Burt C. Hopkins: The Origin of the Logic of SymbolicMathematics. Edmund Husserl and Jacob Klein

Bloomington and Indianapolis, Indiana University Press, 2011,559 pp., ISBN 978-0-253-35671-0

Mirja Hartimo

Published online: 10 March 2013

� Springer Science+Business Media Dordrecht 2013

Burt Hopkins’ magnum opus details the conceptual origination of the logic of

symbolic mathematics as investigated by Edmund Husserl and Jacob Klein, with a

special focus of the latter’s Die griechische Logistik und die Entstehung der Algebra

that appeared in two parts in 1934 and 1936, and in English translation in 1968 as

Greek Mathematical Thought and the Origin of Algebra. The main thesis of Klein’s

book, according to Hopkins, is that

the history of the transformation of what he calls the ‘‘conceptuality’’ of the

most basic concept employed by science, that of number, from a non-

conceptual and non-linguistic multitude of determinate things to a concept that

is identical with a symbolic language, is inseparable from the meaning of the

symbolic employment of letter signs – from the most elementary, such as ‘‘2’’,

to the most universal, say, ‘‘X’’ (p. 4).

According to Klein, ‘‘the transformation took place in the works of Vieta, Stevin,

Descartes, and Wallis in the sixteenth and seventeenth centuries. While in the

ancient writings number was construed as a ‘determinate number’ in the exact sense

of a definite amount of definite items’’ (pp. 5–6), the mode of being of the pure

mathesis universalis after the transformation is identical with the unambiguous

meaning of the symbols employed by its symbolic calculus (p. 6). To establish the

thesis Klein studied the original Greek sources, aiming at reconstructing the Greek

concept of number independently of the conceptuality of the modern symbolic

concept of number (pp. 5–6). Hopkins explains that Klein’s goal is essentially that

of Husserl’s, as expressed in The Crisis of European Sciences and Transcendental

Phenomenology and The Origin of Geometry as an Intentional-Historical Problem

in 1935 and 1936, to reactivate the original evidence that led to the establishment of

modern mathematical physics. According to Hopkins, Klein corrected Husserl’s

M. Hartimo (&)

Helsinki Collegium for Advanced Studies, University of Helsinki, P. O. Box 4, Helsinki, Finland

e-mail: mirja.hartimo@helsinki.fi

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Husserl Stud (2013) 29:239–249

DOI 10.1007/s10743-013-9128-7

account of the content of the historical reflection by adding to it the task of de-

sedimenting and reactivating the original evidence that led to Vieta’s establishment

of modern algebra.

Hopkins’ comparison between Klein and Husserl in the end turns into a rather

severe criticism of Husserl. To make the argument, the book contains several dense

sub-arguments and expositions, ranging from details of Klein’s reading of Plato,

Aristotle, and the neoplatonists to early modern accounts of mathesis universalis

and Husserl’s account of mathematics from the 1890s until the already mentioned

texts from the 1930s. In what follows I will focus on Hopkins’ main argument and

in particular on the comparison between Klein’s and Husserl’s views on

mathematics.

Hopkins’ work consists in four parts, each of which builds upon and deepens the

theses developed in the previous one. The first part provides a detailed explication

of Klein’s interpretation of Husserl’s project in the Crisis, the second elaborates on

Klein’s historical-mathematical study within the framework of the Crisis, the third

part takes up Husserl’s Philosophy of Arithmetic and how it relates to Klein’s views,

and the fourth part provides a comparative analysis of Husserl’s view of

mathematics and Klein’s account of the origination of the logic of symbolic

mathematics. The book ends with a brief consideration of Husserl’s Platonism in

comparison to Plato’s own Platonism. In what follows, I will briefly explain what I

understand to be the main theses of each of these parts.

1 Klein on Husserl’s Phenomenology and the History of Science

Alone among commentators on Husserl’s last texts, Klein appreciated Husserl’s

concern with history and held it to be consistent with Husserl’s lifelong

investigation of the phenomenological origins of the ideal meaning formations

that make both philosophy and science possible. Like Husserl, Klein argued for an

essential connection between historical inquiry and the quest for epistemological

foundations of scientific knowledge. A scholarly curiosity is that Klein argued for

this prior to the first publication of Husserl’s Crisis and The Origin of Geometry.

Hopkins argues that while Husserl is the first to have articulated the methodological

issues involved, Klein was the first to uncover the ‘‘historical apriori’’ of modern

physics in his Greek Mathematical thought and the Origin of Algebra. This is

because on Klein’s reading of Husserl the intentional history is essentially

connected to actual history. Klein goes so far as to hold that Husserl’s

‘‘transcendental inquiry into the origin of these significant formations’’ (e.g.,

mathematical and scientific objects) ‘‘discloses that the significance of these

formations paradoxically ‘appear[s] almost devoid of ‘‘significance,’’’ unless the

‘‘connection to the actual history of their origin is investigated’’ (p. 44). Similarly,

Klein holds that the revival of Greek mathematics in the sixteenth century has to be

understood prospectively rather than retrospectively. Thus Klein’s account of this

history includes the achievements from Plato and Aristotle to an elaboration of

Greek methods of analysis, Diophantine equations, and on to Vieta, Stevin,

Descartes, and Wallis in the sixteenth and seventeenth centuries, all of which makes

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Husserl’s account of it in the Crisis and The Origin of Geometry appear rather

provisional (p. 56).

2 Husserl and Klein on the Method and Task of Desedimentingthe Mathematization of Nature

The second part of the book continues to clarify Klein’s complicated relationship to

Husserl. It starts with a question: Why did Klein not refer to Husserl in his Greek

Mathematical Thought, even though the work is so thoroughly Husserlian?

Hopkins’ answer is that the fundamental critique of Husserl’s phenomenology that

Klein’s work calls for would have taken him too far afield to be included in Greek

Mathematical Thought. In particular, Klein’s book questions Husserl’s conviction

that the proper foundation of formal ontology originates in a modification of

evidence drawn from the experience of individual objects. Klein’s results show,

namely, that the origin of these concepts occurs at a higher conceptual level than the

ontology of individual objects. The critique of Husserl arising from Klein’s account

of the origin of symbolic mathematics is what Hopkins aims to undertake in the rest

of the book.

Hopkins starts from an examination of the shortcomings of Husserl’s Crisis, a

work that Klein’s results uncannily anticipate in many ways. Klein’s research

follows the zig-zag movement that Husserl identifies as characteristic of the method

of historical reflection (p. 77). Further,

while Klein’s articulation of the motive for the turn to history makes no

mention of a crisis in contemporary science, his description of the problem of

modern physics comes remarkably close to stating what Husserl will identify

as the basis of the crisis, namely: the unintelligibility to contemporary

epistemology of the meaning belonging to the formalized concepts that make

physics possible (p. 82).

However, from Klein’s perspective Husserl’s account of the desedimentation of the

origins of mathematical physics is fragmentary. Husserl recognizes the problem, but

he does not actually pursue its desedimentation as Klein does.

What is to be shown next is that Klein, while pursuing a Husserlian goal, reveals

that the required desedimentation ‘‘does not realize Husserl’s aim of tracing their

meaning-genesis to evidence based in something individual’’ (p. 94). This requires

Hopkins, following Klein, to consider the Neoplatonic literature and to investigate

the work of Diophantus from the point of view of its own presuppositions. To better

understand present-day physics, in other words, Klein holds it necessary to go back

to Greek origins and to reactivate the Neoplatonic, Platonic, Pythagorean, and

Aristotelian conceptual steps.

In comparing Klein’s results with Husserl’s Hopkins discusses the latter’s

Philosophy of Arithmetic (1891), where Husserl establishes the division between the

authentic and symbolic concepts of number. The PA notoriously fails to show how

these two concepts of number relate to each other. Curiously, Klein’s historical

reflection reaches the same conclusion, namely, that the logic of symbolic

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mathematics cannot be grounded in the concept of Anzahl. What Hopkins then takes

as ‘‘[t]he guiding aim of the present study is to establish the immense importance of

these different accounts of the ‘unity’ of the units that compose an Anzahl, that is, a

non-symbolic number, for Husserl’s and Klein’s respective accounts of the

origination of the logic of symbolic mathematics’’ (p. 101).

3 Non-Symbolic and Symbolic Numbers in Husserl and Klein

The third part continues the argument by providing a careful and detailed account of

Husserl’s PA. The book is notoriously problematic and contradictory, even on

Husserl’s own admission. Hopkins interprets Husserl’s problems in the PA as

arising from two distinct and incompatible guiding theses. The first, the thesis with

which Husserl starts the book, is that the basic concepts and calculational operations

of universal arithmetic have their foundation in the concept of cardinal number. But,

second, he then shifts to claim that both the numbers and algorithms of universal

arithmetic have their foundation in a system of signs that is not conceptual but rather

formal-logical, in the sense of a symbolic technique. Consequently the PA witnesses

‘‘a linear development from the view of the foundational status of the concept of

cardinal number for universal arithmetic to that of the conceptual independence of

the sign system that defines arithmetic’’ (p. 145). First the symbolic numbers are

seen as surrogates for authentic cardinal numbers and as logically equivalent to

them. But then

both the calculational technique operative in arithmetic and its symbolically

numerical substratum function as signitive surrogates for the inauthentic

conceptual system of (systematic) number formations and the conceptual

operations on these formations. As signitive surrogates for systematically

determined (inauthentic and therefore symbolic) calculational operations upon

them, signitively symbolic numbers manifest a complete conceptual indepen-

dence from the authentic and inauthentic number concepts together with the

concepts involved in the calculation with each (p. 146).

Husserl is quite clear, then, that the contents of the authentic cardinal number

concepts have become superfluous with respect to the foundation of the signitively

symbolic number sequence and the technique of calculation with rule-governed

algorithms (i.e. signitive calculation) (p. 147). But he is not so clear in his final

reflections, which prompts Hopkins to conclude that ‘‘the precise sense of the

numbers that are ‘indirectly’ determined by universal arithmetic remains unclar-

ified’’ (p. 148).

Hopkins then compares Husserl’s notion of authentic numbers to the ancient

notion of number, and the symbolic notion of number to the more modern symbolic

account of it made possible by the invention of algebra. This enables him to hold

that ‘‘Klein’s desedimentation of the origin of algebra provides an account of its

historical context that explains precisely why Husserl’s attempt to establish the

logical origin of the technique of calculating with symbols […] on the foundation of

authentic numbers and their operational relations not only failed but had to fail’’

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(p. 150). Klein’s desedimentation shows that algebra was explicitly understood by

its inventors to be an ‘‘art’’ not a theory. This obscures the theoretical

presuppositions that make algebraic calculation possible and mirrors Husserl’s

problems in the PA. Klein shows that because modern method develops as a praxis,

the innovators of symbolic calculation were unaware of such conceptual presup-

positions. The desedimentation of this incommensurability provides the historical

background for Husserl’s failed attempt to establish systematically the origin of the

symbolic concept of number:

At the theoretical level, Husserl’s failure signifies his inability to establish the

foundation for the logic of the technique proper to signitively symbolic

number calculation, which means that the demonstration of our thesis will

perforce establish the historical continuity of Husserl’s failure to ground such

a logic with the unrealized Platonic demand for a theoretical logistic (p. 151).

Explaining this thesis leads Hopkins to engage in extensive and detailed

elaborations of Klein’s view of the relationship between arithmetic (pure science of

definite amounts) and what he calls ‘‘logistic’’ (the art of calculation). These

elaborations deal with Neoplatonic mathematics, Plato (in various dialogues), and

Aristotle’s critique of Plato, continuing with Klein’s interpretation of Diophantus’s

Arithmetic and Vieta’s reinterpretation of the Diophantine procedure, right up to the

concept of number in Stevin, Descartes, and Wallis. Hopkins’ close reading of

Klein’s writings provides a rich picture of how the concept of number changed. For

example, the change involved replacing the ancient ‘‘supreme’’ discipline—

dialectic in Plato, first philosophy in Aristotle—with mathesis universalis under-

stood as algebra. At the same time, the kind of generality belonging to mathematics

changes. In modern science there is no difference between generality in

mathematics and in philosophy, whereas in ancient philosophy the difference is

significant (p. 262). Moreover, the object of arithmetic and logistic changes from

‘‘number,’’ which is directly related to the things or units, to a symbolically

conceived ratio. The being of symbolic number is no longer a problem because its

being is immediately graspable in the notation. Klein’s claim is related to his view

that the distinction between ‘‘saying’’ and ‘‘thinking’’ is sedimented in the practice

of modern science, and thus no longer encountered. Consequently, the sign and

concept are no longer distinct (p. 257). Signs are perceived as determinate objects,

being unambiguously manifest as readily perceivable letters (p. 288).

While the process was initiated by Diophantus, ‘‘it is ‘Vieta who, by means of the

introduction of a general mathematical symbolism’, transforms Diophantine logistic

into a symbolic logistic and thus should be credited with inventing modern algebra’’

(p. 253). For Klein, the fundamental transformation of the number concept is

finalized by Simon Stevin (1548–1620), who embraced the Arabic positional system

of ciphers. After him, Descartes understood algebra as mathesis universalis. By

‘‘understanding the meaning of the mathesis, the discipline or science of the

ancients, in terms of a general science so articulated and named mathesis

universalis, Descartes accomplishes two portentous things in, as it were, a single

stroke’’ (p. 265). First, ancient first philosophy is collapsed into the modern,

algebraic understanding of mathematical science. With this, second, the mathesis

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universalis, now understood ‘‘as algebra, is first and last an ‘art of finding’

(ars inveniendi) and thus, above all, a ‘practical’ art’’ (p. 265). The transformation

was completed by John Wallis (1616–1703), for whom the object of arithmetic and

logistic is completely consonant with that of algebra, and the material of this science

is furnished by symbolic numbers, whose being is immediately graspable in the

notation (pp. 317, 326).

4 Husserl and Klein on the Origination of the Logic of Symbolic Mathematics

Having discussed Klein’s Greek Mathematical Thought, Hopkins now compares

Husserl and Klein on the authentic and symbolic concepts of number. He starts with

an unprecedentedly rich comparison between Husserl’s authentic number concept

(introduced in the Philosophy of Arithmetic) and Plato’s and Aristotle’s concepts of

number. To get a better view of Husserl’s symbolic number concept, Hopkins traces

the development of Husserl’s thought about the symbolic calculus and the

constitution of collective unity, as well as the phenomenological foundation of the

mathesis universalis, through several works: his review of Schroder’s Vorlesungen

uber die Algebra der Logik (1891), the Logical Investigations, Experience and

Judgment, and Formal and Transcendental Logic. After this detailed excursus

Hopkins returns to the comparison at hand. First, he concludes that ‘‘[b]y ‘logic of

symbolic mathematics’ both thinkers understand the rule-governed method of

designating and manipulating sense-perceptible signs to perform ‘calculations’ with

general mathematical objects’’ (p. 491). However, both find this ‘‘insufficient to

establish their mathematical significance, this significance must somehow be

stipulated by the calculative method employed, only after which the sense-

perceptible signs can become known as ‘symbols’’’ (p. 491). Thus the question

about the origin of the logic of symbolic mathematics for both, according to

Hopkins, involves the question about the origination of the stipulation of the

significance proper to the symbols employed in this mathematics (p. 491). In the

language of present-day discussions on logic, what is at stake, it seems, is that the

logic of symbolic mathematics has a fixed interpretation given by the authentic

conception of number drawn from everyday thought about numbers. According to

Hopkins, Klein and Husserl differ in how they see the relationship between, and the

precise mode of being of, symbolic and authentic numbers.

First, according to Hopkins,

in Husserl’s mature works, the signs belonging to the symbolic calculus are

understood to refer neither to the same logical object as the authentic concept

of cardinal number—namely, determinate pluralities of units—nor to the

indirect ideal representation of this logical object accomplished by inauthentic

systematic number concepts. Rather, the signs of the symbolic calculus are

understood as the non-conceptual game’s symbols, which are manipulated in

accordance with the ‘‘rules of the game’’ of a conceptless calculational

technique that acts as a surrogate for an actual theory of multiplicities (p. 492).

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Moreover, Hopkins distinguishes between the constitution of the categorial unity

proper to formal logic and the collective unity proper to mathematics, and he holds

that this distinction has important implications for Husserl’s formulation of the idea

of the pure mathesis universalis as something that unifies formal logic and formal

mathematics. Husserl’s project to provide a foundation for formal logic and formal

mathematics has its locus in the experience of individual objects (p. 492).

Klein’s account, on the other hand, involves four strands, which originate in the

works of Vieta, Stevin, Descartes, and Wallis. Vieta calculates with species of

magnitudes that are designated with sense-perceptible letter signs ‘‘that are

perceived to have a numerical significance’’ (p. 493). Stevin assimilates the concept

of number to operations on numbers using the Arabic system of ciphers (p. 497).

Descartes’s algebraic reformulation of geometry as the mathesis universalis takes

extension as the ‘‘general object’’ of this science (p. 499), and Wallis goes beyond

Stevin, Vieta, and Descartes in understanding the ‘‘analytical art’’ to be confined

entirely within the bounds of symbolic arithmetic. Klein’s account employs the

scholastic distinction between first and second intention to express the origin of

symbolic numbers. First intention concerns the existence and quiddity of an object,

while second intention concerns an object insofar as it has being in apprehension.

The state of being of an object in cognition is second, while the state of being of an

object in itself is first. The status of the referents of the Greek concept of number is

first intentional, while the conceptuality of the modern concept of number is

‘‘tantamount to the apprehension of the object of a second intention as having the

being of the object of a first intention’’ (p. 508). Thus when we apprehend a

‘‘rational being’’ with the aid of imagination so that intellect can take it up as an

object in the mode of a ‘‘first intention,’’ we are dealing with a symbol as understood

by Descartes. This Klein characterizes as symbolic abstraction. It is what allows the

object of a second intention to be apprehended as the object of a first intention. ‘‘The

indeterminate or general object yielded in ‘symbolic abstraction’ is neither purely a

concept nor purely a ‘sign,’ but precisely the unimaginable and unintelligible

identification of the object of a second intention with the object of a first’’ (pp.

509–510). When the analytic art is understood to be coextensive with symbolic

arithmetic, the logic proper to symbolic mathematics originates. Then, Hopkins

writes, ‘‘numbers become identical with their symbolic character’’ (p. 511). ‘‘What

numbers are is now immediately graspable in their symbolic notation, whose

‘universality’ is tied not to multitudes of units but to the notion of the unit as a ‘1’’’

(p. 511).

Husserl, in contrast, understood the symbolic calculus as a calculational

technique, composed of the signs and ‘‘rules of the game’’ that act as surrogates

for genuine deductive thinking. For him, formal logic is a theoretical discipline that

is more fundamental than such symbolic games. Indeed, Husserl

characterizes formal logic in this sense as ‘‘pure’’ logic and as having as its

conceptual content materially empty ‘‘manifolds’’ conceived as the correlates

of the pure theory forms ‘‘constructed’’ by formal deductive systems. Husserl

therefore comes to understand universal analysis as part of pure logic, as an

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instance of manifold theory, and, as such he no longer strictly identifies it with

a symbolic algorithm (p. 512).

While Husserl’s view of symbolic calculus is purely syntactical, Klein’s account of

symbolic mathematics carries its semantic interpretation in the symbols and the

rules of the calculation. Hopkins holds that

[t]he symbolic calculus that composes the algebra of universal analysis thus

remains identified with a concept-less calculational technique in Husserl’s

mature work. And, on his view, this also holds for the symbolic calculus

associated with the pure (completely formalized) mathesis universalis

(p. 512).

I disagree with the last statement and will return to it below. Hopkins argues that

Husserl does not provide descriptive analyses to show ‘‘that the original thinking

and ‘actual’ or ‘true’ concepts proper to the pure mathesis universalis are excluded

by its technical method of symbolic calculation’’ (p. 513). More specifically, while

Husserl’s phenomenological analyses address the notion of the Etwas-uberhaupt,

and notions of formalization and generalization, eidetic intuition, categorial

thinking, and emptying of contents, his analyses of formalization do not address

‘‘the origination of the symbolic calculus in relation to either the logical structure of

the formal category of ‘anything whatever’ or the genesis of the process of

formalization itself’’ (p. 514).

Hopkins points out that in the Formal and Transcendental Logic Husserl holds

that a proper theory of judgment provides an account of how pure analytics refers

ultimately to the experience of individuals. But he finds this insufficient: ‘‘Husserl’s

answer to the question of the relationship of the method of symbolic calculation to

the ‘true’ nature of formal objectivity, as well as to the logical formalization he

maintains yields this objectivity, necessarily remains phenomenologically unclar-

ified’’ (p. 514). He thus concludes:

Because Husserl’s early phenomenological investigations of formalization do

not explore what his mature investigation (in Formal and Transcendental

Logic) characterizes as its ‘‘genesis,’’ and because the programmatic nature of

the latter investigation marks, at best, only a ‘‘methodological’’ advance over

the imprecision of the earlier investigations, the distinction he makes between

(1) the formally derivative status of the method of the symbolic calculus and

(2) the conceptual formality proper to the ‘‘pure’’ logic that this method

somehow serves has not been established phenomenologically. This means,

among other things, that the guiding problem of Husserl’s first philosophical

work, the origin of the logic of symbolic mathematics, remains strangely

unresolved in his mature phenomenological works (p. 515).

Ultimately, Husserl holds that the logic of symbolic mathematics owes its origin to

the individual objects of the perceptual life-world, which, in the terminology of

Formal and Transcendental Logic, have something to do with each other materially

(p. 516).

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Hopkins’s comparison between Klein and Husserl then focuses on the question

whether ‘‘the formalization that yields the Etwas-uberhaupt is a non-symbolic or

symbolic process, and therefore the answer to the question of whether this ‘concept’

is itself symbolic or whether it is independent of the signs and syntax belonging to

the algebra proper to the symbolic calculus’’ (p. 518). In other words, using Klein’s

results Hopkins plans to ‘‘test’’ the truth of Husserl’s presupposition that the

materially empty formal category Etwas-uberhaupt, the object of the pure mathesis

universalis, is a concept that is more fundamental than the symbolic calculus, and

hence logically independent of it.

More precisely, the question is whether such a concept can be thought apart

from its connection with the letter signs that Vieta’s ‘‘analytic art’’ uses to

represent the species of known and unknown magnitudes, or whether letter

signs, however arbitrary, are a necessary condition for the constitution of a

formalized concept and thus for the ‘‘analytic’’ cognitive intention to be

directed to such a concept (p. 521).

Thus ‘‘at issue […] is the phenomenological status of the novel mode of being that

belongs to what, for Vieta and modern mathematics, is the ‘true concept’ of number,

namely, number ‘formalized’ as ‘mere multitude’ (sola multitude), rather than a

definite amount of definite units’’ (pp. 521–522).

When compared to Klein’s analysis of the origin of symbolic mathematics,

Husserl’s shortcomings, according to Hopkins, are (at least) the following: Husserl’s

account of abstraction is Aristotelian, not symbolic as needed to account for the

modern concept of number; Husserl’s view of categorial intuition is insufficient

compared with Vieta’s method of formalization; and Husserl’s late project of

seeking the origin of logic in the experience of individual objects of the life-world is

incapable of accounting for the origin of a materially indeterminate concept. In

contrast, Klein locates the original impulse toward mathematical formalization that

transcends the perceptual life-world in Plato’s references to techne, where he finds

evidence of an original, pre-theoretical praxis of counting and calculation. Rather

than looking for the origin of logic in judgment, then, Hopkins, with Klein, would

look for that origin in practical calculations.

5 Conclusion

Hopkins’ book is an immense scholarly achievement. It is carefully based on

numerous (perhaps too numerous) quotations from original texts, and it covers an

enormous swath of philosophical and mathematical terrain. Some of Hopkins’

discussions are brilliant—for instance, his account of the role of imagination in

Descartes—and similar gems can be found throughout. Evaluating them must,

however, be left to specialists in early modern and ancient mathematics. Klein’s

criticism of Husserl certainly shows that Husserl did little actual historical work,

and that his account does seem to undermine the importance of the symbolic

conception of mathematics for the subsequent development of mathematics.

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However, Hopkins wants to criticize Husserl’s mature conception of mathematics

for lacking foundations by comparing it to Klein’s symbolic conception, for which

he has provided the detailed historical origin. For him, the problem is that Husserl

identifies the symbolic calculus with a concept-less calculational technique, and he

claims that this is also the case with Husserl’s account of pure mathesis universalis

(p. 512). I cannot agree with the latter claim; Husserl clearly distinguishes between

a mathematics of Spielregeln and pure mathesis universalis (FTL §§33–35). The

latter is non-conceptual only if it is not interpreted to be about objects among which

certain laws hold. But Husserl claims that this is not how we should define it.

Husserl is critical of such formalistic account of mathematics. Hence, when Hopkins

argues that Husserl’s search for the foundations of mathematics is inadequate in

comparison to Klein’s analysis, his criticism goes astray. Hopkins is right to hold

that the non-symbolic approach to mathematics in Husserl’s PA resembles the

ancient ontological conception of mathematics and the symbolic part Klein’s

symbolic mathematics. What Hopkins misses is that Husserl’s subsequent

development goes beyond the conceptualizations Klein offers us. Husserl’s thinking

follow the general development of mathematics in the nineteenth century, and he

arrives in the end at what Ferreiros (2007) calls the abstract conceptual view. As

Husserl points out in the introduction to the Logical Investigations, mathematics has

become qualitative, it is about formal structures. The concept of unity it employs is

not something related to collections of objects but to a theoretical unity emphasized

in the Prolegomena and discussed in Husserl’s Definitheit-lectures, not mentioned

by Hopkins. Thus Husserl’s conception of mathematics goes far beyond Vieta’s or

Klein’s conception, which Hopkins finds so novel but which is not particularly

modern because it conflates syntax and semantics. Thus it views the number concept

as identical with symbolic language (p. 4); as expressed elsewhere, in it the sign and

concept are no longer distinct (p. 257). Or consider the following quote: Hopkins

writes that Klein holds that ‘‘[m]odern mathematics is symbolic in the sense that it

identifies ‘the object represented with the means of its representation’’’ (p. 73).

Husserl, in contrast, would never hold such a view. He distinguished clearly

between the symbols and the objects represented by them. He did so already at the

end of the Philosophy of Arithmetic, where he recognized that the symbolic

calculation can be applied to various different domains. In his Definitheit lectures

from 1901 he eventually arrives at a completely modern notion of a formal

structure. Thus Husserl’s conception is much more modern compared to that of

Klein or Vieta.

Mathematics is, for Husserl, a matter of non-contradiction. When discussing

truth, however, Husserl wants to relate mathematical physics to our empirical

observations and thus to judgments about individuals. Such an approach is rather

theoretical, and if we think about mathematics as it figures in our concrete life-

world, Hopkins’ Kleinian criticism may well have a point. Perhaps in our everyday

life-world we do relate to mathematics as a calculational technique rather than

through theoretical observations. But the problem is that Hopkins clearly wants to

criticize Husserl’s concept of mathematics in general for not having adequate

foundations. The crucial question then is whether Husserl is able to provide a

foundation for abstract conceptual mathematics, not whether he is able to found

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symbolic mathematics, which he criticizes. The answer is not at all obvious, and if it

has to be consistent with the way in which mathematics is given to us in our

everyday life, things get even more complicated. Hopkins, however, does not

address these questions at all.

Generally the book would have been much easier to understand if it had clarified

terms such as ‘‘modern mathematics’’ in reference to typical current usage (cf.

Ferreiros 2007). As used in the book, the term refers to Vieta’s and Klein’s

conception. This conception appears to be, or at least is close to, some kind of

formalist position. What exactly the conception is would have been easier to

understand had there been some explanation of how it differs from, say, formalism

as discussed for example by Detlefsen (2005). In general, the book takes Klein’s

views about mathematics for granted without any attempt at evaluating or

contextualizing them, and it approaches the history of mathematics without

referring to standard accounts of that history. Even though Klein seeks to

complement Husserl’s account of history with an actual history of mathematics, no

such work has been consulted. Consequently, mathematics is seen to be primarily

about conceptual issues related to number and collections. This fails to do justice to

the development of mathematics in terms of structures or theories, in response to

certain problems that mathematicians have attempted to solve. The conceptual

development then appears to take place in a vacuum, which renders it rather

mysterious.

Hopkins’ detailed and careful readings of the texts make his book a source of

numerous insights, and its erudition is breathtaking. Hopkins provides a magisterial

treatment of the genesis of Vieta’s early modern account of mathematics as

discussed by Klein. However, the book does not do justice to Husserl’s view of

mathematics, because it fails to understand Husserl’s conception of mathematics as

abstract and conceptual. In order to warrant the comparison, Klein’s analysis should

have been complemented with an account of the history of mathematics after Vieta,

to bring Klein’s project to the level of Husserl’s conception of mathematics.

References

Detlefsen, M. (2005). ‘‘Formalism.’’ In The Oxford handbook of philosophy of mathematics and logic, S.

Shapiro (Ed.). Oxford: Oxford University Press.

Ferreiros, J. (2007). Labyrinth of thought. A history of set theory and its role in modern mathematics,

second revised edition. Basel, Boston, Berlin: Birkhauser.

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