A critical evaluation of the Frege-Hilbert controversy on the status of geometry axiomes

In the modern period, debates on geometry have deeply affected both mathematical philosophy in particular and philosophical thought in general. Frege, who is one of the founders of the logicist tradition, and Hilbert, who is considered one of the pioneers of modern mathematics with his understanding of axiomatization, are some of the names at the center of these discussions. The correspondence between these two names at the beginning of the twentieth century has revealed the different ideas they have about the status of the axioms of geometry. Although the perspective of Hilbert came to the fore in the dispute between the two and modern mathematics has continued in this direction, the issues subject to the dispute have not been fully clarified. Besides, although there have been various systematic and methodological investigations regarding this controversy, which is important for grounding geometry in particular and mathematics in general, the aim of this study is to make an object-oriented study on the subject. In this context, the process of construction and understanding of the object has been taken with a critical approach to the Kant's transcendental philosophy, and it has been revealed that Hilbert's axiomatization of geometry is built on a thought about the space as an object. The aim of this approach, which reveals the objective ground of a continuous, closed and three-dimensional space as an object, is to explore the possibility of making geometry on an a priori ground. Key Words: Geometry, Axiomatization, Formal System, Logicism, Formalism, Space as an Object, Continuity, Closeedness, Three-dimensionality, Kant, Transcendental Philosophy, Construction and Understanding of the Object