Abstract
According to Kant, “mathematical knowledge is the knowledge gained by reason from the construction of concepts.” In this paper, I shall make a few suggestions as to how this characterization of the mathematical method is to be understood.
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Notes
In referring to the Critique of Pure Reason, I shall use the standard conventions A = first edition (1781), B = second edition (1787). All decent editions and translations give the pagination of one or both of these editions. In rendering passages of the first Critique in English, t shall normally follow Norman Kemp Smith’s translation (Macmillan, London and New York, 1933).
A paradigmatic statement of this view occurs in Bertrand Russell’s Introduction to Mathematical Philosophy (George Allen and Unwin, London, 1919), p. 145: “Kant, having observed that the geometers of his day could not prove their theorems by unaided arguments, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called ‘intuition’.” Needless to say, there does not seem to be a scrap of evidence for attributing to Kant the ‘observation’ Russell mentions.
See, e.g., Kant’s Dissertation of 1770, section 2, § 10; Critique of Pure Reason A 320 = B 376377; Prolegomena §8. Further references are given by H. Vaihinger in his Commentar zu Kants Kritik der reinen Vernunft (W. Spemann, Stuttgart, 1881–1892), Vol. 2, pp. 3, 24. Cf. also C. C. E. Schmid, Kb - ter - burn zum leichteren Gebrauch der Kantischen Schriften (4th ed., Cröker, Jena, 1798) on Anschauung.
“We cannot assert of sensibility that it is the sole possible kind of intuition” (A 254 = B 310). Cf., e.g., A 27 = B 43, A 34–35 = B 51, A 42 = B 59, A 51 = B 75 and the characteristic phrase ‘uns Menschen wenigstens’ at B 33.
The opening remarks of the Transcendental Aesthetic seem to envisage a hard-and-fast connection between all intuitions and sensibility. As Paton points out, however, they have to be taken partly as a statement of what Kant wants to prove. See H. J. Paton, Kant’s Metaphysic of Experience (George Allen and Unwin, London, 1936), Vol. I, pp. 93–94.
This has been brought out clearly and forcefully by E. W. Beth, to whose writings on Kant I am greatly indebted, although I do not fully share Beth’s philosophical evaluation of Kant’s theories. See ‘Kants Einteilung der Urteile in analytische and synthetische’, Algemeen Nederlands Tijdschrift voor Wijshegeerte en Psychologie 46 (1953–54) 253–264; La crise de la raison et la logique (Gauthier-Villars, Paris, 1957); The Foundations of Mathematics (North-Holland Publishing Company, Amsterdam, 1959), pp. 41–47.
Proceedings of the Aristotelian Society 42 (1941–42) 1–24.
See the Academy Edition of Kant’s works, Vol. 2, p. 307. Concerning the Elementa, see Sir Thomas Heath’s translation and commentary The Thirteen Books of Euclid’s Elements (Cambridge University Press, Cambridge, 1926).
Heath, op. cit., Vol. 1, pp. 129–131.
We can see here that according to Kant the peculiarity of mathematics does not lie in the axioms and postulates of the different branches of mathematics, but in the mathematical mode of argumentation and demonstration.
We have to realize, however, that the mere difference of the directions in which one is proceeding in an analysis and in a synthesis, respectively, was sometimes emphasized at the expense of the questions whether constructions are used or not. One could thus distinguish between a ‘directional’ and a ‘constructional’ (or ‘problematic’) sense of analysis and synthesis. Cf. my paper, ‘Kant and the Tradition of Analysis’, in Deskription, Existenz and Analytizität, ed. by P. Weingartner (Pustet, Munich, 1966), reprinted as Chapter 9 of Jaakko Hintikka, Logic, Language-Games, and Information (Clarendon Press, Oxford, 1973).
Cf. also, Prolegomena, §5 (Academy Edition, Vol. 4, p. 276, note). We can also say that Kant’s remarks in effect serve to distinguish between the directional and the constructional (problematic) sense of analysis and synthesis, and to indicate that Kant opts for the latter. (See the preceding note and the article mentioned there.)
See La Géométrie, the first few statements (pp. 297–298 of the first edition).
Concerning the notion of ecthesis in Aristotle, see W. D. Ross, Aristotle’s Prior and Posterior Analytics: A Revised Text with Introduction and Commentary (Clarendon Press, Oxford, 1949), pp. 32–33, 412–414; Jan Lukasiewicz, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic (Clarendon Press, Oxford, 1951), pp. 59–67; Günther Patzig, Die Aristotelische Syllogistik (Vandenhoeck und Ruprecht, Göttingen, 1959), pp. 166–178; B. Einarson, ‘On Certain Mathematical Terms in Aristotle’s Logic’, American Journal of Philology, 57 (1936) 34–54, 151–172, esp. p. 161. As will be seen from these discussions, the precise interpretation of the Aristotelian notion of ecthesis (as used in his logic) is a controversial problem to which no unambiguous solution may be available. The interpretation which I prefer (and which I shall rely on here) assimilates logical ecthesis to the ‘existential instantiation’ of modem logic. I cannot argue for this interpretation as fully here as it deserves. For Aristotle’s use of the term ecthesis in geometry, which seems to me to be closely related to the logical ecthesis, cf., e.g., Analytica Priora I, 41 49b30–50a4 and Analytica Posteriora I, 10, 76b39–77a2.
I am here presupposing the interpretation mentioned in the preceding note. For further remarks on this interpretation, cf. my paper, ‘Are Logical Truths Analytic?’, Philosophical Review 74 (1965) 178–203, reprinted my Knowledge and the Known (Reidel, Dordrecht, 1974) and E. W. Beth’s discussion of the relation of ecthesis and modem logic in ‘Semantic Entailment and Formal Derivability’, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde, N. R., 18, no. 13 (Amsterdam, 1955), pp. 309–342.
Some remarks on these points are contained in my paper, ‘Kant Vindicated’, in Deskription, Existenz und Analytizität, ed. by P. Weingartner (Pustet, Munich, 1966), reprinted as Chapter 8 of Logic, Language-Games, and Information (note 12 above).
Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum Librum I Commentarium, ed. by M. Wallies, in Commentaria in Aristotelem Graeca, Vol. 2(a) (Berlin 1883), p. 32, cf. pp. 32–33, 99–100, 104; Lukasiewicz, op. cit., pp. 60–67. An attempt to explain and to justify the mathematical ecthesis from an Aristotelian point of view also easily gives rise to striking anticipations of Kantian doctrines. Thus we find, for instance, that according to Theophrastus mathematical objects “seem to have been, as it were, devised by us in the act of investing things with figures and shapes and ratios, and to have no nature in and of themselves... ” (Theophrastus, Metaphysica 4a18ff., pp. 308–309 Brandisii). Cf. also Anders Wedberg, Plato’s Philosophy of Mathematics (Almqvist and Wiksell, Stockholm, 1955), p. 89, who emphasizes that Aristotle likewise seems to anticipate some of the most salient features of Kant’s theory of mathematics.
This difficulty was emphasized by Kant’s early critics. For instance, J. G. E. Maas writes in his long paper, ‘Veber die transcendentale Aesthetik’, Philosophisches Magazin 1 (1788) 117–149, as follows, apropos Kant’s notion of an a priori intuition: “Hierbey kann ich (I) die Bemerkung nicht vorbeilassen, dass eine Anschauung a priori... nach Kants eigenen Erklärungen nicht denkbar sey. Eine Anschauung ist eine Vorstellung. Sollte sie a priori seyn, so müsste sie schlechterdings nicht vom Objecte hergenommen werden, und eine Anschauung ist doch nur möglich, sofern uns der Gegenstand gegeben wird, dieses aber ist widerum nur dadurch möglich, dass er das Gemüth auf gewisse Weise afficiere. Eine Anschauung a priori ist demnach unmöglich, und kann mithin auch in Ansehung des Raumes nicht zum Grunde liegen” (pp. 134–135). Maas does not realize, however, that the possibility of a successful use of a priori intuitions is precisely the problem Kant was trying to solve in the Transcendental Aesthetic.
In B xviii Kant says that he is “adopting as our new method of thought... the principle that we can know a priori of things only what we ourselves put into them.” Cf. also B xii-xiv. I have commented briefly on the historical background of this Kantian assumption in ‘Kant’s “New Method of Thought” and his Theory of Mathematics’, Ajatus 27 (1965) 37–47, reprinted in Knowledge and the Known (note 16), and in ‘Tieto on valtaa’, Valvoja 84 (1964) 185–196.
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Hintikka, J. (1992). Kant on the Mathematical Method. In: Posy, C.J. (eds) Kant’s Philosophy of Mathematics. Synthese Library, vol 219. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8046-5_2
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