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Podstata matematiky

Slovenský filozofický časopis, 11 (1956), 1, 47-65.
Typ článku: Články
I define mathematics as the science investigating into quantitative relations of the real world. In the analysis and logical foundation of this definition I demonstrate above all that mathematics actually deals always with quantitative relations. The fundamental ideas of mathematics themselves — such as number, position, direction, form —- have in mathematics the significance of quantitative relations and arose in human mind at the process of social production as the result of the need to give exact expression to certain quantitative relations. The character of quantitative relations is seemingly hidden by the axiomatic structure and procedure of mathematics, but the axioms themselves are in most cases nothing else hut quantitative relations proved by experience. The axiomatic system as a whole, however, makes us forget its origin as long as there is no need to complete or change the system. The very essence of mathematics being the research of quantitative relations, this science doesn’t fail to comprise also qualitative differences. In doing so mathematics states the new quantitative relations necessarily established with the change of quantity. Then I treat the process of abstraction, whereby mathematics seeks to find the most general quantitative relations. I prove that this is a dialectical process, that by mathematical abstraction ideas do not become poorer, but in the opposite, by the aid of this process of abstraction the actual relations of the real world are comprehended more profoundly and with greater veracity. Finally I treat the relation of mathematics to the real world. I prove that the ideas of mathematics and their relations are modelled according to the real world. They arose due to the capacity of human mind to comprehend and express quantitatively differenciated state of the real world. This capacity developed in men as a result of the urgent need of human society. Also where mathematics seems to separate from the real world, and somehow transgresses the horizon of immediat insight, it remains in close connection with reality. In such cases the immediat criterium of its exactness (as e. g. in the application of complex numbers, quaterniones, and of non-euclidean geometry) lies in its immanent logical consistency. Logical consistency, however, means accordance with certain laws of logics, the latter being many times verified abstract expressions of actual relations in the real world. Further evidence of this connection is the applicability of mathematics to reality. I prove that there is nothing accidental nor any element a priori; the approximativity of this application can be fully explained by dialectics of the abstract and the concrete. In the end I examine the relations between mathematics, logics and epistemology. This relation — so I conclude — is the same as with other disciplines. Mathematics has its specific contents and cannot be reduced to logics; it is based, however, on fundamental gnoseologic and logical premises. The most fundamental ideas of mathematics are taken directly from the sphere of the epistemology. I close by stating that only the method of dialectical materialism is able to explain clearly the essence of mathematics, while all idealistic experiments finally prove to be failures.
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