LOGICAL AND ANALYTICAL RECONSTRUCTION OF FORMATION OF MATHEMATICAL PARADIGM (ON EXAMPLE OF CREATION OF NON-EUCLIDEAN GEOMETRIES)
Journal Title: Філософські обрії - Year 2017, Vol 0, Issue 37
Abstract
In the article on the creation and study non-Euclidean geometries as a fundamental discovery there is made a logical and analytical reconstruction of the formation of a mathematical paradigm. It is shown that non-Euclidean geometries are inherent in all essential features of a fundamental discovery. First, non-Euclidean geometries as any fundamental discoveries serve as a solution of the fundamental problem of mathematical knowledge. Fundamental discoveries are ideological in nature, requiring separation of qualitatively new principles on which they are based. It is shown that Euclidean geometry has been a perfect geometric system since the works of ancient Greek geometers to the early nineteenth century. combining the real and the ideal features and having axioms consistent with the empirical experience and «common sense». The problem of the fifth postulate of Euclid, which was waiting for its solution for two thousand years becomes a fundamental problem in mathematics on the background of blurring of core mathematical concepts, especially «infinitely small value», «number», «probability», uncertainty of important mathematical operations as «differentiation», «integration», «adding numerical series» and so on. Second, discovery and components of non-Euclidean geometries as a fundamental problem are prepared by historical development of mathematical knowledge. For two thousand years there has been made mathematicians’ attempts to clarify the nature or prove the fifth postulate of Euclid. Theoretical aspects of the idea of creating of geometries other than Euclidean geometries are found in works of Kant, John. Sakkeri and others. All of them are inherent in a problem of interpretation of Euclid’s fifth postulate as a non-fundamental one, making it impossible for an adequate interpretation of the results. Awareness of the fundamental problem of the fifth postulate of Euclid is the key to its solution. It is proved that the ideas of non-Euclidean geometry systems arise from purely logical speculation about the nature of the axioms of Euclid. The existence of non-Euclidean geometries aslogically and correctly constructed systems prove that our intuitive understanding of space is not a purely logical necessity
Authors and Affiliations
Людмила Миколаївна ШЕНГЕРІЙ
Keywords
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- EP ID EP496221
- DOI 10.5281/zenodo.808772
- Views 82
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