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SELECTIONS FROM: FERNANDO ZALAMEA, SYNTHETIC PHILOSOPHY OF CONTEMPORARY MATHEMATICS, TRANS. Z.L. FRASER (FORTHCOMING FROM URBANOMIC PRESS, 2012). This text has been circulated for the purposes of Fernando Zalamea’s seminar for the Versus Laboratory project at the Jan van Eyck Academie, to be held on Thursday, September 29th, 2011, at 14:00 in the JVE Auditorium, Maastricht.

A NOTE TO THE READER

This translation is still an early draft. This leaves, at times, the rhythm of the text a bit rough in certain patches, and the terminology in a state of flux. On a perhaps excessive number of occasions, the reader will find the original Spanish text given in square brackets like so [así]. While I expect a few of these occurrences to survive in the final manuscript, most are there for my own editing purposes. Of the terms that are in a clear state of flux, I may give two very prominent examples: ámbito, which I have variously translated as horizon, scope, realm, arena, and, when the set of values of a function is what is meant, as range. This variety will not be altogether eliminated in the final draft, but I expect it to be reduced. Another particularly difficult word to translate univocally is acotar, which I have rendered, variously, as to restrict, to restrain, to circumscribe, to pinpoint, to delimit, to sharpen. Another temporary flaw in this draft is that footnotes are often omitted. This was done purely for reasons of expedience, so as to not break the momentum of translating. For the time being, absent footnotes are indicated by double square brackets and an asterisk, like so [[320*]]. The page numbers of the original Spanish text are indicated in single square brackets thus [175]. Zalamea’s own placements of page numbers in square brackets to refer back to past instances in his text are here rendered in boldface like this [45]. Many diagrams have been left in Spanish, simply cut and pasted from the original text. Where their meaning is not immediately clear to the Anglophone reader, I have translated the vocabulary they employ and noted it beneath the diagram. Since this is a draft translation, I ask that it not be circulated without my permission. I am distributing it solely for use in this seminar. If you wish to quote it for whatever purpose, please send me an email at zlfraser@gmail.com so that I can make any corrections I see fit to make. I would also like to say that I welcome any suggestions or corrections, which can be sent to the same email address. Finally, a short word of encouragement concerning the text: throughout, Zalamea discusses the ins and outs of advanced contemporary mathematics. No mathematical training or knowledge, however, is presupposed of the reader (though there’s no doubt that such knowledge and such training might open up a few aspects of Zalamea’s argument that would otherwise remain hermetically sealed). It is enough, for the purposes of this text, that the reader gets a ‘feel’ for what is at stake in contemporary mathematics. I say this so that the moments of mystification the non-mathematical reader may experience, when hearing Zalamea invoke this or that great mathematical achievement, should not become impassable obstacles to her or his movement through the text. The ‘gist’ of what’s going on in the mathematics is sufficient to understand the main content of the philosophical argument, and Zalamea’s own descriptions of the mathematics behind this book will, I believe, suffice to convey that gist. (Often, too, you will find Zalamea referring back to the case studies carried out in Part II of the book, which are not reproduced here. Those wishing to read the case studies may obtain a copy by emailing me at the address provided above.) Of course, the ideal outcome of Zalamea’s evocative descriptions of these various mathematical achievements will be a heightened curiosity in the mathematics themselves, and a desire to study the mathematics for their own sake. Doing so will immensely deepen the reader’s apprehension of the philosophical problematics at stake in this text. In the meantime, don’t panic.

Thank you,

Zachary Luke Fraser, translator.

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Epistemology

Realism

Idealism

Ont

olog

y

Realism

Maddy Resnik Shapiro

Tennant

Idealism

Chihara Hellman

Dummett Field

Figure 1: Contemporary Tendencies in the philosophy of mathematics, according to Shapiro. [THIS IS ‘SHAPIRO’S SQUARE’, WHICH WILL BE REFERRED TO – AND SUBJECTED TO CRITICISM – ON SEVERAL OCCASIONS THROUGHOUT THE TEXT. SHAPIRO PRESENTS THE SQUARE AS A MEANS FOR CLASSIFYING THE VARIOUS POSSIBLE POSITIONS ONE CAN OCCUPY AS A ‘PHILOSOPHER OF MATHEMATICS.’]

Chapter 3

Towards a Synthetic Philosophy

Of Contemporary Mathematics

We have seen in the introduction and in the preceding chapters that a contrasting (and often contradictory) multiplicity of points of view traverse the field of the philosophy of mathematics. Also, we have delineated (on first approximation [primera instancia] which we will continue to refine throughout this work) at least five characteristics that separate modern mathematics from classical mathematics, and another five characteristics that distinguish contemporary mathematics from modern mathematics. In that attempt at a global conceptualization of certain mathematical tendencies of well-defined historical epochs [épocas], the immense variety of the technical spectrum that had to be traversed seemed obvious. Nevertheless, various reductionisms have sought to restrict [acotar] the philosophical multiplicity no less than the mathematical variety at stake [en juego]. Far from a one kind of all-consuming [omniabarcador] philosophical wager, or of one given reorganization of mathematics, that one would try to put into a univalent correlation with one another, what seems fundamental is that we must now consider the necessity of constructing multivalent correspondences between philosophy and mathematics, or rather between philosophies and mathematics in the plural [entre la filosofía o las

filosofías y la matmática o las matemáticas] [CHECK] In a manner consistent with this situation, we will not assume any a priori philosophical position at the outset, before having carefully observed the landscape of contemporary mathematics. We will, however, adopt a precise methodological framework following which, we believe, will be able to help us better observe that landscape. Of course, that methodological schematization will also influence our forms of cognition [modos de conocer], but we trust that the distortions can be controlled, since the methodology of investigation [de la mirada] that we will adopt and the spectrum we presume to observe are drawn sufficiently near to one another [se acercan bastante entre sí]. The philosophical and mathematical consciousness of multiplicity at stake in fact requires a minimal instrumentarium [instrumentario] that would be particularly sensitive to the transit of multiple, that could take adequate stock of [registrar] that multiplicity, and that would let us understand its processes of translation and

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transformation. To those ends [a estos efectos], we will adopt certain minimal epistemological guidelines, furnished, in philosophy, by Peirce’s pragmati(ci)sm, and in mathematics, by category theory. A vision moderately congruent with the multiformity of the world should integrate at least three orders of approximations: a [64] diagrammatic level – schematic and reticular [reticular] – on which the skeletons of the multple correlations between phenomena are sketched out, a modal level – gradual and mixed – on which the relational skeletons acquire the various ‘hues’ [tinturas] of time, place and interpretation, and a frontier [fronterizo] level – continuous – on which webs and mixes and progressively combined. In this ‘architecture’ of vision, these levels never find themselves fixed or completely determined; in Lautman’s sense, they are articulate themselves in form of [se articulan] diverse contextual saturations (since something mixted and saturated on a given level may appear to be skeletal and in the process of saturation in another context of greater complexity) and knowledge’s dynamic frontier reflects the undulating [ondulante] frontier of the world. An adequate integration of diagrams, correlations, modalities, contexts and frontiers between the world and its various interpretants [intérpretes] is the primordial object of pragmatics. Far from being reduced to the study of utilitarian correlations in practical contexts of action-reaction – a degeneration of the term ‘pragmatics’ which corresponds to disparaging way in which it gets used these days [su despectivo uso actual] – pragmatics aims to reintegrate the differential fibres of the world, explicitly inserting the broad relational and modal spectrum of fibres into the entire investigation [la integral buscada]. The technical attention to contextualizations, modulations and frontiers affords to pragmatics – in the sense of Peirce, its founder – a fine and peculiar methodological timbre. Just as vision, like music, benefits from an integral modulation in which tones and tonalities are interlaced in order to realize its texture, pragmatics benefits from an attentive examination of contaminations and osmoses between categories and frontiers of knowledge in order to articulate diversity coherently. Various natural obstructions are encountered on the way to any architectonic system of vision seeking [que pretenda] the many [multiple] and the one without losing the multivalent richness of the differential. One of such obvious obstruction consists in the impossibility that such a system should be stable and definitive, since no given angulation [angulación] can capture everything that remains [todas

las demás]. In effect, from a logical point of view, each time a system observes itself – a necessary perspective if it seeks to capture the ‘whole’ that includes it – sets loose [se desata] a self-referential dynamic that endlessly hierarchizes the universe. As such, a pragmatic architectonic of vision can only be asymptotic, in a very specific sense in which evolution, approximation and convergence are interlaced, but without requiring a possibly inexistent limit. That an ‘internal’ accumulation of neighbourhoods can indicate an orientation without having to invoke an ‘external’ entity representing a supposed ‘finality’ [final] – the power to orient ourselves within the relative without having to have recourse to the absolute – is a fact that bears enormous consequences, whose full creative and pedagogical force is just beginning to be appreciated in the contemporary world. The maxim of pragmatism – or ‘pragmaticism’ (‘a name sufficiently ugly to escape the plagiarists’ [COMPARE WITH ORIGINAL]) as Peirce would later name it in order to distinguish it from other behaviouralistic, [65] utilitarian or psychologistic interpretations – appears to have been formulated several times throughout the intellectual development of the multifaceted North American sage [sabio]. The statement usually cited is from 1878, but other more precise statements appear in 1903 and 1905:

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Consider what effects that might conceivably have practical bearings, we conceive the object of our conception to have. Then our conception of those effects is the whole of our conception of the object. 1 Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood. 2

The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. 3

What is emphasized in the statement of 1905 is that knowledge of symbols is obtained by following certain ‘general modes’, and traversing a spectrum of ‘different possible circumstances’. This modalization of the maxim (underscored in the awkward [torpe] repetiion of the ‘conceivable’ in 1878) introduces into the Peircean system the problematic of the ‘interlacings’ between the possible contexts of interpretation that a given symbol can receive [pueden tenerse para un símbolo dado]. In the statement of 1903 we observe, on the one hand, that every practical maxim should be able to be expressed in the form of a conditional whose necessary consequent should be adequately contrasted, and on the other hand, that any indicative theoretical judgment, in the actual, can only be made precise by way of a series of diverse practices associated with that judgment. Expanding these precepts to the general field of semiotics, to know a given sign (horizon [ámbito] of the actual) we should then traverse the multiple contexts of interpretation susceptible of interpreting that sign (horizon of the possible) and, within each context, study the practical imperative consequences associated with each one of those interpretations (horizon of the necessary). Within that general landscape, the incessant and concrete transit between the possible, the actual and the necessary turns out to be one of the specificities of mathematical thought, as we will repeatedly underscore through the remainder of this work. In that transit, the relations between possible contexts (situated in a global space) and the relation between the fragments of necessary contradistinction [contrastación] (situated in a local space) take on a primordial relevance, something that, of course, [66] finds itself in perfect tune [en plena sintonía] with the conceptual importance of the logic of relations that the same Peirce systematized. In this way, the pragmaticist maxim indicates that knowledge, seen as a logico-semiotic process, is pre-eminently contextual (versus absoluto), relational (versus substantial), modal (versus determined), synthetic (versus analytic). The maxim filters the world through three complex webs that let us differentiate the one into the many and, inversely, integrate the many into the one: the modal web already indicated, a representational web and a relational web. Indeed [en efecto], beyond opening up to the world of possibilities, the signs of the world should above all be able to be represented within the languages (linguistic or diagrammatic) manipulated [manejan] the communities of interpretants. And so the problems of representation (fidelity, distance, reflexivity, partiality, etc.) find themselves immediately bound to the differentiation of the one and the many: the reading of a selfsame fact [un mismo hecho], or a

1 Charles S. Peirce, “How to Make Our Ideas Clear” (1878), in C.S. Peirce, Collected Papers, Harvard: Harvard University Press, 1931-1958 (republication: Bristol: Thoemmes Press, 1998). Cited: CP 5.402. 2 Charles S. Peirce, “Harvard Lectures on Pragmatism” (1903). Cited: CP 5.18. 3 Charles S. Peirce, “Issues of Pragmaticism” (1905) [Z. HAS (1903), BUT THIS MAY BE A TYPO]. Cited: CP 5.438.

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selfsame concept, dispersed through many languages, through many ‘general modes’ of handling [manejo] information, and through many rules for the organization and stratification of information. One of the strengths of Peircean pragmatism, and in particular of the fully modalized pragmatist maxim, is how it lets us reintegrate anew the many and the one, thanks to the third web it puts into play [en

juego]: the relational web. In face, after decomposing a sign into subfragments in the various possible contexts of interpretation, the correlations between fragments give rise to new forms of knowledge, which were entombed [sepultas] in the initial perception of the sign. The pragmatic dimension emphasizes the coalition of some possible correlation, discovering analogies and transfusions [transvases] between structural strata that, prior to effecting that differentiation, had not been discovered. In this way, though the maxim registers the fundamental importance of local interpretations, it also insists upon the reconstruction of global approximations by means of adequate gluings [pegamientos] of the local. We will later see how the tools of the mathematical theory of categories endows these initial vague and general ideas with a great technical precision. The pragmaticist maxim will then emerge as a sort of abstract differential and integral calculus, which we will be able to apply to the general theory of representation, that is to say, to logic and semiotic in these sciences’ most generic sense, the sense foreseen by Peirce. Below, we present a diagrammatic schematization of the pragmaticist maxim, in which we synthetically condense the comments expounded so far. This diagram (figure 4) will be indispensible to naturally [en forma natural] capturing the structuration of the maxim from the perspective offered to us by the mathematical theory of categories. Reading from left to right, the diagram displays an actual sign, which is multiply represented (that is, underdetermined) in possible contexts of interpretation, and whose necessary actions-reactions in each context give rise partial comprehensions of that sign. The terms ‘pragmatic differentials’ and ‘modulations’ evoke the initial process of differentiation; the latter term reminds us of how a single [mismo] motif can be extensively altered throughout the development of a musical composition. The process of reintegration proper to Peircean pragmatics is evoked by the terms ‘pragmatic integral’, ‘correlations, gluings, transferences’, which remind us of the desire to return to unity what has been fragmented. The pragmatic dimension seeaks the coalition of all possible contexts and the integration of all the differential modulations obtaining in each context, a synthetic effort which has constituted the fundamental task [labor primordial] of model theory and category theory in contemporary logic.

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[KEY: signo = sign, esquema = scheme, pegamientos = gluings. Everything else is what it looks like.] The pragmaticist maxim thus serves as a sophisticated ‘sheaf of filters’ for the distillation of reality. The crucial role of the ‘sheaf’ secures an amplified multiplicity of perspectives and, for that matter, one which does not filter information in just one way [68], thereby establishing, at the outset, a certain plausibility [[for the claim]] that our knowledge will be sufficiently rich and multivalent. The role of the Peircean pragmaticist maxim in the philosophy of contemporary mathematics may come to be extraordinarily useful. The first upshot [punto resultado] of the maxim consists in not privileging any point of view or any fragment of language over the other and in thereby opening the possibility of considering, in Susan Haack’s works, “alternative truths without having to reduce them all to one single privileged vocabulary or one privileged theory” [FIND ORIGINAL QUOTE]. In a pendular fashion [pendularmente], the second crucial strength of the maxim consists in the power to compare very diverse levels within that multiplicity of perspectives, lenguages and contexts of truth which it has succeeded in opening; indeed, not wishing to restrictively [taxativamente] assume any privileged ‘foundation’ does not oblige us adopt an extreme relativism, without hierarchies of value. So, for example, to not reduce things, in questions of foundations, to discussions referring to the supposedly ‘absolute’ base ZF or the dominant force of first-order classical logic, but to open things to discussions in other deductive fragments of ZF (‘reverse mathematics’), semantic fragments (abstract model theory) or structural fragments (category theory), is a strategy that amplifies the contexts of contradistinction [contrastación] – and, therefore, the mathematical richness at stake – without for all that spilling [abocarse] into epistemological disorder. Our contention, in reality, is just the contrary: it is thanks to being able to escape a few ‘ultimate’ foundations, and thanks to situating ourselves in a relative fabric [tejido] of

contradistinctions, obstructions, residues and glueings, that a genuine epistemological order for mathematics evolutionally and asymptotically arises.

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The Peircean pragmaticist maxim can be seen as a sophisticated way of weaving between analysis/differentiation and synthesis/integration. The contemporary world requires new conceptual instruments of ‘collation’ [encuadernación] or ‘gluing’ – which respond with new arguments to the ‘primordial’ differentiation/integration dialectic4 -- and, to a large extent, the Peircean pragmaticist maxim provides one of those gluing-instruments. To our understanding, if there exists a mathematical concept that can serve as a threshold between modern and contemporary mathematics, it is that of a mathematical sheaf (faisceau, haz), an indispensible mathematical concept for reintegrating adequate local compatibilities into a global gluing.5 Correlatively, within the scope of [69] epistemology, we believe that the Peircean pragmaticist maxim can serve as a remarkable methodological sheaf to compare and, afterwards, attempt to interlace the diverse [para comparar lo diverso y, luego, intentar entrelazarlo]. The Peircean web of webs [red de redes] effectively opens onto full modal horizons [ámbitos plenos de lo

modal], and systematically attends to contrasting given facts [hechos] (within the phenomenological world) and necessary behaviours (within well-defined contextual systems), in order to then reintegratethem within an extended spectrum of possible signs. Differentiation and reintegration reach a high degree of methodological precision in the mathematical theory of categories. As a counterpart to the set-theoretical analytic championed by the heirs of Cantor,6 category theory no longer dissects objects from within and analyses them in terms of their elements [deja de desmenuzar los objectos por dentro y analizarlos mediante sus elementos], but goes on [pasa] to elaborate synthetic approaches by which objects are studied through their external behaviour, in correlation with their ambient milieu. Conceived as ‘black boxes’, categorial objects cease to be [dejan

de ser] treated analytically, and by way of the significant accumulative effort of synthetic characterizations one observes the objects’ movement through variable contexts. Category theory detects, for certain classes of structures (logical, algebraic, ordered, topological, differentiable, etc.), some general synthetic invariants within those classes, and defines them by way of certain ‘universal properties’. Those properties may hold [valer], in the first instance, for given universes of classes of structures ( = ‘concrete’ categories), but may at least be extended to more general fields in which the minimal generic properties of those classes are axiomatized ( = ‘abstract’ categories). Multiple weaves [vaivenes] of information ( = ‘functors’) are then established between abstract and concrete categories. An incessant process of differentiation diversifies the universal constructions that are given [se dan] in abstract categories and ‘incarnates’ them in contrasting forms in multiple concrete categories. Inversely – in a pendular fashion, we could say [diríamos pendularmente] – an incessant process of integration seeks out common constructs [constructos = constructors?] and roots, at the level of abstract categories, for a great variety of special constructions that continually present themselves [van dándose] in concrete categories. In this way, a quadruple synthetic strategy takes form [va adquiriendo forma] within category theory. First of all, internally, within each concrete category, one seeks the characterization of certain

4 A good presentation of the analysis/synthesis polarity and its subsumption into the ‘greater’ polarity, differentiation/integration, can be found in Gerald Holton’s article, ‘Analisi/sintesi’, for the Enciclopedia Einaudi, Torino: Einaudi, 1977, vol. I, pp. 491-522. 5 We will study, in detail, the multiple facets of sheaves in topology, algebraic geometry and logic in the second part of this work. The mobile plasticity of sheaf’s not only lets us pass from the local to the global, but, in a natural fashion, allows for multiple osmoses between very diverse subfields of mathematics. In a certain way, since their very genesis, sheaves have acquired an incisive reflexive richess that has rendered them extraordinarily malleable. 6 Cantor himself is better situated in terms of a sort of general organicism (with considerable and surprising hopes that his alephs would help us understand the world and the living), where analytic and synthetic considerations relate to one another [se entroncan].

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special constructions [ciertas construcciones especiales intentan ser caracterizadas] in terms of their environmental [medioambientales] properties in the given class. Next, externally, in the general field of abstract categories, one seeks certain universal constructions that can account for the characterizations obtained in the concrete categories. In the third [70] stage, in a remarkable weaving between concrete and abstract categories, one goes on to define adequate functors of differentiation and reintegration. Finally, the same functors go on to become the object of study from a synthetic point of view, and their osmoses and obstructions ( = ‘natural transformations’) are systematically studied. Category theory, as we will see in Chapters 4-7, has aquired considerable mathematical value in itself, but for the moment we are interested only in accentuating its methodological interest for a philosophy of mathematics open to incessant pendular processes of differentiation and reintegration. Indeed, if the philosophy of mathematics could make use of the synthetic lessons on differentiation and

reintegration codified in both the Peircean pragmaticist maxim and in the functorial processes of category theory, many of the fundamental problems of the philosophy of mathematics might acquire new glints and twists [visos y giros] which – we believe – could enrich philosophical dialogue. The objective of the third part of this essay will consist precisely in the discussion of those problems, in view of the contributions of contemporary mathematics, and in view of the synthetic grafting [entronque] [[splicing?]] of the Peircean pragmaticist maxim and the methodological lineaments of category theory. That said, for now, to pose the same problematics from the complementary perspectives of analysis and synthesis, we can indicate a few fundamental inversions (see figure 5) in the requirements that an analytic or synthetic perspective thrust upon us [llevan a forzar] [CHECK]. [TABLE FROM P. 71]

problematic

analytic vision (philosophy

of language + set-theoretical foundations)

synthetic vision

(pragmaticist maxim +

category-theoretic

contexts)

realist ontology: mathematical

objects exist in a real world

requires us to postulate the real existence of the universe

of sets, to which we are granted access by a reliable [fiel] form of intuition

requires us to postulate the existence of a covering of the real by means of progressive

hierarchies of structural contexts which asymptotically

approximate it

idealist ontology: mathematical

objects are linguistic subterfuges

requires us to postulate a dissociation between

mathematical constructs and their physical environments

requires us to postulate a dissociation between classes of

linguistic categories and classes of categories from

mathematical physics

realist epistemology: truth values

reflect objective forms of

knowledge

requires us to postulate the existence of a set-theoretic semantics as an adequate transposition of semantic

correlations in the real world

requires us to postulate the existence of categorial

semantic adjunctions and skeletal invariants [invarianzas

de esqueletos] through functorial weaves

idealist epistemology: truth values requires us to postulate a requires us to postulate the

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are subjective forms of control variability of modalization of sets, and assume the existence of stable transitions between

'compossible' worlds

impossibility of 'archetypical' initial categories capable of generically classifying the

truths of their derived categories

realist metaphysics: 'to ti en einai'

('the essential of essence') exists

mathematically

requires us to postulate the existence of a 'monstrous'

model and reflexion schemas that would accommodate

every universe of sets [donde

quepan todos los universos de

conjuntos]

requires us to postulate the existence of multiple classifier

toposes and ulterior inverse limits where the classifiers can

be 'glued together'

idealist metaphysics: 'to ti en einai'

does not exist mathematically

requires us to postulate the necessity of towers of set-

theoretical universes controllable only by means of

relative consistencies

requires us to postulate the necessity of functorial iterations ad infinitum,

irreducible to projections from a supposedly 'final'

classifier Figure 5. Complementary perspectives on the 'pure' posing of problematics in the philosophy of mathematics. As we will see and discuss in the third part of this word, some of the above requirements seem too strong, and go against the grain [en contravía] of various advances achieved in contemporary mathematics. For example, seen from a synthetic approach to mathematics – more appropriate than an analytic approach if we value mathematical practice – an idealist ontology that dissociates linguistic categories (a la Lambek) from categories of mathematical physics (a la Lawvere) seems unviable, because it places itself in immediate contradiction with advances in n-categories (a la Baez) that let us account for complex linguistic and physical torsions [complejos torsores lingüisticos y físicos] simultaneously. [CHECK] Another example, brought into focus once again from a synthetic perspective, seems to show that an idealist epistemology is equally unviable, because it would enter into conflict with the (already realized) possibility of constructing classifier toposes and initial allegories [CHECK] (a la Freyd). In this way, it is already easy to intuit how our double alternative strategy – to exploit [72] 'synthetic' methodological foci and to approach [acercarnos] contemporary mathematics – can bear considerable philosophical fruit. Later, we expect we will be able to show that steering oneself in this way towards more mathematics (and not necessarily 'more philosophy') represents a reasonable strategy for seeking to open attractive and unsuspected floodgates [compuertas] in philosophical dialogue. The Peircean pragmaticist maxim and the methodological lineaments of category theory help to provide a vision of mathematical practice that is fuller and more faithful [más plena y fiel] than an analytical vision can provide. The reasons are various and are related to the customary meaning [sentido natural] that the terms 'full' and 'faithful' acquire, if we extend their reach beyond [a partir de] what they have been given in the technical context of functors in category theory.i Observing that every construction that is realized in a given mathematical environment (topological, algrebraic, geometrical, differential,

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logical, etc.) is necessarily local within an adequate context,7 we will call minimal context of adequation a context in which the construction can be locally realized, but which does not in addition invoke redundant global axioms. We will then say that a vision of a determinate mathematical environment is full when it lets us associate, with every mathematical construction in the environment, a minimal context of adequation for that construction, and we will say that a vision of a mathematical environment is faithful when it lets us reconstruct every mathematical construction of the environment within a minimal context of adequation. The fullness of the vision ensures that the local richess of the theories will not be diluted in a global magma; the faithfulness of the vision ensures that that local richness really be sufficient for its full development. For example, the usual analytic vision of mathematics – based on ZF set theory and the underlying first-order classical logic – turns out to be neither full nor faithful in the above sense. By the very reason of the ZF axioms' broad global reach, the vision is not full since the contexts of minimal adequationa are obligatorially lost (a situation of information loss – that is to say, a loss of fullness – for which reverse mathematics proposes a palliative), nor is it faithful, given that that majority of the constructions are realized by invoking the full force of the axioms without control. By contrast, mathematical practice turns out to be much closer to a vision that genuinely and persistently attempts to detect [realmente atenta a ir detectando] transferences no less than obstructions between minimal contexts of adequation. The notions of obstruction and residue are fundamental here, because mathematical inventiveness, no less than its subsequent demonstrative regulation, is incessantly surveying [registrando] obstructions and [73] reconstructing entire maps of mathematics on the basis of certain residues stuck to those obstructions.8 Now, the obstructions and residues acquire meaning only locally, with respect to certain contexts of adequation, something that is often lost by the usual analytic vision and which, by contrast, a synthetic vision helps us realize. As we saw in the 'map' of the Peircean pragmaticist maxim (figure 4), we are also dealing with a situation that is particularly inclined to be detected by the maxim, attentive to local differentials and contextual singularities no less than to a subsequent modal reintegration of local fractures. A synthetic philosophy of contemporary mathematics will therefore have to attend to capturing at least the following minimal characteristics that naturally arise in a sort of generic 'differential and integral methodology' in which mathematics, philosophy and history are interlaced:

(1) contextual and relational delimitation of the field of contemporary matheamtics with respect to the fields of modern and classical mathematics;

(2) differentiation of the plural interlacings between mathematics and philosophy, and, afterwards, a reintegration of those distinction within partial, unitary perspectives;

7 The context of adequation can be very large: if the mathematical construction is, for example, the cumulative universe of sets, a context rendering the cumulative hierarchy local will have to reach some inaccessible cardinal or other. Nevertheless, the majority of 'real' mathematical constructions (in Hardy's sense, taken up again by Corfield) live mathematical contexts that are under far greater control, with respect to both cardinal and structural requirements. 8 Riemann's !(s) function provides an examplary case, here. From its very definition (by analytic extension, surrounding its singularities in the line Re(s) = 1) [CHECK], to its still mysterious applicatibility in number theory (clustered [bloqueos] around the demonstration that the zeros of the ! function lie in the line Re(s) = 1/2), the ! function provides a notable example of how mathematics extends its domains of invention and proof in virtue of the obstructions – as much definitional as structural – onto which certain mixted constructions of great draught are dashed (here, the ! function as a 'hinge' between number theory, complex analysis and algebraic geometry).

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(3) presentation of a full and faithful vision of mathematical practice, particularly sensistive to a pendular weaving between transferences and obstructions, between softenings [suavizaciones] and residues;

(4) diagramming the multivalences, ramifications and torsions between spectra of mathematical theorems and spectra of philosophical interpretations.

In what follows, we will enter into precise case studies inside contemporary mathematics, ones which will be able to throw the above points into sharp relief [con los que podremos ir resaltando los punto

anteriores], before returning, in the third part of the work, to other 'skeletal'9 considerations regarding the philosophy of mathematics.

[...]

Chapter 8

Fragments of a Transitory Ontology

In this third part of the work, we will elaborate a reflection (philosophical, methodological and cultural) on the case studies that we have presented in the second part. Accordingly, when we refer to

‘mathematics’ (and its derivative adjectives) what we will mean by this is ‘contemporary mathematics’, unless we specifically restrict ourselves [se acote] to the contrary [CHECK]. Now, right away we should note that this essay cannot cover all forms of mathematical activity [formas de hacer matemáticas] [[mathematical forms of activity??]], and, in particular, will not dwell on [no contempla] the practices peculiar to elementary mathematics. We therefore do not intend to produce anything like an all-encompassing philosophy of mathematics, but only to call attention to a very broad mathematical spectrum has rarely been

accounted for in philosophical discussions, and which should no longer be neglected [no debería seguirse

evitando]. In the final chapter we will try to provide an intrinsic characterization of the 1950–2000 diachrony (open on both extremes) with reference to ‘contemporary mathematics’, but, for the moment, we we will merely ground ourselves on the concrete cases of mathematical practice reviewed in the second part. We will try to provide an extensive number of cross references [referencias cruzadas] to those case studies; to that end, we will systematically use references between square parentheses such as [x], [x–y], or [x, y, ...] in the body of the text, which will direct the reader to the pages x, x–y, or x, y, ... of our essay. The case studies of the second part should made it clear that contemporary mathematics incessantly occupies itself with processes of transit in exact thought, with multiple networks of

contradistinction [contrastación], both internal and external, for those processes [CHECK SENTENCE]. From this it immediately follows that the questions concerning the content and place of mathematical objects – the ontological ‘what’ and ‘where’ – by which we aspire [pretendemos] to describe and situate those objects, cannot be given absolute answers, and cannot be fixed in advance. The relativity of the ‘what’ and the ‘where’ are indispensible to contemporary mathematics, in which everything tends to be transformation and flux. In this sense, the great paradigm of Grothendieck’s work, with its profound

9 In our strategy one can observe an approximate analogue [símil] to the practice in category theory, where, first, a category is delimited from other neighbouring categories, second, the category's various concrete constructions of are studied in detail, and third, its skeleton and its free constructions are finally characterized. The three parts of our work correspond – by a not overly-stretched analogy [no muy lejana analogía] – to the study of the 'category' of contemporary mathematics and its various adjunctions with respect to the various 'categories' of philosophical interpretations.

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conception of a relative mathematics [81] interspersed [entreverada] with changes of base [cambios de

base] of every sort within very general topoi [81-82], deserves to be fully understood [154] as an ‘Einsteinian turn’ in mathematics. As we have seen, this has to do [se trata de] a vision has ramifications [se ramifica] through all the mathematics of the age, and which is capable of giving rise to a genuine Einsteinian turn in the philosophy of mathematics as well. Now, the point of Einstein’s theory of relativity, once we assume the movement of the observers, amounts to finding adequate invariants (no longer Euclidean or Galilean) behind that movement. Likewise, the point of a relative mathematics à la Grothendieck, once we assume the transit of mathematical objects, amounts to finding adequate invariants (no longer elementary or classical) behind that transit. This is the case with many of the archeal situations in mathematics that we have come to review: motifs {83], pcf theory [114], intermediate allegories [138], Zilber’s extended alternative [alternativa extendida] [CHECK] [145], h-principle [149], etc. A sceptical relativism, which leads to disorientation and which permits an isotropy of values, in the style of certain postmodern subrelativisms or the infamous pensiero debole, thus finds itself very far from the Einsteinian or Grothendieckian projects, where, though there cannot be absolute foundations or fixed objects, not everything turns out to be comparable [equiparable] or equivalent, and where one can calculate correlative archeal structures – that is, invariants with to respect to a given context and a given series of correlations – which precisely allow us to detect and reintegrate differences. The first important point in the specification of ‘what’ mathematical objects are is that we really take the relativity and the transit in contemporary mathematics seriously. Within this horizon [ámbito], objects cease to be fixed, stable, classical, well-founded – in sum, they cease to be ‘ones’ – but rather approach the mobile, the unstable, the non-classical, the merely contextually-founded – in sum, the ‘multiple’. Multiplicity everywhere underlies contemporary transit, and the objects of mathematics basically become webs and processes. Determinate ‘entities’, firmly [sólidamente] situated in one absolute, hard and fast [firme y rocoso] universe, do not exist; instead we have complex signic webs interlaced amongst one another in various relative, plastic and fluid universes. These ‘complex signic webs’, in which mathematical objects are constituted, contemplate a multitude of levels, and no fixed level can

afford the richness of the object (web). This is obvious, for instance, with the mathematical ‘object’, group; we have seen how that object appears and captures disparate information (under the most diverse representation theorems) in the most distant realms [ámbitos] of mathematics: homology and cohomology groups [82-84, 102], Galois groups [86, 88, 128], group actions [92, 103], Abelian groups [93], homotopy groups [101], algebraic groups [105], the Grothendieck-Teichmüller group [128, 132], Lie groups [127], quantum groups [127], Zilber groups [144], hyperbolic groups [149], etc. It is not that we are confronted here, ontologically, with a [155] universal structure, submitted to supplementary properties on each supposed level of reading (logical, algebraic, topological, differential, etc.); what happens is rather that the diverse webs of mathematical information codified under the structure of group overlap one another (‘pre-synthesis’) and compose with one another [se componen] (‘synthesis’) in order to transmit information coherently. It is not that there exists ‘one’ stable [sólido] mathematical object that could be given life [cobrar vida] independently of the others, in a supposed primordial universe, but rather that there exist (plurally) webs that incessantly evolve as they connect with new universes of mathematical interpretation. This is particularly evident in the inequality webs [147] studied by Gromov, or in the equideductive theorem webs [140] that Simpson has shown us; the progressive advances [avances y adelantos] the webs go on to

13

configure the global landscape, and this modifies in turn the entities locally internalized in the global environment. Given that the objects of mathematics ‘are’ not stable sums but reintegrations of relative differentials, the question concerning their situation (‘where they live’) takes on an aspect that is almost orthogonal to the posing of that same question from an analytic perspective (grounded [fundamentada] in set theory). In effect, if mathematics finds itself in perpetual transit and evolution, the situation of an object cannot be anything but relative, with respect to a realm [ámbito] (‘geography’) and with respect to the moment of that area’s evolution (‘history’). This is no more than to countersign Cavaillès’ position – an understanding of mathematics as gesture [gestualidad] – which has echoed throughout the century, all the way to Gromov, who pointed out how we should come to ‘admit the influence of historical and sociological factors’ [[297*]] in its evolution through Hilbert’s tree [148]. [CHECK] The reading of mathematics as a historical science goes against the grain, of course, of the readings proposed by the analytic philosophy of mathematics. In those readings, fragments of edifice [de edificación] intemporally rise over absolute backgrounds, codified in the various analytic ‘isms’ [60], with which each commentator pretends to undermine contrary positions and propose his version as the most ‘adequate’, which is to say, as the most potential to resolve the problematics at stake. Curiously, however, the supposed local reconstructions of ‘mathematics’ [la ‘matematica’] – studied a hundred times over in the analytic texts – go clearly against what mathematical logic has been discovering in the period 1950-2000. In fact, as we have seen – in the style of Tarski (logics as fragments of algebras and topologies) and Lindström (logics as coordinated systems of classes of models) – the most eminent mathematical logicians of the last decades of the 20th century (Shelah, Zilber, Hrushovski) have emphasized the emergence of deep, underlying geometrical kernels [núcleos] [111, 143] behind logical manipulations. [156] Just as Jean Petitot – in his mixed studies [[298*]] on neurogeometry, Riemannian varieties and sheaf logic (Petitot declares himself a great admirer of Lautman’s ‘mixes’) – has begun to defend the idea of geometrical proto-objects underlying neuronal activity, and, moreover, the heuristic

precedence of a proto-geometry over a language, contemporary mathematical logic has likewise come to demonstrate a necessary precedence of a proto-geometry over a logic. What is at stake, then, is a situation that leads us to completely overturn [girar], again almost orthogonally, the usual approaches of analytic philosophy. Within contemporary mathematics, and following the ‘double orthogonality’ that we have pointed out, objects ‘are’ not, but are in the process of being [están siendo], and these occurrences are not situated in a logical warp [urdimbre], but in an initial spectrum of proto-geometries. The consequences for an ontology of mathematics are immense and radically innovative. On the one hand, from an internalist point of view, the ‘what’ involves webs and evolutive transits, while the ‘where’ involves proto-geometrical structures anterior to logic itself. On the other hand, from an externalist point of view, the ‘what’ takes us back to the unexpected presence of those proto-geometries in the physical world (the actions of the Grothendieck-Teichmüller group on the universal constants of physics [128, 132], the Atiyah index [119], the Lax-Phillips semigroup [124], the interlacing of Zilber’s pseudo-analyticity with the physical models of non-commutative geometry [145], etc.), while the ‘where’ conceals a profound dialectic of relative correlations between concrete phenomena and their theoretical representations. In fact, what comes to light in these readings is that the questions concerning an absolute ‘what’ or ‘where’ – whose answers would supposedly describe or situate mathematical objects once and for all (whether in a world of ‘ideas’ or in a ‘real’ physical world, for example) – are poorly posed

questions.

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The richness of mathematics in general (and of contemporary mathematics in particular) consists precisely in liberating and not restricting the experience [vivencia] and influence of its objects. In a certain sense, a base from which we may to better understand mathematics’ indispensable transitability could be furnished by the mixture of (i) structuralism, (ii) categories and (iii) modalization, that Hellman proposes [62], but in reality the situation is more complex, as comes to light in the multiple oscillations, hierarchizations and ramifications that we have been able to cast in relief in the second part of this work. In fact, if a categorical reading (ii) seems indispensable in contemporary mathematics – something we have emphasized [refrendado] with multiple case studies [157] beneath which, implicitly or explicitily, lies the Grothendieckian categorial and relativistic programme – and if, beyond a certain threshold of complexity, mathematical modalization (iii) seems equally indispensable (the multiform

transit between hypotheses, models, calculations and contradistinctions, carefully hidden in the classical formalization of set theory, and carefully ignored by so many of the ‘hard’ currents of the analytic philosophy of mathematics), contemporary mathematics nevertheless underscores the importance of processes over structures (i), since the latter emerge on a second level as the invariants of suitable [adecuados] processes. The relative ontologizatin of objects in contemporary mathematics thus forces us to dynamize Hellman’s base even further (which appears fixed in a quadrant of Shapiro’s square [14]), in order to be able to give a better account of the contemporary processes. At bottom, faced with contemporary mathematics, we can not therefore escape a certain ‘transitory ontology’ that, at first [en ciernes], would seem self-contradictory, terminologically speaking. Nevertheless, though the Greek ontotetês sends us, through Latin translations, to a supposedly intemporal ‘entity’ or an ‘essence’ that ‘ontology’ would study [[299*]], there is no reason (besides tradition) to believe that those entities or essences should be absolute and not asymptotic, governed by

partial gluings in a correlative bimodal evolution, of both the world and knowledge [conocimiento]. The philosophical supports of that ‘transitory ontology’ are found in Merleau-Ponty’s theory of shifting [desliz] [[300*]] [CHECK M-P TERMINOLOGY: GLISSEMENT] (in the general realm [ámbito] of knowledge [conocimiento] and the particular realm of visuality), and in the specific transitory ontology proposed by Badiou [[301*]] (in the field of mathematics). For Merleau-Ponty, the ‘highest point of reason’ [[302*]] consists in feeling the shifting of the soil, in detecting the movement of our beliefs and supposed claims to knowledge [saberes]: ‘each creation changes, alters, clarifies, deepens, confirms, exhalts, recreates or creates by anticipation all the others’. A complex and mobile fabric [tejido] of creation surges into his view, full of ‘detours, transgressions, slow encroachments and sudden drives’, and in the entire force of creation emerges in the contradictory strata of sediment. This is exactly what we have come to detect in contemporary mathematics, such as we have presented in it the second part of this work. For Badiou, mathematics is a ‘quasi-thinking of a quasi-being,’ distributed in ‘quasi-objects’ [[303*]] that reflect strata [158] of knowledge and world, and whose simple correlations (harmonic, aesthetic) grow in time. Those quasi-objects not only escape fixed and determinate identities: further still, they evolve and distribute themselves between warps [urdimbres] of ideality and reality. Through the dozens [decenas] of situations that we have taken stock of in this work’s second part, we have indeed been able to evince the distribution, the grafting [entronque] and the gluing of those quasi-objects not only between fragments internal to mathematics, but between theoretical mathematics (arena [ámbito] of the eidal) and the concrete physical world (arena [ámbito] of the quiddital). Badiou’s transitory ontology allows us to reintroduce, with new perspectives and with new force, an untrivialized Plato into the landscape of mathematical philosophy. In the style of Lautman’s dynamic

Platonism – attentive to the composition, hierarchization and evolution of the ‘mixes’ in the Philebus

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[[304*]] – Badiou likewise insists on a Platonism essentially open to a ‘cobelonging of the known and the knowing mind’, through which one derives an ‘ontological commensurability’ [[305*]] that incorporates movement and transit. We thus immerse ourselves in a Platonism that is not static, not fixed to a supposed transcendent world of Ideas, and very far from the image with which the doctrine is usually summed up.[[306*]] Badiou’s dynamic Platonic reading opens entirely different philosophical perspectives – particularly, an understanding of mathematics as evolutive thought [[307*]] and a study of the problems of saturation, maximization and invariance in thought’s movements – which falls in the asymptotic order of ‘shifting’ following Merleau-Ponty, and which covers some of the deeply mathematical problematics that we have been able to underscore in the case studies in this essay’s second part. The process of being [estar siendo], commensurably between the webs of cognition and the webs of phenomena, is a fundamental characteristic of a ‘transitory ontology’ required by contemporary mathematics. At issue is an apparently inoccuous change of grammatical category in ontological questions [plantamientos], which nevertheless afford them a whole new dimensionality and enrich the problematics at stake: the ‘what’ and the ‘where’ – which were initially taken up in an absolute and actual present, which led us to ask poorly posed questions – come to be understood in a modal and relative present perfect. The major consequence of this [159] opening onto transit consists in immediately shattering the dualisms and pigeonholes of Shapiro’s table [14]. As we have seen, the objects – that is, in reality, the processes and quasi-objects – of mathematics ceaselessly transit between eidal, quiddital and archeal webs, whether in the interior mathematical world, or in their perpetual contradistinction with the physical exterior. What in one context turns out to be eidal (Grothendieck-Teichmüller (GT), tied to the Euler numbers [128], for example), in another context appears as quiddital (GT acting on the universal formulae of physics [132]), and in yet another context even appears as archeal (GT in dessins d’enfant [128]). The classical dissociations and exclusions (of the either... or... variety, in the style of Benacerraf’s dillema [15]) have caused excessive and unnecessary damage in the philosophy of mathematics, and it is time to begin to overcome them. To that end, we can already count on at least three great tendencies in contemporary mathematics – where binary positionings are cast aside and where perpectives of continuity between diverse webs are opened up – that should start being seriously considered in philosophical reflection: (i) the understanding of the ‘positive’ (classical, commutative, linear, elementary, structured, etc.) as a limit of ‘negative’ mediations (intuitionism, non-commutativity, non-linearity, non-elementarity, quantization, etc.); (ii) the theory of sheaves, with its continuous handling [manejo] of coherence and gluing between the local and the global; (iii) the mathematical theory of categories, with its handling of differentiation and reintegration between the particular and the universal. In what follows, we will clarify [desbrozaremos], from a conceptual and philosophical perspective, these three deep technical tendencies, and we will explain how they allow us to reinforce a transitory ontology of mathematics. Amongst the specific characteristics of mathematics from 1950 to 2000, we have indicated the act [hecho] of working with the fluxion and deformation of the usual boundaries of structures [31]. This is obvious, for example, in Grothendieck’s K-theory [78-79, 119-120], where one studies the perturbations of morphisms over classical fibres; in Shelah’s abstract elementary classes [111-113], where one studies limites of algebraic invariants beyond first-order classical logic; in the Lax-Phillips semigroup [124], where non-Euclidean geometry lets the scattering of waves be understood; in Connes’ non-commutative geometry [125-127], where the dispersion and fluxion of quantum mechanics and thermodynamics are unveiled; in Kontsevich’s quantizations [131-132], where classical structures obtain as limits of quantized ones; in Freyd’s allegories [137-138], where topoi are seen as

16

limits of intermediate categories; in Zilber’s extended alternative [145], where counterexamples to the trichotomy arise as deformations of non-commutative groups; or in Gromov’s theory of large-scale groups [149], where one studies the non-linear asymptotic behaviour of finitely generated groups. These are all examples of high mathematics, with great conceptual and concrete content (and so far from being able [160] to be reduced to mere ‘language games’), in which a new understanding of mathematical (quasi-)objects is demonstrated, in virtue of fluxions, deformations and limits, passing through intermediate non-positive strata (non-classical, non-commutative, non-linear, non-elementary, quantized). A result due to Caicedo [[308*]] demonstrates that classical logic in a ‘generic’ fibre of a sheaf of first-order structures is no more than an adequate limit of intuitionistic logic in the ‘real’ fibres of the sheaf. This remarkable situation shows that the constructions of the classical and the positive as ‘limit idealizations’ exhibited in the mathematical examples above are likewise reflected, in exactly the same way, in the arena [ámbito] of logic. What emerges, once again, is the continuity of mathematical knowledge [saber], where watertight compartments are worthless [donde no valen los compartimientos estancos]. The richness of Caicedo’s sheaf logic is precisely due to its elaboration in an intermediate zone [franja] between Kripke models [[309*]] and Grothendieck topoi [81-82], benefiting from the many concrete examples of the former and the abstract general concepts of the latter. In a full intersection [cruce] of

algebraic, geometrical, topological and logical techniques, Caicedo constructs an instrumentarium that incites transit and transference. The result is his ‘fundamental’ theoremof model theory, in which central statements in logic such as the Loz theorem for ultraproducts, and completeness theorem for first-order logic, forcing constructions in sets, theorems of type omissions in fragments of infinitary logic, can all be seen, uniformally, as constructions of generic structions in suitable sheaves. All of this implies a striking consequence from a philosophical point of view. In effect, through fluxions and transit we observe that the classical perspectives are no more than idealities, which can be

reconstructed as limits of non-classical perspectives that are much more real. A particular case of this situation is a new synthetic understanding of the point-neighbourhood dialectic, where – contrary to the analytic set-theoretic perspective according to which neighbourhoods are constructed on the basis of points – the classical ideal ‘points’, never seen in nature, are constructed as limits of real non-classical [161] neighbourhoods, which by contrast are linked to visible physical deformations. From this perspective, the ontology of the (quasi-)objects at stake again undergoes a radical turn [giro]: an ‘analytic’ ontology, linked to the study of set-theoretic classes of points, cannot be anything but an idealized fiction, which forgets an underlying ‘synthetic’ / ‘transitory’ ontology, which is much more real, linked to the study of continuous neighbourhoods that can be physically contrasted. The transitory and continuous (quasi-)objects of non-classical mathematics are thus interlaced with transitory and continuous phenomena in nature, through webs of informative correlation that are gradual, non-binary and non-dichtomous. The principal paradigms of sheaf theory [[310*]] confirm this situation. The ancient philosophical problematic, ‘how do we go from the many to the one?’ (corresponding to a phenomenological transit) becomes the mathematical problematic, ‘how do we go from the local to the global?’ (technical transit), which subdivides in turn into the questions (a) ‘how do we differentially register the local?’ and (b) ‘how do we globally integrate those registers?’ The natural mathematical concepts of neighbourhood, covering, coherence and gluing emerge in order to analytically tackle question (a), while the natural mathematical concepts of restrictions, projections, preservations and sections emerge in order to synthetically tackle question (b). Presheaves (a term due to Grothendieck) cover the combinatorics of the discrete interlacings between neighbourhood/restriction and covering/projection, while sheaves

17

cover the continuous combinatorics tied to the couplings of coherence-preservation and gluing-section (figure 13). In this way, the general concept of sheaf is capable of integrating a profound web of correlations in which aspects both analytic and synthetic, both local and global, and both discrete and continuous are all incorporated. [162]

Figure 13: The concept of sheaf. General transitoriness between analysis/synthesis, local/global, discrete/continuous.

[KEY: prescisiones = prescissions, secciones = sections, proyecciones = projections, pegamiento = gluing, cubrimiento = covering, vecinidad = neighbourhood, haz = sheaf, prehaz = presheaf]

In ‘prescinding’ [[311*]] the fundamental notion of ‘gluing’ in the sheaf, the notions of coherence, covering and neighbourhood progressively and necessarily emerge. These last two notions are of the greatest importance in an ontological discussion. On the one hand, if, as we have seen, it turns out that the ‘real’ cannot be anchored in the absolute, and if the ‘real’ can therefore only be understood by way of asymptotic conditions, the strategies of covering fragments of the real takes on a central ontological importance. It is in this sense that we should recall Rota’s exhortations to escape the ‘comedy of existence’ of mathematical objects staged by analytic philosophy, and examine instead a ‘primacy of identity’ [[312*]] tied to a grafting [entronque] of modal coverings of the real that will let us classify the possible identities between ideas and physical objects, identities that evolve – and in the most interesting cases remain invariant – over large-scale periods (here we should remember Gromov). On the other hand, a turn towards a real logic of neighbourhoods in counterpoint to an ideal punctual logic – that is to say, a turn towards a sheaf logic in counterpoint to classical logic – likewise leads to a radical ontological turn. Many of the supposed analytic and classical exclusions are thereby undermined [se

quedan entonces sin piso] [CHECK] in synthetic environments, both physical and cognitive, where mediations are the law. In particular, in such synthetic/transitory/continuous fields [ámbitos], there is no reason why the disjunctive question between the ‘ideality’ or ‘reality’ of mathematical (quasi-)objects should be answered by means of an exclusion [13-16], as reflected in the watertight compartments of Shapiro’s table [14]. In concrete terms, sheaves, those (quasi-)objects indispensible to contemporary

mathematics, acquire all their richness thanks to their double status as ideal/real, analytic/synthetic, local/global, discrete/continuous; [163] the inclusion (and not the exclusion) [[313*]] between opposites – an incessant exercise of mediation – secures their technical, conceptual and philosophical force. Category theory axiomatizes regions of mathematical practice, according to the structural similarities of the objects in play and the modes of transmission of information between those objects,

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in harmony [sintonía] with Peircean pragmaticism [64-68], which is equally sensitive to the problems of

transference. In an inversion of set theory, where objects are internally analysed as a conglomerate of elements, category theory studies objects through their external, synthetic behaviour, in virtue of the object’s relations with its environment. A morphism is universal with respect to a given property if its behaviour with respect to other similar morphisms in the category possesses certain characteristics of unicity that distinguish it within the categorial environment. The basic notions of category theory associated with universality – the notions of free object and adjunction – answer to problematics tied to the search for relative archetypes and relative dialectics. Within the multiplicity, in the wide and variable spectrum of mathematics’ regions, category theory succeeds in finding certain patterns [patrones] of universality that allow it to overtake [adelantar] [ACCELARATE?] the processes of the splintering [desgajamiento] of the local and the surmounting of concrete particulars. In a category, for example, a free object can be projected into any object of the adequately large subclasses of the category: it is therefore something of a primordial sign, which is incarnated [encarna] in all of the correlated contexts of interpretation. Beyond relative localizations, certain relative universals thereby surface, giving an entirely new technical impulse to the classical notions of universality. Though we can no longer situate ourselves in a supposed absolute, nor believe in concepts that are universally stable in space and time, category theory has redimensioned the notions of universality, coupling them with a series of transferences relative to the universal-free-generic, where, affording itself transit, comes back to encounter remarkable invariants behind the latter. In this way, category theory explores the structure of certain generic entities (‘generalities’ [‘generals’]) by a route similar to the ‘scholastic realism’ of the later Peirce. In effect, categorial thought contemplates a dialectic between universal definitions in abstract categories (generic morphisms) and realizations of those universal definitions in concrete categories (classes of structured sets [conjuntos con

estructura]), and within the abstract categories we can behold [entender] universals that are real but not existent (that is to say, that are not incarnated [no encarnan] in concrete categories: think, for example, of an initial object, definable in abstract categories, but whose reality is not incarnate in the category of infinite sets [164], where initial objects do not exist). The transitory ontology of mathematical (quasi-)objects therefore opens onto an intermediate hierarchy of modes of universality and modes of existence, beyond restrictive binary demarcations. In particular, as Lawvere has pointed out, the objects of elementary topoi (where we find presheaves and sheaves) [111] should be seen as variable sets, with modes of belonging that fluctuate over time. The ontological transitoriness of those ‘entities’ could not be more obvious [patente]. In the spectrum of pure possibilities, the Peircean pragmaticist maxim [64-68] must be confronted with the idea of concepts that are universal, logically correct, but which cannot be adequately incarnated in restricted [acotados] contexts of existence. The thought of the mathematical theory of categories recaptures [recupera] this kind of situation with great precision. Following, for example, tendencies in universal algebra and abstract model theory, category theory has been able to define very general notions of relative universal semantics [semánticas relativas universales] as appropriate invariants of given classes of logics. Behind the multiplication of logical systems and varieties of truth, therefore, persist certain universal patterns. Indeed, behind the back and forth [ir y venir] of mathematical information, the control of those transferences of information (via functors, natural transformations, adjunctions, equivalences, etc.) constitutes on the the crucial reasons for categorial thought. A subtle technical calculation over adjunctions gives rise to various complex systems of gluing

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between mathematical objects and lets us better understand the what Thom called the ‘fundamental aporia of mathematics’. From a pragmaticist point of view, we can amplify our conception of logic and open ourselves to a lattice of partial flows of truth over a synechistic ground (‘syneches’: the glued, the continuous; ‘synechism’: the continuity operative in the field [ámbito] of nature, hypothesis of the Peircean architectonic). This takes place, in fact, in Yoneda’s lemma [139], where, in trying to capture a given reality (the category C, or, equivalently, its representable functors), the forced emergence of ideal objects (non-representable functors) necessarily

amplifies [amplía] the channels through which the functors at stake may flow [los cauces de flujo de los

functores en juego]. The ideal-real amplification [ampliatividad] constitutes one of the strong points of a transitory ontology, in consonance with the mixture of realism and idealism present in Peirce’s philosophy. If we return to the pragmaticist maxim [67] and its expression in figure 4, the diagram places before all else, on the left, a sign in an abstract category, and, on the right, the same sign partially incarnated in various concrete categories. The various ‘modulations’ and ‘pragmatic differentials’ let us pinpoint [acotar] a sign that is one, abstract and general, and convert it into the multiple, concrete and particular. It is a task that, in category theory, is achieved by means of the various functors in play, which, depending on the axiomatic richness of each of the [165] categorial environments on the right, incarnate the general concepts in mathematical objects of greater or lesser richness. This first process of specialization to the particular, of concretization of the general, of differentiation of the one, can therefore be understood as an abstract differential calculus, in the most natural possible sense: to study a sign, one first introduces its differential variations in adequate contexts of interpretation. But, from the point of view of the pragmaticist maxim, and from that of category theory, this is only the first step in a pendular dialectical process. In effect, once the variations of the sign/concept/object are known, the pragmaticist maxim then urges us to reintegrate those various pieces of information [informaciones partiales] in a whole that constitutes [conforma] the knowledge of the sign itself. Category theory likewise tends to show that behind the concrete knowledge [conocimiento] of certain mathematical objects, there exist, between these objects, strong functorial correlations (adjunctions, in particular), which are what actually and profoundly inform us about the concepts at stake. In either of these two approaches, we are urged to complete our forms of knowing, following the lineas of an abstract integral calculus, the pendular counterpart of differentiation, which lets us detect certain approximations between certain concrete particulars that may seem distant, but which respond to natural proximities [cercanías] over an archeal ground that is not at first perceptible. The differential/integral back and forth, present in both the pragmaticist maxim and the theory of categories, is not only situated on the epistemological level that we have just mentioned in the last paragraph, but continuously extends to the ‘what’ and the ‘where’ of the (quasi-)objects at stake. The vertical interlacings to the right of figure 4 – denoted by ‘correlations, gluings, transferences’, and situated under the general sign of the ‘pragmatic integral’ ( ! ) – codify some of the most original contributions, of both a broad modal pragmatism and the theory of categories, which is something that we have already corroborated with the theory of sheaves. Both (mathematical) (quasi-)objects and (Peircean) signs live as vibrant and evolving webs in those environments of differentiation and integration. Sheaves, categories and pragmaticism therefore seem to answer to a complex regime [ordenamiento] of the prefix TRANS, on both the ontological and the epistemological level. We will now make our way into that second dimension.

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Chapter 9

Comparative Epistemology and Sheafification We have described the (quasi-)objects of contemporary mathematics as webs of structured information and as (in Leibniz’s sense) ‘compossible’ deformations within those webs, open to relative and

asymptotic composition in variable contexts. The dynamism of those webs and deformations straddles the eidal and the quddital, in an iterated weaving [vaivén] of conceptual and material approximations. These are bimodal objects in Petitot’s sense, situated in both physical and morphological-structural space, acting and reacting along spectra of formal and structural correlations with the many mobile environments where their partial transit takes place. In this way, the ontological ‘what’ and ‘where’ become blurred [se desdibujan], and their frontiers turn out to be much murkier [borrosas]. We thereby confront an ontological fluctuation that may come to provoke an understandable horror vacui in certain analytic approaches to the philosophy of mathematics, which seek to delimit and restrain [acotar] their perspectives in the clearest possible way, fleeing from blotches [borrones] and ambiguities [vaguedades], and situating those tidy delimitations over fragments of the absolute. Nevertheless, to a contrapuntal [contrapuntística] synthetic approach, open to relative transit, as signalled by contemporary mathematics, it becomes imperative to take into consideration the mobile frontiers between the conceptual and the material [[314*]]. A transitory ontology, such as we have described in the previous chapter, therefore gives rise to a fluctuating ontology, not easily pigeonholed into the watertight compartments of Shapiro’s square [14]: a natural variation of the ‘what’ and the ‘where’ gives rise to an associated variation of the ‘how’. Once the correlative spectrum of disparate epistemological perspectives [168] is opened – within what we could call a comparative epistemology – we will, in this chapter, to nevertheless reintegrate several of those perspectives under a sort of epistemological sheaf, sensitive to the inevitable complementary dialectic of variety and unity that contemporary mathematics requires. Beyond the differentiated, we have repeatedly seen, in the second part of this work, how it is that many central constructions in mathematics put pendular processes of differentiation and reintegration into play, behind which emerge invariant archeal structures. Recalling, for example, Grothendieck’s motifs behind the variations of cohomologies [83-84], Freyd’s classifier topoi behind the variations of relative categories [138], Simpson’s arithmetical kernels [núcleos] behind the theorematic variations of ‘ordinary’ mathematics [140-141], Zilber’s proto-geometric kernels behind the variations of strongly minimal theories [144-145], the Lax-Phillips semigroup behind the non-euclidean variations of the wave equations [124-125], the Langlands group behind the variations of the theory of representations [105], the Grothendieck-Teichmüller group behind arithmetical, combinatoric and cosmological variations [128, 132], Gromov’s h-principle behind the variations of partial differentials, etc. They are all examples in which, through the webs and deformations, the knowledge of mathematical processes advances

itself by means of series of iterations in correlative triadic fields [ámbitos]: differentiation-integration-invariance,

eidos-quidditas-arkhê, abduction-induction-deduction, possibility-actuality-necessity, locality-globality-mediation. Advanced mathematics invokes this incessant ‘triangulation’, the most striking reflection (as much technical as conceptual) of which perhaps appears in the notion of sheaf [161-162]. A fact of tremendous importance in contemporary mathematics is the necessity of situating oneself in a full thirdness

and not reducing it (not ‘degenerating it’, Peirce would say) to secondnesses or firstnesses. In the analytic modes of understanding, however, thirdness disappears because accomplished [realizado] (classical) analysis is basically dual, in the style of either... or... exclusions à la Benacerraf [15]. This means that the apparatus of analytic vision finds itself intrinsically limited in observing the functioning of advanced

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mathematics, if it considers them on its own, independently of any synthetic contrast. A pendular combination of the analytic and the synthetic, the differential and the integral, the ideal and the real, seems to be the epistemological road [camino] to follow. Observe in passing [en camino] that, on a ‘meta-epistemological’ level, what is needed is a precedence of the synthetic, not the analytic, in order to allow flows, oscillations and gluings between analytic and synthetic subfragments on lower levels. The usual ‘idealist’ or ‘realist’ epistemologies, as they have been separately presented in the first part [13-14, 71], are burdened [llevan] with exclusions [169] that do not suit the ideal-real decantation [al trasegar] of contemporary mathematics. Neither an analytic-ideal differentiation (which would impede, for example, the emergence of Grothendieck’s schemes [95]), nor a synthetic-real integration (which would impede, for example, the emergence of Gromov’s inequality webs [147]), separately serve to cover the world. When it is a question of approaching a collection of ever-finer [[315*]] recoverings of the ideal/real junction, a ‘comparative epistemology’ then finds itself to be the order of the day. Articulation, dialectical balance, correlativity, and pragmatics here turn out to be indispensible. If each epistemological perspective gives rise to an interpretive cut, to a peculiar environment of projectivity, a differential modulation of knowing [conocer], then the next step consists in articulating the coherent projectivities, balancing the polarities and gluing the modulations, just as Peirce’s pragmaticist maxim

postulates [65-68]. Entering into this process, we thus immerse ourselves in a sort of epistemological ‘sheafification’, where the local differential multiplicity is recomposed into an integral global unity. Before considering the theoretical obstructions and advances that arise in that enterprise of the

sheafification of a comparative epistemology, a detailed example may be of some use. Consider the processes of ‘smoothing’ and ‘globalization’ introduced by Gromov in Riemannian geometry [146]. On the one hand, Gromov studies infinitesimal deformations of inequalities, in well-defined local contexts; on the other, he studies the set of those deformations according to many possible metrics over the variety, and calculates global invariants tied to the consideration of all the metrics together. In this to and fro [ir y

venir], the knowledge of the objects is situated neither in the process of deforming and circumscribing [acotar] the inequalities locally nor in the process of constructing global invariants, but in an indispensable dialectic between the two approaches: without the web of analytic inequalities the synthetic invariants do not emerge [147], without the invariants the web of inequalities drifts about pointlessly [sin razón de ser] (whether this be from a technical or a conceptual point of view). The knowledge of Riemannian geometry thus incorporates analytic elements and synthetic configurations, and situates itself simultaneously in both ‘idealist’ (set of all metrics) and ‘realist’ (local physical deformations) perspectives. In the case in question, the sheaf of the different local metrics gives rise to continuous ections tied to the invariants at stake. But, going beyond this case, the same Gromov points out how, in ‘Hilbert’s tree’ [148], the contrast between ‘exponentials’, ‘clouds’ and ‘shafts’ [‘pozos’] [???] [170] governs the multidimensionality of mathematical knowledge, never reducible to just one of its dimensions. A ‘sheafification’ in comparative epistemology requires – as we have seen in the generic analysis of the notion of sheaf [161-162] – first prescinds [prescisar] notions of neighbourhood, covering and coherence, before passing to possible [eventual] gluings. A neighbourhood [vecinidad] between

epistemological perspectives requires us to postulate before all else a certain commensurability between them; here, we explicitly position ourselves against the supposed incommensurabilities between ‘paradigms’ (Kuhn), which, at least in mathematics, rarely occur [se dan] [[316*]]. In this context, an epistemological neighbourhood should bring together [acercar] the different perspectives whether in their

methods (for example, the ‘synthetic’ method applied to a realist or idealist stance: pigeonholes 23 and 24

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in figure 5 regarding the ‘pure’ posing of problematics in the philosophy of mathematics [71], where ij

denotes the jth pigeonhole in column i), with in their objectives (for example, the approach [acercamiento] to the real: pigeonholes 13 and 23 in figure 5), or in their mediations through the warps in play (for example, asymptotic mediation: pigeonholes 16 and 25 in figure 5). This third option is the most rich and renovative [renovadora] from conceptual a point of view, since it can help us eliminate dualistic exclusions by working with epistemic webs of approximation. Once a certain neighbourhood has been established (for example, pursuing the cases above, a neighbourhood of ‘synthetic methodology’, ‘realist finality’ or ‘asymptotic mediation’), we pose the question of how to cover that neighbourhood. Here, the neighbourhoods could be covered in a binary fashion (in the cases above, for example, by means of the pairwise-indicated pigeonholes), but the most interesting cases could correspond to non-binary coverings, as happens, for example, in the third situation: the ‘asymptotic mediation’ can be covered not only with a ‘diagonal’ binary (pigeonholes 16, 25 of figure 5), but with a ternary ‘corner’ (pigeonholes 16, 25, 26 of figure 5). The emergence of this ‘corner’, beyond a merely dual counterpoint, leads us to observe the importance of the limiting [limitrofe] and the

asymptotic, a crucial condition not only in the extrinsic knowledge of objects, but in the very intrinsic characteristics of the objects under investigation (condition (ix) concerning certain specificities of contemporary mathematics [31]). In some cases, the coverings could be coherent (that is to say, not locally contradictory) and could therefore give rise to adequate gluings (‘epistemological sections’ in the sheaf). These gluings would open new epistemological perspectives, capable of locally responding to certain problematics, in one way, and other [171] problematics in another way, so as to preserve coherences between the responses. This would thereby give impetus to a partial and asymptotic use of relative epistemological strategies (that change in correspondence [de acuerdo] with a change of neighbourhood), going beyond the restrictive

[taxativa] or absolute epistemologies like those proposed in Shaprio’s square [14]. Let us consider a concrete case of this situation, by confronting the problem of how we can know the structure of groups. Its archeal root [raigambre], as a phenomenological invariant in the world (and not only as a descriptor in a language), emerges thanks to the works of Grothendieck [83], Connes [128], Kontsevich [132], Gromov [149], Zilber [144], etc. Nevertheless, genetically, the notion of group appears as an eidal organization of certain symmetries (with Galois), which later gives rise (with Jordan) to enormous quiddital facilitations in calculation. A group is therefore, at once, an archetypical (quasi-)object, both ideal and real. The apparent confusion of these three perspectives is nevertheless eliminated by reading them as fragments of a section in a sheaf. On the one hand, if we identify the topological base of a sheaf with a collection of neighbourhoods in a temporal tree (‘history’), we see how a group – locally symmetrical in one neighbourhood (Galois: 1830), locally combinatorial in another (Jordan: 1870), locally structural in another (Noether: 1930), locally cohomological in yet another (Grothendieck: 1960), or locally cosmological in still yet another (Connes, Kontsevich: 2000) – can perfectly well live in multiple registers of knowledge, coherent with one another. On the other hand, if we take the base of the sheaf to be a collections of neighbourhoods in a conceptual map (‘geography’), we see how a group – locally linear in one neighbourhood (Galois, Jordan, Noether), locally differential in another (Lie, Borel, Connes), locally arithmetical in yet another (Dedekind, Artin, Langlands), or locally categorial in still yet another (Grothendieck, Serre, Freyd) – can then be projected onto all of those apparently divergent manifestations. ‘History’ and ‘geography’ can, in turn, be founded in a sort of squared sheaf that traces out all of the richness – both extrinsic and intrinsic – of the concept.

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In accordance with the multidimensionality of mathematical vision, with the depth of ‘Hilbert’s tree’, with relativity of the webs of categorial perspectives, and with the ‘sheafification’ of those webs, we can intuit the existence of a complex protogeometry underlying a comparative mathematical epistemology. Lying behind these considerations of ours, let us make explicit the hypothesis of a continuity [[317*]] between the [172] world of phenomena, the world of mathematical (quasi-)objects associated with those phenomena, and the world of the knowledge of those objects – which is to say, the hypothesis of a continuity between the phenomenal, the ontic and the epistemic. The mathematical constructions (and discoveries) that we have extensively reviewed in the second part of this work show that contemporary mathematics offers new support for the possible correctness of that continuity hypothesis. Here, advanced mathematics offers a finer illustration than elementary mathematics, since the latter’s low thresholds of complexity impede the emergence of the continuous/discrete dialectic, which ceaselessly traverses the contemporary space of mathematics. From an epistemological point of view, the distinct perspectives are therefore only breaks [quiebres] of continuity. In those breaks (just as in Peircean abduction) new bodies of knowledge [conocimientos] are generated [[318*]], and – in an epistemology open to transit – those bodies of knowledge, when they are coherent, can later be adequately reintegrated. The protogeometry underlying a comparative epistemology of mathematics presents various peculiar characteristics, tied to a combinatorics of coherent couplings between the webs in play (deep, multidimensional, iterative webs). On the one hand, in fact, the inverse bipolar tensions between ‘prescission’ and ‘deduction’ [162] show that, in many mathematical cases of a high threshold of complexity (among which we may situated the case of sheaves, in any of their geometrical, algebraic or logical expressions), there emerges a (‘horizonal’) hierarchy of partial couplings, whose resolutions strip

by strip [franja por franja] give rise to important forms of mathematical knowledge. This is the case, for example, with Gromov’s h-principle [149], where an ‘inverse bipolar tension’ between local and holonomic sections gives rise to an entire series of homotopic mediations, with calculatory fragments of enormous practical interest in the resolution of partial differential equations. On the other hand, the self-referential [autorreferentes] exponential nodes in ‘Hilbert’s tree’ [148] gives rise to another hierarchy (this time ‘vertical’) of partial couplings whose resolutions level by level likewise yield remarkable advances. This is the case, for example, with the fundamental notion of space, whose breadth continually expands as it transits [cuya amplitud se extiende al ir transitando] between sets, topological spaces, Grothendieck topoi [81] and elementary topoi [111], on each level generating distinct (though mutually coherent) conglomerates of new mathematical results. But there also, at least, exists a third hierarchy (now ‘diagonal’) of partial couplings between mathematical webs, directly influenced by the transgessive mind of Grothendieck. In fact, beyond the horizontal or vertical displacement, and beyond their simple

combination, the exist diagonal mediations with archeal traits [rasgos] [173] (‘non-degenerate’ Peircean thirdnesses) between decidedly far-flung realms [ámbitos] of mathematical knowledge (dessins d’enfants between combinatorics and complex analysis [128], the Grothendieck-Teichmüller group between arithmetic and cosmology [128], non-commutative groups between logic and physics [145], etc.). The protogeometry of those couplings incorporates, therefore, an entire complex connection [trabazón] of multidimensional elements, in concordance with the ‘intuitive’ multidimensionality of mathematical knowledge. The image of that mathematical knowledge is far removed from its logical foundation [[319*]], and a new pre-eminence of geometry enters into the map (item (vii) in the distinctive tendencies of contemporary mathematics [31]). Indeed, the beginning of the 21st century may perhaps be a good time [epoca] to begin to seriously consider a geometricization of epistemology – such

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as we are proposing [[320*]] – that would help us to overcome (or, at least, to complement) the ‘logicization of epistemology’ undertaken throughout the 20th century. The influence of analytic

philosophy, whose logical support comes down to [se reducía] first order classical logic, deserves to be countered by a synthetic philosophy, much closer to the emerging logic of sheaves. What is at issue here is an important paradigm change in mathematics and in logic (the fundamental results of Caicedo [160]), which should begin to be reflected in philosophy (ontology, epistemology, phenomenology) as well. Indeed, the changes in the logical base codify mathematical (quasi-)objects’ deformability, through relative transits, and let us overcome the objects’ classical ‘rigidity’, in a supposed absolute universe. Approaching Grassmann’s work in detail, Châtelet lucidly describes one of the fundamental problematics of mathematical knowledge:

Like the contrast continuous/discrete, the equal and the diverse are the result of a polarization; it is thus that algebraic forms ‘become through the equal’ and combinatory forms ‘become through the diverse’ can be distinguished. It is a matter of finding the articulation that makes it possible to pass continuously from the equal to

the diverse. [[321* BUT USING SHORE & ZAGHA’S TRANS., p.110]] In that search for a continous articulation, Châtelet’s mathematico-philosophico-metaphorical instruments include ‘dialectical balances’, ‘diagrammatic cuts’, ‘screwdrivers’ [‘destornilladores’]], ‘torsions’, and ‘articulating incisions of the successive and the lateral’, which is to say, an entire series of gestures attentive to movement, which ‘inaugurate dynasties of problems’ [[322* CORRELATE W/ TRANS.]] and correspond to a certain fluid electrodynamics [[323*]] of knowledge [saber]. Here, we once again find ourselves faced with the multidimensionality of mathematical (quasi-)objects, a multiplicity that can only be apprehended by way of ‘gestures’, that is, by way of articulations in motion [en

movimiento] that allow for partial overlaps between the ‘what’ and the ‘how’. Mathematical epistemology – at least if it wishes to be able to incorporate the objects of contemporary mathematics (and in many cases, those of modern mathematics) into its spectrum – should therefore be essenially mobile, lending itself to torsion [presta a la torsión], capable of reintegrating cuts and discontinuities, sensitive to fleeting [pasajeras] articulations, in sum, genuinely in tune [afín] to the objects it aspires to catch sight of [vislumbrar]. No fixed position, determined a priori, will therefore be sufficient for understanding the TRANS-form-ability of the mathematical world, with its elastic transits, its unstoppable [inatajable] weavings between diverse forms, its zigzagging pathways between modal horizons [ámbitos]. In the case studies in the second part of this work, we can concretely detect various properties of the ‘epistemic protogeometry’ that we have just been discussing. Both in the (quasi-)objects at stake, and in the forms by which they are known [las formas de conocerlos], we observe common protogeometrical features, amongst which we must emphasize: (a) multidimensional cuts and reintegrations, (b) triadic iterations, (c) deformations of objects and perspectives, (d) processes of passage to the limit by means of non-classical approximations, (e) asymptotic interlacings and couplings, (f) fragments of sheafification. The ‘what’ and the ‘how’ continuously overlap in contemporary mathematics, and the very protogeometry of those overlappings tends to rise up over a unitary common base. Grothendieck’s schemes [80] combine many of the above properties: ‘ontically’ the schemes are constructed by (a) introducing structural representations of rings, (c) viewing points as prime ideals and topologizing the spectrum, (f) locally defining fibres as structural spaces of rings and gluing them together in an adequate fashion; ‘epistemically’, schemes involve (a) a process of generalization and specification between [175] multidimensional objects, (b) an iterated threeness [triadicidad] [[324*]] between a base, an

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amplification [ampliación] and a mediation (projection) between them, (c) an understanding [comprensión] of objects by means of their relative positions, etc. Other similar registers can also be specified in Grothendieck’s other great constructions: topoi [81-82] or motifs [83-84]. In the same way, the Langlands correspondence [103-106] includes, ‘ontically’, (a) representations of groups, (b) iterated transits between the modular, the automorphic and the L-representable, (c) mixed differential and arithmetical structures, and (d) manipulations of non-commutative groups, while ‘epistemically’, the strategy of the programme is based on [se asienta en] (a) understanding arithmetical objects as ‘cuts’ (projections) of more complex geometrical objects, (b) systematically seeking geometrical mediations between (discrete) arithmetic and (continuous) complex analysis, (c)-(d) observing the objects’ analytical deformations, etc. Another similar situation occurs with the general theory of structure/non-structure according to Shelah [111-113]: ‘ontically’ there emerges (a) amalgams in high finite dimensions, (d) cardinal accumulations by means of non-exponential objects (pcf theory), and (e) monstrous models and asymptotic warps of submodels, while, ‘epistemically’, Shelah’s vision integrates (a) a celebration of mathematics’ multidimensionality, (b) an incessant moderation between the structured and the non-structured, (d)-(e) a deep understanding of the objects at high levels of the set-theoretic hierarchy as limits of ‘moderate’ and ‘wild’ fragments, etc. In this way, certain common protogeometric characteristics are naturally interlaced between ontic webs and the epistemic processes in play [[325*]]. Throughout the second part of this work, we have implicitly carried out other analyses, and could explicate other examples here (the works of Connes, Kontsevich, Zilber, and Gromov especially lend themselves to this), but perhaps the cases above provide sufficient illustration. [176] The principal limitation that an ‘analytic’ epistemology would seem to have – in contrast to the ‘synthetic’ comparative epistemology that we are hinting at here – revolves around [se centra en] the analytic difficulty in confronting certain inherently vague environments, certain penumbral zones [zonas de

penumbra], or, in Châtelet’s terms [[326*]], certain ‘outposts of the obscure’, certain elastic places of ‘spacial negativity’, certain ‘hinge-horizons’ [‘bisagras-horizontes’] where complex mixtures emerge which resist every sort of strict [taxativa] decomposition. We will study the problematic of the penumbra with greater care in the next chapter, where we take up the (sinuous, non-linear) dynamics of mathematical creativity, but in what remains of this chapter we will approach the dialectic of the obscure and the luminous in conjunction with the three ubiquitous polarities, analysis/synthesis, idealism/realism and intensionality/extensionality, which appear in any epistemological approach. Our objective consists in mediating (‘moderating’: Grothendieck, Shelah) between them, and proposing reasonable couplings from the ‘outposts’ of contemporary mathematics. The analysis/synthesis polarity has always been a source of equivocations and advances, in philosophy [[327*]] no less than mathematics [[328]]. Dialectically [forma dialéctica] (of the thesis-antithesis type, " -" [theta, theta-with-a-bar-over-it]), the polarity opposes [contrapone] knowledge by descomposition and knowledge by composition [el conocimiento por descomposición o por composición], relatively, since ‘there exist no absolute criteria, either for processes of analysis or for processes of synthesis’ [[329*]]. Indeed, nature always presents itself by way of mixes that the understanding decomposes and recomposes in iterated oscillations. Analytic differentiation (typical in the mathematical theory of sets: objects known by their ‘elements’) can lead to a better resolution of certain problematics, though subdivision in itself (Descartes) does not assure us of any result (Leibniz) [[330*]]. Synthetic integration (typical of the mathematical theory of categories: objects known by their relations with the environment), for its part, helps to recompose a unity that is closer to the mixes of nature, but – as with its counterpart – inevitable obstructions arise in its use. One should therefore try

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to bring about a [177] ‘continuous effort of balancing analysis and synthesis’ (Bohr) [[331*]], something that is achieved with particular success in Peircean pragmaticism, and, in particular, with its pragmaticist maxim [64-68]. An epistemological orientation can then surface thanks to an abstract differential/integral (analytic/synthetic) ‘calculus’, situated over a relative fabric [tejido] of contradistinctions, obstructions,

residuals and gluings [67]. The case studies that we have carried out in contemporary mathematics show that such an orientation is never absolute, but only asymptotic, that it depends on multiple relativizations – eidal ascents and quiddital descents, ‘first, reason climbs up [sube] with analysis, and then down [desciende] with synthesis’ [[332*]] – but relies on having sufficient points of reference (archeal

invariants) to calibrate the relative movements. Lying still further beyond the pendular analysis/synthesis equilibrium invoked by Bohr, contemporary mathematics hints [sugiere pistas] at how that equlibrium can be effectuated [efectuar]. We have seen how the notion of sheaf, in a very subtle manner, combines the analytic and the synthetic, the local and the global, the discrete and the continous, the differential and the integral [161-162]. A ‘sheafification’ of the analysis/synthesis polarity thus gives rise to a new web of epistemological perspectives, in accordance with [en forma acorde con] the directions of [direccionamientos de] contemporary mathematics. If, in fact, we look at the diagrammatization [diagramación] of the general concept of sheaf that appears in figure 13, we can see how, behind the double opposition [contraposición], analysis/synthesis (the vertical arrangement in the diagram) and local/global (horizontal arrangement), lies a very interesting diagonal mediation that is rarely made manifest. Taking the pivotal analytic concept of ‘covering’ and modulating it by means of a synthetic ‘section-preservation-projection-restriction’ hierarchy (figure 13), the natural notion of a transversal transform [transformada] of the covering emerges, in which the open and closed [[333*]] strata of the various representations (or ‘covers’) involved are superposed on one another. The transform – tied to a sort of protogeometry of position (‘pre-topos’) – combines the analytic capacity to cover through decompositions (whether with elements, neighbourhoods or asymptotic approximations), and the synthetic capacity to recombine the fragmented through variable contexts (whether with partial couplings or more stable structural gluings). We will call this type [178] of sweeping mediator between the analytic and the synthetic a Grothendieck

transform, a sweeping reticulation [un barrido reticular] particularly attentive to the relative mixes between the concepts in play. In its very definition, the Grothendieck transform incorporates peculiar modes of knowing. Through the transverse, it introduces a reticular warp over what contradistinctions, couplings and asymptotes stabilize. Through the covering, it introduces a fluid dynamics (Merleau-Ponty’s ‘shifting’ [[glissement]]), incarnated in the gerund itself: the ‘to be covering’ [‘estar cubriendo’] over the course of a situation (‘geography’) and a duration (‘history’). The Grothendieck transform thus changes our epistemological perspectives by the simple fact of smoothing them [suavizarlas] as a whole [en su conjunto] (Gromov [146, 148]): integrating them into an evolving fabric [tejido evolutivo], the punctual perspectives are desingularized [se desingularizan], in favour of a comparability that emphasizes the regularities, mediations and mixtures between them. This is made manifest in the mathematical practice that we have reviewed in the second part of this essay. The transversality and ‘marvelous mixture’ in Serre’s ways of working [101], the transversal contamination in Langlands’ programme [105], the omnipresent dialectic of adjunctions and evolution of objects in Lawvere [109], the covering moderation in Shelah’s pcf theory [114], Atiyah’s index theorem with its medley [mezcla] of eidal transversality (equilibrium between transits and obstructions) and quiddital coverage (from algebraic geometry to physics) [121], the harmonic analysis (precise technical form of covering transversality)

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applied to the non-Euclidean wave equation in Lax [124], the non-commutativity that runs [recorre] transversally over quantum mechanics in Connes [126], the quantizations whose transversal recoverings let us reconstruct classical structures in Kontsevich [132], the intermediate categories between regular categories and topoi, produced as transversal cuts (Map, Split, Cor) over free allegories in Freyd [138], the logical kernels [núcleos] covering second order arithmetic in Simpson [141], the proto-geometrical kernels covering strongly minimal theories in Zilber [144], and the transversality of the h-principle in Gromov [149] are all very subtle and concrete examples in which the mediations and smoothings of Grothendieck’s ‘transversal covering transform’ are lurking [están al acecho]. In all of these constructions, the epistemological vision of mathematics is one which is flexible enough to cover a

hierarchy of strata of ideality and reality, dynamically [en forma dinámica] interchanging, when the technical environment or the creative impulse requires it, the contexts of interpretation or adequation of the (quasi-)objects. These considerations help us to better understand how – in contemporary mathematics and, more broadly, in advanced mathematics [23] – the polarities of idealism and realism should not [179] be considered as separate (discrete exclusion, through an analytic approach), but in full interrelation (continuous conjuction, through a synthetic approach). Indeed, a good way [forma] of understanding the ideal/real dialectic in advanced mathematics can be achieved through the notion of an epistemological back-and-forth [[335*]][ENGLISH IN ORIGINAL]. The back-and-forth postulates not only a pendular weaving [vaivén] between strata of ideality and reality, but, above all, a coherent covering of various partial approximations. A direct example of this situation is found in the Lindström-style back-and-forth in abstract model theory: the semantic (stratum of reality, constituted by a collection of models with certain structural properties) is understood through a series of syntactic invariants (strata of ideality, constituted by languages with other partial reflection properties), and more specifically, the elementary (real) equivalence is reconstructed through (ideal) combinatory coherences in a collection of partial homomorphisms articulated in the back-and-forth. Other indirect examples appear in the case studies that we have carried out: the progressive back and forth [ir y venir] in the elucidated of the functorial properties of the Teichmüller space according to Grothendieck [96], the structural amalgams by strata according to Shelah [112], the approximations of hyperbolic groups and the polynomial growth of groups according to Gromov [149], etc. In these processes, mathematical (quasi-)objects’ modes of creation, modes of existence, and the modes by which they are known are interlaced and reflected in one another (general tranitoriness between

phenomenology, ontology and epistemology). The relative knowledge – partial, hierarchized, distributed – of those transits therefore becomes an indispensible task for mathematical epistemology. Beyond trying to define the ideal or the real in an absolute manner (a definition that, from our perspective, corresponds to a poorly posed problem), the crucial task of mathematical epistemology should instead consist in describing, pinpointing [acotar], hierarchizing, decomposing and recomposing the diverse forms of transit between the many strata of ideality and reality of mathematical (quasi-)objects. Through the same forces that impel the internal development of the mathematical world, we have explicated here, for example, [180] some contemporary forms of transit that possess a great expressive and cognitive [cognoscitivo] power: proto-geometrization, non-classical approximation [aproximación], sheafification, Grothendieck transformation, back-and-forth modulation. Another important epistemological opening is found in mediating the usual ‘extensional versus intensional’ dichotomy, which grosso modo corresponds to the ‘sets versus categories’ dichotomy. One of the credos of Cantorian set-theoretic mathematics – and, for that matter, of traditional analytic

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philosophy – has been the symmetry of Frege’s abstraction principle, introduced locally by Zermelo with his axiom of separation: given a set A and a formula #(x), there exists a subset B = {a $ A : #(a)}, and, therefore, an equivalence obtains (locally, within the restricted [acotado] universe A) between #(a) (intensionality) and a $ B (extensionality). But, beyond the credo and an indisputable technical convenience, there is no philosophical or mathematical reason to prevent an asymmetrization of Frege’s principle of abstraction [[336*]]. A non-Cantorian continuum, for example, seems to involve profound intensional characteristics that are impossible to realize through an extensional modelization [[337*]]. In the same manner, many of the intensional characterizations of mathematical objects and processes obtained in category theory offer new perspectives (‘relative universals’) that do not coincide [[338*]] with extensional set-theoretical descriptions. Though the extensional/analytic/set-theoretic influence has, until now, been preponderant in mathematics, its intensional/synthetic/categorial counterpart every day becomes more relevant, and a new ‘synthesis of the analysis/synthesis duality’ is the order of the [181] day. Of course, the self-reference just mentioned in quotation marks [corchetes] is not only not contradictory, but multiplicative in a hierarchy of knowledge. Throughout Chapters 8 and 9, we have tried to forge some mediating perspectives within that ‘synthesis of the analysis/synthesis duality’. Expanding our spectrum to more general cultural horizons [ámbitos], we will see in the two final chapters how the incessant mediations of contemporary mathematics are grafted [se entroncan], on the one hand, to the (local) ‘creative spirit’ of the mathematician, and, on the other, to the the complex and oscillating (global) ‘spirit of the age’ in which we find ourselves immersed.

i 'Full' and 'faithful' are, as Zalamea indicates, technical terms in category theory, and are used to describe certain kinds of functors. A full functor is surjective (i.e. covers its entire target, or codomain), a faithful functor is injective (i.e., preserves the distinctness of its arguments, or elements of its domain), and a full and faithful functor, being both, is bijective (a one-to-one correspondence), when restricted to a set of morphisms with a given source and target.