Fri Dec 2, 2005
727 Evans, 4:10–6 PM
Charles Chihara (UC Berkeley)
Burgess’s “Scientific” Arguments for the Existence of Mathematical Objects
This paper addresses John Burgess’s answer to the “Benacerraf Problem” (given in his review of my book “A Structural Account of Mathematics”): How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess’s response is summed up in the motto: “Don’t think, look!” In particular, look at how mathematicians come to accept:
[*] There are prime numbers greater than 10 to the tenth.
That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers.
What underlies this argument can be seen more clearly by studying another recent work of Burgess (co-written with Gideon Rosen), viz. “Nominalism Reconsidered”. That work presents an argument with three premises:
(1) Mathematics abounds in theorems that assert the existence of mathematical objects.
(2) Mathematicians accept these existence theorems and rely on them in both theoretical and practical contexts.
(3) These theorems are proved in an acceptable way. Ergo, there are mathematical objects.
My paper is a rebuttal of the Burgess arguments.