A Comparative Analysis Of
Charles S. Peirce's Philosophy Of Mathematics
Stephen Harry Levy
Source: DAI, 43, no. 01A, (1982): 0185
This dissertation, relying upon Peirce's New Elements of Mathematics, his Collected Papers, and unpublished manuscripts, analyzes the classical philosophies of mathematics, and argues that Peirce shares their insights, but deftly avoids the snares entrapping them.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's 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's analysis of the role of diagrams in the mathematician's acts of construction. Finally, the chapter examines Peirce's 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's 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's 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's view, mathematical existence differs from mathematical possibility; that Peirce's broad concept of truth applies to mathematical truth in particular; and that mathematical knowledge is knowledge of deductive relations. It is concluded that Peirce's philosophy is superior to the classical views.
The first appendix analyzes Peirce's argument that the ordinal conception of number is primary; the second extends his ideas on continuity, and proves some original theorems concerning infinitesimals.
Accession No: AAG8213246