L.E.J. Brouwer's `Unreliability of the logical principles'. A new translation, with an introduction (original) (raw)
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L.E.J. Brouwer and H. Weyl both recognized that there is the intuitive continuum in mathematics. However their researches encountered with the concept of incommensurable pieces, which is connected to the Hippasus's theorem and a principle «tertium non datur». Therefore the followers of intuitionism could not compete to the Cantor's theory of transfinite numbers. Nevertheless, the logic of intuitionism allows criticizing the Hippasus's proof which is based on an axiom of indivisibility of unit because we use the concept of infinite decimal fractions today. Various classes of axioms in antique and modern arithmetics generate contradictions in the foundations of mathematics (2nd Hilbert's problem). In the given paper the constructive decision is offered, which allows to look at a problem of quadratic irrational numbers in a new way. We can effectively build such sequences within the framework of the Brouwer's concept about a continuum consisting of intervals. As it is known, the Dutch mathematician Luitzen Egbertus Jan Brouwer rejected the universal application of the logical law «tertium non datur» in mathematics. In other words: «If one proves that not all integers of an infinite set of whole numbers are even, the conclusion that there exists at least one integer which is odd was denied by Brouwer, because this argument applies the law of excluded middle to infinite sets». 1 For this reason Hermann Weyl wrote: «According to his [Brouwer's] view and reading of history, classical logic was abstracted from the mathematics of finite sets and their subsets... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antinomies. It is not that such, contradictions showed up that is surprising, but they they showed up at such a late stage of the game». 2 However, the history of mathematics allows to detect incorrect application of the law of the excluded middle to infinite sets not only «at the last stage of the game» in a Сantor's theory of sets. We find this argument in the antique proof of the incommensurability of the diagonal of the square relating to Hippasus from Metapont (or Hippasus of Metapontum, 5th century B.C.).
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