A COMPARATIVE ANALYSIS OF CHARLES S. PEIRCE\u27S PHILOSOPHY OF MATHEMATICS

This dissertation, relying upon Peirce\u27s New Elements of Mathematics, his Collected Papers, and unpublished manuscripts, analyzes the classical philosophies of mathematics, and argues that Peirce shares their insights, but deftly avoids the snares entrapping them. The first chapter states Peirce\u27s definition of mathematics as the science which draws necessary conclusions, analyzes the concept of necessity and hypothetical states of things which Peirce deems equivalent; and considers Peirce\u27s analysis of the role of diagrams in the mathematician\u27s acts of construction. Finally, the chapter examines Peirce\u27s view that mathematics is wholly independent of logic. The second chapter on the formalists shows that Peirce agrees that mathematical sentences can be viewed as purely meaningless, but disagrees that this constitutes the essential nature of mathematics. Meaning, he argues, arises from the very manipulation of sentences and their applications. Moreover, the truth of the sentences of mathematics does not depend upon its applications. The third chapter on the logicists shows that Peirce agrees on the value of logic, but to the thesis that mathematics is logic objects that it does not appreciate the hypothetical nature of mathematics and misconstrues the nature of number. The fourth chapter on the intuitionists argues that Peirce agrees that constructions are crucial for mathematics, but differs as to the precise sense in which they exist. Moreover, the intuitionist\u27s restrictive conception of the law of excluded middle is proved unwarranted. The fifth chapter on the logical positivists argues that both maintain that mathematics is analytic, but Peirce\u27s view that mathematical truths depend upon hypothetical states implies that they do not, contrary to the positivists, also depend upon definitions. Definitions vary, truth grounds do not. The last chapter argues that, in accord with Peirce\u27s view, mathematical existence differs from mathematical possibility; that Peirce\u27s broad concept of truth applies to mathematical truth in particular; and that mathematical knowledge is knowledge of deductive relations. It is concluded that Peirce\u27s philosophy is superior to the classical views. The first appendix analyzes Peirce\u27s argument that the ordinal conception of number is primary; the second extends his ideas on continuity, and proves some original theorems concerning infinitesimals