Kant on Intuition Essay Example for Free

Kant on Intuition Essay Introduction Kant seems to have adapted the Spinozan trichotomy of spiritual activity. (Rocca, 77) In addition to sensible (empirical) intuition and understanding, Kant introduces pure intuition. The principles of this a priori, supra-empirical sensibility are dealt with by the transcendental aesthetic, a discipline which establishes that there are two pure forms of sensible intuition, serving as principles of a priori knowledge, namely, space and time. (Hayward, 1) Space is a necessary a priori representation, which underlies all outer intuitions (Hayward, 1); in particular, in order to perceive a thing, we must be in the possession of the a priori notion of space. Nor is time an empirical concept: it is the form of the inner sense, and is a necessary representation that underlies all intuitions. (Ewing 24) Pure intuition, unaided by the senses and, moreover, constituting the very possibility of sense experience, is for Kant the source of all synthetic a priori judgments. These include the synthetic judgments of geometry, which is for Kant the a priori science of physical space, and arithmetic, which he regards as based on counting, a process that takes time. Moreover, if for Aristotle, Descartes and Spinoza intuition was a mode of knowing first truths, it is for Kant no less than the possibility of outer experience. The faculty by means of which man creates geometries and theories is reason certainly sustained in some cases by sensible intuition, though not by any mysterious pure intuition. However, the products of reason are not all of them self-evident and definitive. Kantian time had a similar fate. We now consider that the characterization of time as the a priori form of the inner sense is psychologistic, and we reject the radical separation between time and physical space. The theories of relativity have taught us that the concepts of physical space and time are neither a priori nor independent from one another and from the concepts of matter and field. Infallibilism is, of course, one of the sources of Kantian intuitionism. Further sources are psychologism and the correct acknowledgment that sensible experience is insufficient for building categories (e. g. , the category of space). Instead of supposing that man builds concepts which enable him to understand the raw experience he like other animals has, Kant holds dogmatically and, as we now know, in opposition to contemporary animal and child psychology, that outer experience is possible only by the representation that has been thought. (Hahn, 89) Of all the influential contributions of Kant, his idea of pure intuition has proved to be the least valuable, but not, unfortunately, the least influential. Contemporary Intuitionism If Cartesian and Spinozan intuitions are forms of reason, Kantian intuition transcends reason, and this is why it constitutes the germ of contemporary intuitionism, in turn a gateway to irrationalism. There are, to be sure, important differences. While Kant admitted the value of sensible experience and of reason, which he regarded as insufficient but not as impotent, contemporary intuitionists tend to revile both. Whereas Kant fell into intuitionism because he realized the limitation of sensibility and the exaggerations of traditional rationalism, and because he misunderstood the nature of mathematics, intuitionists nowadays do not attempt to solve a single serious problem with the help of either intuition or its concepts; rather, they are anxious to eliminate intellectual problems, to cut down reason and planned experience, and to fight rationalism, empiricism, and materialism. This anti-intellectualist brand of intuitionism grew during the Romantic period (roughly, the first half of the nineteenth century) directly from the Kantian seed, but it did not exert a substantial influence until the end of the century, when it ceased being a sickness of isolated professors and became a disease of culture. Sensible intuition and geometrical intuition, or the capacity for spatial representation or visual imagination, have very few defenders in mathematics nowadays, because it has been shown once and for all that they are as deceptive logically as they are fertile heuristically and didactically. Therefore, what is usually called mathematical intuitionism does not rely on sensible intuition. It is now well understood that mathematical entities, relations, and operations, do not all originate in sensible intuition; it is realized that they are conceptual constructions that may altogether lack empirical correlates, even though some of them may serve as auxiliaries in theories about the world, such as physics. It is also recognized that self-evidence does not work as a criterion of truth, and that proofs cannot be shown by figures alone, because arguments are invisible. In particular, it is no longer required that axioms be self-evident; on the contrary, because they are almost always richer than the theorems they are designed to explain, axioms are often less evident than the theorems they give rise to, and are therefore apt to appear later than the theorems in the historical development of theories. Thus it is easier to obtain theorems on equilateral triangles than to establish general propositions about triangles. Mathematical intuitionism is best understood if it is regarded as a current that originated among mathematicians (a) as a reaction against the exaggerations of logicism and formalism; (b) as an attempt to rescue mathematics from the shipwreck that, at the beginning of our century, the discovery of the paradoxes in set theory seemed to forecast; (c) as a minor product of Kantian philosophy of pure intuition. It is only indebted to Kant, who was as much a rationalist and an empiricist as he was an intuitionist; and even what mathematical intuitionism owes to Kant may be left aside without fear of seriously misunderstanding the theory as has been recognized by Heyting, (Heyting 13) although Brouwer might not agree. The debt of mathematical intuitionism to Kant boils down to two ideas: (a) time though not space according to neointuitionists is an a priori form of intuition and is essentially involved in the number concept, which is generated by the operation of counting; (b) mathematical concepts are essentially constructible: they are neither mere marks (formalism) nor are they apprehensible by their being ready-made (Platonic realism of ideas); they are the work of human minds. The first assertion is unmistakably Kantian, but the second will be granted by many non-Kantian thinkers. Those mathematicians who are sympathetic with mathematical intuitionism tend to accept the second thesis while ignoring the first. Since a large part of mathematics may be built on the arithmetic of natural numbers, which would be generated by the intuition of time, it follows that the apriority of time does not only qualify the properties of arithmetic as synthetic a priori judgments, but it does the same for those of geometry, though certainly along an extended conceptual chain. The sole basal intuition would, then, suffice to engender step by step and in a constructive or recursive form not merely by means of creative definitions or by resorting to indirect proof the whole of mathematics or, rather, the mathematics allowed by mathematical intuitionism, which is only a portion of classical (pre-intuitionist) mathematics. It is true that Kant maintained that mathematics is the rational knowledge obtained from the construction of concepts. But what Kant meant by construction was not, for instance, the formation of an algorithm for the effective computation or construction of an expression like 100 100100, but rather the exhibition of the pure intuition corresponding to the concept in question. (Black 190) For Kant, to build a concept means to give its corresponding a priori intuition which, if possible, would be a psychological operation whereas, for mathematical intuitionism, the construction may be entirely logical, to the point that it may consist in the deduction of a contradiction. The ultimate foundation of all mathematical concepts, which for Kant and Brouwer alike must be intuitive, is quite another matter. Unlike Kant, the mathematical intuitionist will require that only the basic ideas be intuitive. With regard to the assertion that the basic intuition is prelinguistic, it seems definitely inconsistent with the findings of contemporary psychology, according to which every thought is symbolical, i. e. , accompanied by visual or verbal signs. Finally, the existence of Brouwers basic intuition (Stigt, 45) is at least as problematic as the existence of mathematical objects. (Curry 6) Mathematical intuitionism has both positive and negative elements. The former, the realistic elements, concern logic and the psychology of mathematics; the negative constituents are aprioristic and limiting, concern the foundations and methods of mathematics. Conclusion The debt of mathematical intuitionism to philosophical intuitionism is not large and, at any rate, what is involved is Kant’s intuitionism and not the anti-intellectualist intuitionism of many Romantics and post-Romantics. Besides, the contacts between mathematical and philosophical intuitionism are precisely those which the majority of mathematicians would not accept. The working mathematician, if he is concerned with the philosophy of mathematics at all, does not sympathize with intuitionism, because it looks for an a priori foundation or justification, or because it praises an obscure basic intuition as the source of mathematical creation, or because it claims that such an intuitive foundation is the sole warrant of certainty. Mathematical and logical intuitionisms are prized to some extent despite their peculiar dogmas, because they have contributed to the disintegration of alternative dogmas, particularly the formalist and the logicist ones. Works Cited Black Max. The Nature of Mathematics: London: Routledge Kegan Paul, 1933. 191 Curry Haskell B. Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North-Holland, 1951. Ewing A. C. Reason and Intuition, Proceedings of the British Academy, XXVII (1941) Hahn Hans. The Crisis of Intuition in The World of Mathematics. Edited by J R Newman New York: Simon Schuster, 1956 Hayward, Malcolm: The Geopolitics of Colonial Space: Kant and Mapmaking. Article accessed on 12/04/2007 from http://www. english. iup. edu/mhayward/Recent/Kant. htm Heyting A. â€Å"Heyting, Intuitionism in Mathematics, ( 1958), 13. Kant Immanuel. Kritik der reinen Vernunft (1781, 1787). Edited by R. Schmidt. Hamburg: Meiner, 1952. Translated by N. Kemp Smith . Immanuel Kants Critique of Pure Reason. London: Macmillan, 1929. Rocca, Della Michael. 1996. Representation and the Mind-Body Problem in Spinoza. Oxford University Press. Stigt, W. P. van 1990, Brouwers Intuitionism, Amsterdam: North-Holland, 1990.