In Defense of the Ideal 5th DRAFT

I will outline an argument that intends to undermine a certain form of skepticism about mathematics. This skepticism hangs on the fact that mathematics is about ideal things—structures and their elements, such as numbers, sets, functions, etc.—concerning the objective knowledge of which our sensory experience is entirely irrelevant.1 Precisely because of this irrelevance, the skeptic holds that we cannot meaningfully speak of and therefore cannot know anything about ideal things. Prominent expressions of this skeptical argument in relatively recent times are to be found in Paul Benacerraf’s “Mathematical truth” [1973] and Michael Dummett’s “Platonism” [1978a]. I first wrote on this topic in 1986, in the first part of a paper “Truth and proof: the Platonism of mathematics” [1986]. In writing it, I had the sense of diving into deep waters, well beyond my initial interest in foundations of mathematics. When I retired ten years later, further pursuit of the topic was first on my agenda. It, alas, still is. There have been those, from Aristotle in fourth century B.C. to Willard V.O. Quine and his followers in the twentieth, who have denied that mathematical objects are ideal in this sense, who believe that in one way or another truth about the natural world is relevant to mathematical truth. But, in all of its forms, this view has squared badly with mathematics, both as it was in Aristotle’s time and as it has been ever since, and it seems fair to see it as a failed attempt to avoid the skepticism as applied to mathematics while accepting the skeptic’s argument. Others, such as Poincaré and Brouwer, in more recent times and at least partially influenced by the skeptic’s argument, have attempted to place the objects of mathematics, if not in the domain of our outer experience, at least in the domain of our inner experience, in the mind. There are also those who accept that mathematical objects cannot