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Kant on Intuition in Geometry

Published online by Cambridge University Press:  01 January 2020

Emily Carson
Emily Carson*
Affiliation:
McGill University, Montreal, PQ, CanadaH3A 2T7
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Extract

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’

This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 27 , Issue 4 , December 1997 , pp. 489 - 512
Copyright
Copyright © The Authors 1997

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References

2 Kant and the Exact Sciences (Cambridge, MA: Harvard University Press 1992), chs. 1 And 2.

3 As an anonymous referee pointed out, this is not Kant's argument for the claim that a concept can contain within itself only finitely many concepts. He argues rather that it is because there is a highest genus (Ak.9:59), that is, the concept of an object (Ak.24:755) or something (Ak.24:911), which we can arrive at it in a finite number of steps by omitting everything (ibid.). But it doesn't seem to follow from the existence of a highest genus that we can arrive at it in a finite number of steps without some additional assumption about the finiteness of our minds.

4 Friedman's disregard of these two distinctions reflects his attempt, which I mentioned earlier, to show that Kant's reasons for introducing intuition were primarily mathematical, and not philosophical.

5 For this reason, it strikes me as somewhat misleading to talk, as Friedman does, of ‘capturing’ the infinite divisibility of space, as this suggests that the feature is available to us independently of the means of capturing it, i.e., the representation of space. It's not clear to me that Kant would accept this, and indeed, this may be at the heart of the matter, as we shall see below.

6 Ak.20:419-21. Friedman himself cites this passage in this connection in an unpublished paper entitled ‘Geometry, construction and intuition in Kant and his successors.’

7 Parsons, Charles ‘The Transcendental Aesthetic,’ in Guyer, Paul ed., The Cambridge Companion to Kant (Cambridge: Cambridge University Press 1992), 72Google Scholar

8 Kant describes the continuity of time in the same terms in §14.4 of the Inaugural Dissertation: ‘any part whatever of time is itself a time. And the things which are in time, simple things, namely moments are not parts of time, but limits with time between them.’ Curiously, he does not provide a parallel argument there for the continuity of space, but he does claim in a footnote that this is easily demonstrated (Ak.2:403).

9 This argument also constitutes a reply to an objection first raised by Parsons, (‘Infinity and Kant's Conception of the “Possibility of Experience,”’ Philosophical Review 73 [1964]CrossRefGoogle Scholar. Reprinted in Mathematics in Philosophy [Ithaca: Cornell University Press 1983]95-109) that Kant's attempt to explain how we have synthetic a priori knowledge of certain features of space — in particular, the infinite divisibility of space — is ad hoc. The problem is that Kant limits the extent of our synthetic knowledge to objects of possible experience; but, Parsons argued, when we try to give a concrete intuitive meaning to the notion of possible experience, we see that the limits of possible experience are narrower than the extent of the geometric knowledge which Kant wants to account for. The only alternative is to define the possibility of experience by what, on mathematical grounds, we take to be the form of our intuition. But if the form of intuition is determined by our knowledge of geometry, it cannot be called upon to provide an explanation of that knowledge. On the present account, however, the nature of the form of intuition is determined independently of geometry, and thus can be called upon to explain our knowledge of geometry, as I have just argued. This is not to say that Kant's argument for the infinity of space is successful. All I have tried to show is that Kant sought to explain our knowledge of geometry in this way, not that he succeeded in doing so. That would require a much more detailed analysis of the argument of the Exposition than has been given here.

10 It also seems to me that taking the ‘cognitive grasp’ determining the nature of our representation of space as given primarily by geometry renders uninteresting any argument against the Wolff-Leibniz view of geometry that, as Friedman himself makes clear, Kant opposed throughout his writing, that is, the view that geometrical concepts are, in some sense, ‘imaginary.’ Against this, Kant is determined to show how geometrical concepts are grounded in the world of experience. But then, to assume that the essential features of our representation of space are to be determined by what is required for geometry begs the question. In particular, Friedman's focus on infinite divisibility seems misplaced, since it was the infinite divisibility of geometrical space which resulted in what Leibniz called ‘the labyrinth of the continuum,’ which in turn contributed to the Leibnizean view that geometric space is ideal (see, e.g., Die philosophischen Schriften von Gottfried Wilhelm Leibniz. Hrsg. C. Gerhardt. (Georg Olms, 1960-61: reprint of 1875-1890 Berlin edition), 2:282). These considerations seem to me to provide some answer to Friedman's objections to ‘anti-formalist’ views like the one put forward here, that is, the objection that they cannot explain ‘the role of Kant's conception of the syntheticity of mathematics in motivating his rejection of the dogmatic metaphysics of the Leibnizean-Wolffian philosophy’ (Friedman, 104). This issue is discussed in detail in my doctoral dissertation, ‘Mathematics, Metaphysics and Intuition in Kant’ (Harvard, 1996).

11 See for example, Beck, Lewis White ‘Can Kant's Synthetic Judgments be Made Analytic?’ in Wolff, Robert Paul ed., Kant: A Collection of Critical Essays (London: MacMillan 1968) 3–22Google Scholar; Brittan, Gordon Kant's Theory of Science (Princeton: Princeton University Press 1978)Google Scholar; Parsons, Charles ‘Kant's Philosophy of Arithmetic,’ in Morgenbesser, S. Suppes, P. and White, M. eds., Philosophy, Science and Method: Essays in Honor of Ernest Nagel (New York: St. Martin's Press 1969)Google Scholar, reprinted in Mathematics in Philosophy (Ithaca: Cornell University Press 1983) 110-51.

12 Ak.24:919-20. See also Ak.24:268ff and Ak.9:143-4. Kant's theory of definition deserves much more attention than can be given to it here.

13 It should be noted that the nominal definition is, strictly speaking, not a definition at all for Kant.

14 Thompson, Manley ‘Singular Terms and Intuitions in Kant's Epistemology,’ Review of Metaphysics 26 (1972-3) 314–43Google Scholar

15 This also seems to support the view that Kant's objection to the Leibniz-Wolff account of mathematics is philosophical (anti-formalist) and not mathematical (based on the impossibility of proving theorems with only monadic logic); cf. n. 10 above.

16 I am grateful to John Carriero, Janet Folina, Warren Goldfarb, Michael Hallett, Alison Laywine, Charles Parsons, and two anonymous referees for this journal for comments on various drafts of this paper, to Peter Clark for making me write it, and to the Social Sciences and Humanities Research Council of Canada for generous support of my doctoral studies, during which this paper was written.