Richard von Mises
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More books by Richard von Mises…
“The construction of a "problem calculus" in the sense of Heyting and Kolmogoroff yields a model of logic in which the theorem of the excluded middle does not appear among the basic formulas. The study of such a logic widens our insight into the basic elements of mathematics and, in particular, points out the special position of the so-called indirect proofs within mathematics.”
― Positivism: A Study in Human Understanding
― Positivism: A Study in Human Understanding
“In agreement with the empiristic conception of science, intuitionism holds that the source of mathematics is the insight which we intuitively comprehend from experience of the external world, but which cannot once and for all be collected in a closed system of axioms.”
― Positivism: A Study in Human Understanding
― Positivism: A Study in Human Understanding
“According to L. E. J . Brouwer, the founder of the intuitionist school, the simplest mathematical ideas are implied in the customary lines of thought of everyday life and all sciences make use of them; the mathematician is distinguished by the fact that he is conscious of these ideas, points them out clearly, and completes them. The only source of mathematical knowledge is, in Brouwer's opinion, the intuition that makes us recognize certain concepts and conclusions as absolutely evident, clear, and indubitable.
However, he does not assume that it is possible to list in a precise and complete way all basic fundamental concepts and elementary methods of deduction, which in this sense are to serve as a basis of mathematical derivations. It should always be possible to supplement the once fixed set of assumptions by accepting new ones, if a further intuition leads that way.”
― Positivism: A Study in Human Understanding
However, he does not assume that it is possible to list in a precise and complete way all basic fundamental concepts and elementary methods of deduction, which in this sense are to serve as a basis of mathematical derivations. It should always be possible to supplement the once fixed set of assumptions by accepting new ones, if a further intuition leads that way.”
― Positivism: A Study in Human Understanding
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