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- W1528863826 abstract "David Hilbert's Foundations of Geometry (1899) contain nineteen statements, labelled axioms, from which every theorem in Euclid's Elements can be derived by deductive inference, according to the classical rules of logic. The axioms use three property words -'point', 'straight' and 'plane'and three relation words -'incident', 'between' and 'congruent'forwhich no definition is given. These words have, of course, a so-called intuitive meaning in English (as the German equivalents actually used by Hilbert have in his language). But Hilbert believed they ought to be understood in whatever sense was compatible with the constraints prescribed by the axioms themselves. l To show that some of his axioms were not logical consequences of the others he unhesitatingly bestowed unorthodox meanings on the undefined terms. This enabled him to produce models that satisfied all the axioms but one, plus the negation of the excluded axiom. The mathematician-philosopher Gottlob Frege showed little understanding for Hilbert's procedure. Frege thought that the undefined terms stood for properties and relations that Hilbert assumed to be well-known and that the axioms were intended as true statements about them. Hilbert disabused him: do not wish to presuppose anything as knownj I see in my declaration in § 1 the definition of the" @default.
- W1528863826 created "2016-06-24" @default.
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- W1528863826 date "2001-05-15" @default.
- W1528863826 modified "2023-09-25" @default.
- W1528863826 title "A model for Smarandache's anti-geometry" @default.
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- W1528863826 doi "https://doi.org/10.5281/zenodo.9378" @default.
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