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<ul><li><p>Hegeler Institute</p><p>KANT ON THE MATHEMATICAL METHODAuthor(s): Jaakko HintikkaSource: The Monist, Vol. 51, No. 3, Kant Today: Part I (JULY, 1967), pp. 352-375Published by: Hegeler InstituteStable URL: http://www.jstor.org/stable/27902038 .Accessed: 16/09/2013 16:29</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .</p><p>JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>Hegeler Institute is collaborating with JSTOR to digitize, preserve and extend access to The Monist.</p><p>http://www.jstor.org </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>KANT ON THE MATHEMATICAL METHOD </p><p>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. </p><p>The characterization is given at the end of the Critique of Pure Reason in the first chapter of the Transcendental Doctrine of Method (A 713 = B 741) In this chapter Kant proffers a number of further observations on the subject of the mathematical method. These remarks have not been examined very intensively by most students of Kant's writings. Usually they have been dealt with as a sort of appendix to Kant's better known views on space and time, </p><p>presented in the Transcendental Aesthetic. In this paper, I also want to call attention to the fact that the relation of the two parts of the first Critique is to a considerable extent quite different from the usual conception of it. </p><p>To come back to Kant's characterization: the first important term it contains is the word 'construction*. This term is explained by Kant by saying that to construct a concept is the same as to </p><p>exhibit, a priori, an intuition which corresponds to the concept.2 Construction, in other words, is tantamount to the transition from a general concept to an intuition which represents the </p><p>concept, provided that this is done without recourse to experience. How is this term 'construction* to be understood? It is not </p><p>surprising to meet it in a theory of mathematics, for it had in Kant's time an established use in at least one part of mathematics, </p><p>lin referring to the Critique of Pure Reason, I shall use the standard con </p><p>ventions A = first edition (1781), B = second edition (1787). All decent edi </p><p>tions and translations give the pagination of one or both of these editions. </p><p>In rendering passages of the first Critique in English, I shall normally follow </p><p>Norman Kemp Smith's translation (London and New York: Macmillan Co., </p><p>1929). 2Loc. cit. </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>KANT ON THE MATHEMATICAL METHOD 353 </p><p>viz. in geometry. It is therefore natural to assume that what Kant </p><p>primarily has in mind in the passage just quoted are the con structions of geometers. And it may also seem plausible to say that the reference to intuition in the definition of construction is calculated to prepare the ground for the justification of the use of such constructions which Kant gives in the Transcendental Aesthetic. What guarantee, if any, is there to make sure that the </p><p>geometrical constructions are always possible? Newton had seen the only foundation of geometrical constructions in what he called 'mechanical practice* (see the preface to Principia). But if this is so, then the certainty of geometry is no greater than the certainty of more or less crude 'mechanical practice*. It may seem natural that Kant's appeal to intuition is designed to furnish a better foundation to the geometrical constructions. There is no need to construct a figure on a piece of paper or on the blackboard, Kant may seem to be saying. All we have to do is to represent the </p><p>required figure by means of imagination. This procedure would be </p><p>justified by the outcome of the Transcendental Aesthetic, if this can be accepted. For what is allegedly shown there is that all the </p><p>geometrical relations are due to the structure of our sensibility (our perceptual apparatus, if you prefer the term) ; for this reason </p><p>they can be represented completely in imagination without any </p><p>help of sense-impressions. This interpretation is the basis of a frequent criticism of Kant's </p><p>theory of mathematics. It is said, or taken for granted, that con structions in the geometrical sense of the word can be dispensed with in mathematics. All we have to do there is to carry out certain logical arguments which may be completely formalized in terms of modern logic. The only reason why Kant thought that </p><p>mathematics is based on the use of constructions was that con </p><p>structions were necessary in the elementary geometry of his day, derived in most cases almost directly from Euclid's Elementa. But this was only an accidental peculiarity of that system of ge ometry. It was due to the fact that Euclid's set of axioms and </p><p>postulates was incomplete. In order to prove all the theorems he wanted to prove, it was therefore not sufficient for Euclid to </p><p>carry out a logical argument. He had to set out a diagram or </p><p>figure so that he could tacitly appeal to our geometrical intuition which in this way could supply the missing assumptions which he </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>354 THE MONIST </p><p>had omitted. Kant's theory of mathematics, it is thus alleged, arose by taking as an essential feature of all mathematics some </p><p>thing which only was a consequence of a defect in Euclid's par ticular axiomatization of geometry.3 </p><p>This interpretation, and the criticism based on it, is not with out relevance as an objection to Kant's full-fledged theory of space, time, and mathematics as it appears in the Transcendental Aesthet ic. It seems to me, however, that it does less than justice to the way in which Kant actually arrived at this theory. It does not take a sufficient account of Kant's precritical views on mathe </p><p>matics, and it even seems to fail to make sense of the arguments by means of which Kant tried to prove his theory. Therefore it does not give us a chance of expounding fully Kant's real argu ments for his views on space, time and mathematics, or of criticiz </p><p>ing them fairly. It is not so much false, however, as too narrow. We begin to become aware of the insufficiency of the above </p><p>interpretation when we examine the notion of construction somewhat more closely. The definition of this term makes use of the notion of intuition. We have to ask, therefore: What did Kant </p><p>mean by the term 'intuition'? How did he define the term? What is the relation of his notion of intuition to what we are accustomed to associate with the term? </p><p>The interpretation which I briefly sketched above assimilates Kant's notion of an a priori intuition to what we may call mental </p><p>pictures. Intuition is something you can put before your mind's </p><p>eye, something you can visualize, something you can represent to your imagination. This is not at all the basic meaning Kant himself wanted to give to the word, however. According to his definition, presented in the first paragraph of his lectures on logic, every particular idea as distinguished from general concepts is an intuition. Everything, in other words, which in the human mind </p><p>3 A paradigmatic statement of this view occurs in Bertrand Russell's Introduc tion to Mathematical Philosophy (London: George Allen and Unwin, 1919) , </p><p>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 'intui tion'." Needless to say, there does not seem to be a scrap of evidence for attrib </p><p>uting to Kant the 'observation* Russell mentions. </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>KANT ON THE MATHEMATICAL METHOD 355 </p><p>represents an individual is an intuition. There is, we may say, nothing 'intuitive* about intuitions so defined. Intuitivity means </p><p>simply individuality.4 Of course, it remains true that later in his system Kant came to </p><p>make intuitions intuitive again, viz. by arguing that all our human intuitions are bound up with our sensibility, i.e., with our faculty of sensible perception. But we have to keep in mind that this connection between intuitions and sensibility was never taken </p><p>by Kant as a mere logical consequence of the definition of intuition. On the contrary, Kant insists all through the Critique of Pure Reason that it is not incomprehensible that other beings might have intuitions by means other than senses.5 The connection be tween sensibility and intuition was for Kant something to be </p><p>proved, not something to be assumed.6 The proofs he gave for </p><p>assuming the connection (in the case of human beings) are pre sented in the Transcendental Aesthetic. Therefore, we are entitled to assume the connection between sensibility and intuitions only in those parts of Kant's system which are logically posterior to the Transcendental Aesthetic. </p><p>My main suggestion towards an interpretation of Kant's theory of the mathematical method, as presented at the end of the first </p><p>Critique, is that this theory is not posterior but rather systematic ally prior to the Transcendental Aesthetic. If so, it follows that, within this theory, the term 'intuition* should be taken in the 'unintuitive* sense which Kant gave to it in his definition of the </p><p>4 See e.g. Kant's Dissertation of 1770, section 2, 10; Critique of Pure Reason A 320 = B 376-377; Prolegomena 8. Further references are given by H. </p><p>Vaihinger in his Commentar zu Kants Kritik der reinen Vernunft (Stuttgart: W. Spemann, 1881-1892), Vol. 2, pp. 3, 24. Cf. also c.ch. E. Schmid, W rterbuch zum leichteren Gebrauch der Kantischen Schriften (4th ed., Jena: Croker, </p><p>1798) on Anschauung. 5 "We cannot assert of sensibility that it is the sole possible kind of intuition" </p><p>(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. </p><p>6 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 (London: George Allen and Unwin, 1936), Vol. I, pp. 93-94. </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>356 THE MONIST </p><p>notion. In particular, Kant's characterization of mathematics as based on the use of constructions has to be taken to mean merely that, in mathematics, one is all the time introducing particular representatives of general concepts and carrying out arguments in terms of such particular representatives, arguments which cannot be carried out by the sole means of general concepts. For if Kant's </p><p>methodology of mathematics is independent of his proofs for con </p><p>necting intuitions and sensibility in the Aesthetic and even prior to it, then we have, within Kant's theory of the method of mathe* </p><p>matics, no justification whatsoever for assuming such a connec </p><p>tion, i.e., no justification for giving the notion of intuition any meaning other than the one given to it by Kant's own definitions. </p><p>There are, in fact, very good reasons for concluding that the discussion of the mathematical method in the Doctrine of Method is prior to, and presupposed by, Kant's typically critical discussion of space and time in the Transcendental Aesthetic. One of them should be enough: in the Prolegomena, in the work in which Kant wanted to make clear the structure of his argument, he explicitly appeals to his discussions of the methodology of mathematics at the end of the Critique of Pure Reason in the beginning and dur </p><p>ing the argument which corresponds to the Transcendental Aesthet </p><p>ic, thus making the dependence of the latter on the former </p><p>explicit. This happens both when Kant discusses the syntheticity of mathematics (Academy edition of Kant's works, Vol. 4, p. 272) and when he discusses its intuitivity (ibid. p. 281; cf. p. 266). </p><p>Another persuasive reason is that at critical junctures Kant in the Transcendental Aesthetic means by intuitions precisely what his own definitions tell us. For instance, he argues about space as follows: "Space is not a. .. general concept of relations of things in </p><p>general, but a pure intuition. For ... we can represent to ourselves </p><p>only one space. . . . Space is essentially one; the manifold in it, and therefore the general concept of spaces, depends solely on the introduction of limitations. Hence it follows that an . . . intuition underlies all concepts of space" (A 24-25 = B 39). Here intuitivity is inferred directly from individuality, and clearly means nothing </p><p>more than the latter. But I am afraid that, however excellent reasons there may be </p><p>for reversing the order of Kant's exposition in the first Critique and for putting the discussion of mathematics in the Methodenlehre </p><p>This content downloaded from on Mon, 16 Sep 2013 16:29:39 PMAll use subject to JSTOR Terms and Conditions</p></li><li><p>KANT ON THE MATHEMATICAL METHOD 357 </p><p>before the Transcendental Aesthetic, my readers are still likely to be incredulous. Could Kant really have meant nothing more than this by his characterization of the mathematical method? </p><p>Could he have thought that it is an important peculiarity of the method of mathematicians as distinguished from the method of </p><p>philosophers that the mathematicians make use of special cases of general concepts while philosophers do not? Isn't suggesting this to press Kant's definition of intuition too far? </p><p>The answer to this is, I think, that there was a time when Kant did believe that one of the main peculiarities of the mathe matical method is to consider particular representatives of general concepts. 7 This view was presented in the precritical prize essay of the year 1764. Its interpretation is quite independent of the </p><p>interpretation of Kant's critical writings. In particular, the formu lation of this precritical theory of Kant's does not turn on the notion of intuition at all. It follows, therefore, that the idea of the mathematical method as being based on the use of general con </p><p>cept in concreto, i.e., in the form of individ...</p></li></ul>