Elegi Menggapai "Kant on the Construction of Mathematical Concepts and Cognition"

By Marsigit
Yogyakarta State University

In his Critic of Pure Reason, Kant ascribes that mathematics deals with conceptions applied to intuition. Mathematics is a theoretical sciences which have to determine their objects a priori. To demonstrate the properties of the isosceles triangle, it is not sufficient to meditate on the figure but that it is necessary to


produce these properties by a positive a priori construction. According to Kant, in order to arrive with certainty at a priori cognition, we must not attribute to the object any other properties than those which necessarily followed from that which he had himself placed in the object. Mathematician 1 occupies himself with objects and cognitions only in so far as they can be represented by means of intuition; but this circumstance is easily overlooked, because the said intuition can itself be given a priori, and therefore is hardly to be distinguished from a mere pure conception.
The conception of twelve 2 is by no means obtained by merely cogitating the union of seven and five; and we may analyze our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. Kant 3 says that we must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two-our five fingers, add the units contained in the five given in the intuition, to the conception of seven.
Further Kant states:

For I first take the number 7, and, for the conception of 5 calling in the aid of the fingers of my hand as objects of intuition, I add the units, which I before took together to make up the number 5, gradually now by means of the material image my hand, to the number 7, and by this process, I at length see the number 12 arise. That 7 should be added to 5, I have certainly cogitated in my conception of a sum = 7 + 5, but not that this sum was equal to 12. 4


Arithmetical propositions 5 are therefore always synthetical, of which we may become more clearly convinced by trying large numbers. For it 6 will thus become quite evident that it is impossible, without having recourse to intuition, to arrive at the sum total or product by means of the mere analysis of our conceptions, just as little is any principle of pure geometry analytical.
In a straight line between two points 7, the conception of the shortest is therefore more wholly an addition, and by no analysis can it be extracted from our conception of a straight line. Kant 8 sums up that intuition must therefore here lend its aid in which our synthesis is possible. Some few principles expounded by geometricians are, indeed, really analytical, and depend on the principle of contradiction. Further, Kant says:
They serve, however, like identical propositions, as links in the chain of method, not as principles- for example, a = a, the whole is equal to itself, or (a+b) > a, the whole is greater than its part. And yet even these principles themselves, though they derive their validity from pure conceptions, are only admitted in mathematics because they can be presented in intuition. 9

Kant (1781), in “The Critic Of Pure Reason: Transcendental Analytic, Book I, Analytic Of Conceptions. Ss 2” , claims that through the determination of pure intuition we obtain a priori cognitions of mathematical objects, but only as regards their form as phenomena. According to Kant, all mathematical conceptions, therefore, are not per se cognition, except in so far as we presuppose that there exist things which can only be represented conformably to the form of our pure sensuous intuition. Things 10, in space and time are given only in so far as they are perceptions i.e. only by empirical representation. Kant insists that the pure conceptions of the understanding of mathematics, even when they are applied to intuitions a priori , produce mathematical cognition only in so far as these can be applied to empirical intuitions. Consequently 11, in the cognition of mathematics, their application to objects of experience is the only legitimate use of the categories.
In “The Critic of Pure Reason: Appendix”, Kant (1781) elaborates that in the conceptions of mathematics, in its pure intuitions, space has three dimensions, and between two points there can be only one straight line, etc. They 12 would nevertheless have no significance if we were not always able to exhibit their significance in and by means of phenomena. It 13 is requisite that an abstract conception be made sensuous, that is, that an object corresponding to it in intuition be forth coming, otherwise the conception remains without sense i.e. without meaning. Mathematics 14 fulfils this requirement by the construction of the figure, which is a phenomenon evident to the senses; the same science finds support and significance in number; this in its turn finds it in the fingers, or in counters, or in lines and points. The mathematical 15 conception itself is always produced a priori, together with the synthetical principles or formulas from such conceptions; but the proper employment of them, and their application to objects, can exist nowhere but in experience, the possibility of which, as regards its form, they contain a priori.
Kant in “The Critic Of Pure Reason: SECTION I. The Discipline of Pure Reason in the Sphere of Dogmatism.”, propounds that, without the aid of experience, the synthesis in mathematical conception cannot proceed a priori to the intuition which corresponds to the conception. For this reason, none of these conceptions can produce a determinative synthetical proposition. They can never present more than a principle of the synthesis of possible empirical intuitions. Kant 16 avows that a transcendental proposition is, therefore, a synthetical cognition of reason by means of pure conceptions and the discursive method. Iit renders possible all synthetical unity in empirical cognition, though it cannot present us with any intuition a priori. Further, Kant 17 explains that the mathematical conception of a triangle we should construct, present a priori in intuition and attain to rational-synthetical cognition. Kant emphasizes the following:
But when the transcendental conception of reality, or substance, or power is presented to my mind, we find that it does not relate to or indicate either an empirical or pure intuition, but that it indicates merely the synthesis of empirical intuitions, which cannot of course be given a priori. 18

To make clear the notions, Kant sets forth the following:
Suppose that the conception of a triangle is given to a philosopher and that he is required to discover, by the philosophical method, what relation the sum of its angles bears to a right angle. He has nothing before him but the conception of a figure enclosed within three right lines, and, consequently, with the same number of angles. He may analyze the conception of a right line, of an angle, or of the number three as long as he pleases, but he will not discover any properties not contained in these conceptions. But, if this question is proposed to a geometrician, he at once begins by constructing a triangle. He knows that two right angles are equal to the sum of all the contiguous angles which proceed from one point in a straight line; and he goes on to produce one side of his triangle, thus forming two adjacent angles which are together equal to two right angles. 19

Mathematical cognition 20 is cognition by means of the construction of conceptions. The construction of a conception is the presentation a priori of the intuition which corresponds to the conception. Mathematics 21 does not confine itself to the construction of quantities, as in the case of geometry. It occupies itself with pure quantity also, as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity. In algebra 22, a certain method of notation by signs is adopted, and these indicate the different possible constructions of quantities, the extraction of roots, and so on. Mathematical cognition 23 can relate only to quantity in which it is to be found in its form alone, because the conception of quantities only that is capable of being constructed, that is, presented a priori in intuition; while qualities cannot be given in any other than an empirical intuition.

References:

Kant, I., 1781, “The Critic Of Pure Reason: Preface To The Second Edition”, Translated By J. M. D. Meiklejohn, Retrieved 2003
2 Ibid.
3 Ibid.
4 Ibid.
5 Ibid.
6 Ibid.
7 Ibid.
8 Ibid.
9 Ibid.
10 Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Analytic, Book I, Analytic Of Conceptions. Ss 2”, Translated By J. M. D. Meiklejohn, Retrieved 2003).
11Ibid.
12Kant, I., 1781, “The Critic Of Pure Reason: Appendix.”, Translated By J. M. D. Meiklejohn, Retrieved 2003
13 Ibid.
14Ibid.
15Ibid.
16Kant, I., 1781, “The Critic Of Pure Reason: SECTION I. The Discipline of Pure Reason in the Sphere of Dogmatism.”, Translated By J. M. D. Meiklejohn, Retrieved 2003
17Ibid.
18Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method; Chapter I. The Discipline Of Pure Reason, Section I. The Discipline Of Pure Reason In The Sphere Of Dogmatism”, Translated By J. M. D. Meiklejohn, Retrieved 2003
19Ibid.
20Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method, Chapter I, Section I .”, Translated By J. M. D. Meiklejohn, Retrieved 2003).
21Ibid.
22Ibid.
23Kant, I., 1781, “The Critic Of Pure Reason: SECTION I. The Discipline of Pure Reason in the Sphere of Dogmatism.”, Translated By J. M. D. Meiklejohn, Retrieved 2003)