Pinkard, Terry (1981) - Hegel's Philosophy of Mathematics (Philosophy and Phenomenological Research, 41.4)

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Hegel's Philosophy of Mathematics Author(s): Terry Pinkard Source: Philosophy and Phenomenological Research, Vol. 41, No. 4 (Jun., 1981), pp. 452-464Published by: International Phenomenological SocietyStable URL: http://www.jstor.org/stable/2107251Accessed: 28-06-2015 08:34 UTC

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HEGEL'S PHILOSOPHY OF MATHEMATICS

Hegel's philosophy is often cited in the Anglo-American philosophical world as a good example of just what can go wrong in philosophy. Besides the many errors he is said to have made, he is supposed to have done particularly badly in his philosophy of mathematics. Russell's comments to the effect that it was Hegel's stupidity in mathematics which drove him away from Hegel's philosophy are by now part of the Hegel legend. In being passed down from one generation to another, this legend has now assumed the form of something like a dogma. What is unfortunate about this particular legend is that Russell's specific comments on Hegel's philosophy of mathematics are often either misleading or simply false. In the Principles of Mathematics1 the only work to which Russell alludes is Hegel's 'Encyclopedia of Logic' which contains even less than the outline of Hegel's philosophy of mathematics; the 'En- cyclopedia of Logic' has no arguments or examples and was intended to be used as an aid for Hegel's lectures.2 Hegel's real treatment of the subject comes in the much longer Wissenschaft der Logik (Science of Logic). Moreover, in his Introduction to Mathematical Philosophy, Russell suggests that Hegel is one of those philosophers who ignored developments in mathematics and clung to the belief that the dif- ferential and integral calculus required the postulation of infinitesimal quantities.3 One could certainly not get that idea from Hegel's Wissenschaft der Logik, where the largest section of tile book is spent attacking the notion of the infinitesimal. But myths die hard.

Given these oversights by Hegel's perhaps most prominent critic, a reassessment of Hegel's philosophy of mathematics is in order. A fresh understanding of that might also perhaps pave the way for a

'Bertrand Russell, The Principles of Mathematics (New York: W.W. Norton, 1903). (Hereafter: Principles).

2This is the version of Hegel's general ontology sometimes called the 'Lesser Logic.' It is only a section of a longer book by Hegel Enzyklopddie der philosophischen Wissenschaften, which consists of short summaries of Hegel's philosophical position. It is translated into English by William Wallace as The Logic of Hegel (London: Oxford University Press, 1873).

3Bertrand Russell, Introduction to Mathematical Philosophy (New York: Simon and Schuster). p. 107.

452


HEGEL'S PHILOSOPHY OF MATHEMATICS 453

fresher understanding of his whole body of thought. In this paper, I would like to present an outline for such a reassessment of Hegel's philosophy of mathematics. My strategy will be to present first a very brief outline of what I take Hegel's general program to be; second, to reconstruct what are the basics of his theory of mathematics; and then finally to sketch out how Hegel's theory could be consistently modified so as to bring it up to date with current thought on the subject.

I. Hegel's Program

In his Wissenschaft der Logik (hereafter WdL), Hegel attempts to provide the underpinning for the rest of his philosophy. His program involves minimally the ordering of a set of categories in such a way that a step by step analysis and justification of each category can be given according to a small set of basic principles.4 The program involves a reconstruction according to this set of basic principles of the concepts of everyday experience (i.e., of the basic notions of those things with which we have an 'acquaintance,' Bekanntschaft),5 of the concepts of natural science,6 and of past philosophical theories. One of the basic aims in Hegel's program (although certainly not the only one) was the construction of what could be called a thoroughgoing compatibilist philosophy. That is, a basic tenet of Hegelianism (at least as Hegel saw it) was that many apparent contradictions in our

4This is a somewhat revisionist interpretation of Hegel. It differs from more usual readings of Hegel in that it sees him as a categorial philosopher rather than primarily as a metaphysician of 'Spirit' working its way out in history. In a paper of this type a full defense of this kind of interpretation of Hegel cannot be given. For similar readings, however, cf. Klaus Hartmann, "Hegel: A Non-Metaphysical View" in A. MacIntyre (ed.). Hegel: A Collection of Critical Essays (Garden City: Doubleday 1972). pp. 101-124; G. Maluschke, Kritik und absolute Methode in Hegels Dialektik (Bonn: Bouvier, 1974); J. Heinrichs, Die Logik der Phdnomenolo- gie des Geistes (Bonn: Bouvier, 1974); Gerd Buchdahl. "Hegel's Philosophy of Nature and the Structure of Science" Ratio (xv), 1973, pp. 1-27.

5G.W.F. Hegel, Enzyklopddie der philosophischen Wissenschaften (Hamburg: Felix Meiner, 1969). 1, 6.

6Ibid., 246.


454 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH

categorial framework or many apparent incompatibilities between competing categorial frameworks were only that: apparent and not real contradictions and incompatibilities. They could be 'overcome' (aufgehoben) if the conceptual framework was sufficiently expanded and ordered correctly. Examples of such are notions like freedom and necessity which prima facie may appear to be incompatible ideas and which therefore give rise to the competing categorial frameworks of determinism and libertarianism. A more refined and expanded schema, however, would show that freedom and necessity were indeed compatible notions. Indeed, one of the most persistent metaphors animating all of Hegel's philosophy is that of the tension between the understanding and reason, Verstand and Vernunft, with 'the understanding' being that 'faculty' which holds on to apparent incompatibilities, and reason being the 'faculty' which expands and reorders the schema so as to make the apparent contradictions compatible.

The way to do this, according to Hegel, is to begin with a mass of concepts from experience, science, and the history of philosophy and order them according to their "immanent development"7 or "movement." The idea of the 'movement' of things is one of the most often mentioned and hotly debated points in Hegel's philosophy. The understanding of this is crucial to an understanding of Hegel's theory and consequently to a rational assessment of his whole theory. In- deed, to explicate Hegel as it has so far been done here may seem to be a gross misrepresentation of Hegel, for is not Hegel par excellence the philosopher of the dialectical movement of the cosmos and not one who analyzes and orders mere concepts?

It is important first to look at what Hegel says. The object of such a study is pure thoughts (Gedanken, which might be taken to be like Fregean senses).8 That is, the object of such a reconstruction is not things (Dinge) but "facts, the concepts of things" ("Sache, der Begrijf der Dinge').9 The "facts themselves" ("Sachen an sich selbst') are, however, "pure thoughts."10 By saying the facts (Sache) are thoughts (Gedanken), Hegel presumably means that facts are not theory-free items but are constituted by one's conceptual framework;

'G.W.F. Hegel, Wissenschaft der Logik (Hamburg: Felix Meiner, 1971), p. 7. (Hereafter WdL).

8Ibid. 9Ibid., p. 18. 10Ibid., p. 30.



the facts we encounter in experience therefore would be conceptual unities (i.e., Gedanken). If the method is the "movement of the facts themselves,"11 and the facts are thoughts, then as a consequence of Hegel's own words, the so-called movement in the WdL should be one of concepts. Not surprisingly, Hegel also characterizes WdL as "the development of thought."12

However, how is this to be understood? As an interpretative model for making sense of Hegel's theory, perhaps the familiar model of a game will help. The WdL can be seen as a constructive theory, that is, one which generates the determinateness of its notions by a small set of construction rules; in Hegel's theory these are themselves given in the opening section of WdL and in the section on 'Quality.' The concepts in the theory are thus like pieces in a game. A piece, we might say, has its meaning in terms of the role it plays in the game. A piece in a game, that is, is a normative kind. 13 A normative kind is an entity whose 'being' is constituted entirely by prescriptive rules, i.e., those that specify that an action ought or must be done. Any piece in a game has two components: (1) a descriptive, accidental component - a description of the material of which the piece is made; (2) a prescriptive, essential component - the rules which constitute its being a normative kind. Concepts may be thought of as such normative kinds. Their 'logical meaning' is that set of rules which constitute them, which in their case would be inference rules. The movement of concepts may be thus conceived in analogy to a game.

The game analogy somewhat breaks down, however, when one asks what is moving. What moves in Hegel's theory would not really be the pieces, i.e., the concepts themselves, but thought itself. That is (to take the game analogy a bit further), each concept is a position in the game, and thought moves from one position to another. The meaning of all the particular concepts lies thus in where they are in the game. In light of this analogy, the WdL can be seen in two ways: (a) in a static way - arrangement of all the pieces taken at once; (b) in a dynamic way - the movement from one position to another. The goal of the movement is the construction of a categorial scheme which will be rich enough to render compatible what were only apparent contradictions between other competing categorial schemes.

"Ibid., p. 36. 1 2Ibid., p. 19. 1 Jay Rosenberg, Linguistic Representation (Dordrecht: D. Reidel, 1974). cf.

pp. 30-48.



II. The One, The Many, And Class Concepts

Hegel's philosophy of mathematics proceeds in two distinct stages, although the discussion of it mostly occurs in the section of the WdL called "Quantity." His strategy is first to develop a series of nonquantitative concepts in terms of which he can then reconstruct the quantitative ones. He then wishes to develop a procedure for generating in terms of his own dialectical theory the basic categories of mathematical thought and to contrast this with other modes of thought.

At the end of the section on 'Quality' Hegel claims to have generated the concept of totality, which he calls Being-for self (Fur- sichsein). Within this concept as a 'moment' of it is the notion of a simple unit, which Hegel claims is an abstract entity (this is, I am taking it, what Hegel means by calling it 'ideal). 14 Hegel characterizes it by coining a term for it, das Eins, the one, which is a nonquantitative unit in distinction from the quantitative one, das Eins. In fact, Hegel specifically uses the neuter form to make this point (Charles Taylor apparently confuses the German words on this point claiming that das Eins is an ordinal number).15 Thus, the transitional section on das Emns is to be an elaboration of the logic of discourse about pluralities of such abstract units.

The actual mechanics of the transition from this concept of a plurality on nonquantitative units to the concept of quantity are given in Hegel's discussion of the 'One and the Many,' and in his ex- tension of the discussion to that of 'repulsion' and 'attraction.'16 This section goes (in outline) in the following way. The one (das Eins) is just the concept of being-for-self considered simply as a self-identical unit (this is what Hegel means by calling it being-for-self in its 'simple relation to self).17 That is, it is the notion of an ideal totality of units (being-for-self is an ideality, an abstract entity).18 As such it is the unity of being - the moment of self-identity - and determinate be-

4Hegel, WdL, p. 150. 15Cf. Charles Taylor, Hegel (London: Cambridge University Press, 1975). p.

245. (Klaus Hartmann pointed out this use of the German to me.) 16Ibid., pp. 154-176. This section on Repulsion and Attraction belongs,

technically, in the Naturphiloophie. Cf. Buchdahl, op. cit. on this point. "7Hegel, WdL, p. 154. 18Ibid.



ing (Dasein) - the moment of relation-to-others.19 But it is a purely abstract, 'empty' notion; it expresses only the concept of an abstract x which is self-identical. Having laid this groundwork, Hegel goes on to speak of a plurality of such empty, self-identical units. He does not argue specifically for this move but justifies it by appeal to an earlier figure in the WdL, viz., the move from the concept of a determinate being to that of determinate beings. That transition was justified (roughly) by the claim that to say of anything that it is something or another is to contrast it with other different things. Having done that Hegel goes on to speak of how the one is identical with the many, yet is not identical with the many. Somehow, this is supposed to generate the notion of quantitatively distinct units.

How are we to make sense of Hegel here? It is certainly an obscure section, with Hegel speaking of the 'repulsion' of the many in- to a plurality and the corresponding 'attraction' of them back into the 'One,' with the relation between them being the void. As an inter- pretive hypothesis, I propose that this section be read in Russellian terms as a consideration on Hegel's part of classes and their members, specifically, of the class as one and as many, i.e., of the class as distinct from its members and as identical with them. Classes are abstract entities (idealities) which are not equivalent to their members; the class, e.g., of green books is not equivalent to the heap of green books, if for no other than that it has logical or mathematical properties which the heap does not have. But two classes are identical if and only if their members are identical; this is the class as many. The class as one would be the nonquantitative unit, das Eins; the class as many is the members which it has - die Eins. There is, of course, the danger of anachronism in interpreting Hegel this way. But in defense of this interpretation, two points need to be noted. First, Hegel is, after all, talking of the relation of totalities to that of which they are the totalities, and one finds the same language of wholes and members in Russell's discussion of this issue.20 More- over that Hegel, because of his place in history, did not have the modern language of classes to use should not be held against him. Second, this interpretation does, I think, shed light on an otherwise boisterously obscure section of Hegel's work and helps to clarify the transition from the section on Quality into the section on Quantity.

19Ibid., p. 155. 20Russell, Principles, p. 68.



On this second ground, therefore, it should be a preferred interpretation since it makes sense of what otherwise might appear to be unintelligible.

On this reading, therefore, the so-called contradiction of 'repulsion' and 'attraction' in this section is really that of the tension of regarding the class as one and the class as many. The tension lies in taking the class as an entity like its members, which would make it just one more entity 'alongside' its members instead of including them as members. The talk, e.g., about the many ones repelling and excluding one another in the void may then be read as simply the for- mal self-identity of the members 'excluding' all the others. More pro- saically, it just means that the members are separated only by their relations of nonidentity, e.g., x # y and y # z, etc., without any account being given of in what respect they are not identical. Indeed, Hegel's talk of the many repelling their ideality may be seen as taking the members as many, as a 'heap' and not as members of a class. 'Repulsion' then merely denotes the Realitdt of the one (i.e., its exten- sional equivalence to its members), while 'attraction' denotes the Idealitdt of the one (its status as an abstract entity).21 In fact, Hegel even uses the term, Menge (set), to characterize this.22 A fuller explanation of the whole section is not needed here. The important point is that Hegel is claiming that classes (die Eins, the ones) are necessary for thinking about pluralities, for to say of any set of units that they are distinct is to employ the notion of a class for doing so. I.e., to say x # y, is to say that (30 c ) (x tcc &iXec), where 0c denotes a class. The concept of purely quantitative difference can thus be generated out of the discussion of classes.

III. Quantity and Number

A. Hegel and the Tradition

Hegel's full blown discussion of mathematics is, it must be ad- mitted, a bit outdated. He follows the tradition of his time in suppos- ing that objects of mathematics are the related notions of quantity

21Hegel, WdL, p. 164. 22Ibid.



and number. Russell is helpful on this point, pointing out that certain modern researches demonstrate that a whole set of mathematical notions must be defined without reference to quantity.23 Moreover, for algebra and analysis one does not need the notion of quantity but only integers, which can be defined in set theory. Further, there are new branches of mathematics which deal neither with quantity nor with number, such as projective geometry and group theory.24 Finally the traditional quantitative notion of measurement (to which Hegel also appeals) is not necessarily tied up with quantity.

What then is left to be said of Hegel's philosophy of mathematics? The point of trying to reach an understanding of Hegel's mathematics is twofold. First, if it can be shown that Hegel's work in this field is not that of an incompetent or is not composed of absolutely silly assertions, then one major traditional obstacle to the fair assessment of Hegel would be removed, and the way would thus be cleared perhaps for a more rational appraisal of other areas of his thought. Second, we ought to attempt to see if some of his mistakes can be righted - that is, if a reconstruction of mathematical categories in more contemporary terms can be consistently done in the Hegelian system (this would also serve the first point). Therefore, the task of understanding this portion of Hegel's thought is also twofold: (1) it is interpretative - what did Hegel say? (2) it is reconstructive - what could he consistently be made to say which would be of contemporary relevance?

B. The Definition of Number

Hegel defines 'quantity' through the notions of continuity and discreteness. The first of these terms, it must be noted, is not in any way equivalent to the modern mathematical notion of continuity. Russell himself remarks on this, giving Hegel the benefit of the doubt - in one of the few places he does so.25 Russell notes that 'continuity' and 'discreteness' in Hegel's case refer to the opposition of unity and diversity in a collection. This interpretation by Russell is important, for it gives added credence to our earlier interpretation of das Eins as the class as one and die Eins as the class as many. Continuity would then be the class as one, and discreteness would be the class as many.

23Russell, Principles, p. 157. 24Ibid., p. 158. 25Russell, Principles, p. 348.



Hegel uses these notions to define the concept of number. The one (das Eins) is the "principle of the quantum,"26 i.e., the model by which Hegel interprets the notions involved in the quantitative units. Number is the amount of such units in a class - "Amount (Anzahl) and the unity (Eznheit) constitute the moments of number."27 Number is therefore the 'amount' of units (classes) in another unit (itself a 'one,' i.e., a class). Thus, Hegel was at least close (although it would be fatuous to say he achieved it) to defining numbers as classes of classes. He is, of course, basing this on the more traditional notion of numbers as the amount of units which one can count. Husserl, e.g., in his early Philosophie der Arithmetik held this, giving it a psychologistic twist.

One could, however, reformulate this without any psychologism. Number would be defined then via rules for counting, which would make the concept of number dependent on that of order or series. But while this may seem to be the rational, indeed, sensible thing to do, and while it may also seem to be in keeping with Hegel's program of constructing the determinateness of categories from their relations to one another, it unfortunately goes against what Hegel said. Hegel, true to his tradition, defines numbers as amounts of units, whereas this latter way would have numbers being defined in terms of ordering principles. The least one could do is reformulate Hegel's doctrine into saying that the two concepts defining numbers are those of unity and multiciplicity; numbers would then be multiplicities of units which we count.

If Hegel is to make sense, therefore, he must be reformed. The question is, of course, whether reform is what is needed and not a complete revolution. In this light, it is well to keep in mind Hegel's program: (1) categories are defined via their 'positions' in a larger framework; (2) the opening concepts of the section on mathematics concern those of classes and their members. As a proposal for a new Hegelian notion of number we may offer the following. He should begin with the notion of units (die Emns, the ones) as members of classes and then proceed to show how construction rules which involve these units can be given for numbers. Numbers (better: integers) should be the opening section of this part and not continuous and discrete magnitudes. One would thus define integers via rules of

26Hegel, WdL, p. 197. 27Ibid.



counting, e.g., 1, 11, 111, then define magnitude (i.e., 'greater or less') by these counting rules. This would have the form: x < y d X1, then y1 is assertible within the system.28 To defend mathematical truths, one would have to perform a construction according to rules. One could then use the categorial notion of a unit (a member of a class, represented by a variable), proceed to counting units (i.e., adopt construction rules), thus introducing the concept of numbers, then define magnitude in the way mentioned, and then one could define quantity (i.e., that which is capable of relations of quantitative equality) at the end of the series, not at the beginning. This would not be circular, since quantitative equality could be defined using the categorially prior notion of magnitude; i.e., m ? n df (m < n) or (n < m); and m = n df not-(m # n). This could all be in keeping with Hegel's program. The point here is not to actually provide such a construction but merely show that Hegel's program may not be as silly as it is by legend supposed to have been. If nothing else, such a reading would at least allow us to situate Hegel within what for contemporary philosophers is a respectable philosophical tradition.

C. The Infinite

Hegel's understanding of the infinite is one of the most fruitful notions in his philosophy of mathematics, for it, more than his other ideas on the subject, should dispel the notion that he was totally out of touch with mathematical thought. He claims to find in 'modern' notions of the mathematical infinite a vindication of both his program and his own understanding of the 'qualitative' infinite found in the earlier section of the WdL. Therefore, it will perhaps pay to look, however briefly, at Hegel's first formulation of the infinite in the section on Quality.

Hegel distinguishes between two ways in which the infinite way be conceptualized, viz., the 'bad' infinite and the 'affirmative' infinite. The bad infinite is conceived as an entity, a Daseiende. In the earlier sections of the WdL, Hegel dealt with the logic as 'entity- concepts,' of the logic of the various external and internal relations in which entities (Daseiende) can stand with regard to each other. The infinite, however, is not such an entity which might be reached by following a series out; it is not, that is, an entity to be reached which is

28Paul Lorenzen has done something like this. Cf. Paul Lorenzen, Oswald Schwemmer, Konstruktive Logik, Ethik und Wissenschaftstheorie (Mannheim: Bibliographisches Institut, 1973).



beyond (Yenseits) the finite. To see the infinite in this way is to 'finitize' it, to make it a finite infinite - cleary a contradiction.29

The contradiction occurs when one tries to use a certain set of categorial distinctions to talk about that to which the categories do not apply. Nor is it the potential infinite, i.e., a limiting condition, something becoming and never completed (what Hegel calls the 'pro- gress into infinity').30 Both of these two modes of conceptualizing the infinite constitute the 'bad' infinite; one way (taking it as an entity) is just contradictory, the other (regarding it as the potential infinite) is for Hegel unsatisfactory (since the infinite, if it is to be rational, must be actual).31

The affirmative infinite, the 'rational' one, would be thus the actual infinite, i.e., the infinite regarded as a complete whole. Hegel characterizes the affirmative infinite as "being" - "it is and is there, present."32 The affirmative infinite is, morevoer, the unity of the finite and potential infinite;33 it is the movement (Bewegung) of these concepts. I take Hegel to be saying here that the actual and the potential infinites are compatible notions. It is so because the actual infinite is an ideality,34 it is simply the representation (something ideal) of a sequence (a movement, Bewegung, in Hegel's terminology) by a rule which shows what would result if the sequence were followed out. The affirmative infinite thus is the potential infinite represented by a rule which shows what would happen were the process to be car- ried through.

One finds a parallel situation with the mathematical infinite. In the section treating this, Hegel is, it must be noted, not concerned with all the problems of the infinite which one finds nowadays treated in mathematical analysis but only with what was a burning issue in his own time: the notion of the infinitesimal. The mathematical infinite is found, so Hegel claims, where one has a representation of a numerical sequence which seems to proceed to an infinitely small amount. Proponents of the notion of the infinitesimal claimed that such an infinite series thus would culminate in a quantity which is in-

29Hegel, WdL, p. 128. 30Ibid., pp. 130-31. 31G.W.F. Hegel, Grundlinien der Philosophie des Rechts (Hamburg: Felix

Meiner, 1955), p. 14. 32Hegel, WdL, p. 138. 33Ibid., p. 134. 31Ibid., pp. 139-140.



finitely small, i.e., a number, n, such that n is greater than zero but smaller than any other finite number. Hegel heaps nothing but scorn on this view, calling it "Bilder der Vorstellung,'15 only 'fog' and 'shadows' of thought. It is a notion, besides, with too much inpreci- sion (Ungenauigkeit).36 To hold to the doctrine of the infinitesimal would be like holding that there is a midpoint between being and nothing.37 Russell's statement, therefore, to the effect that philosophers influenced by Hegel would be compelled to accept the notion of the infinitesimal is not only false but seriously misleading on the nature of Hegel's philosophy. The notion of the infinitesimal is only another example of one form of the bad infinite, i.e., treating the infinite as an 'entity' which is reached by following out an infinite series.

The affirmative mathematical infinite would, like the affirmative qualitative infinite, be a relation between two notions - in his terms two quanta, each determined by the other and determined only in this relation (Verhdltnis) to one another. The problem with understanding what Hegel means by this lies in the fact that Hegel supplied his full explanation of what he meant by his talk of 'relations of quanta' in an extended note (Anmerkung) rather than in the full body of the text. In the Anmerkung, Hegel gives a brief philosophical history of the notion of the mathematical infinite and sides with Lagrange in taking the quantitative relation (Verhdltnis) to be ex- pressed as a function, e.g., y = f(x) or y = x2. The concept of the infinitesimal should be replaced by a notion of limits. 38 It is in functions of variable magnitudes that the true mathematical infinite is to be found, not in the notion of an infinitesimal. Lagrange's idea was, for Hegel (and for subsequent mathematicians) the correct one: the point was to devise a method in which the limit could be made as arbitrarily small as one pleased. He does give Newton some credit for his doctrine of 'fluxions,' i.e., vanishing divisibles, which Newton called limits; but Newton incorrectly inferred, according to Hegel, from final proportions to proportions of final magnitudes. Hegel would, so it would seem, accept the interpretation which moderns like Weierstrass gave to the infinite. Weierstrass claimed that the idea of a value tending toward another value was just a metaphor; i.e., that the idea that '2x + h' tends to 2x as h tends towards zero is a

35Ibid., p. 236. 36Ibid., p. 241. 37Ibid., p. 235. 38Ibid., cf. pp. 257, 268, 269.


464 HEGEL'S PHILOSOPHY OF MATHEMATICS

metaphorical way of expressing the by now accepted notion of the infinite as a limit, that is, as being capable of being read as arbitrarily small as one pleases. Hegel finds, therefore, in modern mathematical analysis as practiced by Lagrange a vindication of his constructivist program overall and in particular of his notion of infinity. The actual infinite is not a 'thing,' not even an infinitesimally small number but is a representation of a sequence by a rule which shows what would happen if the sequence were followed out. The actual and potential infinite are thus in mathematics compatible notions.

IV.

The purpose of this 'revisionist' interpretation of Hegels thoughts on mathematics is to remove, it is to be hoped, some of the obstacles to understanding this section of Hegel's philosophy. Cer- tainly the subject deserves a more extended treatment than that given to it here. Hegel had much more to say on the topic (not all of it lucid.) If nothing else, this way of reinterpreting Hegel might help to reintegrate him into the philosophical tradition and perhaps to save him from those 'friendly' interpreters of his thought who see in Hegel their champion for the right to contradict oneself.

TERRY PINKARD. GEORGETOWN UNIVERSITY.


Article Contentsp. 452p. 453p. 454p. 455p. 456p. 457p. 458p. 459p. 460p. 461p. 462p. 463p. 464

Issue Table of ContentsPhilosophy and Phenomenological Research, Vol. 41, No. 4 (Jun., 1981) pp. 419-574Volume InformationFront MatterPhilosophy as an Agent of Civilization [pp. 419-436]Contextualistic Realism [pp. 437-451]Hegel's Philosophy of Mathematics [pp. 452-464]Development and Criticism of a Behaviorist Analysis of Perception [pp. 465-486]Recognizing Clear and Distinct Perceptions [pp. 487-507]How to Complete the Compatibilist Account of Free Action [pp. 508-523]The Specification of Facts in Linguistic Contexts [pp. 524-531]A Theory of Tarka Sentences [pp. 532-546]DiscussionThe Concept of Evidence in Edmund Husserl's Genealogy of Logic [pp. 547-555]The Use of Language and its Objects in Literature and Society [pp. 556-560]

ReviewsReview: untitled [pp. 561-562]Review: untitled [pp. 562-563]Review: untitled [pp. 563-565]Review: untitled [pp. 565-566]Review: untitled [pp. 566-567]Review: untitled [pp. 567-568]

Notes and News [pp. 569]Recent Publications [pp. 570-574]Back Matter

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Pinkard, Terry (1981) - Hegel's Philosophy of Mathematics (Philosophy and Phenomenological Research, 41.4)