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A Philosophical Study on the Foundations of Projective Geometry as A Metageometry
(Beytulhikme Felsefe Cevresi, 2024) Birlik, Busra asik; Ekren, Ugur
Throughout history, research on the foundations of mathematics, particularly the debates surrounding Euclid's fifth postulate, has led to the emergence of the concept of metageometry. This paper examines the problems created by Euclid's fifth postulate in a historical and philosophical context, the contributions to the development of non-Euclidean geometries, and how projective geometry was established as a metageometry during this process. The article notes that the first steps of projective geometry began with the Renaissance architect and artist Leon Battista Alberti and were mathematically grounded through the contributions of the French mathematician Girard Desargues. Additionally, the works of mathematicians like Jean Victor Poncelet and Felix Klein facilitated the development of projective geometry and the acceptance of non-Euclidean geometries. The paper evaluates the philosophical background of these developments through Kant's transcendental philosophy and discusses Russell's thoughts and contributions to metageometry. In this context, the article provides a deep analysis of both the historical development and philosophical foundations of projective geometry.