computing and Philosophy Conference, (Malardalen

University, Vasteras, Sweden, 2005), 2-3. 13 Noam Chomsky, Aspects of the Theory of Syntax (Boston: The Massachusetts Institute of Technology,

1965). 14 Lacan makes this claim, in several different versions, throughout his career. It is perhaps most famously

put in Four Fundamental Concepts of Psychoanalysis. See: Jacques Lacan, The Seminar of Jacques Lacan: Book XI

The Four Fundamental Concepts of Psychoanalysis, ed. Jacques-Alain Miller, trans. Alan Sheridan (New York

and London: W. W. Norton and Company, 1998), 20. 15 Coetzee quoted in: Johnston, ‘‘Presences of the Infinite’: J.M. Coetzee and Mathematics,’ 71.

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autonomous systems would even extend to the theories of authorship that he would

wrestle with, in the coming decades, in his fiction. In the 1984 essay ‘A Note on Writing’

Coetzee summarises such a notion of authorship, a notion redundant by the time of his

writing: ‘One might also want to think,’ he says, ‘of A is-written-by X (passive) as a

linguistic metaphor for a particular kind of writing, writing in stereotyped forms and

genres and characterological systems and narrative orderings, where the machine runs

the operator.’ It is this figuring of writing as an output system, and the structuralist

notion of language as a system capable of running itself, that would form the heart of

Coetzee’s fictional enquiries: each novel enquires into its status as just such an output

system, as well as the status of linguistic expression as a kind of autonomous rather than

agential system.16

It is precisely this question of authorship, and its implications for style, that

prompted Coetzee to write his dissertation on Beckett’s novels, considering Beckett’s

attempts to write without style (to write against the idiosyncracies of language, to write

X as A: an algebraic authorship). The doctoral work, written from 1965 to 1968

occupies a perplexing, transitional place between structuralism and post-structuralism.

Coetzee sought to bring two different methods of working with (or against) ‘style’ into

interaction: the discipline of stylistics, and Samuel Beckett’s notions of language and

style.17 It is Beckett’s own resistance to style, famously exemplified by his move from

English to French, which inspires Coetzee to pursue the possibility of a literature

relieved of the burden of style.

16 Coetzee, ‘A Note on Writing,’ 95. 17 Whilst Coetzee’s work here is, obviously, on Beckett, Coetzee was also a reader of Roland Barthes, and

Barthes’ reflections on style are a pertinent influence here. The English translation of Barthes’ Writing

Degree Zero would be published the year that Coetzee completed his doctorate. See: Roland Barthes,

Writing Degree Zero, trans. Annette Lavers and Colin Smith (New York: Hill and Wang, 2012).

230

Beckett’s classic tirade against style summarises the position that interested

Coetzee:

Grammar and Style! They appear to me to have become just as obsolete as a

Biedermeier bathing suit of the imperturbability of a gentlemen. A mask . . . Is

there any reason why that terribly arbitrary materiality of the word surface

should not be dissolved [...]18

Beckett felt that in French the burden of style was not so heavy as in English. Coetzee

seems to interpret Beckett’s rejection of English as stemming from the fact that form

dominates in English, creating content, where in French ‘form remains subordinate,’

not least because in English, his native tongue, Beckett’s writing was subject to his own

tics and mannerisms.19 Coetzee attempted, in this work, to measure whether one could

successfully rid a text of style. If stylistics (or, its purely quantitative sibling, stylostatistics)

can measure this then this may afford something for literary criticism, rather than just

linguistics. A writer’s style could be measured, as could the evacuation of style. Coetzee

notes that there are similarities between the claims of the preeminent stylostatician,

Bernard Bloch, and Beckett. On a superficial level, each retains an obsession with a

statistically- verifiable equilibrium in language: ‘Beckett’s ‘writing without style’ could be

interpreted as writing with the statistical features of the language as a whole, whatever

that may be.’20 However, Bloch sees the text

18 Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929-1940, ed. George Craig and Dan Gunn

(Cambridge and New York: Cambridge University Press, 2009), 518. 19 J.M. Coetzee, ‘The English Fiction of Samuel Beckett: An Essay in Stylistic Analysis’ (University of

Texas at Austin, 1968), 5. 20 Ibid. 2.

231

as a collection of sets of linguistic features (phonemes, morphemes, words, etc.)

which can be treated like members of statistical populations; and a statistical

population is only a metaphor for a set of points in probabilistic space. For

Bloch, a word can be conveniently reduced, for the purposes of study, to a

dimensionless and immaterial point.21

This utilitarian convenience is not shared by Beckett, for whom ‘on the other hand, the

‘terribly arbitrary materialist of the word surface’ was, we infer, at least in 1937, a

burden.’ 22 Beckett’s far more contradictory relation to the word as point without

dimension, without materiality, would extend his work beyond the endeavour to write

without style to a much more radical geometricised poetics, such as we see in All Strange

Away and Imagine Dead Imagine, for instance, texts composed during the years that

Coetzee was working on this dissertation.

Coetzee’s doctoral project meshes a modernist artistic endeavour – an essentially

qualitative task – and the quantitative capacities of logic, programming and statistics

deployed by stylostatistics. Or, depending on how one considers this, it is a confusion

between a qualitative and a quantitative drive towards an objectification of the artwork.

The notion of language and signs as a medium for construction of the world had

currency at least since the publication of Saussure’s Course in General Linguistics in 1916.

Saussure’s vision is summarised by his translator, Roy Harris:

[...] instead of men’s words being seen as peripheral to men’s understanding of

reality, men’s understanding of reality came to be seen as revolving about their

social use of verbal signs. [...] Words are not vocal labels which have come to be

21 Ibid. 2-3. 22 Ibid.

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attached to things and qualities already given in advance by Nature, or to ideas

already grasped independently by the human mind. On the contrary languages

themselves, collective products of social interaction, supply the essential

conceptual frameworks for men’s analysis of reality and, simultaneously, the

verbal equipment for their description of it.23

The wider structuralism that took hold in the late 1960s and early 1970s was

particularly concerned with culture and discourse as formed through underlying

linguistic structures. In some senses, this preserves the notion of a form of presentation

prior to representation, a matheme beneath material. Crucial to this was the Saussurean

ambition of a ‘general science of signs’ – signs being an object of study –named

‘semiology.’ 24 Saussure’s vision of semiology came from the notion that signs are

produced from a common faculty – a kind of proto-language ability in human animals

that knits each signifier to the signified.25 There is thus a supposed split generic form of

each word, the sign and its signifier. Coetzee would ratify these premises, but only to a

certain extent. It was the vision of linearity behind this that he took issue with in his

doctorate:

Stylostatistics oversimplifies because it is dominated by a metaphor of linearity, a

conception of language as a one-dimensional stream extending in time. The

origin of the metaphor probably lies in our alphabet; it has been fortified by

23 Roy Harris, ‘Introduction to the Bloomsbury Revelations Edition,’ in Course in General Linguistics

(London and New York: Bloomsbury, 2013), xiv. 24 Ibid. xv. 25 For an elaboration of this contention, see: Julian Warner, ‘Semiotics, Information Science, Documents

and Computers,’ Journal of Documentation 46, no. 1 (1990), 16.

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printing technology and by the twentieth-century metaphor of the mind as a

computer with an input system which reads linear strips of coded information.26

The statistics behind the methodology come to facilitate a supposedly objective (rather

than impressionistic) designation of style, and realize a kind of ‘code form’ of style. Here,

Coetzee diagnoses this objectification of style as reliant upon essentially contingent

material facts: the alphabet, the printer. The theories of language and art are reducible

here to the technologies of the mark, much like the strips of input paper for elementary

computers.

Stylostatistics dares to measure the extent to which the automatism of language

and style (that numerical instrumentality that Coetzee encountered at IBM) may hold

force in a domain that seeks to reject such instrumentality, namely Samuel Beckett’s

modernism. Coetzee’s structuralism seeks out a numerical index of a language without

style, a position that can wriggle out of the confines of stylistic grooves. This is the

linguistic instantiation of a wider crisis: how to speak some kind of truth when the

conditions of one’s language condemn one to a fate of misrepresentation. The legacy of

stylostatistics is something of a caricature of its beginnings. Like European structuralism,

stylostatistics could have been open to a range of numerical formulations that were not

necessarily blandly deterministic or quantitative. Indeed, Coetzee, with his knowledge of

Gödel and Dedekind, would hardly have adopted quite so naïve a view. David Atwell

notes that, whilst repellent on a surface level, this pursuit was in fact also an enquiry into

‘the ontology of fictional discourse,’ a topic that would remain a fascination for Coetzee

and one which is clearly manifest in the two novels that I am about to analyse here.27

For Attwell, Coetzee’s ‘reflexive narrative’ (the fact that his novels typically enquire into 26 Coetzee, ‘The English Fiction of Samuel Beckett: An Essay in Stylistic Analysis,’ 160. 27 David Attwell, ‘Editor’s Introduction,’ in Doubling the Point: Essays and Interviews by J.M. Coetzee,

(Cambridge: Harvard University Press, 1992), 1.

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the conditions of their own composition) emerges directly out of mathematics. In

Attwell’s words, it was the time at IBM, ‘Coetzee’s professional pursuit before his

decision in the mid-1960’s to return to literature’ that paved the way for ‘the later

interest in the rule-conditioned character of discourse.’28 The combination of the early

computer programming and poems and the research in stylistics suggest, here, that the

ultimate ‘rule’ conditioning language for Coetzee might just be the numerical rule: what

counts, how to give an account, and what constitutes an adequate enumeration.

In the analysis below, I elucidate Coetzee’s development of a reciprocity between

prose and mathematics in two of his novels. This reciprocity emerges in each case from

a confrontation between instrumental number and pure number: the stylistic effects of

prose accountancy and the mathematical foundations of this and the syllogistic form of a

discursive operation that formalises a ‘one.’

*

4.2 IN THE HEART OF THE COUNTRY: FREEDOM AND EQUALITY

In the Heart of the Country, Coetzee’s second novel, is written in response to the South

African plaasroman, or farm novel. The plaasroman was chiefly an Afrikaans endeavour

that celebrated personal connection with the land. It is suggestive of a kind of European

pastoral tradition, although by virtue of its agricultural emphasis can be considered a

distinct regional form.29 Coetzee’s plaasroman constitutes an interrogation of the genre’s

linguistic conditions through a careful manipulation of the grammar within that form.

The novel consists of only one voice: a monologue by Magda, a young woman who lives

on a farm with her widowed father and two labourers, Hendrik and Jakob, and Jakob’s 28 Ibid. 5. 29 For an articulation of the distinctiveness of the genre, see Coetzee’s own essay on the plaasroman: J.M.

Coetzee, ‘Farm Novel and ‘Plaasroman’ in South Africa,’ English in Africa 13, no. 2 (1986).

235

wife, Anna, who works in the house. Magda identifies herself by virtue of her self-

constituted ‘exceptionality,’ or, to put this in other words, as a figure whose method of

withdrawing from the world, whose mode of affliction, is one common to a literary

tradition of the isolated and mad woman:

I was in my room, in the emerald semi-dark of the shuttered late afternoon,

reading a book or, more likely, supine with a damp towel over my eyes fighting a

migraine. I am the one who stays in her room reading or writing or fighting

migraines. The colonies are full of girls like that, but none, I think, so extreme as

I. My father is the one who paces the floorboards back and forth, back and forth

in his slow black boots. And then, for a third, there is the new wife, who lies late

abed. Those are the antagonists.30

Alongside this simple, matter of fact outline of Magda’s situation, there is both a

graphological and narrative count. Each of the paragraphs is numbered. This

paragraph, for instance, is number 1 in a count that will run to 269. The ‘matter- of-fact

outline’ presenting the antagonists takes the form of a count too: Magda coming first,

her father second (although he too is ‘the one’) and the new wife being the third. After

the first person opening of the paragraph, in the following sentences the first person

singular becomes an object. In this minor manoeuvre Coetzee signals the perversion of

the plaasroman to come: the ‘I,’ here, will transition back and forth from subject to object,

inserting a disruptive, implicit self-reflexivity into Magda’s testimonial as well as

initiating the linguistic demands of Magda’s unstable consciousness. In this monologue,

Magda’s own status as subject or object of and in language, and the status of the other

characters, is never secure.

30 J.M. Coetzee, In The Heart of the Country (London: Vintage, 1999), 1.

236

This brief presentation of both Magda’s own extremity of character and the

introduction of the other two antagonists is quickly compromised. We learn that

Magda’s lucid account of her existence is far exceeded by a near hallucinatory relation

to the world. Whilst her multiple narratives and interpretations of her world do become

increasingly disturbed as she encounters opposition and difficulty throughout the novel,

from the outset she possesses a bitter, warped relation to reality. Magda’s voice only

realises a single coherent narrative for a few pages before a departure from a single

account of events and the world. This forking of narrative events comes from an

uncertainty in the identity of the new wife. Either Magda speaks of the new wife of

Henrik – the farm labourer – who eventually comes to work as a maid in the house, or

she speaks of her father’s new wife, who also intrudes upon Magda’s domestic sphere.

These two narratives are never simultaneous, and quickly become a confusion that

suggests that Magda is not narrating an actuality, but processing a trauma: her lonely,

withdrawn father has used his power over Hendrik and his young wife to start a sexual

relationship with her.

The paragraph quoted above is itself an enumeration of certain conventions of the

novel. We are presented with characters with attributes, who are also antagonists and

will set the plot in motion. After this initial setting of the scene, Magda plunges into an

introverted world that quickly overturns any factuality or order this enumeration might

have established. The novel is thus modified by the notation that accompanies it, which

comes to serve a function of contrast. In no way is the monologic narrative consistent,

objective, or progressive as the numbers that follow the paragraphs are. The numbers

thus seem to dutifully enumerate a narrative progression that never occurs. This

omission thus departs from the literary convention of numbering chapters, evoking

instead the conventions of witness statements and legal documents, which fits with

Magda’s monologic account of events, her ‘witnessing’ of what goes on at the farm

237

recorded for purposes of verification. This graphological suggestion of legal convention,

as well as the implied logic of the consistent numerical stream, serves to bring into relief

the contradictions and reformulations in Magda’s narrative that obscure rather than

order a coherent narrative. As the narrative turns back upon itself what is illuminated

are not events but the processes of repression and sublation that produce Magda’s

reality. In other words, the annotated paragraphs create a tension between number and

narrative sequence; the consistent and completed order of the numbers exists in contrast

to the narrative failure to ‘count.’

Whilst this numerical accompaniment to the monologue seems to be isolated –

the numbers that are supposed to track progress cannot hope to do so – it also reflects a

process of exchange, distance and reciprocity between the words and numbers that

preoccupies Magda throughout her narration. There are several different forms of

numeracy that preoccupy Magda, all of which involve counting, but very different sorts

of counting to that which indexes the paragraphs. Like the numbers alongside the

paragraphs, these forms of counting are also mechanisms of ordering language. In the

first instance Magda’s conception of her own identity and alienation from the world

presents itself as an overarching count. This invokes both a clear self/other distinction

that might remedy her isolation from others, as well as an emphasis on rationality or

explication as the means by which to remedy this. She identifies herself with zero:

To my father, I have been an absence all my life. Therefore instead of being the

womanly warmth at the heart of this house I have been zero, null, a vacuum

towards which all collapses inward, a turbulence, muffled, grey, like a chill draft

eddying through the corridors, neglected, vengeful.31

31 Ibid. 2.

238

This identification with zero is an absent or negative identification, an identification that

is a contradiction, indeed a negation of identity. It is from this negative identity that

many of Magda’s problems with language stem. If Magda is not a one but instead a

zero, how is she to address and communicate with others, who are cancelled out

grammatically and socially by this negative identity.

The question of whether zero is a number has been a persistent conflict in

philosophy and mathematics. If zero has no value, then can it still be considered to be a

number? In Fibonacci’s work on Arabic numerals, for instance, zero is listed not as a

number but as a sign.32 In the Liber Abaci: ‘These are the nine figures of the Indians: 9

8 7 6 5 4 3 2 1.’33 In this text, these signs, taken together with the ‘sign 0, which in

Arabic is called zephirum, any number can be represented, as will be demonstrated.’34

Zero’s unique standing predicates a set of relations that would not be possible with any

other number, for instance in the equation x + y = x, y must be zero. As zero, Magda

does not ‘add’ anything to the world she lives in, she is the marker of insubstantiality.

She has identity here, but her identity is without value; similarly, she is not a figure but a

sign. And, as a sign, Magda’s function is to supplement the figures (1, 2, 3) with other

signs: her ‘absent’ presence produces precisely this monologue, precisely the textual

extension of these other figures. In this sense, ‘zero’ is the numerical placeholder, in

Magda’s metaphoric system, for the first person pronoun, revealing one of the common

numerical contradictions in her capacities for narrative.

Magda possesses a desire for intimacy that she believes would solve this breach

between her absence and the presence of others:

32 Carl B. Boyer, History of Analytic Geometry (Mineola, New York: Dover Publications, 2004), 44. 33 L.E. Sigler, Fibonacci’s Liber Abaci / Leonardo Pisano’s Book of Calculation (New York: Springer-

Verlag, 2003), 17. 34 Ibid.

239

If my father had been a weaker man he would have had a better daughter. But

he has never needed anything. Enthralled by my need to be needed, I circle him

like a moon. Such is my sole risible venture into the psychology of our debacle.

To explain is to forgive, to be explained is to be forgiven, but I, I hope and fear,

am inexplicable, unforgivable.35

In Emile Beneviste’s formulation, a first count is set up in the polarity created by

language – the I and the you – the latter being not only other to the subject but also

inherently plural. Here there is already a count between Magda and the others she

attempts to communicate with, which is mediated by words. Closing the distance

between herself and others (curiously coupled with the salvation of her soul) would be

achieved by the explicable, by an ‘unfolding’ of a situation. The sense here is that this

unfolding is also numerical: with reasoned explication Magda could travel from zero

upwards into numerical existence though, notably, only by addition, not multiplication.

Her monologue in its entirety can be read as an attempt at explication and this is where

the numerical progression can be seen to have a function, even if the narrative that

accompanies this fails to ‘unfold’ anything. The numbers that accompany the

paragraphs thus seem to allegorise Magda’s own desires for what language can achieve.

The process of explanation is thwarted by Magda’s own personal theories of

language. The progression facilitated by explication, by unfolding, is impossible when

the language is in fact circular. Magda’s reflections on language continually run up

against what she takes to be its tautological, captivating form: words function to create

the speaker, who in turn creates these words. In one gloss, she responds to the problem

of equality that is at the heart of this tautology: ‘I create myself in words that create me,

I who living among the downcast have never beheld myself in the equal regard of

35 Coetzee, In The Heart of the Country, 6.

240

another’s eye, have never held another in equal regard of mine.’36 Here, language

creates creation. Here we have pure reciprocity between performative capacities of

words and the subject that uses or produces those words. This is Magda’s affliction: she

feels trapped in a system that is utterly non-contradictory and absolute, which completes

itself. This compounds the isolation that Magda experiences in identifying with zero. As

zero, Magda cannot be equal to another, nor can another be an equal to her. Her

identification with zero means that no equation can be correct with a positive numeracy

on one side and zero on the other: 1 = 0, and so on. The metaphor of Madga as zero

carries through to semiotic numerical coimplication: Madga cannot balance equations

socially; she cannot lend a meaningful existence insofar as she cannot participate in

operations of numerical equality.

The theme of equality in language, and in relation to language, is taken up

broadly and explicitly by Magda in several times: ‘Of course, the truth is that I am equal

to anything.’37 She will proceed to repeatedly evoke or establish equality or lack thereof

between herself, others, words or objects, or lack of equality based on this initial

presumption about the pronoun. Ultimately, the use of words seems to present a

problem of equality; nothing (precisely, zero) is equal to Magda. Communication is

possible dependent on a rule of inequality (the colonial rule, which produces an

adequate mediation that allows a certain form of communication, but also, more

generally, difference in address) but is also the solution to equality (a direct communion,

which provides non-mediated access to other objects, whose otherness in fact collapses

under the law of exchangeability). Of course, what an ‘equation’ is, formally, is a

balancing act. Any legitimate equation has two numerical or algebraic formulations that

are, when placed on either side of the equals sign, equivalent. Magda is thus engaged in

36 Ibid. 8. 37 Ibid. 17.

241

a process of literary equations, which resonate with the mathematical equation insofar

as both have stakes in a balancing system regulated by an intransitive count, be it

through words or numbers. When Magda understands herself to embody zero – thus

grammatically cancelling whatever she may encounter in a radical ‘forbidding’ of

equation, she is both equal to anything and equal to nothing.38

The perspectives on language already encountered in Magda’s attempts at

communication can be transmuted into a sexual problem. Where language creates

creation, where it is isolated and without communion, it can be said to reproduce

asexually. Magda’s ‘hysteria’ comes in part from the sexual transgression of her father

(be it with a new wife, or with Hendrik’s wife) but also in part her own sexual and social

isolation. In her virgin state, Magda presumes herself excluded from communion, and

trapped in an economy of unfulfilled desire. She imagines that Hendrik and Klein Anna

have lain naked all night, waking and sleeping, giving off their complex odours:

the smoky sourness of brown people, I know that by heart, I must have had a

brown nurse though I cannot recall her; [...] The question to ask is not, How do

I, a lonely spinster, come to know such things? It is not for nothing that I spend

evenings humped over the dictionary. Words are words. I have never pretended

to embrace that night’s experience. A factor, I deal in signs merely.39

Magda considers herself relegated to a life of signs, a life of abstraction. Her confronting,

voyeuristic passages on sex revolve around words and signs, both in terms of her

capacity to understand what she has not experienced as well as her own existential status

and that of the environment around her. This constitutes a metafictional nod to the 38 Peter Johnston has also noted Magda’s paradoxical relation to infinity: she is both nothing and the

infinite at the same time. See Johnston, ‘‘Presences of the Infinite’: J.M. Coetzee and Mathematics,’ 200. 39 Coetzee, In The Heart of the Country. 29.

242

sensory qualities of printed prose. How do we, as readers, extrapolate from these pages

of printed words to a knowledge of people, events and places that we have never actually

encountered? Again, the very constitution of the narrative, the content of Magda’s

monologue, is bound up with an allegory of the formal process that it relies on: the

transmutation from printed words to experience.

These ‘signs’ that Magda deals in are, crucially, not differentiated and unique

units of meaning but, again echoing her own identification as zero, identical points.

Magda expresses this in a numerical register: ‘Seated here I hold the goats and stones,

the entire farm and even its environs, as far as I know them, suspended in this cool,

alienating medium of mine, exchanging them item by item for my word counters.’ This

process, it seems, is ostensibly infinite:

The world is full of people who want to make their own lives, but to few outside

the desert is such freedom granted. Here in the middle of nowhere I can expand

to infinity just as I can shrivel to the size of an ant. Many things I lack, but

freedom is not one of them.40

The absence of any identity or stability for the sign, here, rests on the conflict between

the infinite and the finite. Magda’s ‘infinite’ operations, whereby all is equal to all,

contradict a world and a language that requires finitude to enable difference,

communion, communication. It is precisely the oscillation of Magda’s equations that

indicates this lack of finitude. She can never settle on a form of communication because

there is no necessity to the equality. Either she could be equal to all else she encounters on

the farm, or traumatically separated from everything because there is no numerical

identity for words. On both sides of the equation, numerical materialism or numerical

40 Ibid. 55.

243

identity collapse: the numbers are evacuated of whatever it is that they signify. Magda’s

coruscating subjectivity, caught in the circle of tautology, erases numbers of their

identity through a process of frictionless exchange. (All numbers are rendered indistinct

as pebbles are rendered indistinct – each pebble does not have a distinct identity.)

Magda thus exists in the unbounded freedom of the imagination, where all numbers are

zero, because these acts of thought cannot modify the actuality of what is. Her language

skims the surface of the world, without being able to possess or bore into its material

invariance.

The excessive intertextuality in In the Heart of the Country continues this rolling

process of equivalence. At the same time as Magda’s personal and solipsistic language

excludes her from civilisation, it also inscribes her in literary traditions. In spite of

herself, Magda’s vision of herself outside of language is peculiarly culturally refined. She

goes on:

While I am free to be I, nothing is impossible. In the cloister of my room I am

the mad hag I am destined to be. My clothes cackle with dribble, I hunch and

twist, my feet blossom with horny callouses, this prim voice, spinning out

sentences without occasion, gaping with boredom because nothing ever happens

on the farm, cracks and oozes with the peevish loony sentiments that belong to

the dead of the night when the censor snores, to the crazy hornpipe I dance with

myself.41

The last image invokes Pan, the crazed Greek god who is associated with hornpipes and

mad dances. The hunched, twisted figure, and the calloused feet are typical of witches,

with comparable images found in Macbeth and many other texts. Here again we see the

41 Ibid. 8.

244

problematic of ‘equation’ standing in for communication and authenticity, Magda’s own

self-descriptions are littered with images that are not her own. These images are taken

from the canon of Western art and inscribe her self-consciously in a tradition. In this

intertextuality Magda is both created and creator, as well as both alienated from the

world and inscribed in it, dispossessed and possessed simultaneously. The sheer variety

of allusion, which appears much more densely at the end of the text, generates

something of a maelstrom of influence, where the references accrue meaning not

through their specificity but through their number.

The book ends with a slew of intertextuality even more explicit and intense than

these first signals. Towards the end of the novel, when Magda lives isolated, in madness,

on her own on the farm, she begins to hear voices:

It is my commerce with the voices that has kept me from becoming a beast. For I

am sure that if the voices did not speak to me I would long ago have given up

this articulated chip-chop and begun to howl or bellow or squawk. [...] The

voices speak to me out of machines that fly in the sky. They speak to me in

Spanish. I know no Spanish whatsoever. However, it is characteristic of the

Spanish that is spoken to me out of the flying machines that I find it immediately

comprehensible.42

These flying machines (which seem to be planes flying overhead) send Magda messages,

to which she responds by using stones to create words. These voices are not only

completely comprehensible to Magda but, more than this, use words that are ‘tied to

universal meanings.’43 The messages that they send her are all quotes from literary

42 Ibid. 136. 43 Ibid. 137.

245

history. The first message that she reports is a line by Novalis: ‘When we dream that we

are dreaming, the moment of awakening is at hand.’44 As with the earlier, unquoted

intertextuality, there appears to be no necessity to these quotes. The phrases address a

variety of issues, sentiments and positions, and come from a variety of authors. The

majority of the quotes sent by the machines overhead come from very well known

European and American writers: G.W.F. Hegel, Jacques Lacan (‘it is a world of words

that creates a world of things’), Simone Weil, Friedrich Nietzsche, Jean-Jacques

Rousseau (‘every man born in slavery is born for slavery’), and Luis Cernuda. The sole

exception to this is the sentence ‘A blind man dancing seems not to observe his period of

mourning,’ which comes from a book entitled Southwest Native American Oral Literature,

which anthologises a selection of oral stories. Magda attempts to ignore or disregard the

messages, but seems not to succeed, eventually sending her own messages back. Magda

gathers stones from the veld and ‘painted [them], one by one, with whitewash left over

from the old days [...] Forming the stones into letters twelve feet high I began to spell

out messages to my saviours: CINDRLA ES MI; and the next day: VENE AL TERRA;

and: QUIERO UN AUTR; and again: SON ISOLADO.’45 When Magda’s pebbles

finally cluster to make signs, they are snatches of meaning, which respond to the

catchphrases from literary and philosophical classics. Just as Samuel Beckett used stones

to function as points and ciphers of words in Molloy, so too do we see Coetzee’s Magda

construct an idiosyncratic linguistic system using pebbles which she circulates to create

different, truncated messages. In this action we might say that Coetzee ‘doubles the

point’ – to reference the title of one of his first books of collected essays. Here the

counters have become points or pixels that are not signs themselves – not placeholders

for nouns – but parts of adjectives and pronouns in various languages or shades of 44 Ibid. 138. This quote is wrongly attributed, in the Oxford Dictionary of Quotations, to Coetzee rather

than Novalis. 45 Ibid. 144.

246

languages. Many of Magda’s phrases are not quite translatable – seemingly a mix of

Spanish, Latin and Esperanto – but appeal to the objects in the sky to remedy her

isolation. Even Magda, despite her preposterous access to a universal language, must

admit a lack of understanding:

How I cursed my lot on the sixth of these days for denying me what of all things

I needed most, a lexicon of the true Spanish language! To rack one’s innate store

for a mere conjunction when the word lay sleeping in a book somewhere! Why

will no one speak to me in the true language of the heart?46

The ‘true language of the heart’ seeks yet another form of equality: a pure language with

no need for words, one in which communication is a pure transaction without medium.

The ‘Spanish’ of the planes is also the other language that accompanies Coetzee’s most

recent novel. Although The Childhood of Jesus is written in English, we are told that the

characters are speaking a newly acquired language, which is Spanish.47 As in The

Childhood of Jesus, in Magda’s monologue we never ‘hear’ or ‘see’ Spanish. All the quotes

are rendered in English and Spanish remains only a referent through which we never

gain textual access to, just as Magda does not gain full access to it either.

The theoretical context in which In the Heart of the Country was composed is crucial

to a reading of Magda’s problems with language. Magda’s troubles echo the theoretical

46 Ibid. 145. 47 A full elaboration of the significance of Spanish in these two novels is beyond the scope of this thesis,

although it is an urgent and fascinating aspect of Coetzee’s work. In lieu of a full analysis of this, I can

merely point here to the existence of Spanish as a ‘silent’ language: a language referred to, a language that

conditions all the speech of the novel, but which is never fully or realistically ‘presented.’ This of course

resonates as an allegory of authorial conditions that Coetzee was preoccupied with: the idea of languages

‘speaking us’ rather than being spoken by us. Any language that properly speaks through a subject (or

someone subjected to that language) would, of course, be ‘silent’: inherently unheard, its identity

acknowledged but never fully present to the speaking self.

247

preoccupation with authorship and referentiality. Julia Kristeva’s formulation of

intertextuality as a ‘replacement’ for intersubjectivity is perhaps typical of this

connection and crucial to an understanding of the depersonalisation implicit in this

scattered intertexuality affronting Magda from the skies. Kristeva, indeed, is responsible

for the very term intertextuality: she defines it as ‘the transposition of one or more systems

of signs into another, accompanied by a new articulation of the enunciative and

denotative position.’48 This is not a matter of authorial intentionality: Kristeva’s system

suggests a more textual and personal existence that is not rooted in the simple concepts

of agency and intentionality. Rather, in a direct intervention with Bakhtin’s ‘dialogism,’

Kristeva claims that the writing subject is replaceable with or by a text. For Kristeva, the

‘person-subject of writing’ (the ‘A that writes X,’ in Coetzee’s formulation) becomes

blurred by the co-production of intertextuality and subjectivity.49 The multiple levels of

referentiality in Magda’s interactions with the planes suggest an internalised

intertextuality, almost as if Kristeva’s theory is doubled over here. Magda has failed at

intersubjectivity, so she now seems to have commuted into an intertextual

communication. If she cannot speak between the I and the you, she will speak between

texts, and address possibly benevolent machines. She does so, naturally, in a version of a

(purported) ‘universal’ language. Where the planes speak to Magda with lines from the

‘universal’ exchange of the Western canon, she enters into dialogue with a version of

Esperanto: another language based in a potentially universal communicability. The

universal communicability is, for the reader of In the Heart of the Country, invisible. We are

told, by Magda, that it is there, but we never see the textual evidence of this. Although

we do not witness this universal language described as ‘Spanish’ there is, of course,

48 Leon Roudiez, ‘Introduction,’ in Desire in Language: A Semiotic Approach to Literature, by Julia Kristeva (New

York: Columbia University Press, 1980), 16. 49 Julia Kristeva, Desire in Language: A Semiotic Approach to Literature (New York: Columbia

University Press, 1980), 37.

248

another ‘universal’ language that is present right beside Magda’s monologue: the

number system. Like ‘Spanish,’ its presence is also silent: it accompanies the narrative

but is not expressed within the narrative.

In the Heart of the Country is thus a story about agreement in language and the

numerical allegory of this. Magda is preoccupied with the pragmatics of language, of

how communication is constituted in terms of the abstractions of number that sit beside

narrative. How does one resolve the discrepancy between person pronouns and the

generic, universal pronouns, between ‘Hendrik’, ‘you’ and ‘one’? What numerical

revolution in her own identity is required for Magda to regard another as ‘equal’? She

‘knows’ brown people only through the dictionary, or some memory of contact, but has

no immediate experience of them. There is both a striking kinship between numbers,

then, and between pronouns – Hendrik can never be a ‘one’ in Magda’s language, or,

more acutely, the complex of distance and intimacy, violence and vulnerability, in

Magda’s situation mean that she retains only zero and cannot progress to name, or

embody, a one. Her pragmatics are formulated using numbers and forms of equivalence

mediated through ciphers of objects, most notably the pebbles. Each of the objects that

she gives a word to is made equal with other objects, is turned into a placeholder, a

stone. Magda’s ‘counters’ are abstractions of words stripped of their ‘identity’ or

particular signification. In this sense, each thing is exchangeable for all else. Her

‘counters’ are also not properly numbers – they are merely placeholders. In this sense,

each stone is not a number but must be a zero. They are ‘sign’ as opposed to ‘signs.’

This is where Magda’s efforts become problematic, and indeed, this is where Magda

inadvertently comes up against a persistent and galling problem in the theory of

numbers, which is perhaps not uncoincidentally a problem that involves an intersection

between language and numerical identity. Numbers are not immune from the problems

of identity, and to proceed with numerical equation with the assumed quotidian surety,

there must be a provable base for the identity of numbers, the spaces between them.

249

Take, for example, the number eight. Does this number ‘contain’ all the numbers that

precede it in the number line? Or does this rather signify something distinct? Moreover,

what exactly is eight? We agree that the number has a specificity – to use it to describe

seven objects or no objects would be incorrect. This is precisely the problem that Magda

confronts with words: radical equality exists, and one word counter could easily replace

another, stymieing progression and understanding, where numbers are bereft of their

singular identity; again, where they become ‘sign’ rather than ‘signs.’ A classic essay in

number theory that considers precisely this problem is Paul Benaceraff’s ‘What

Numbers Could Not Be.’ The title alludes to both the dilemma – the identity,

explication and reduction of numbers – as well as the solution. Benacerraf concludes,

after a thorough logical interrogation of what a number might be, that ‘if truth be

known, there are no such things as numbers; which is not to say that there are not at

least two prime numbers between 15 and 20.’50 Benacerraf shows how the problem of

numerical identity actually arises from a problem internal to the set theoretical

understanding of numbers. Benacerraf begins with a thought experiment, one whereby

the ‘pedagogical order’ becomes the ‘epistemological order.’51 Rather than learning to

count objects, two children – Ernie and Johnny – are taught set theory. Ernie is taught

about a set of numbers N with a ‘less than’ relation R. For Ernie, ‘the assumptions made

by ordinary mortals about numbers were in fact theorems.’52 Ernie learns that there are

two types of counting: intransitive and transitive, where the former ‘admits of a direct

object,’ whereas the latter does not. 53 Transitive and intransitive counting are

distinguished in part by their relation to the objects counted:

50 Paul Benacerraf, ‘What Numbers Could Not Be,’ The Philosophical Review 74, no. 1 (1965), 73. 51 Ibid. 48. 52 Ibid. 49. 53 Ibid. 49.

250

There are two kinds of counting, corresponding to transitive and intransitive uses

of the verb ‘to count.’ In one, ‘counting’ admits of a direct object, as in ‘counting

the marbles’; in the other it does not. The case I have in mind is that of the

preoperative patient being prepared for the operating room. The ether mask is

placed over his face and he is told to count, as far as he can. He has not been

instructed to count anything at all.54

All goes well with the educational experiment until Ernie and Johnny discuss their

respective proofs and discover a problem: ‘Comparing notes, they soon became aware

that something was wrong, for a dispute immediately ensued about whether or not 3

belonged to 17.’55 This dispute arises from different conceptions of the identity of

numbers as and within sets. For Ernie, sets of numbers look like this:

[Ø], [Ø,[Ø]], [Ø,[Ø],[Ø,[Ø]]],...

The ‘number’ that succeeds the previous number in the set N under the recursive law R,

here, is a set consisting of that number and all the numbers it contained (thereby making

for alarmingly complex notation). For Ernie, as is illustrated in the above notation, two

is clearly a member of three, and three a member of other numbers that it is less than.

Whereas for Johnny, a set of numbers will look like this:

[Ø], [[Ø]], [[[Ø]]], ...56

54 Ibid. 49-50. 55 Ibid. 54. 56 Ibid. 55.

251

This dispute is set along the divergent forms of the identity of numbers, and whether

identity can be expressed in terms of one number’s relation to or possession of other

numbers; for Ernie it is the latter, for Johnny the former. Benacerraf tries and fails to

prove the necessity and sufficiency of one set theoretical understanding of numerical

identity over the other. Each mode of arriving at numbers is correct: ‘The two accounts

agree in over-all structure. They disagree when it comes to fixing the referents for the

terms in question.’57 Benacerraf’s question revolves around which set has the privileged

relation to number. Where exactly do we find the things that are numbers in Ernie and

Johnny’s models? What constitutes a number, and is its identity singular or does it

consist of component parts. Can things individuate as numbers? Or, are numbers

individuated as things? Benacerraf, ultimately, concludes that numbers do not exist.

Numbers are not entities nor are they predicators or quantifiers. They cannot be

individuated but must be understood not as objects but as an abstract structure and

arithmetic must be understood as a ‘science that elaborates the abstract structure that all

progressions have in common merely in virtue of being progressions.’58 This is perfectly

true of Magda’s predicament. Numbers, ultimately, sit outside or beside the progressions

that she attempts or perceives. This is quite literally the case: the numbers accompany

but do not intrude upon her paragraphs, enumerating the novel without providing any

semblance of structure or order to Magda’s increasingly disordered tale. Numbers are

that which we cannot know other than by abstract progression: we are excluded, as

Magda’s paragraphs are, from the actual count, from knowing the being of numbers,

which accompany but are not included in the narrative. The ‘one’ that signals the hope

of a unit of measure is never resolved into either an abstract unit or a singular entity, but

exists as both of these, or either of these, in an irrational quantity that cannot be

57 Ibid. 56. 58 Ibid. 70.

252

resolved into a progressive set. One could claim that the count Magda does evoke is a

count of literature, peppering her observations with artworks and texts from the canon:

but it is precisely this excessive evocation which dissolves the relation between these

classics, and the possibility of either Magda’s monologue or In the Heart of the Country

responding formally rather than indexically to these works. Moreover, it is in Magda’s

failings to articulate herself as a person and assert a proper numerical identity that we

find a literary allegory for precisely Benacerraf’s problem: is Magda, the zero, even a

number? Or is she relegated to belong only beside the numbers, in the paragraphs full of

signs, thwarted universal languages, and quotes copied from canonical culture, issued by

machines from overhead? The connection between these machines that emit symbols

from overhead and the universal languages that Magda perceives and attempts to

communicate in makes the machines seem to be more than simply planes. They also

strike one as computers, almost quite literally as the ‘head’ on a Turing machine,

printing symbols on the blank strip below. The words that come from Magda, from

zero, from the sign that is not yet a signifier, appropriately exist beside the numbers.

There is no zero contained in the numerical progression alongside the paragraphs.

Coetzee’s interest in stylistics and his work in mathematics had indeed anticipated this

fiction: this novel is formed as an enquiry into the role of language as a rule bound

system. This text allegorises its own numerical processes: the capacity of Magda to use

the pronoun, to assert her own voice through which narrative events take shape, to

name and communicate with others, to count herself and others in language. In other

words, the formal allegory that occurs here is fundamentally an allegory of the

numerical processes of the text and the novel. This creates, then, a generic literature:

these numerical processes that constitute the most fundamental rules of the linguistic

system are staged in the reciprocity between number – the notation next to each

paragraph – and its other: the attempts in prose to count and to make oneself and others

count.

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*

4.3 THE CHILDHOOD OF JESUS AND MATHEMATICAL NOMINALISM

Coetzee’s recent novel, The Childhood of Jesus (2013), is preoccupied with the problems of

initiating a sequence where there is no origin story for identity, or, in other words, the

capacity to continue to count when the first, founding number has been forgotten. The

novel revolves around an unusual child, David, and the man who cares for him, Simón.

David and Simón are refugees in a city whose inhabitants have been ‘washed clean’ of

their memories and desires, given new identities and a new language, Spanish. I will

argue here that David’s ‘divinity’ emerges in this novel out of his unwittingly nominalist

mind. David can comprehend singularity but not sequence, probability or linearity. It is

in David’s simultaneous embodiment of and sense of singularity that Coetzee produces a

profound intersection between name and number: in this novel David embodies both

the number without name and the name without number. It is through this reciprocal

exchange between name and number that Coetzee achieves a broader reciprocity

between mathematics and literature, indeed, this novel realises what I will call a

‘transfinite exchange’ between literature and mathematics.

The novel opens with a boy, David, and a man, Simón, arriving at what appears

to be a welfare centre in a city named Novilla. The sign at the centre reads Centro de

Reubicación Novilla; Simón does not know the word Reubicación. It transpires that these

new arrivals have come to this city having undertaken intensive Spanish lessons for

several weeks at a processing camp called ‘Belstar,’ where they were also assigned names

and ages. They had arrived at this camp by boat, having no memories of a past, having

been ‘washed clean.’ They are thus in the literal sense new arrivals in a new place,

without origin stories and only the vaguest sense of a life before Novilla. It is this erasure

of history and identity that precipitates the strange search for the boy’s mother.

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According to Simón, the boy’s mother had travelled by ship to Novilla ahead of the boy,

expecting to be reunited with David at a later stage. David had papers relating to his

mother but he lost them on the ship, an unusual predicament given that once his

mother had undergone the rehabilitation process at Novilla, and been given a new

name, an approximate age, and allocated housing, the papers would not have been

useful.

There are many subtle but intricate connections between The Childhood of Jesus

and the biblical infancy gospels. Simón and David have arrived at Novilla (which shares

its first letter with Nazareth) and, when there is no accommodation available at the

relocation centre, they are forced to reside not quite in a stable but in an improvised

shack at the back of a house. Like the infant Jesus, David only has a mother and his

father figure is a ‘god-father’ or a kind of uncle, as Simón describes his own role. But of

course, David does not remember or resemble his mother (once she has been found) and

she does not know him: she will be a mother by nomination rather than biology or even

proper adoption (and, perhaps appropriately, is always referred to by her name, Inés,

rather than directly as ‘mother’). Simón supposes from the outset that he and the boy

will recognise her as soon as they see her, relying on an intuitive notion that mother and

child have some form of transcendental bond, and foregoing the usual biological or legal

necessity that would determine motherhood. Inés is thus, potentially, a virgin mother

and indeed Saint Inés, or Saint Agnes, is the patron saint of virgins.59

The common situation shared by Jesus’ mother Mary and Inés is that they

become mothers through nomination, and it is this nomination that will be the origin

story for a child who has no traceable origins or, in another set of terms, no (earthly)

59 The names, under the title, are suggestive without establishing allegory. David is a king and is suggested

to be an ancestor of Jesus, before becoming King he also triumphs over Goliath, defying the giant’s might

by killing him with five small stones. ‘Simón’ was the first name of St Peter, and ‘Simón the Zealot’ was

also an apostle of Jesus; Jesus also had a brother named Simón.

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father. In this novel, there is no message from the heavens, or any other world outside of

Novilla, and the society is one based on pragmatic management without any apparent

transcendental, religious or spiritual foundation: without any foundation based on

another world. The absence of necessity or verifiability for such an origin story has

arisen several times in Coetzee’s recent fiction, including in his Nobel Prize acceptance

speech, which raises the connection between the recognition of a miracle and the

naming of an allegory. In He and His Man – yet another text that revolves around the

foundations of personal and the generic pronoun – ‘his man’ comes across a scene

where a form (a cloudy one, at that) is nominated as an allegory:

I came upon a crowd in the street, he writes, and a woman in their midst

pointing to the heavens. See, she cries, an angel in white brandishing a flaming sword!

And the crowd all nod among themselves, Indeed it is so, they say: an angel with a

sword! But he, the saddler, can see no angel, no sword. All he can see is a strange-

shaped cloud brighter on the one side than the other, from the shining of the

sun.

It is an allegory! cries the woman in the street, but he can see no allegory for the

life of him. Thus is his report.60

The angel with the flaming sword appears again in The Childhood of Jesus as a statue in

the garden of the housing estate that Inés originally lives in. The nomination of Inés as

the child’s mother occurs in the same way that the designation of the miracle occurs in

the above passage: a recognition of an allegory without a necessary link to the attributed

meaning. In the short scene from He and His Man the woman’s vision and the

60 J.M. Coetzee. ‘He and His Man: The 2003 Nobel Lecture,’ World Literature Today, Vol. 78, No. 2 (May-

Aug 2004), 19.

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surrounding crowd’s acceptance of this vision is founded upon the naming of an allegory:

another act of nomination that traverses a transcendental gap which presents no

assurance of its existence or character. This is not to presume that there exists any

allegory that is properly without a name or exists prior to nomination. Indeed, any

proper allegory exists without immediate dependency on its ‘other’ tale. Rather, what is

emphasised here is very creation of allegory as a process of symbolic attribution. The

suggestion in this scene is that the divine communication is delivered by an appearance

that must be named, an appearance that presents without offering its own name. This

process of nomination also reflects the key tenet of allegory: the form or text or image

deemed allegorical is named for something other than itself. This process of nomination

does not involve a description of the cloudy form by those that deem it an allegory (only

his man witnessing but not participating in the scene stops to describe the shape of the

clouds). Instead, it depicts the giving of another name to the text or shape and the

recognition that the other name applies to or resonates with this once amorphous event.

Simón’s process of ‘nominating’ Inés as David’s mother is not an act of pure

chance or a refutation of reason. It is an appeal to a different reason that resides in

intuition, a form of reason that follows not logic but allegory. From the moment of his

arrival, Simón resists the absence of sexual desire and passionate attachment in Novilla,

as well as more generally the absence of sensual pleasure. In Novilla food is nutritious

and basic (bean paste and crackers) and Simón craves meat. Simón’s appeals to the

sexual passions fall equally flat in the face of a world for whom these are unsolicited and

absurd. His insistent attempts at becoming involved with women appear misguided and

pathetic. Simón finds work as a stevedore, even though loading and unloading cargo

everyday is physically demanding on his ageing body. He is initially perplexed by his

integration into the group of labourers:

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The fellow stevedores are friendly enough but strangely incurious. No one asks

where they come from or where they are staying. He guesses that they take him

to be the boy’s father – or perhaps, like Ana at the Centre, his grandfather. El

Viejo. No one asks where the boy’s mother is or why he has to spend all day

hanging around the docks.61

The men are not unfriendly, but rather seem immune to the question of origins or the

necessity of situations, precisely that which Simón – in part because of his obligation to

find David’s mother – seems preoccupied by. When Simón questions whether their

world is the best possible world, a fellow dock worker, Álvaro, assures him that this is

not a possible world, it is the only world. Simón notes that to Álvaro there is no irony in

that statement. Simón finds a similar problem in Elena, the mother of one of David’s

friends: ‘Elena is an intelligent woman but she does not see any doubleness in the world,

any difference between the way things seem and the way things are.’62 For Simón, this

means that the world ‘lacks weight.’ ‘The music we hear lacks weight. Our lovemaking

lacks weight,’ he complains,

The food we eat, our dreary diet of bread, lacks substance – lacks the

substantiality of animal flesh, with all the gravity of bloodletting and sacrifice

behind it. Our very words lack weight, these Spanish words that do not come

from our heart.63

For Simón irony, desire, and an interrogation of the necessity of labour and habit lend

‘weight’ to the components of that world. Of course, this notion of ‘weight’ is 61 J.M. Coetzee, The Childhood of Jesus (London: Harvill Secker, 2013), 122. 62 Ibid. 64. 63 Ibid. 64-65.

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irremediably, maddeningly vague because it operates on a certain structure that is

neither induction nor deduction but tropology. Here, Simón seeks a ‘weight’ that

emerges from the doubleness of words that can associate flesh (his heart) and gravity (a

sincere use of language). This ‘doubling’ of words adds a currency to language that

exists properly outside of the linguistic system: an association, a resonance that comes

not from the mind at all but from the pre-linguistic, somatic beating heart.

Simón’s desire to lend weight and value to his world lies in stark contradiction to

the organisation of Novilla and the disposition of its inhabitants. A society like Novilla,

one that involves an exceptional consistency of biopolitical organisation, has a peculiar

relation to value: a numerical one. Novilla seems to be a society that, in providing all

basic necessary services for its citizens, has mostly conquered the desire that leads to

inequality and competition. Work is readily available, goods seem cheap, housing and

education are free, and people are generally open minded and amiable if incurious and

somewhat passive. This is a society of consolidated biopolitical control. Biopolitics works

with a curious numerical form: the population, which is a ‘global mass that is affected by

the overall characteristics specific to life [...] like birth, death, production, illness, and so

on,’ a number indistinct in its numerical form.64 This is a society whose founding myth

lies in the consistency of the count. The intransitive progression of numbers continues

unbroken: there is no new number, or a number without a name, rather there is a

sequence of numbers whose identity is their name, a sequence that continues unbroken

for no one knows how long, but certainly long enough to generously provide for the

calculus of human life.

64 This is Foucault’s later definition of biopolitics. His earlier definition referenced ‘a generalised

disciplinary society.’ For a discussion of this distinction see: Michell Senelart, ‘Course Context,’ in Security,

Territory, Population by Michel Foucault, trans. Graham Burchill (New York: Picador, 2007), 378.

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This is, more generally, also the numerical regime of the ‘camp,’ which, in

Giorgio Agamben’s words is ‘the fundamental biopolitical paradigm of the West.’65 The

camp is, in Agamben’s definition, a ‘state of exception’ from normal law that

nonetheless, despite this status, becomes the enduring norm. This is an apt description

of Novilla. As a state of refugees, who reside under its ‘protective custody,’ Novilla is a

perfect example of some normalised state of exception.66 The most important aspect of

Agamben’s definition of the camp is the claim that ‘the camp is a hybrid of law and fact

in which the two terms have become indistinguishable.’67 Here, in Novilla, the relation

to number formalises this: number exists only insofar as it institutes a norm, a law of

order, and emphatically not the appearance of singularity and novelty.

What Simón seems to be looking for is an aspect to life in Novilla which will lend

value that exceeds economic terms, a value that does not already have a place in the

economy of number; a transcendental necessity, or a transcendental ideal. Anthony

Uhlmann has suggested that this book may be a response to Coetzee’s Nobel prize win,

which recognises authors based on ‘the most outstanding work in an ideal direction.’68

The question of idealism is most certainly under scrutiny, here, and Coetzee takes one of

the dominant idealistic socio-economic models of the twentieth century, welfare state

socialism, and recreates a version of it. This society may be described as ideal where

‘ideal’ is taken to imply the ‘best situation’ in biopolitical terms, that is in terms of the

successful management of life. Rather than being an ideally managed society, then,

Novilla seems to be a place that has no ideals left, no ideals beyond the affirmation of

quotidian reality, which might exceed or transcend the successful management of life.

65 Giorgio Agamben, Homo Sacer: Sovereign Power and Bare Life, trans. Daniel Heller-Roazen (Stanford,

California: Stanford University Press, 1998), 181. 66 Ibid. 167. 67 Ibid. 170. 68 Anthony Uhlmann, ‘Signs for the Soul,’ Sydney Review of Books, July 9, 2013, accessed 12/12/2014,

http://www.sydneyreviewofbooks.com/signs-for-the-soul/.

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Simón is looking for a necessity or an ideal that is qualitatively rather than quantitatively

different from the Novillan conception of these terms. Simón uses the same words as his

companions in Novilla but means the exact opposite. In Novilla necessity implies the

minimum requirement whereas for Simón necessity implies imperative, and whereas

‘ideal’ implies the ‘suitable or practical’ in Novilla, so ‘ideal’ for Simón is an elusive

standard of perfection. In each case, we see the split between a pragmatic and a

transcendental side of the coin. Likewise, for Simón meaning is delivered not through a

pragmatic rationality of cause and effect but though the ‘doubling’ necessary to a

consciousness of the transcendental: through the aporetic leaps of metaphor and symbol.

This is exemplified by the traversal of signs that leads Simón to Inés. Simón and

David are on a walk, following a path that will take them to a ‘scenic spot’ that is

signified on their map by a starburst. With the title of the book in the back of readers’

minds, this will allude to the Star of Bethlehem. Simón stumbles upon Inés playing

tennis with her brothers at the end of the walk, when they have reached the point on the

map that was supposed to be ‘scenic.’ Simón, suspecting Inés is David’s mother,

introduces himself and implores her to consider mothering David. He will later reflect

on this failed conversation using a metaphor of the star: ‘But alas, it came too suddenly

for her, this great moment, as it had come too suddenly for him. It had burst on him like

a star, and he had failed it.’69 This evokes the famous line from T.S. Eliot’s Gerontion:

‘Signs are taken for wonders.’70 For Simón, signs, here, indicate wonders indeed,

whereas in Novilla they do not. As in the unreasonable and unverifiable ‘recognition’ of

Inés as David’s mother, we see Simón link otherwise unconnected signs together

associatively, joining together the three separate stars and their significations together to

lead to a sort of miracle: the discovery of Inés.

69 J.M. Coetzee, The Childhood of Jesus (London: Harvill Secker, 2013), 77. 70 T.S. Eliot, ‘Gerontion,’ in Collected Poems 1909-1962 (Orlando: Harcourt, Brace and Company, 1991).

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Whilst Simón struggles with his new life in Novilla, and the absence of

‘doubleness’ in this newfound home, it is David who will most fully – and

unintentionally – resist the founding elements of that society. David’s resistance is also

rooted in his use of language, in particular his capacity to name. David cannot think in

terms of either universals or economics, but only singularity. David is unwittingly a

nominalist and in his struggles in school with numbers and stories will undermine the

notion of ‘possible worlds’ held by Simón as well as the conviction in ‘one world’ held by

other members of Novilla. David is acutely aware of the singular, and oblivious to the

natural or habitual law or accepted interpretation of ‘the way things are.’ This is most

directly illustrated through a conversation between David and Simón as they are

walking home one evening:

‘Come on, hurry up,’ [Simón] says irritably. ‘Keep your game for another day.’

‘No. I don’t want to fall into a crack.’

‘That’s nonsense. How can a big boy like you fall down a little crack like that?’

‘Not that crack. Another crack.’

‘Which crack? Point to the crack.’

‘I don’t know! I don’t know which crack. Nobody knows.’

‘Nobody knows because nobody can fall through a crack in the paving. Now

hurry up.’

‘I can! You can! Anyone can! You don’t know!’71

Whilst it initially appears that David is being facetious here, this behaviour will come to

appear genuine. David’s way of seeing the world repeatedly flaunts probability for what

others take to be contingency. Above all, David’s sense of possibility operates according

71 Coetzee, The Childhood of Jesus. 35.

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to an approach to situations and objects as singular rather than general. Where Simón

occupies what might be described as a ‘Platonic’ position, aware of the discrepancy

between appearance and reality, the imminent and the ideal, so David is mired

cognitively and behaviourally in a nominalism that recognises only singularities. It is

true on one level that David does recognise some form of law here. Somewhere there is

a crack that will swallow him up. But this law is that of contingency, possibility and

singularity, rather than the law of norms or ‘reasonableness.’

David’s disruptive behaviour will come to a head in his inability to learn to read

and write. There are other laws of numeracy – and indeed of literacy – that this

nominalist child struggles with. Long before his time has come to go to school, Simón

has already begun to introduce David to books and counting. He acquires a copy of Don

Quixote from a library and reads David the story. But David refuses – or is unable – to

accept the ‘doubleness’ necessary for a traditional reading of Don Quixote. He takes Don

Quixote’s perspective, not Sancho’s, convinced that Quixote’s observation of a giant is

correct and Sancho’s view of a windmill is incorrect.72 David’s problems with literature

seem to be an extension of his problems with numeracy. His grasp on the narrative is

not checked by any necessity of natural law, just as his experience with the cracks in the

pavement was not checked by any attribute of size, gravity or the finitude of the crack.

Giants and windmills have equal purchase in David’s mind because he attributes

identity to objects on the basis of possibility rather than probability, putting Quixote on

the side of singularity, and Sancho on the side of probability.

Further signs of David’s numerical dissonance appear in his interpretations of

the stories that Inés tells him. His interpretations link the strange blank origin narratives

of all the citizens of Novilla – which disrupt what we consider to be the natural law of

progeny – with a wider problem of order. Inés tells David a story of three brothers, who

72 Ibid. 152-153.

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leave home in turn to seek out a remedy for their mother’s illness from the ‘Wise

Woman who guards the precious herb of cure.’73 The first two brothers leave and each

meets an animal along the way, who offers to show each brother the way to the Wise

Woman in exchange for food. Both brothers dismiss the animals and are never heard of

again. The third brother meets a bear on his way to the Wise Woman, and agrees to

give the bear food. The bear asks for his heart, he assents, and the bear leads him to the

Wise Woman where he procures the necessary herb. Upon returning to the mother,

who is healed by the herb, the third son turns into a star (he cannot continue to live with

no heart), leaving his mother alone. David has decided that he wants to be like the third

brother, which is significant in the first instance because it reinforces David’s

idiosyncratic relation to the world: the third brother does not operate according to types

or probability (that the bear will attack him) but rather takes his situation at face value,

giving the bear food. Again, this exchange happens within the terms of a certain law,

but the very conditions of the exchange exceed what we would consider to be a ‘law’

that operates within and enforces the stability of norms. When Simón disputes the

possibility of David becoming like the third brother, claiming that he can only be the

first, because he has no siblings, David objects and tells him Inés will give him more

brothers. The boy is demanding and uncompromising, and whilst he understands that

numbers occur, he has no comprehension of sequence, no comprehension that, even if

two brothers are born, he will still be first in the familial sequence. David is typically

unresponsive to the requests and reasoning of his parents and others and counters

Simón with intensity: ‘I want to be the third son! She promised me!’ ‘One comes before

two, David, and two before three,’ Simón retorts, ‘Inés can make promises until she is

blue in the face but she can’t change that. One-two-three. It’s a law even stronger than a

73 Ibid. 146.

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law of nature. It is called the law of numbers.’74 Despite Simón’s reasoning, David will

not, or cannot, be convinced. And here we have the most significant instance of number

in the book: David effectively identifies himself as that singular number, a number not

yet named ‘one,’ or, to use the words of Christ (or God) from Revelation 22:13: ‘I am

the Alpha and the Omega.’ The Alpha and the Omega: a beginning and an end, a

number outside of sequence. The only number in mathematical history that may be

akin to this ‘singular number’ is Cantor’s transfinite ‘aleph-null’ (0א): the cardinal

number that measures the infinity of all definable numbers. This, of course, echoes

Magda’s identification with ‘zero’: the singular number that is the locus of the personal

pronoun, which cancels the other numbers when they are multiplied or divided by it.

The position in mathematical epistemology that exists in antithesis to the

numerical regime of Novilla is nominalism: a philosophical position – espoused most

famously in the twentieth century by Willard Quine – that enforces a system of what

can and cannot be known, and seeks to clarify language and thought by injecting such a

distinction into linguistics and the sciences. Nominalism is the philosophical position

whereby universal entities are denied existence, though the existence of particular or

singular entities is affirmed. Whilst nominalism is an operative version of metaphysics, in

The Childhood of Jesus we encounter nominalism as epistemological condition. In

mathematics, nominalism denies the existence of mathematical objects or entities; these

are, rather, merely tools to assist in the description of particulars. This is opposed to

Platonism, which posits that such mathematical objects do have a positive existence in

an ideal realm. On one level, nominalism can involve only a minor amendment to

mathematics, which involves the clause that, although mathematics involves conceptual

entities, any claim to the actual existence of a universal entity such as a number would

be a transgression against the proper metaphysical implication of mathematics. On 74 Ibid. 148.

265

another level, nominalism can imply a much greater transformation in the way that

objects in the worlds are recognised. If numbers are taken as objects, they cannot exist

within a sequence but must be encountered individually, as proper nouns or names.

David does not choose nominalism as a philosophical position but cognitively

cannot comprehend universals, only particulars. In a society that runs on sequence, and

identity that emerges from sequence, this contrary numerical cognition will be a

disability. The nominalism that we encounter in the character of David will require a

much more radical renunciation of mathematical tenets, with enormous upheavals in

social and personal existence. The problem that precedes a successful nominalism is that

of individuation which, as defined by Ray Brassier, is the question of ‘how it is that

something comes to be counted as one.’75 In Brassier’s account, Quinean individuation is

linguistic: it bears not on a reality but on what is deemed a ‘one’ in language. The initial

means of clarifying this is through grammatical significance: ‘We shall not forego all use

of predicates and other words that are often taken to name abstract objects. We may still

write ‘x is a dog,’ or ‘x is between y and z’; for here ‘is a dog’ and ‘is between ... and’ can

be construed as syncategorematic: significant in context but naming nothing.’76 In this

sense, universals or abstractions can be qualifiers but not the variables that establish

meaning. Where Simón has a conception of value of numbers that differs from the

narrative that structures Novilla (and hence has a resistance that is largely affective),

David’s dissent arises from a different conception of the identity of numbers (and hence

a resistance that is cognitive). Simón understands the disjunction between forms of

measure and the definition and import or objects of measure. As such, his view of

numbers is tempered by a consciousness of that which exceeds quantification. David

appears as kind of radical nominalist because he takes numbers to have an existence and 75 Ray Brassier, ‘Behold the Non-Rabbit: Kant, Quine, Laurelle,’ Pli 12 (2001), 50. 76 Nelson Goodman and Willard Van Orman Quine, ‘Steps Toward a Constructive Nominalism,’ The

Journal of Symbolic Logic 12, no. 4 (1947), 105.

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identity that is not sequential. In other words, for David, each number is a ‘one’ in

language and thought: his application of nominalism takes abstractions as existent,

reversing the Quinean allocation of abstraction and singularity.

When David explains to Simón that Inés believes he should not have to go to

school because he is clever (and thus will be bored by school), Simón asks David what it

is that he thinks makes him so clever. David answers that he is clever because he knows

all numbers:

‘I know all the numbers. Do you want to hear them? I know 134 and I know 7

and I know’ – he draws a deep breath – ‘4623551 and I know 888 and I know

92 and I know -’

‘Stop! That’s not knowing numbers, David. Knowing numbers means being able

to count. It means knowing the order of the numbers [...]’77

When Simón challenges the boy to tell him the next number after 888, and corrects the

boy when he produces a number nowhere near 888, David retorts that Simón cannot

know which number comes after ‘888’ because he has never ‘been there.’ This is a

direct contradiction to knowing all the numbers, although David’s response to Simón –

that he hasn’t ‘been there’ – suggests that David’s idea of knowing numbers differs very

significantly from Simón’s: that numbers are encountered or arrived at rather than

simply extracted from their sequence. Moreover, there is an implicit nod to the infinite

in Simón’s choice of arbitrarily large number. 888 presents the upright (and finite) mark

of the lemnsicate three times over. If David is reading for the singularity of the marks,

rather than sequential numerical identity, his claim that he cannot know what comes

77 Coetzee, The Childhood of Jesus, 149.

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‘after’ 888 seems surprisingly astute. David, here, reveals himself as ‘unfallen’ in Biblical

terms because he cannot comprehend sequence.

Eugenio, who is also a stevedore, and a friend of Simón’s, has his own point of

view on David’s predicament. ‘An apple is an apple is an apple’, explains Eugenio,

An apple and another apple make two apples. One Simón and one Eugenio

make two passengers in a car. A child doesn’t find statements like that hard to

accept – an ordinary child. He doesn’t find them hard because they are true,

because from birth we are, so to speak, attuned to their truth. As for being afraid

of the empty spaces between numbers, have you ever pointed out to David that

the number of numbers is infinite?78

Eugenio’s characterisation of David seems to be correct, here. David is not attuned to

the basic truth that guides others and the subsequent question will revolve around

whether David is attuned to a different truth (and subsequently how many truths then

might exist) and what the implications might be for society should wildly different

perceptions of the way reality is structured exist. And yet Eugenio reveals himself as also

fundamentally incapable of truly comprehending David’s singularity. Eugenio seems to

have no conception that the number of infinities is infinite, and that infinities can take

on numerical value and be different ‘sizes.’ Eugenio’s point regarding the infinity of

numbers suggests a ‘whole’ infinity that exists as a single totality without gaps. Eugenio,

along with several of the other stevedores, attends a philosophy course at a place called

‘The Institute’ that offers a wide range of evening courses and seems popular with the

residents of Novilla. Simón declines to join, but is given an account of one of the

conversations that group engaged in:

78 Coetzee, The Childhood of Jesus, 250.

268

The discussion grew more interesting after you left. We talked about infinity and

the perils of infinity. What if, beyond the ideal chair, there is yet more ideal

chair, and so forth for ever and ever?79

Eugenio does not dwell on what the perils of infinity might exactly be, though his

conception here is revealing enough. The danger of infinity, it seems, is that any concept

of matter never really stops: the ideal chair will always withdraw from the grasp of

thought and knowledge. Eugenio then goes on to present a novel theory of the infinite:

There are good infinities and bad infinities, Simón. [...] A bad infinity is like

finding yourself in a dream within a dream within yet another dream, and so

forth endlessly. [...] But the numbers aren’t like that. The numbers constitute a

good infinity. Why? Because, like being infinite in number, they fill all the spaces

in the universe, packed one against another tight as bricks. So we are all safe.

There is nowhere to fall. Point that out to the boy. It will reassure him.80

Again, Eugenio’s analytic philosophy misses the point and he seems oblivious to any

theory of numbers that may include the gaps between numbers or multiple infinities.

There is no conceptual or philosophical acknowledgment of an entity like an irrational

number, here. Eugenio describes a totality that is infinite, consistent and whole. Eugenio

envisages numbers as bricks: each standardised, solid and strong but ultimately identical,

distinguishable only in their existence in a sequence. This, it would seem, is the

79 Ibid. 123. 80 Ibid. 250.

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mathematics of ‘one world’ rather than possible worlds or worlds of singularities. And

this mathematics of ‘one world’ is, equally, a mathematics of the camp.

David’s notion does not so much comprehend the infinite but rather is itself

inflected with the infinite. In this sense, David is both closer to any sense of the infinite

than anyone else, including and especially Eugenio with his analytic philosophy, and

further away. He cannot speak of what characterises or creates him, and he is above all

unable to articulate infinity, because his way of relating to the world is nominalist and he

is fixated on singularity (David would be no more capable of comprehending a

transfinite number than Eugenio). This is not to say that Eugenio is without a

conception of numerical identity or has a naïve concept of number. Quite the contrary.

Eugenio’s statement belies the fact that he does have a concept of number, indeed,

Eugenio’s statement reveals a singular number. Eugenio’s unique number is the brick,

the algorithm from which all other numbers and the structure of the universe extends,

which is not only an extrapolation from his choice of metaphor but also resonates with

his notions of labour and purpose. David’s number, on the other hand, is the wand,

which he becomes obsessed with in the end, wearing a cloak and carrying a wand and

purporting to be a magician. And what is the wand? The number one that isn’t a

number one: the properly singular number without a name.

As in Coetzee’s earlier work, there is an extended and complex engagement with

sequence, with what constitutes a unit and how it is naturalised in some continuing

count. During the composition of Dusklands Coetzee made notes from Henri Poincaré’s

Mathematical Creativity. The first quote that Coetzee transcribed in his notes relates to the

syllogism. ‘Imagine a long series of syllogisms,’ Poincaré writes,

and that the conclusions of the first serve as premises of the following. We shall

be able to catch each of these syllogisms, and it is not in passing from premises to

conclusions that we are in danger of deceiving ourselves. But between the

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moment in which we first meet a proposition as conclusion of one syllogism, and

that in which we reencounter it as premise of another syllogism occasionally

some time will elapse, several links of the chain will have unrolled; so it may

happen that we have forgotten it, or worse, that we have forgotten its meaning.

So it may happen that we replace it by a slightly different proposition, or that,

while retaining the same enunciation, we attribute to it a slightly different

meaning, and thus it is that we are exposed to error.81

According to Poincaré, the true danger of a syllogism is in the forgetting of the origin in

the chain of logic. The syllogistic peril indeed relates to the chain that language forms,

but what Poincaré emphasises, here, is the fact that the falsity of a sequence may come

not from the structure in itself but the acceptance of some original statement as true in

itself. Such an original statement may be the assertion of an allegory, as we saw in the

naming of the cloud as an angel with a flaming sword, a conviction that this is the only

world, or a concept of number or the infinite. Poincaré made a famous distinction

between intuitionist mathematicians (who he also aligned with geometry) and logicist

mathematicians (who he aligned with analysis).82 The latter do not think spatially, but

the former, who jump to conclusions faster, at times with an absence of reasoning, do

think spatially. Intuitionist mathematicians do not arrive through due process at their

conclusions, but instead see their conclusions. We might call David an intuitionist, rather

than logicist. In David’s world the logic of probability (perhaps the most important

81 This quote by Henri Poincaré from Mathematical Creation is found in: J.M. Coetzee, Reading notes

including materials for Dusklands. 1960s, Subseries A: Long Works 1960s-2012, Folder 99.3. Harry

Ransom Centre, University of Austin, Texas, 34. 82 The second Poincaré quote from Coetzee’s notes runs as follows: ‘When a sudden illumination seizes

upon the mind of a mathematician, it usually happens that it does not deceive him; ...[if false] we almost

always notice that this false idea, had it been true, would have gratified our natural feeling for

mathematical elegance.’ Ibid. 40.

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normative logic) is absent and all depends on intuition, a fact that is equally applicable

to Eugenio for whom numerical truth extends not from logic but a pre-rational

apprehension of truth. Each opposing worldview and epistemology contains its own

‘unique number’ that founds not only all other numbers but a whole metaphysics of

individuation. A syllogism occurs not simply when one extracts a conclusion from a

piece of information in an illogical or erroneous process, Poincaré reminds us, but only

where the original piece of information or the original ‘jump’ is forgotten. The overall

effect of this is not to necessarily validate intuitionism over analysis but instead to

illustrate the means by which various numerical regimes emerge based on conceptions

of the nature of an initial ‘jump,’ the forgetting that facilitates a ‘naturalised’ world.

The variant intuitions of numerical identity – and the truth of number – held by

David and Eugenio suggest that the origin of the numerical structure of societies rests in

a kind of ‘forgetting’ of the origins of number, just as the residents of Novilla too have

arrived with only the shadiest notions of their prior lives. This of course resonates with

the story of the biblical Jesus that the title refers to: the man who is both finite and

infinite, human and divine, mortal and immortal. Jesus is also, in quotidian terms, the

origin of our chronological count. The birth of Jesus sets the measure of years in play: it

is his birth that enables the jump from zero (and perhaps away from some, now

forgotten count) to one. Numerical identity makes a transcendental and unavowed jump

that finds its echoes in allegory but never presents itself, or, in other words, never reveals

its own name.

*

4.4 COUNTING AS ONE, RATHER THAN COUNTING TO ONE

These two novels were written in vastly different circumstances, more than three

decades apart, at either end of Coetzee’s career. Yet, in each, we can detect a point of

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comparison: the preoccupation with what is countable as ‘one,’ a mathematical

enterprise that pertains to the roots of the linguistic, the numerical and the subjective.

This is not so much a question of learning to count to one as an issue of finding the

originary, self-contained unit from which one can institute the very concept of unit, and

from which one can hope to count upwards. This issue is the fundamental point at

which numerical identity is abstract and utterly personal, rational and potentially

aphasic. The concept of the ‘count-as one’ is deployed by the French philosopher Alain

Badiou to describe the relation between a unit and a broader structure, and the means

by which such an operation comes to define a situation. Badiou does not imply a

nominalist or universalist perspective on what counts as one. Rather, Badiou is referring

to a very specific action in set theory. The ‘one’ is not a stable unit for Badiou, but is,

rather, an operation.83 Each ‘situation’ ‘admits its own particular operator of the count-

as-one.84 This is the most general definition of a structure; it is what prescribes, for a

83 Sigi Jottkandt provides an excellent definition of Badiou’s concept of the ‘count-as-one’ which bears

quoting at length: ‘Briefly, the count-as-one is founded upon the structuring that, in presenting the void

through a nominal decision, originarily specifies which elements of a ‘set’ are in a relation of belonging.

The name founds the law of the situation, although Badiou is always quick to point out that, even as it

purports to name the void, every eventual naming is inevitably an illegal misnaming. In emphasizing that

what passes as a complete representation or identity is only ever a semblance, an ‘as if,’ Badiou reminds us

that every identity, in so far as it is tied inextricably to that original evental misnaming, can only be

counted ‘as’ one – rather than actually being a one.’ See: Sigi Jottkandt, ‘Love,’ in Alain Badiou: Key Concepts,

ed. A.J. Bartlett and Justin Clemens (Oxford and New York: Routledge, 2010), 75. 84 This is expertly summarised by Peter Hallward:

First, whatever is presented must be presented as one, that is, it must be structured in such a way that it

can be counted as a one: Badiou defines a situation, in the most general sense, as the result of any such

structuring or counting operation. To exist is to belong to a situation, and within any situation there is

normally no chance of encountering anything unstructured, that is, anything that cannot be counted as a

one. Second, we know nonetheless that this operation is a result, and that whatever was thus structured or

counted as one is not itself one, but multiple. Although the being of what was thus counted cannot be

presented as the inconsistent multiplicity that it is, its multiplicity continues to hover like a shadowy

‘phantom’ or remainder on the horizon of every situation. But because it is unpresentable, this multiplicity

must figure from within the situation purely and simply as nothing. As far as any situation is concerned,

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presented multiple, the regime of its count-as-one.’85 Lorenzo Chiesa has pointed out

that Badiou’s ‘count-as-one’ is bound up with ‘the relationship between structure and

metastructure, presentation and representation.’86 The count-as-one is an operator that

inscribes the structure of a situation. It is this count that Coetzee is preoccupied with in

both of these very different novels: the significance of a ‘one’ in traversing presentation

and representation, as an operator that emerges from a certain situation.

In In the Heart of the Country we see the incommensurability between the pure

‘Esperanto’ of numbers and the capacities for language to form equations and to count.

The answer to the question ‘who counts?’ is connected to the question of how to

communicate, and, subsequently, Magda’s monologue is repeatedly confounded by the

untranslatability of numbers into situations of singularity, or, in other words, the absence

of a true conversion between the pronoun and the numeral. In The Childhood of Jesus we

are again confronted with a problem of translation between a singularity and a generic

figure: David exists in a dangerous border-zone where he can recognise numbers, but

only as actually existent and hence as singularities rather than as generic placeholders in

a stable schema. David’s unique brand of nominalism has significant repercussions in

terms of his ability to respond to rules and conventions. These reverberations express a

comparable condition to that revealed in Magda’s failings with language: the conception

there is simply nothing that resists the operation of the count, nothing that cannot be presented as a one,

as a particular person or thing. (Hallward, ‘Generic Sovereignty: The Philosophy of Alain Badiou,’ 64.)

It is for this reason that, in Badiou’s philosophy, one can then ‘affirm the being of nothing’ (Ibid. 65).

Despite this fact seeming more intuitively useful for a reading of Coetzee in terms of this operation of the

count, I do not believe, ultimately, that we find an affirmative gesture as regards nothing, multiplicity, or

some figure of ‘zero’ and all description, here. Rather, this novel simply observes the traversal of that

operation of the one, remaining silent on the ‘affirmation’ of nothing.

85 Alain Badiou, Being and Event, trans. Oliver Feltham (London and New York: Continuum, 2007), 24. 86 Lorenzo Chiesa, ‘Count-As-One, Forming-Into-One, Unary Trait, S1,’ Cosmos and History: The Journal of

Natural and Social Philosophy 2, no. 1–2 (2006), 69.

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of numerical identity that affirms numbers as generic abstractions and thus inexistent is

compatible with and operates as an extension of a certain social system.

This link between numeracy and social structure pertains to one of the most

important socio-cultural issues in the twentieth century: the idea of an originary

language – the language that Magda, perversely, lusts after. ‘Like the North African

Augustine, who resolved to write in Latin, in order to spread his thoughts throughout an

unholy empire, the South African Coetzee became a student and master of English,’

Stephen Kellman writes, ‘the greatest imperial language of them all.’87 There are a

series of questions that the split in the nature of the ‘land of the poet’ – a linguistic versus

the cultural homeland, a form of ‘originary’ speech – presents for the post-colonial

artist, educated in the classics of European and North American literature. How does an

artistic language travel into contexts where there is no recognition of the predicates of

that artwork? And what becomes of the artist emerging from ‘the Provinces,’ the further

reaches of Empire for whom the natural language that literature experiments with is

absent? In these questions about the postcolonial form and response to European

modernism, Coetzee will be preoccupied with what constitutes literary inheritance and

the disruption of this inheritance. Paul Sheehan situates Coetzee’s writing in terms of

‘geomodernism,’ a turn which describes ‘a way of putting classical modernism into

dialogue with postcolonialism, through the politics of place.’88 Sheehan’s reading of

Coetzee as ‘geomodernist’ involves a reconfiguration of the notion of provincialism and

the relation of the provinces to art. This revisioning of modernism’s relation between the

‘margin and the metropole’89 allows for a recognition of a modernism that stems

87 Steven G. Kellman, ‘J.M. Coetzee and Samuel Beckett: The Translingual Link,’ Comparative Literature

Studies 33, no. 2 (1996), 162. 88 Paul Sheehan, ‘The Disasters of ‘Youth’: Coetzee and Geomodernism,’ Twentieth Century Literature 57,

no. 1 (2011), 26. 89 Ibid.

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precisely from the position of the colonial subject, who is ‘drawn to the heart of Empire

yet at the same time repelled by it.’90 The ‘geoliterary spirit’ of Coetzee’s South African

origins is, then, the ‘fulfillment’ of modernism rather than its ‘betrayal.’91 Much like the

young Borges, Coetzee’s preoccupation with the space or split between centre and

province, and the intermingling between these two spheres of artistic possibility,

structurally mirrors a search for the transmissibility of language between A and B,

communication as a kind of conduction but also particular pattern that may be

interpreted (or mis- interpreted, or not even considered for interpretation) by the

interlocutor in a whole host of different ways, dependent upon unique linguistic make

up. Of course, the notion of communication from A-B, and the conduction that

facilitates this, is also the question of what computers can do to poetry. This as much an

issue of how computers might mimic humans and the human art of poetry, as it is of the

extent that a human is like a computer. To what extent do humans possess a capacity to

recognise and share universal art? And how, then, would we recognise such universality,

when our operative concept of the classic is imbricated in a construction of centre and

periphery that originates from conquest? South African fiction becomes a retrospective

hermeneutic endeavour – the issue of what it means to write, and to communicate,

becomes the necessary subject. In Watt Beckett consciously works through such a task,

unfolding the surface through permutative monologues that make the ‘dimensionless

and immaterial point[s]’ of language appear. In Coetzee’s In the Heart of the Country and

The Childhood of Jesus number is, again, is at the heart of a formal allegory, a writing of

writing. It is that mediator between structure and metastructure, presentation and

representation. What constitutes the ‘One’, or, the failure to constitute a ‘One’ renders

90 Ibid. 29. 91 Ibid.

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the form of the narrative as well as the implications for the characters’ existence within a

community.

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∞ CONCLUSION ACTUAL LITERARY INFINITIES

In this dissertation, I have argued that mathematics occupies a position of core

significance in the literary innovations of these three writers. The work of Jorge Luis

Borges, Samuel Beckett and J.M. Coetzee deliver impressive and substantial examples of

a certain type of literary enquiry, an aesthetic endeavour that I have called a generic

literature. This literature can be described as generic by virtue of its stakes in formal self-

referentiality, whereby the text enquires into its own processes of composition, even as

that composition – its plot, dialogue, imagery and structure – takes shape. This self-

referentiality is, most often, a version of formal allegory as it has been described by Paul

de Man. In these instances of generic literature, the text stages, interrogates and

intervenes in the very formulae that underpin the extensity and continuity of description

and the consistency of relation between the world, the fictional world, and language. In

the work of Borges, Beckett and Coetzee, this is achieved by the appearance of number

in the text and indeed, conversely, by the use of number to explore the numerical

apparatuses that undergird fiction. It is this claim to a generic fiction, and for the ontic

significance of numeracy to representational regimes, which brings this preoccupation

with number into a wider modernist project.

Prose fiction can think with and through its own numbers. Literary studies now

needs to develop a relation to mathematics that can comprehend these literary

transfinites too. The occlusion of number in literary criticism is perhaps abetted by the

paradoxes and impasses that have historically plagued the identity of the numeral.

Numbers are the foundation of the system of arithmetic (a term derived from the Greek

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portmanteau of arithmos – number – and techne – art) and yet the very definition of

numerical identity has remained controversial throughout modern and ancient

mathematics: how can a number be defined within the system of arithmetic? As marks

of presentation numbers exist only as their notations, not as objects that can be

subjected to empirical investigation. Numbers never function the way a standard sign in

natural language does, as a signifier with a relation to that which is signified. Rather, a

number presents nothing other than itself. In other words, a number does not possess a

referent or signified, but rather is itself the signified. This is the tautologous havoc that

presentation wrecks upon representation. In this regard, mathematical signs are still

regarded as properly outside of ‘natural language’ and not subject to the laws of natural

language; their problematic status regarding standard representation in natural

language belies their traditionally vexed status as objects of study in mathematics.

There is an irony in the fact that it is only through ‘doubling’ the ‘sense’ of

numbers that Cantor can prove the uncountability of the reals and the countability of

natural numbers. The proof of countable and uncountable numbers, which is the first

step in the development of transfinite numbers, relies on a subversion of the purity of the

count: what happens when counting numbers are used to count numbers? An actual

infinity is produced by virtue of a system of measure being applied to itself. In a

formulation that Borges, Beckett and Coetzee would each have enjoyed, this is an

instance of a measure measuring the measure. The exact nature of the infinitude of a

system is extracted from a truth that resides within the system’s presentation and not

elsewhere, not at the end of a number line, or at any end whatsoever. Instead, it is the

measure of measure through one-to-one correspondence that allows for the

development of a transfinite number: a number that presents the size of an infinity

without being inaccessible to finite mathematics. I have argued here that this doubling is

echoed in literature through a kind of internal allegory that has been best articulated by

Paul de Man in his theory of allegory and rhetorical form. This morphological link is the

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first step en route to an idea called ‘transfinite allegory,’ which describes a form through

which literature presents an actual infinity, rather than a transcendental or exclusively

metaphorical infinity.

I have used a morphological metaphor to model the relation between literary

transfinites and mathematical transfinites: the lemniscate. The lemniscate models

separation and connection between mathematics and literature through the form of a

reciprocity as well as a reflection. Of course, a morphology pertains to shape, to a mark,

to the materiality of a sign. In this sense a morphology necessarily sits outside the

bounds of tolerance of both literature and mathematics. It is the absent third category of

textual existence: the mark. To a certain extent, suturing metaphor to morphology

echoes the techniques of a generic literature itself by locating significance in the

arbitrary materialisation of the mark. This metaphor between literature and

mathematics produces its own visual allegory of separation and reciprocity.

The formal conduit of what is metaphorically depicted in the lemniscate is

‘transfinite allegory.’ What is a transfinite allegory? In simple terms it is an allegory for

which the a-logos – the other ‘order’ or text contained within a fiction – is not some other

story or moral truth but instead a representational, descriptive or imaginative limit or a

conditioning factor for language. Importantly, this is not some romantic ‘never-

attainable’ infinitude, but one that is materially available within the text: the a logos is

manifest in the stories or novels, not referred to or felt. The most obvious example,

perhaps, is the ‘Golden Book’ in ‘The Library of Babel’ or the memory of Ireneo Funes

in ‘Funes, His Memory.’ In the second chapter I argued that Funes’ attempt to create a

number system, and indeed his very unusual paralysis itself, contribute to a story that

revolves around the possibility of presentation, without necessarily purporting to

incorporate that presentation within the bounds of the story. This story incorporates the

limits of its own medium, revolving around a world of infinite detail and infinite recall

and the consequences of such capacities. Funes embodies both the possibility that gives

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rise to fictional worlds (imagined realities, access to a tangible world beyond the one we

exist in, potentially infinite worlds accessed through the mind) as well as the numerical

limitations that underlie the possibilities available to fiction: a simultaneously genetic

and generative kernel. The story is brought to life by two forms of consciousness: one

finite and one infinite, one belonging to the narrator and the other belonging to a

paralysed, extraordinary Uruguayan youth. I have argued that this constitutes one mode

of transfinite allegory: the construction of a fiction that points outside of its own borders,

to an autonomous existence, which is embedded but not contained by the boundaries of

the stories. This story achieves generic form because it evokes measure without

definition on two different levels: Funes’ embodies an uncountable consciousness, one

which is presented in a finite narrative that actualises an infinite fictional world without

totalising that world, or commuting it to a form of ‘endlessness.’ In other words, it

indexes the numerical potentials undergirding the possibilities of representation and

yields an image of infinite imagination in the smallest possible form: through a recursive

short story.

Number in Beckett’s work also achieves a generic status for the fiction.

Mathematics in Molloy provides the textual material in which Beckett traces the non-

semiotic jump between reason and material (partial, recurrent negation), mind and

extension (proprioception), and language and referent. As in Borges’ stories, Beckett’s

novels index the numerical stakes in description through a kind of radicalization of

naturalism. In Watt, this happens through cycles not of description or narration but

permutation. In Beckett’s later short prose works we find a transition from measure to

magnitude, and a transition from a concern with content of phenomena to conditions for

phenomena. This enables Beckett to radicalise the tenets of naturalism and approach a

fiction that allegorises the emergence of images and fantasies of the essential literary

domain: the imagination (a short story is set in a skull, a television play presents the

material production of its own images).

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Number is the enigma of prose, because it is its allegorical ‘other.’ It has the

capacity to index a novel by virtue of the fact that it is exempt from the text and

representational economy that composes that novel. Equally, number is the enigma of

prose because it is the constitutive force behind the development of narrative insofar as

it cleaves to the genre of the novel from the early eighteenth century onwards. The fact

that literary criticism sees number as ‘obviously’ related to poetry but not prose is a

blindness that emerges from a failure to grasp both the broad formal and thematic

significance of number to narrative. Prose narrative depends on forward motion, on

enumeration and resolution, and the role of syntax as well as novelistic length rely on

continuity and cumulation. This is not the same form of number that is relevant to

poetic work. Moretti’s formulation demarcating poetry to symmetrical form and prose

to forward motion illustrates the different stakes in number for poetry and for prose

quite simply: issues surrounding the number line are inherent to prose form, even where

it does not have the same metrical investment of poetry. In the representational

currencies of narrative words constrain number, which bursts the seams of referentiality,

a very different effect from the expressive currencies of poetry, where number as metric

can constrain the word.

This is seen perhaps most literally in Coetzee’s In the Heart of the Country, where

the numbers index the paragraphs of the story, one of several modes of counting that

bring order or disorder to Magda’s narrative. Indeed, these numbers that accompany

the paragraphs index Magda’s own linguistic and representational predicament:

numbers are never properly incorporated into her own attempts to communicate and

relate through language. This novel questions the impact of zero in language: zero as

the placeholder for ‘no-one’ or one that cannot be ‘counted.’ A similar problem of the

constitution of the ‘one’ is found in The Childhood of Jesus, where we witness the subjective

and social consequences of a child who, like Magda, quite literally embodies a

singularity, by virtue of existing prior to ‘one,’ prior to the social, biopolitical and

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pedagogical regime of countable numbers. As the allegorical child Jesus, David initiates

a chronological count that extends not from some objective foundation for numbers but

from his own being, in a generic account of the foundations of thought and being in

concepts of number.

*

Transfinite allegory comes good on the claim by Friedrich Nietzsche in Writings From The

Late Notebooks cited as the epigraph to this dissertation:

Since Copernicus, man has been rolling from the center towards X.1

The fiction of ‘transfinite allegory’ or in other terms a ‘generic fiction’ is a literature of

what Nietzsche refers to as x.

These literary transfinites are the textual form appropriate to the age that

Nietzsche describes. It is the form of story and the form of imaginative and artistic

endeavor that no longer has a proper end but is not – crucially – endless, or does not

involve the experience of endlessness. X – in Nietzsche’s formulation – is the algebraic

substitute for the new displacement of the human after Copernicus and, indeed, after so

many of the core modern revolutions, Cantor’s development of the transfinites

preeminent among these. It is fascinating that the displacement of the human, into a

vast universe wherein there is no clear sacral, and no narrative of transcendental order,

is represented as an algebraic mark in this aphorism. The absence of a clear end or

trajectory is registered by a mark that stands in for a potentially delimitable variable. X

1 Friedrich Nietzsche, Writings from the Late Notebooks, ed. Rüdiger Bittner, trans. Kate Sturge (Cambridge

and New York: Cambridge University Press, 2003), 84.

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is not a future deity, a different ideal, or a nihilist future. Instead it is a placeholder for

the generic. A fiction that replaces predicates, locations and progressive narrative

enumeration with allegory of its own composition, an allegory that measures the

measure of fiction, is a generic literature. The work of Borges, Beckett and Coetzee

fulfils this generic literature through a composition attentive to its own mechanisms of

numeration, and this allegory that ‘measures measure,’ a transfinite allegory.

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ACKNOWLEDGMENTS

The past few years have been intellectually enriching and challenging beyond measure. Thanks

go first and foremost to my supervisor, Julian Murphet. It has been an honour to work with

Julian at the Centre for Modernism Studies over the last few years and I am grateful for the

time, insight and direction he has afforded me.

I have been the grateful recipient of a Research Excellence Scholarship from the School

of English at the University of New South Wales. Grants from the Postgraduate Student

Support Scheme and from faculty funding enabled trips to conferences and three weeks of

archival research at the Harry Ransom Centre at the University of Austin, Texas.

I would like to thank Helen Rydstrand, Jacinta Kelly Khemchandani, Kate Montague,

Penelope Hone, Tam Avery, Sigi Jöttkandt and Sean Pryor for their feedback on drafts.

In 1883 Cantor wrote that ‘Apart from the journey which strives to be carried out in the

imagination [Phantasie] or in dreams, I say that a solid ground and base as well as a smooth path

are absolutely necessary for secure traveling or wandering, a path which never breaks off, but

one which must be and remain passable wherever the journey leads.’1 This was the path that

Cantor formulated for mathematics: he was convinced that transfinite numbers were the sure

road to the infinite. Here I would like to give thanks to Jayne and Marius Brits, and to Maegan

Brits, who have given me just such a ‘solid ground and base’ for wandering. Without their

support I would surely not attempt things like writing doctoral theses.

Cantor elaborated his metaphor of the path in order to claim the necessity of transfinite

numbers for mathematics: ‘every potential infinity (the wandering limit) leads to a Transfinitum

(the sure path for wandering), and cannot be thought of without the latter.’2 Just as there are

roads to finite destinations, here Cantor envisages a road unlike any other we have encountered,

a road that always leads to an infinite destination. I thank R for sharing such a journey with me

the past few years, and for his clarity of vision along the way ∞.

1 Cantor quoted in Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, 127. 2 Ibid. 127.