Internalization of Logic in Category Theory

This paper attempts to provide a philosophical analysis of the phenomenon of internalization of mathematical structures, including logical ones, in category theory. Using the methods of category theory, as well as guided by the philosophy of mathematical structuralism, we show how the set-theoretic approach to the foundation of mathematics differs from the category-theoretic one. The main part of the work, firstly, is the analysis of the use of topoi theory to formalize set theory within the corresponding category. Secondly, the paper presents two ways of category-theoretic formalization of quantifiers as functors between categories. Using conceptual and formal presentation, a distinction is made between different ways of internalizing logic. Thirdly, a formalization of such model theory concepts as model, signature, and theory is presented using category theory. Based on the work done, it becomes clear why, from the point of view of mathematical structuralism and the paradigm of practically oriented foundations of mathematics, type theory and category theory have several philosophical advantages over set theory. Finally, a definition of the internal language of topoi is presented and its conceptual significance is shown in the context of distinguishing between internal and external reasoning in mathematics.