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Published online by Cambridge University Press: 01 January 2020
While not paramount among Kant scholars, issues in the philosophy of mathematics have maintained a position of importance in writings about Kant’s philosophy, and recent years have witnessed a rejuvenation of interest and real progress in interpreting his views on the nature of mathematics. My hope here is to contribute to this recent progress by expanding upon the general tacks taken by Jaakko Hintikka concerning Kant’s writings on geometry.
Let me begin by making a vile suggestion: Kant did not have a philosophy of mathematics. When Kant was writing about mathematics, essentially he was reporting the views of others. The texts provide sufficient evidence to make this suggestion plausible. Generally, when Kant writes about mathematics in his mature works, he does so in order to illustrate or argue for a philosophical point. There are important references to mathematical method in the preface to the 1787 edition of Critique of Pure Reason; however, Kant’s purpose is to describe those basic features of a method that he intended to incorporate in his theory of philosophical method: ‘our new method of thought, namely, that we can know a priori of things only what we ourselves put into them.’ Indeed, Kant makes it clear in this preface that he thought there to be no extant problems to be solved in mathematical methodology; such was the state of the science, he thought. It was for this reason that Kant felt some confidence in borrowing from this method to improve the state of metaphysics; it is also for this reason that one should not expect to find Kant engaging in basic research in mathematical methodology. Similarly, the material on syntheticity added to the second edition Introduction to Critique of Pure Reason occurs in the context of a discussion of the syntheticity of metaphysical principles; that the propositions of both disciplines are synthetic a priori lends credence to the extrapolation of some features from the mathematical method for use in developing a metaphysics. Many writers find a philosophy of mathematics in the ‘Transcendental Aesthetic’; it is clear, however, that in this section his concern is to support his theory of the nature of space, time, and sensation. What is said about geometry, for example, is restricted to those of its features relevant to the subjectivity of space. The other major discussion of mathematics and its method is found in the section, ‘Doctrine of Method.’ Here we find Kant’s fullest account of the mathematical method and of constructions. It must be borne in mind, though, that his purpose is to argue against views that the proper methods of mathematics and metaphysics (philosophy generally) are identical, that the disciplines differ in subject matter alone. The result of the discussion is not a theory of mathematical method, but an account of the method proper to the philosopher.2 Kant simply is mentioning certain features of mathematical method sufficient to support the claim that the philosopher cannot incorporate it lock, stock, and barrel.’ In short, we do not find a systematic theory of mathematics or its method described by Kant in the first Critique, nor do we find discussions of mathematics other than in contexts where philosophical positions are being developed. This holds for Kant’s other works, too.
1 I have relied most heavily on his ‘Kant on the Mathematical Method’ in Kant Studies Today, ed. Beck, L.W. (La Salle, Illinois: Open Court 1969)Google Scholar; Logic, Language Games and Information (Oxford: Oarendon Press 1973); The Method of Analysis, co-authored with Unto Remes (Dordrecht: Reidel 1974); ‘Kant's Theory of Mathematics Revisited’ in Philosophical Topics 12 (1981) 201-15; ‘Kant's Transcendental Method and his Theory of Mathematics,’ Topoi 3 (1984) 99-108. The tacks that I have in mind are those where Hintikka takes Euclidean methodology as a clue for interpreting Kant. This is a bit of a departure from the norm: Hintikka's reconstruction of Kant’s theory of constructions in terms of the logical operation of existential instantiation in natural deduction systems is usually the focus of discussion. Though this is important, I take the most important insight to be of the Euclidean roots of Kant's thought. The reconstruction utilizing existential instantiation applies equally to Euclid as to Kant.
2 The point here is very similar to that in Kant's 1763 Prize Essay (‘Enquiry concerning the clarity of the principles of natural theology and ethics’). Kant's descriptions of the mathematical method are virtually identical in the Prize Essay and Critique of Pure Reason; here, I will concentrate on the Critique.
3 Aristotle, Posterior Analytics, tr. Mure, G.R.G. in The Basic Works of AristotleGoogle Scholar, ed. McKeon, R. (New York: Random House 1941) Book l, Ch. 2, 71bGoogle Scholar. In his important The Beginnings of Greek Mathematics (Dordrecht: Reidel 1978), Arpad Szabo argues that Euclid is in the Eleatic-Platonic tradition rather than in the Aristotelian. Szabo’s work has stimulated a considerable controvery, and Hintikka has been a key player here. See, for example, The Method of Analysis, which includes a reply by Szabo and a counter by the authors; see also Theory Change, Ancient Axiomatics and Galileo's Methodology, Hintikka, J. Gruender, D. Agazzi, E. eds., (Dordrecht: Reidel 1981)Google Scholar, Section II. This section is primarily a set of responses to Szabo's hypothesis. My account is the conservative Aristotelian account. I recognize the obvious departures from Aristotle by Euclid: for example, Euclidean proofs are not syllogistic (though they can be expanded to constitute a series of syllogisms). However, the similarities to which I appeal are well-established, and, leaving the scholarly issues aside, Kant's Aristotelian roots, especially concerning logic, make it likely that he would have considered Euclid from an Aristotelian point of view.
4 Aristotle, I, 2, 72a: ‘If a thesis assumes one part or another of an enunciation, i.e., asserts either the existence or the non-existence of a subject, it is a hypothesis; if it does not so assert, it is a definition’ and I, 10, 76b: ‘The definitions — viz. those which are not expressed as statements that anything is or is not — are not hypotheses: but it is in the premises of a science that its hypotheses are contained. Definitions require only to be understood.’
5 This must, though, constitute an implicit assumption of the existence of points, given that points are by definition the extremities of lines of finite length. See Definition 3.
6 Euclid, The Elements, tr. Heath, T. (New York: Dover 1956) 154Google Scholar
7 Thus, while one cannot, for example, use geometrical definitions in proofs of arithmetic, one can use the axioms of geometry in arithmetic; the same set of axioms is fruitful in all sciences.
8 Euclid, 129
9 Euclid, 138
10 , Hintikka Logic Language Games and Information, 201Google Scholar
11 Hintikka, ibid, 202. On the issue of the interpretation of ‘analysis’ and ‘synthesis,’ I generally side with Szabo's directional interpretation. See his discussion in Hintikka and Remes, 1974.
12 Hintikka, ibid, 202
13 Hintikka, ibid, 203. Hintikka presents his views here more fully in chapter V of The Method of Analysis. The distinction between analysis and synthesis seems nar rower here than in the Kant essays. ‘Thus in the most literal sense constructions are not presupposed in analysis, but are rather sought after in it, and carried out in synthesis. It is a subtle characteristic of the analytical method, however, that it can succeed in finding these actual constructions only if enough as it were hypothetical ones were already anticipated in it’ (46).
14 See above, 8
15 See Euclid, 129.
16 , Hintikka Logic, language Games and Information, 207Google Scholar
17 Hintikka does not completely restrict his interpretations of constructions to kataskeue. At times, he makes references to figures accompanying the setting out (ekthesis). But the latter are clearly secondary, and it is kataskeue which does the interpretive work for Hintikka.
18 See Szabo's The Beginning of Greek Mathematics, Part III. His Eleatic hypothesis construes Euclidean proofs as not involving visualization or appeal to concrete entities as data or evidence; rather, proofs are abstract. I think Szabo is right, but I doubt that this claim requires the Eleatic hypothesis. I also think that these points hold for Kant.
19 See Heath in Euclid, p. 146: The transition from the subjective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated).’ See also his footnote 2, also on p. 146.
20 Hintikka, ‘Kant's Theory of Mathematics Revisited,’ 205. Hintikka specifies the logical function of construction as a precursor to existential instantiation in natural deductive logic. Again, I will not comment on this aspect of his interpretation. Analyses of this can be found in Gordon Brittan's Kant's Theory of Science (Princeton: Princeton University Press 1978) and Butts’, Robert ‘Rules, Examples and Constructions: Kant's Theory of Mathematics’ in Synthese 47 (1981) 257–88.CrossRefGoogle Scholar
21 , Kant Critique of Pure ReasonGoogle Scholar, tr. Smith, N.K. (New York: St. Martin's Press 1929)Google Scholar (hereafter CPR) A727 = B755.
22 Kant, CPR, A729=B757. It follows that ‘mathematical definitions can never be in error’ (CPR, A731 = B759).
23 Kant, CPR, A729=B757
24 I have said that I do not wish to become involved in the issue of the syntheticity of mathematical judgements in this essay. However, one might notice Kant's claim at CPR, A730=B758 that mathematical concepts are produced synthetically, as opposed to discovered by analysing given experiences. The production of mathematical concepts in a synthetic fashion is logically independent of the establishment of the objectivity through constructions; this might support the suggestion that the syntheticity of mathematics is independent of the function of constructions, a position contrary to Hintikka's.
25 Kant, CPR, A240 = B299
26 See Kant, CPR, A164=B205.
27 Kant, CPR, A734=B762
28 In addition to the works listed in Note 1, see also Hintikka's ‘On Kant's Notion of Intuition (Anschauung)’ in Penelhum, T. and Macintosh, J.J. eds., The First Critique (Belmont, CA: Wadsworth 1969)Google Scholar and Robert Butts’ article of 1981.
29 Kant, CPR, A734=8762
30 Kant, CPR, A239=B298. For an excellent account of the Kantian interpretation of mathematical objects, see Robert Butts’ 1981 article.
31 Kant, CPR, A714=B742
32 Aristotle, I, 10, 76b
33 Kant, CPR A730=B758
34 Kant, CPR Bxviii
35 Kant, CPR Bxii. Hintikka rightly places much emphasis on this passage.
36 Kant, CPR, A713-B741
37 Kant, CPR, A716=B744
38 Kant, CPR, A716=B744
39 See Kant, CPR, A734 = B762: ‘The concepts … are presented in intuition; and this method, in addition to its heuristic advantages, secures all inferences against error by setting each one before our eyes.’
40 Kant, CPR, A734=B762
41 Prize Essay, tr. Walford, D.E. in Kant: Selected Pre-Critical Writings (Manchester: Manchester University Press 1968), 24Google Scholar
42 Kant, Prize Essay, 23. Compare p. 25: ‘It is known from experience, that even outside mathematics, we can in many cases be perfectly certain, to the extent of conviction, by means of rational argument.’
43 Kant, CPR, A734=B762
44 Russell, Bertrand Introduction to Mathematical Philosophy (London: Allen and Unwin 1919), 145.Google Scholar
45 Walker, Ralph kant (London: Routledge and Kegan Paul 1978), 65Google Scholar
46 There are other possibilities. Consider Strawson's appeal to Kant's so-called phenomenal space in his Bounds of Sense, Part 5 (London: Methuen 1966). Frequently, the interpretation of Kant's theory of construction in geometry is guided by an interpretation of his theory of space. I cannot see the logical connections as being sufficiently strong for this strategy. Hintikka's work, too, makes no appeal to the theory of space in this context.
47 One must wonder how this mode of interpretation can handle the problems issuing from the particularity of the intuitions. Kant cannot allow the intuitions to be like Lockean abstract ideas. And, recalling Walker's example of the parallels postulate, one wonders what might happen if another point were chosen; some guarantee is required that future choices of points will not produce a plurality of parallel lines, and this guarantee cannot be got by inspecting individual intuitions.
48 Kant, CPR, A729=B757f
49 Recall Kant's claims about the certainty of definitions — see note 22. One might object to the philosophical point using as a counterexample the history of Euler’s theorem concerning polyhedra. The controversy had much to do with the refutation of counterexamples on the grounds that they were not polyhedra, but were some other sort of creature. Kant would have to say that the concept had not yet been defined, and this would pose problems for his view that concepts are defined at the outset, and not arrived at by investigation, such as that magnificently laid out by lmre Lakatos in Proofs and Refutations (Cambridge: Cambridge University Press 1976).
50 Kant, CPR, Bxii
51 See note 41 and quotation.
52 See the preface to Metaphysical Foundations of Natural Science (Indianapolis, IN: Bobbs-Merrill, 1970), 6: ‘I maintain, however, that in every special doctrine of nature only so much natural science proper can be found as there is mathematics in it.’
