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International Phenomenological Society
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
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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 in- finitesimal 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
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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 sub- ject.
I. Hegel's Program
In his Wissenschaft der Logik (hereafter WdL), Hegel attempts to
provide the underpinning for the rest of his philosophy. His pro-
gram 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 in- volves 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.
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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 in- deed 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 com- patible.
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 "move-
ment." 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.
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HEGEL'S PHILOSOPHY OF MATHEMATICS 455
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
be- ing 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.
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456 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH
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 non- quantitative 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 tak- ing 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.
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HEGEL'S PHILOSOPHY OF MATHEMATICS 457
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 dis- tinct 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.
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458 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH
On this second ground, therefore, it should be a preferred
interpreta- tion since it makes sense of what otherwise might
appear to be unintelligible.
On this reading, therefore, the so-called contradiction of
'repul- sion' 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 ex- cluding 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 ac- count 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 ex- planation 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.
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HEGEL'S PHILOSOPHY OF MATHEMATICS 459
and number. Russell is helpful on this point, pointing out that
certain modern researches demonstrate that a whole set of
mathematical no- tions 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.
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460 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH
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 order- ing 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.
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HEGEL'S PHILOSOPHY OF MATHEMATICS 461
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 con-
struction 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 pro- gram 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 sec- tion on Quality.
Hegel distinguishes between two ways in which the infinite way
be conceptualized, viz., the 'bad' infinite and the 'affirmative'
in- finite. 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).
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462 PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH
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
ac- tual 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.
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HEGEL'S PHILOSOPHY OF MATHEMATICS 463
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 in- finite as an 'entity'
which is reached by following out an infinite series.
The affirmative mathematical infinite would, like the affir-
mative 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 in- finitesimal 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 doc- trine 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.
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464 HEGEL'S PHILOSOPHY OF MATHEMATICS
metaphorical way of expressing the by now accepted notion of the
in- finite 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.
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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