Volume 17, 1992
The Argument from Agreement and Mathematical Realism
Traditionally, in the philosophy of mathematics realists claim that mathematical objects exist independently of the human mind, whereas idealists regard them as mental constructions dependent upon human thought.
It is tempting for realists to support their view by appeal to our widespread agreement on mathematical results. Roughly speaking, our agreement is explained by the fact that these results are about the same mathematical objects. It is alleged that the idealist’s appeal to mental constructions precludes any such explanation. I argue that realism and idealism, as above characterized, are equally effective (or problematic) in accounting for our widespread mathematical agreement.
Both accounts are descriptivist for they take mathematical statements to be true if and only if they correctly describe mathematical objects. By contrast, non-descriptivist accounts take mathematical statements to be rule-like and mathematical symbols to be non-referential. I suggest that non-descriptivism provides a simpler and more natural explanation for our widespread agreement on mathematical results than any descriptivist account.