The Research Approach To The Mathematical Foundation Based On Category Theory

"The Mathematical foundation" is the most important part of the mathematics.Almost all the mathematical knowledge is based on mathematical foundation,so the research to the mathematical foundation is of great importance.The emergence of the paradox of the set theory directly leads to the outbreak of crisis on the mathematical foundation,which has resulted in the long lasting debate on the mathematical foundation.Therefore,to solve the crisis of mathematical foundation and to find a suitable mathematical foundation have become an urgent problem for the mathematical philosophers.As the research trend of mathematical philosophy in the 20 th century,the combination of structuralism with category theory gives rise to the research thought of categorical structuralism,it is on this basis that we put forward the research approach of the mathematical foundation based on category theory,which has opened up a new method and provided new possibilities for the research to the mathematical foundation.This paper systematically analyzes the research approach of the mathematical foundation based on category theory,and discusses about the feasibility of category theory as a mathematical foundation.The first chapter points out the dilemma of some mathematical foundation approaches,including naive set theory,axiomatic set theory and three major mathematical schools,then,by emphasizing the trend of structuralism in mathematical philosophy,it indicates a structuralism turn of the mathematical foundation research,which finally shows the research approach of mathematical foundation based on category theory.The second chapter analyzes the theoretical connotation of the mathematical foundation of category theory.Along the route of "mathematics-structure-category",it explains the path of category theory as the mathematical foundation,specifically explores the essence of mathematics,the exposition about mathematical structure by category theory,and the meaning construction of the mathematical foundation of category theory.The third chapter analyzes the ETCS axiomatic system and CCAF axiomatic system by the method of contextual analysis,which firstly illuminates the universality between the categories and the context,and then interprets how to build mathematical wholeness on the two axiomatic systems from a historical,social,academic,psychological and other non-linguistic level and the linguistic level of form,semantics and pragmatics.The fourth chapter analyzes the challenges and doubts faced by the mathematical foundation of category theory,then makes a strong defense on whether category theory presupposes the concept of set theory and whether the axiomatic system of category theory asserts existence.The fifth chapter makes a comprehensive investigation of the mathematical foundation of category theory from the whole point of view.Firstly,it demonstrates the autonomy of category theory as the mathematical foundation,and then focuses on how the category theory can serve as a mathematical foundation.Finally,it discusses the research advantages of the mathematical foundation of category theory compared with the mathematical foundation of the set theory.The sixth chapter analyzes the significance of the research on the mathematical foundation of category theory from the aspects of the advancements of mathematical philosophy,the promotion of scientific research,and the application of context analysis method.The conclusion reviews the whole elaboration of the research approach to the mathematical foundation based on category theory,affirms the research value of this approach,and prospects the development of the mathematical philosophy in the future.To sum up,this paper puts forward the research approach to the mathematical foundation based on category theory in view of the difficulties faced by the research of mathematical foundation,expounds the research path of the category theory as a mathematical foundation,and analyzes the two axiomatic systems of category theory with the context analysis method.Some mathematical philosophers hold the attitude of questioning or even opposing about the mathematical foundation of category theory,in the process of defending the mathematical foundation of category theory,the approach to the mathematical foundation based on category theory has been interpreted in more detail.Through a comprehensive study of the mathematical foundation of category theory,which further enriches the rationality of the mathematical foundation of category theory,and analyzes the significance of the research on the mathematical foundation of category theory from multiple perspectives finally.