Mathematics Without Numbers-on Hellman's Modal Structuralism

Under the influence of the Bourbaki school's structural movement in the early 20th century,the mathematical structuralism which advocated "the nature of mathematics is structure" gradually rose,and formed a detailed demonstration system.Based on different ontological stance,there appeared two major interpretation frameworks of eliminative and non-eliminative structuralism.As one of the representatives of the eliminative structuralism,Hellman put forward the idea of the modal structuralism,and developed it to an highly influential theoretical framework of mathematical structuralism.Modal structuralism emphasized avoiding the quantification on structure or position individually,but founding the structuralism based on some domain and the domain on the relationship between implicit definitions given by the axiom system condition on the possibility of a second order logic.This paper elaborates systematically the theory of Hellman's modal structuralism,including an introduction,five chapters of systematical discourses and a conclusion.Introduction introduces the background of Hellman selecting modal structuralism and the domestic and foreign research progress of Hellman's modal structuralism.Chapter 1 mainly states the process of Hellman building his modal structuralism on the basis of Putnam's modal realism and Field's fictionalism,and introduces in detail components of its foundational framework.Chapter 2 expounds Hellman's modal structural analysis towards the number theory,and emphasizes mainly the framework of the Nature Numbers' modal structuralism and the justification of this translation scheme.It also discusses feasibility of extending the modal structural analysis of Natural Numbers to Real Analysis.Chapter 3 discusses Hellman's modal structural interpretation about the set theory,taking the Zermelo-Fraenkel set theory for instance.It also demonstrates systematically feasibility of its eliminating the set theory as the foundation of mathematics.Chapter 4 discusses mainly Hellman's solution about the problems of mathematical applicability.Through elaborating the relationship between mathematics and physical reality,thus Hellman researches the modal structural interpretation for the material worlds based on the mathematical theories.The conclusion is a brief summary of this paper.It also points out the problems which still need to be further researched or solved:the problems of presupposing the second order logic and original modality and the applied problem of mathematics.