On Mathematical Naturalism Of Penelope Maddy

Mathematical naturalism is proposed by Penelope Maddy in order to defend the axioms of set theory. Maddy’s whole ideas has been presented in her book《Realism in Mathematics》 which was printed in 1990; 《Naturalism in Mathematics》 which was printed in 1997; 《Second Philosophy:A Naturalistic Method》 which was printed in 2007 and the 《Defending the Axioms:On the Philosophical Foundations of Set Theory》 which was printed in 2011.By putting philosophy of mathematics、 mathematical practice and the practice of science together, Maddy’s mathematical naturalism tries to turn philosophers’focus from discussing issues about Mathematical existence and the truth of the traditional metaphysics to problems related to the method of mathematical practice and mathematical objectivity. The work Maddy has done have significant influence on the direction of the development of the philosophy of mathematics. In analyzing the background, basis, train of thought and strategy of Maddy’s mathematical naturalism, the thesis is aimed at investigating its weakness, inf luence and significance, hoping to comprehend the development tendency an d the research methods of contemporary philosophy of mathematics.This thesis consists of introduction, five systematic argumentations and conclusion.The introduction mainly elaborates the reason for the thesis to research the topic of Maddy’s mathematical naturalism and analysis on the research status at home and abroad. On the basis of this, the thesis wants to clarify the problem which should be solved, the basic idea of writing and the main content, the research significance of the thesis.Chapter one is mainly to analyze the background of Maddy’s mathematical naturalism. First, in order to understand how to generate the beliefs about Maddy’s mathematical naturalism, we will discuss the origin of her idea. Second, in order to understand the basic framework of Maddy’s mathematical naturalism, we will discuss the foundational thought of her mathematical naturalism. Third, in order to make a clear outline of Maddy’s mathematical naturalistic ideas, we will discuss the evolution thought of her mathematical naturalism.Chapter Two is mainly to discuss the naturalistic set theoretic realism. In discussing the motivation of naturalistic set theoretic realism and its strategy, we will elaborate the naturalistic thought which Maddy has defended for mathematical realism. And then, we will point out the weakness of naturalistic set theoretic realism and Maddy’s towards a complete mathematical naturalism.Chapter Three mainly discusses the complete mathematical naturalism. First, this chapter is designed to demonstrate what Maddy’s mathematical naturalism is by explaining the routine that the theory shifts from set theory to mathematical naturalism. Second, by discussing the philosophical problems in the practice of set theory, Maddy tries to make the philosophical problems which have no business to the methodology of set theory. Third, this chapter makes a good explanation of mathematical naturalism of set theory.Chapter Four is "The naturalism of second philosophy". On the basis of mathematical practice of set theory, we will discuss how to turn to second philosophical naturalism from mathematical naturalism. By elaborating what is "second philosophy" and how the second philosophical position of set theoretic is to defend the mathematical ontology and epistemology, we will elaborate the relationships between the naturalism of second philosophy and the mathematical practice. In this way, we can see that Maddy has finished her defense for the axioms of set theory.Chapter Five mainly discusses the weakness of Maddy’s mathematical naturalism and its influence on the contemporary philosophy of mathematics. Thus, we can make a comprehensive evaluation on Maddy’s mathematical naturalism.In conclusion, we will summarize the main work of the thesis, and point out the innovation and deficiency, and make prospects for the mathematical naturalism of further research.