Intuitionism - Philosophy of Science


His creator was the Dutch mathematician Leitzen Egbert Jan Brower. Brouwer's entire academic career was associated with the University of Amsterdam. Here, from 1897 to 1904, he studied mathematics and natural science, then prepared and in 1907 defended his thesis "On the foundations of mathematics."

For a correct understanding of Brouwer's general views and specific features of intuitionism, the pamphlet Life, Art and Mysticism, written by him during his thesis, is of great importance. (1905) [53]. This is a worldview manifesto, representing a romantic rebellion against rationalism and against the traditional understanding of intellect and science. In addition, moods are very strong in the spirit of universal mysticism (Brower quotes M. Eckhart, J. Boehme and the Bhagavad Gita abundantly), including the call to turn inward, and voluntaristic motives that recall Schopenhauer with his teaching about the will to live . A number of themes of this outrageous youthful work have become part of the philosophy of intuitionism. Initially Brouwer did not think his studies of the basics of mathematics in isolation from moral, social and philosophical questions. To a certain extent, the scientific adviser managed to inspire him with the need to separate mathematics from his mystical insights. However, later works of Brouwer impressed his contemporaries with his prophetic tone and inclination to analyze the depths of consciousness.

In the thesis of Brouwer in 1907, "On the foundations of mathematics" contains the main ideas of intuition: 1) mathematics is mental construction ; 2) mathematics is extralinguistic activity. Mathematics according to Brouwer can not deal with anything that she herself did not construct in accordance with intuitively clear requirements. Therefore, in such constructions, the only possible basis for mathematics is, and Brower does not accept other attempts to justify mathematics.

Already in the pamphlet of 1905 he had a clear separation of the constructive activity of consciousness from the means of language [53, p. 4011. In the dissertation, he wrote about it this way: "People seek by means of sounds and symbols to cause in other people copies of mathematical constructions and reasonings that they themselves produced; by the same means they are trying to help their own memory. In this way a mathematical language appears, and as its particular case is the language of logical reasoning. & lt; ... & gt; Thus, it is easy to imagine that with the same organization of human intellect and, consequently, the same mathematics, another language could be formed, to which the well-known language of logical reasoning would not correspond. Perhaps, there are still peoples living isolated from our culture, for which this actually takes place. And the possibility is no more excluded that at a later stage of development, logical thinking will lose its present role in the languages ​​of cultural peoples. " [52, p. 73-74].

The mathematics for Brouwer - protection the very foundations of human thought and consciousness. It is here that we find the basic pre-intuition of mathematics (the Brownian version of the Kantian intuition of pure time), "in which the combined and divided, continuous and discrete are combined" and which generates, on the one hand, the intuition of ordinal natural numbers, and on the other hand, the linear continuum [49, p. 78-81]. If the logic for Brauer is empirical science, then mathematics is a priori science.

Mathematics for the intuitionist is a special kind of mental activity, lying deeper than the level of the language. The existence with which a mathematician deals is an introspective sphere of mental constructive processes that is verified in its truth. The genuine subject of mathematics is mental constructs, exist in mathematics means be built [9, p. 10-11]. For Brauer here we have an absolute criterion of real mathematics (mental evidence), and he considers it possible to challenge the right to be considered mathematics for everything that can not pass such a verification of truth.

As a result, the paradoxes of set theory find the most unexpected resolution in intuitionism. They simply become combinations of words devoid of mathematical meaning. Mathematical resolution of them is impossible for the simple reason that they lie outside the field of what is permissible to call mathematics.

For an intuitionist, a true mathematical statement must have the form: "I performed in my mind the construction of A & quot ;, and its mathematical (not logical!) negation is this: I performed in my mind the construction of In, which leads to the contradiction of the assumption that you can complete the construction of L [9, p. 28-291. In connection with this understanding of denial, intuitionists, of course, do not recognize the unrestricted application of the logical law of the excluded third, and consequently - and indirect evidence based on it (pure evidence of existence) in mathematics. An alternative in the formulation of the law of the excluded third means for the intuitionist the realization of one of the two mental constructions. Therefore, the third option is quite possible - none of these constructions were implemented.

In the examples given by intuitionists in support of their views, it is very important that we are talking about an infinite subject area. In mathematics, the main interest is precisely such objects. The supporter and propagandist of intuitionist ideas in the 1920s, Herman Weil, even identified (1925) mathematics as the "science of the infinite" [7, p. 9, 90]. Belief in the universal applicability of the law of the excluded third, Brower wrote in the article "Intuitionistic Set Theory" (1919), historically was due to the fact that originally classical logic was abstracted from the mathematics of subsets of a certain finite set, then an a priori independent of mathematics was attributed to this logic, and finally, on the basis of this imaginary a priori, applied it in an unjustified manner to the mathematics of infinite sets [7, p. 77-78]. The interpretation of mathematical infinity as the infinity actual (which the creator of set theory, G. Cantor insisted) supports this false analogy between finite and infinite sets. A constructive understanding of existence in mathematics does not allow intuitionists to recognize actual infinity as a legitimate mathematical object, the infinity with which a mathematician deals, there is always an infinity [20, p. 49-50].

Perhaps most vividly this attitude toward infinity manifested itself in the intuitionistic interpretation of the continuous, ie, in the continuum theory. The continuum can not be thought of as composed of separate parts, it is the "medium of free becoming" [7, p. 22-26, 76-80, 100-128]. The points on the line (real numbers) are defined through potentially infinitely continuing sequences of rational numbers not connected by a certain continuation law (in German Wahlfolge, a freely becoming sequence, a sequence of free choice). "It does not matter," writes Geyting, "how the terms of the sequence are determined, whether by the law, by free choice, by lot or otherwise" [9, p. 43].

The radicalism of the position of intuitionism led to the fact that intuitionistic mathematics required a revision not only of the methods, but also of a number of results of classical mathematics. Moreover, this concerned not only the Cantor theory of transfinite numbers, but also mathematical analysis. Some concepts of classical analysis break up into several different concepts (for example, the notion of convergence of a series), and some theorems cease to exist (for example, the Bolzano-Szllierstrass theorem).

In addition, in the original (Brown) version of intuitionism, there is a pronounced solipsism trend . In the work "Will, Knowledge, Language (1933) Brouwer wrote: & lt; ... & gt; arising from the self-disclosure of the original intuition, extralinguistic constructions are accurate and correct solely due to the presence in memory; & lt; ... & gt; However, the human memory that these constructions have to survey is, by its very nature, limited and subject to errors, even when it calls for help signs of language. For the human consciousness, which would be armed with unlimited memory, pure mathematics, practiced alone and without the help of signs of language, would be accurate. This accuracy, however, would again be lost in the exchange between human beings, even those with unlimited memory, because they would remain doomed to use language as a means of communication [90, p. 580]. The ability to bring mathematical constructs beyond the limits of individual consciousness, without loss in accuracy and reliability, seemed to Brouwer very, very problematic. This side of the browser's view of mathematics has also provoked criticism.

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