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Elements of Mathematics, his Collected Papers, and unpublished manuscripts,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
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
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