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HINTIKKA ON KANT'S MATHEMATICAL METHOD Emily Carson De Boeck Supérieur | « Revu

HINTIKKA ON KANT'S MATHEMATICAL METHOD Emily Carson De Boeck Supérieur | « Revue internationale de philosophie » 2009/4 n° 250 | pages 435 à 449 ISSN 0048-8143 ISBN 9782930560014 DOI 10.3917/rip.250.0435 Article disponible en ligne à l'adresse : -------------------------------------------------------------------------------------------------------------------- https://www.cairn.info/revue-internationale-de-philosophie-2009-4-page-435.htm -------------------------------------------------------------------------------------------------------------------- Distribution électronique Cairn.info pour De Boeck Supérieur. © De Boeck Supérieur. Tous droits réservés pour tous pays. La reproduction ou représentation de cet article, notamment par photocopie, n'est autorisée que dans les limites des conditions générales d'utilisation du site ou, le cas échéant, des conditions générales de la licence souscrite par votre établissement. Toute autre reproduction ou représentation, en tout ou partie, sous quelque forme et de quelque manière que ce soit, est interdite sauf accord préalable et écrit de l'éditeur, en dehors des cas prévus par la législation en vigueur en France. Il est précisé que son stockage dans une base de données est également interdit. Powered by TCPDF (www.tcpdf.org) © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) Hintikka on Kant’s mathematical method CARSON With a series of important papers in the 1960s, Jaakko Hintikka initiated a surge of interest in Kant’s philosophy of mathematics, developing an interpreta- tion which has in some ways deﬁ ned the ﬁ eld. Because I can’t do justice to all of the details of Hintikka’s view here, I propose to stand back and consider the broader context. In a later paper revisiting the topic, Hintikka complained that subsequent discussions of his interpretation of Kant paid no attention to “the overall picture of Kant’s thinking about mathematics in its historical setting or to my view of the role of Kant’s theory of space, time, and mathematics (including the whole of his transcendental aesthetics) within the structure of his philosophical system”1. In the intervening years, Kant scholars have taken up these two tasks, so it seems worthwhile to consider the implications of this work for Hintikka’s interpretation of Kant’s theory of mathematics. The key question at issue is the role of intuition in Kant’s philosophy of mathematics. On the logical approach developed by Hintikka, the primary role of intuition is formal or inferential. By following Hintikka’s suggestion and reconsidering the historical setting — both philosophical and mathematical — and the structure of Kant’s philosophical system, I want to suggest that we can make a case for a richer role for intuition in Kant’s philosophy of mathematics than the merely formal one.2 1. Hintikka’s view Hintikka’s reading takes as its starting point the account of mathematical method as presented in the Prize Essay of 1763 where Kant compares the certainty of metaphysical cognition with that of mathematical cognition. Hintikka takes Kant’s claim that “mathematics, in its analyses, proofs and inferences exam- ines the universal under signs in concreto” whereas “philosophy examines the 1. Hintikka [1981], pp.201-215. 2. As Hintikka recognizes, the formal role of intuition is compatible with its playing other roles (ibid., p. 213), so the following considerations are not intended to undermine his positive arguments for the formal role for intuition. © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) CARSON 436 universal by means of signs in abstracto” [2:279]3 to mean that for Kant “one of the main peculiarities of the mathematical method is to consider particular repre- sentatives of general concepts”.4 To say that mathematics uses general concepts ‘in concreto’ is, for Hintikka, just to say that it uses general concepts “in the form of individual instances”; more speciﬁ cally, it is to say that mathematical reasoning involves the introduction of individuals. The auxiliary constructions in Euclidean geometrical proofs — Kant’s favourite examples of mathematical demonstrations — are paradigmatic examples of such singular representations resulting from existential instantiations. According to Hintikka, Kant holds this view of the distinctive feature of mathematics even in the Critique of Pure Reason. The claim that mathematics proceeds by constructing its concepts in pure intuition is simply another way of saying that in mathematical reasoning, one must introduce particular representa- tives of general concepts. To construct a concept is, Kant says, to exhibit apriori an intuition corresponding to the concept; this, on Hintikka’s view, “is tanta- mount to the transition from a general concept to an intuition which represents the concept, provided this is done without recourse to experience”.5 The salient feature of intuitions for this reading is that they are individual representations; indeed, Hintikka takes this to be the essential feature of intuitions for Kant: In Kant and his immediate predecessors, the term ‘intuition’ did not neces- sarily have anything to do with appeal to imagination or to direct perceptual evidence. In the form of a paradox, we may perhaps say that the ‘intuitions’ Kant contemplated were not necessarily very ‘intuitive’. For Kant, an intui- tion is simply anything which represents or stands for an individual object as distinguished from general concepts.6 Hintikka believes that this is precisely how Kant deﬁ ned the notion of intui- tion: for example, at A320/B376-7, Kant says that “a cognition…is either an intuition or a concept”; an intuition “is immediately related to the object and is singular”, a concept is “mediate, by means of a mark which can be common 3. All references to Kant’s writings, except references to the Critique of Pure Reason, are given by volume and page number of the Akademie edition of Kant’s gesammelte Schriften, Georg Reimer, 1910-; the Critique of Pure Reason is cited by the standard A and B pagination of the ﬁ rst (1781) and second (1787) editions respectively. Translations of the pre-Critical works are from Kant: Theoretical Philosophy 1755-1790, David Walford (ed. and tr.) in collaboration with Ralf Meerbote, Cambridge: Cambridge University Press 1992. Translations of the Critique are from Critique of Pure Reason, Paul Guyer and Allen W. Wood (ed. and tr.), Cambridge University Press, 1998. 4. Hintikka [1967], p.25. 5. Ibid., p.21. 6. Hintikka [1965], p.130. © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) HINTIKKA ON KANT’S MATHEMATICAL METHOD 437 to several things”. Although Kant here seems to present two criteria which an intuition must satisfy, singularity and immediacy, Hintikka argues that the immediacy condition is merely a corollary of the singularity condition. Since what makes a concept mediate is that it refers to its object only by means of characteristics which several objects may share, any representation which does not refer to its object in this way is immediate; singular representations do not refer to objects in this way; therefore singular representations are immediate. Kant’s notion of intuition is then, on this view, “not very far from what we would call a singular term”.7 Construction in pure intuition in the Critique amounts to the introduction of a singular term into an argument. So in both the Prize Essay and the ﬁ rst Critique, the distinctive feature of mathematics is simply that mathematical reasoning involves the introduction of particular representatives of general concepts. This is all Kant means when in the Prize Essay he refers to the “examination of the universal under signs in concreto”, and it is all that he means when in the ﬁ rst Critique he speaks of “construction in pure intuition”. This reading of Kant’s discussion in the Prize Essay of the distinctive role of signs in mathematics derives some support from Kant’s claim there that he appeals “ﬁ rst of all...to arithmetic, both the general arithmetic of indeterminate magnitudes, and the arithmetic of numbers” [2:278, my emphasis]: In both kinds of arithmetic, there are posited ﬁ rst of all not things them- selves but their signs, together with the special designations of their increase or decrease, their relations, etc. Thereafter, one operates with these signs according to easy and certain rules, by means of substitution, combination, subtraction, and many kinds of transformation, so that the things signi- ﬁ ed are themselves completely forgotten in the process, until eventually, when the conclusion is drawn, the meaning of the symbolic conclusion is deciphered. Hintikka takes Kant in this passage to be focusing on the use of free variables in algebra. The signs stand for individual numbers; by concatenating these signs with the “designations of their relations etc”, we form expressions which also stand for individual numbers. Nonetheless, the derivations of algebraic equations by means of “easy and certain rules” serve to establish general propositions. In this way, as Hintikka puts it, “in mathematical arguments general concepts are considered by means of their individual representatives”, and this is precisely what Hintikka takes the use of intuitions to be: “the use of variables in algebra was for [Kant] an ‘intuitive’ method because these variables were thought of 7. Hintikka [1969], p.43. © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) © De Boeck Supérieur | Téléchargé le 27/06/2022 sur www.cairn.info (IP: 161.9.54.17) CARSON 438 as standing for (unspeciﬁ uploads/Philosophie/ hintikka-on-kant-x27-s-mathematical-method.pdf