On The Dilemma Of Mathematical Truth And Its Realists' Solutions

With the intensive study of mathematic truth, making an analysis of and giving solutions to the dilemma of mathematical truth falls naturally and inevitably on the research of philosophy of mathematics. From the view of realism, the dissertation makes an analysis of the causation of the dilemma of mathematical truth and strategic solutions to it. With a comprehensive analysis of their strengths and weakness and an adoption of the rational strategies, the dissertation puts forward a proposal for it - contextual realism.This dissertation consists of introduction, seven systematic argumentations, conclusion and postscript.The introduction is a review of the history of this research, giving a brief review of the mathematical truth and the philosophical backgrounds of the present research on the dilemma, with the inspection of the relative studies, the dissertation gives an illustration of its basic ideas and clarifies the dilemma of mathematical truth and its strategies, hence the theoretical significance and value of this dissertation.In chapter 1, The Essence of the Dilemma of Mathematical Truth, the essay examines the two constraint conditions of mathematical truth theory put forward by Benacerraf and analyzes the form of the dilemma of mathematical truth firstly. Secondly, the reason of the dilemma of mathematical truth is clarified: Benacerraf took Tarski's semantics and causal theory of knowledge as the standards of his two constraint conditions of mathematical truth theory. Based on these, different kinds of solutions to the dilemma of mathematical truth are presented."The Indispensability Argument Solution to the Dilemma of Mathematical Truth", Chapter Two, analyzes the theoretical foundations of the indispensability argument and its solution to the dilemma of mathematical truth. The indispensability argument tries to confirm the existence of mathematical entity by clarifying the indispensability of mathematics in science. Secondly, it clarifies the problems of this solution and points out that the indispensability could not be equal to confirmation in experience and the existence of mathematical entity could not be deducted from the indispensability of mathematics. On the contrary, the indispensability is entailed by mathematical realism. At last, it clarifies the holism of mathematics and science is the proper requisite for a good truth theory, which is illuminated by the indispensability argument solution to the dilemma of mathematical truth.Chapter Three is "The Naturalistic Realist Solution to the Dilemma of Mathematical Truth". It analyzes the strategy and the main problems of the naturalistic realist solution. By conjoining a compromised Platonism with a two-tiered epistemology naturalistic realists try to reconcile the contradiction between the traditional Platonism and the empiricist epistemology, so as to get the way out of the dilemma. However, the two tiered epistemology has led to two kinds of pictures of ontology. As a result, naturalists could not insist on the abstractness of mathematics, nor could they provide a rational explanation for the sensation faculty of knowing the mathematical entity.In Chapter Four, The Structural Realists Solution to the Dilemma of Mathematical Truth, it discusses the theoretical foundations of structuralism realists' and their strategies for the dilemma of mathematical truth firstly. Insisting that the nature of mathematics is structure, structural realist aims at the transformation of mathematics from didactical into scientific research, thus providing rational epistemology for mathematic study. Secondly, it shows the strengths and weakness of this approach. The structural realist solution realizes the importance of the relationship between entities, but their basic opinion that mathematics is structure could not comply with mathematical practice."The Neo-Fregean Realists Solution to the Dilemma of Mathematical Truth" is Chapter Five. This section discusses the theoretical foundations of Neo-Fregean realists' and strategies for the dilemma of mathematical truth firstly. Neo-Fregean realists take mathematics as logic, emphasizing the importance of contextual principles in confirming the existence of mathematical entity. Correspondingly, they provide a strategy for the dilemma" of mathematical truth by insisting that language is prior to existence. Secondly, it analyzes the problems that this kind of solution must face. Neo-Fregean realists emphasize the abstractness of mathematics, which lead to neglect of the close relation between mathematics and science, could not explain the application of numeral singular terms in science, nor could they provide a uniform truth theory for mathematics and science. At last, it clarifies the inspiration from this kind of solution to the dilemma of mathematical truth: the importance of context principle for answering the epistemological problem.Chapter Six is "The Fictionalism Solution to the Dilemma of Mathematical Truth". Firstly, this Chapter clarifies the theoretical foundations of fictionalism and its response to truth dilemma. Fictionalism denies that mathematics theories have truth value so that they deny the truth dilemma directly. Secondly, analyzing the problems of this kind of solution, and pointing out that the scientific entity realism of fictionalism and mathematical anti-realism dissever the unification of mathematics and science. This kind of solution obviates the nature of the dilemma of mathematical truth. Finally, it discusses the challenges the fictionalists put forward for mathematical realism in epistemology."A New Strategy: Contextual Realism" is Chapter Seven. Firstly, this chapter clarifies the motivation of choosing contextual realism as the Solution to the dilemma of mathematical truth. Secondly, from the point of contextual realism, it elucidates the essential features of mathematical context. That is, being integral and having concrete structure, mathematical context is a recontextualizational process with determinate boundary. At last, it argues the contextual realist solution to the dilemma of mathematical truth. Contextual realism can provide a uniform contextualized semantics and a uniform contextualized epistemology for mathematics and science, which can satisfy the two constraint conditions of Benacerraf's good mathematical truth theory.Summarizing the dissertation, the conclusion part stresses the significance of contextual realist solution to mathematical truth and points out that contextual realism will be a new paradigm in the research of philosophy of mathematics.