Yogyakarta State University
According to Wilder R.L., Kant's philosophy of mathematics can be interpreted in a constructivist manner and constructivist ideas that presented in the nineteenth century-notably by Leopold Kronecker, who was an important for a runner of intuition¬ism-in opposition to the tendency in mathematics toward set-theoretic ideas, long before the paradoxes of set theory were discovered. In his philosophy of mathematics , Kant
supposed that arithmetic and geometry comprise synthetic a priori judgments and that natural science depends on them for its power to explain and predict events. As synthetic a priori judgments , the truths of mathematics are both informative and necessary; and since mathematics derives from our own sensible intuition, we can be absolutely sure that it must apply to everything we perceive, but for the same reason we can have no assurance that it has anything to do with the way things are apart from our perception of them.
Kant believes that synthetic a priori propositions include both geometric propositions arising from innate spatial geometric intuitions and arithmetic propositions arising from innate intuitions about time and number. The belief in innate intuitions about space was discredited by the discovery of non-Euclidean geometry, which showed that alternative geometries were consistent with physical reality. Kant perceives that mathematics is about the empirical world, but it is special in one important way. Necessary properties of the world are found through mathematical proofs. To prove something is wrong, one must show only that the world could be different. While , sciences are basically generalizations from experience, but this can provide only contingent and possible properties of the world. Science simply predicts that the future will mirror the past.
In his Critic of Pure Reason Kant defines mathematics as an operation of reason by means of the construction of conceptions to determine a priori an intuition in space (its figure), to divide time into periods, or merely to cognize the quantity of an intuition in space and time, and to determine it by number. Mathematical rules , current in the field of common experience, and which common sense stamps everywhere with its approval, are regarded by them as mathematical axiomatic. According to Kant , the march of mathematics is pursued from the validity from what source the conceptions of space and time to be examined into the origin of the pure conceptions of the understanding. The essential and distinguishing feature of pure mathematical cognition among all other a priori cognitions is, that it cannot at all proceed from concepts, but only by means of the construction of concepts.
Kant conveys that mathematical judgment must proceed beyond the concept to that which its corresponding visualization contains. Mathematical judgments neither can, nor ought to, arise analytically, by dissecting the concept, but are all synthetical. From the observation on the nature of mathematics, Kant insists that some pure intuition must form mathematical basis, in which all its concepts can be exhibited or constructed, in concreto and yet a priori. Kant concludes that synthetical propositions a priori are possible in pure mathematics, if we can locate this pure intuition and its possibility. The intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodictic and necessary are Space and Time. For mathematics must first have all its concepts in intuition, and pure mathematics in pure intuition, it must construct them. Mathematics proceeds, not analytically by dissection of concepts, but synthetically; however, if pure intuition be wanting, it is impossible for synthetical judgments a priori in mathematics.
The basis of mathematics actually are pure intuitions, which make its synthetical and apodictically valid propositions possible. Pure Mathematics, and especially pure geometry, can only have objective reality on condition that they refer to objects of sense. The propositions of geometry are not the results of a mere creation of our poetic imagination, and that therefore they cannot be referred with assurance to actual objects; but rather that they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing else than the form of all external appearances, and it is this form alone where objects of sense can be given. The space of the geometer is exactly the form of sensuous intuition which we find a priori in us, and contains the ground of the possibility of all external appearances. In this way geometry be made secure, for objective reality of its propositions, from the intrigues of a shallow metaphysics of the un-traced sources of their concepts.
Kant argues that mathematics is a pure product of reason, and moreover is thoroughly synthetical. Next, the question arises: Does not this faculty, which produces mathematics, as it neither is nor can be based upon experience, presuppose some ground of cognition a priori, which lies deeply hidden, but which might reveal itself by these its effects, if their first beginnings were but diligently ferreted out? However, Kant found that all mathematical cognition has this peculiarity: it must first exhibit its concept in a visual intuition and indeed a priori, therefore in an intuition which is not empirical, but pure. Without this mathematics cannot take a single step; hence its judgments are always visual, viz., intuitive; whereas philosophy must be satisfied with discursive judgments from mere concepts, and though it may illustrate its doctrines through a visual figure, can never derive them from it.
Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.205
3 Wegner, P., 2004, “Modeling, Formalization, and Intuition.” Department of Computer Science. Retrieved 2004
4 Posy, C. ,1992, “Philosophy of Mathematics”, Retreived 2004
6 Kant, I., 1781, “The Critic Of Pure Reason: SECTION III. Of Opinion, Knowledge, and Belief; CHAPTER III. The Arehitectonic of Pure Reason” Translated By J. M. D. Meiklejohn, Retrieved 2003
8 Kant, I, 1783, Prolegomena To Any Future Methaphysics, Preamble, p. 19
9 Ibid. p. 21
10 Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect. 7”, Trans. Paul Carus. Retrieved 2003
12Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect.10”, Trans. Paul Carus. Retrieved 2003
15Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect.12 Trans. Paul Carus. Retrieved 2003
16Kant, I, 1783, “Prolegomena to Any Future Metaphysic: REMARK 1 Trans. Paul Carus. Retrieved 2003
19Wikipedia The Free Encyclopedia. Retrieved 2004
21Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect. 6. p. 32
22Immanuel Kant, Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect. 7.p. 32