seminar: Burgess's "Scientific" Argument for the Existence of Mathematical Objects
speaker: Charles Chihara (University of California, Berkeley)abstract: This paper addresses John Burgess's answer to the "Benacerraf Problem": 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:
1 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. Burgess's answer to Benacerraf's Problem is similar to his refutation of a version of the causal theory of knowledge. In "Epistemology and Nominalism", Burgess argues that scientists assert:
2 Avogadro's number is greater than 6 x 10 to the 23rd.
And scientists back such claims with empirical evidence. But 2 also implies that there are numbers, just as 1 above does. Thus, we have strong scientific grounds for believing that there are numbers. Since Burgess holds that mathematicians are scientists, he thinks that the argument based upon a proof of 1 also provides us with a scientific justification for the belief in numbers. What underlies these arguments can be seen more clearly by studying another recent work of Burgess's (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) Mathematician 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. It notes that the implications of the above arguments for the concept of evidence and proof are extremely counter-intuitive. It also brings out crucial ambiguities in premises of the arguments that, when seen, expose flaws in the reasoning.
timetable:
Mon 23 Jan, 15:00 - 16:00, Aula Dini
<< Go back