Oct 13, 2012

Kant’s Theory of Knowledge Contributes to Epistemological Foundation of Mathematics




By Marsigit

As Mayer, F., notes that Kant's fundamental questions concerning epistemology covers how are synthetical judgments a priori possible and the solution of that problem; and comprehending the possibility of the use of pure reason in the foundation and construction of all sciences, including mathematics; as well as concerning the solution of this problem depending on the existence or downfall of the science of metaphysics.


 According to Kant. , in a system of absolute, certain knowledge can be erected only on a foundation of judgments that are synthetical and acquired independently of all experience. While, Hegel, G.W.F (1873) indicates that Kant's epistemology does not seek to obtain knowledge of the object itself, but sought to clarify how objective truthfulness can be obtained, as he named it the "transcendental method."
On the other hand, Distante P. recites that epistemologically, Kant attempts a compromise between empiricism and rationalism. According to Distante P. , Kant agrees with the rationalists that one can have exact and certain knowledge, but he followed the empiricists in holding that such knowledge is more informative about the structure of thought than about the world outside of thought. Further, he indicates that Kant restricts knowledge to the domain of experience, but attributes to the mind a function in incorporating sensations into the structure of experience. This structure could be known a priori without resorting to empirical methods. According to Kant , mathematics has often been presented as a paradigm of precision and certainty. It , therefore, concerns the way to know the truth of mathematical propositions, the applications of abstract mathematics in the real world and the implications of mathematics for the information revolution, as well as the contributions of mathematics. It leads us to examine mathematics as a primary in¬stance of what philosophers have called a priori knowl¬edge.
Steiner R. (2004) thought that in the epistemological sense, Kant has established the a priori nature of mathematical principles, however, all that the Critique of Pure Reason attempts to show that mathematics is a priori sciences. From this, it follows that the form of all experiences must be inherent in the subject itself. Therefore , the only thing left that is empirically given is the material of sensations. This is built up into a system of experiences, the form of which is inherent in the subject. Kant maintains that mathematics is synthetic a priori. If mathematical truths are known, where can we find the basis or grounding of their status as knowledge? The only possibility for knowledge of claims, that are not based on definitions, are universal and go beyond experience as if there is synthetic a priori knowledge.
Hersh R. (1997) assigns that Kant's fundamental presupposition is that contentful knowledge indepen¬dent of experiences, can be established on the basis of universal human intuition. While Mayer, F. (1951) indicates that based on apodictic knowledge forms as the foundation of his philosophy, Kant made it clear that mathematics, as universal scientific knowledge, de¬pends on synthetic judgments a priori; and claims that synthetic a priori judgments are the foundation of mathematics Again, Wilder R.L. (1952) ascertains Kant that mathematical judgments, at least the most characteristic ones, were synthetic, rather than analytic; and argues that mathematics is a pure product of reason, and moreover is thoroughly synthetical. . However, Posy C. indicates that according to Kant, mathematics is about the empirical world; it is special in one important way that 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.
Kant’s theory of knowledge states that mathematics is basically generalizations from experience, but this can provide only contingent of the possible properties of the world. Mathematics is about the empirical world, but usually methods for deriving knowledge give contingent knowledge, not the necessity that pure mathematics gives us. Kant wants necessary knowledge with empirical knowledge, while confirming that the objects in the empirical world are appearances or phenomenon and therefore we come to know them only from experiences. According to Kant , in order to know the properties of mathematical objects we need to build into our minds two forms of intuition and perception in such away that every perception we have is conceived by these forms i.e. space and time. These are, in fact, parts of the mind, and not some-thing the mind picks up from experience; thus, empirical objects are necessarily spatial-temporal objects.
Still, Posy C. (1992) indicates that Kant insists mathematics as the studying of the abstract form of perception or, in other words, mathematics is simply the science that studies the spatial-temporal properties of objects. Bolzano B. learns Kant’s observation that the principle of sufficient reason and the majority of propositions of arithmetic are synthetic propositions; however, who does not feel how artificial it is, has to assert that these propositions are based on intuitions. Kant claims that, in geometry, there are certain underlying intuitions; for in fact, many people may think that the concept of point is the intuition of a point before our eyes. However , the picture accompanying our pure concept of the point is not connected with it but only through the association of ideas; in fact, we have often thought both of them together.
Bolzano B , on the other hand, claims that the nature of this geometrical picture is different with different people; it is determined by thousands of fortuitous circum¬stances. However, Kant adds that if we had always seen just roughly and thickly drawn lines or had always represented a straight line by chains or sticks, we would have in mind with the idea of a line i.e. the image of a chain or a stick. Kant said: “With the word 'triangle' one always has in mind an equilateral triangle, another a right-angled triangle, a third perhaps an obtuse-angled triangle”. According to Kant , mathematical judgments are all synthetical; however he argues that this fact seems hitherto to have altogether escaped the observation of those who have analyzed human reason. It even seems directly opposed to all their conjectures, though incontestably certain, and most important in its consequences.
Kant in “Prolegomena to Any Future Metaphysics”, claims that the conclusions of mathematicians proceed according to the law of contradiction, as is demanded by all apodictic certainty. Kant says that it is a great mistake for men persuaded themselves that the fundamental principles were known from the same law. Further, Kant argues that the reason that for a synthetical proposition can indeed be comprehended according to the law of contradiction but only by presupposing another synthetical proposition from which it follows. Further, Kant argues that all principles of geometry are no less analytical; and that the proposition “a straight line is the shortest path between two points” is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality.
Kant claims that the attribute of shortness is therefore altogether additional, and cannot be obtained by any analysis of the concept; and its visualization must come to aid us; and therefore, it alone makes the synthesis possible. Kant then confronts the previous geometers assumption which claims that other mathematical principles are indeed actually analytical and depend on the law of contradiction. Kant strives to show that identical propositions such as “a=a”, “ the whole is equal to itself”, or “a + b > a”, “the whole is greater than its part”, etc, is a method of concatenation, and not the principles. Kant then claims that although they are recognized as valid from mere concepts, they are only admitted in mathematics, because they can be represented in some visual form. Hersh R. reveals that Kant's theory of spatial intuition means Euclidean geometry was inescapable. But the establishment of non-Euclidean geometry gives us choices. While Körner says Kant didn't deny the abstract conceivability of non-Euclidean geometries; he thought they could never be realized in real time and space
It may need to hold Faller’s notions that Kant's theory of knowledge most significantly contributes to the foundation of mathematics by its recognition that mathematical knowledge holds that synthetic a priori judgments were possible. Kant recognizes that mathematical knowledge seems to bridge the a priori analytic and a posteriori synthetic. According to Kant, mathematical thinking is a priori in the universality, necessity of its results and synthetic in the expansively promise of its inquiry. Particularly, Wilder R.L.(1952) highlights that Kant's view enables us to obtain a more accurate picture of the role of intuition in mathematics. However, at least as de¬veloped above, it is not really satisfying, because it takes more or less as a fact our ability to place our perceptions in a mathematically defined structure and to see truths about this structure by using perceptible objects to symbolize it.
According to Wilder R.L. , Kant’s restriction his discussion to parts of cognition could ground such knowledge to epistemological elaboration of the basis of synthetic a priori knowledge of mathematics. Kant contributes the solution by claiming that geometric propositions are universally valid and must be true of all possible objects of experience. It is not enough that all triangles we have seen have a given property, but all possible triangles we might see must have it as well. According to Kant , epistemologically there are two ways to approach the foundation of mathematics: first, perceiving that there is something about the world that makes it so; second, perceiving that there are something about our experiences that makes it so. The first alone can not produce knowledge because an objective mind-independent fact might be universally true, but we could never verify its universality by experience. So the only source of the foundation for mathematics lies in the second alternative i.e. there is something about our experiencing that makes it so.
Meanwhile, Wilder R.L issues that, in the epistemology of arithmetic, e.g. in Kant’s verification of 7+5=12, one must consider it as an instance i.e. this time in the form of a set of five objects, and add each one in succession to a given set of seven. Al¬though the five objects are arbitrary, they will be repre¬sented by the symbols which are present and which exhibit the same structure; and contemporary, we find this structure involved in the formal proofs of 7+5=12 either within a set theory or directly from axioms for elementary number theory. The proofs in the set theory depend on existential axioms of these theories.
Meanwhile, Shabel L. believes that Kant explores an epistemological explanation whether pure geometry ultimately provides a structural description of certain features of empirical objects. According to Shabel L. , Kant requires his first articulation that space is a pure form of sensible intuition and argues that, in order to explain the pure geometry without paradox, one must take the concept of space to be subjective, such that it has its source in our cognitive constitution. Kant perceives that epistemological foundation of geometry is only possible under the presupposition of a given way of explaining our pure intuition of space as the form of our outer sense. In term of the theory of the epistemology of spatial objects, Kant denies that we use geometric reasoning to access our pure intuition of space, in favor of affirming that we use our pure intuition of space to attain geometric knowledge. Kant claims that pure spatial intuition provides an epistemic starting point for the practice of geometry. Therefore the pure spatial intuition constitutes an epistemological foundation for the mathematical disciplines.
Ultimately, for Kant and his contemporaries, the epistemological foundations of mathematics consists amount of a view to which our a priori mental representation of space-temporal intuition provides us with the original cognitive object for our mathematical investigations, which ultimately produce a mathematical theory of the empirical world. However , Kant’s account of mathematical cognition serves still remains unresolved issues. Shabel L. concludes that the great attraction of Kant’s theory of knowledge comes from the fact that other views seem unable to do any better. Frege, for example, carries the epistemological analysis less than Kant in spite of his enormously more refined logical technique.

References:
In Mayer, F., 1951, “A History of Modern Philosophy”, California: American Book Company, p.295
2Distante, P., 2000-2003, “Epistemology” Retrieved 2004 ).
3 Wilder,R.L., 1952, “Introduction to the Foundation of Mathematics”, New York, p.192
4 Ibid. p.193
5 Ibid. p. 193
6 The Rudolf Steiner Archive. Retrieved 2004
7 -----, 2003, “Kant’s Mathematical Epistemology”, Retrieved 2004
8 Wikipedia The Free Encyclopedia. Retrieved 2004
9 Posy, C. ,1992, “Philosophy of Mathematics”. Retreived 2004
10 Ibid.
11Ibid.
12Ibid.
13Bolzano, B., 1810, “Appendix: On the Kantian Theory of the Construction of Concepts through Intuitions” in Ewald, W., 1996, “From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume I”, Oxford: Clarendon Press, p.223
14Ibid.p. 223
15Ibid. p.223
16Ibid.p.223
17Ibid. p.223
18Kant, I, 1783, “Prolegomena to Any Future Metaphysic: REMARK 1 Trans. Paul Carus. Retrieved 2003
19 Ibid
20Ibid.
21Ibid.
22Ibid
23Kant, I, 1783, “Prolegomena to Any Future Metaphysic: REMARK 1 Trans. Paul Carus. Retrieved 2003
24Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, pp.132
25Faller, M., 2003, “Kant’s Mathematical Mistake”, Retrieved 2004
26Ibid.
27Ibid.
28Ibid.
29Ibid.
30Ibid.
31Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p. 197
32Shabel, L., 1998, “Kant’s “Argument from Geometry”, Journal of the History of Philosophy, The Ohio State University, p.19
33 Ibid. p.20
34Ibid.p.34
35Ibid. p.34
36Ibid.p.34
37Ibid. p.34
38In Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.205

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