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Publication Details

The Discovery of Plurality of Mathematics and its Philosophical Significance

(Original title: Objev plurality matematiky a jeho filozofický dosah)
Filozofia, 23 (1968), 2, 169-178.
Type of work: Papers and Discussions
Publication language: Czech
Idealistic philosophy has been trying since long ago to utilize mathematics for its aims, but the development of the latter deprives it every time of this support. The crisis of the foundations of mathematics at the beginning of our century was provoked by the unrestricted interpretation of the concept of „set“ and by the uncritical transference of logic which arose as an abstract reflection of relations in the finite into the infinite. Mathematical schools which came into being were not capable to provide a logico-philosophical foundation of mathematics because they comprehended it idealistically and gave the problem of its consistency an absolute metaphysical meaning. The narrowness of the axioms of the set theory was not less important than the excessive breadth of the concept of set; this narrowness hindered the solution of many important problems such as: well-ordering principle, Auswahlsprinzip, and especially the continuum-problem. The recent discovery of the independence of these theorems of the rest of the axioms of the set theory yielded in mathematics as a whole a situation analogous to that one which arose in geometry with the creation of non-Euclidean geometries. In the place of one unique mathematics we are faced now with a plurality of equally consistent mathematics. Neither objective- nor subjective-idealistic comprehension of mathematics is capable to cope with the fact of plurality of mathematics, while it corroborates again the materialistic-dialectical theory of reflection according to which arithmetic and geometry are two one-sided, incomplete abstract reflections of the material world, which forms in it an inseparable unity of the discrete and the continuous. At the same time, an important principle of cognition is confirmed, advanced by Lenin: our philosophy ought not bind itself by the provisional notions of matter, space, time, causality etc. as obtained by special sciences, but being a scientific philosophy which leans solely upon the generalizations of their results, it is bound nevertheless only by the acknowledgement of the independence of the material reality of consciousness and its cognoscibility.
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