L. E. J. Brouwer

Intuitionism and formalism

Publisher Summary This chapter discusses the intuitionism and formalism. In Kant, one find an old form of intuitionism almost completely abandoned in which time and space are taken to be the forms of conception inherent in human reason. For Kant, the axioms of arithmetic and geometry were synthetic a priori judgments, that is, judgments independent of experience and not capable of analytical demonstration and this explained their apodictic exactness in the world of experience as well as in abstracto. Diametrically opposed to this is the view of formalism which maintains that human reason does not have at its disposal exact images either of straight lines or of numbers larger than ten, for example; therefore, these mathematical entities do not have existence in conception of nature any more than in nature itself. It is true that from certain relations among mathematical entities, which one assume as axioms, one deduce other relations according to fixed laws, in the conviction that in this way one derive truths from truths by logical reasoning, but this nonmathematical conviction of truth or legitimacy has no exactness.

CONSCIOUSNESS, PHILOSOPHY, AND MATHEMATICS

Publisher Summary This chapter describes the importance of consciousness, philosophy, and mathematics. Consciousness in its deepest home seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation. It seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. By a move of time, a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind. Mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding, creating new mathematical entities in the shape of predeterminately, or more or less freely proceeding infinite sequences of mathematical entities acquired, and in the shape of mathematical species.

HISTORICAL BACKGROUND, PRINCIPLES AND METHODS OF INTUITIONISM

Publisher Summary This chapter discusses the historical background, principles, and methods of intuitionism. The historical development of the mental mechanism of mathematical thought is naturally closely connected with the modifications which, in the course of history, have come about in the prevailing philosophical ideas firstly concerning the origin of mathematical certainty; secondly, concerning the delimitation of the object of mathematical science. The mental mechanism of mathematical thought during so many centuries has undergone a few fundamental changes because of the circumstance that, in spite of all revolutions undergone by philosophy in general, the belief in the existence of properties of time and space, immutable, and independent of language, and experience, remained intact until far into the 19th century. It is found that for the elementary theory of natural numbers, the principle of complete induction, and more or less considerable parts of algebra and theory of numbers, exact existence, absolute reliability, and noncontradictority were universally acknowledged independently of language and without proof.

THE EFFECT OF INTUITIONISM ON CLASSICAL ALGEBRA OF LOGIC

Publisher Summary This chapter discusses the effect of intuitionism on classical algebra of logic. Classical algebra of logic, furnishes a formal image of the laws of common-sensical thought. This common-sensical thought is based on the conscious or subconscious, threefold belief including the belief in a truth existing independently of human thought, and expressible by the means of sentences called true assertions, mainly assigning certain properties to certain objects, or stating that objects possessing certain properties exist, or that certain phenomena behave according to certain laws. Furthermore, in the possibility of extending ones knowledge of truth by the mental process of thinking, in particular thinking accompanied by linguistic operations independent of experience called logical reasoning, which to a limited stock of evidently true assertions mainly founded on experience, and sometimes called axioms, contrives to add an abundance of further truths. Closely connected with the principle of the excluded third, also called principle of judgeability, is the principle of reciprocity of absurdity saying that a mathematical assertion whose noncontradictority has been established is true.

The selected correspondence of L.E.J. Brouwer

Introduction.- 1900 - 1910.- 1911 - 1920.- 1921 - 1930.- 1931 - 1940.- 1941 - 1950.- 1951 - 1965.- Appendices.- List of Enclosures, Editorial Comments and Editorial.- Supplements.-Biographical information.- List of letters.- Abbreviations.- Organizations and journals.

AN EXAMPLE OF CONTRADICTORITY IN CLASSICAL THEORY OF FUNCTIONS

Publisher Summary This chapter presents an example of contradictority in classical theory of functions. The classical theorem asserting that each monotone function of the unity continuum is differentiable almost everywhere cannot be said to be true, neither for the classical nor for the intuitionist continuum. It is assumed that σ is the species of the rationality assertions of the unitary standard number. As is known, the simultaneous assert ion of the principle of testability for the elements of σ is contradictory. On the basis of this supposition, the differential coefficient g of ψ(x) for x = d must either lie apart from 1 or lie apart from 3/2 as well as from 1/2. In the first case, it is impossible that ψ(x) should some time turn out to be ≡ x, hence that α should some time turn out to be contradictory. It is found that in the first case α is noncontradictory.