Parsons’s Mathematical Thought: A footnote on intuition

Qn: “You do seem increasingly out of sympathy with Parsons’s book. So why are you spending all this effort blogging about it?”

Ans: “Well, as I think I said at the outset, I have promised to write a review (indeed, a critical notice) of the book, so this is just my way of forcing myself to read the book pretty carefully. And I’m not so much unsympathetic as puzzled and disappointed: I’m finding the book a much harder read than I was expecting. The fault could well in large part be mine. However, I do think that the prose is too often obscure, and the organization of thoughts unclear, so a bit of impatience may by now be creeping in (and talking to one or two others, I don’t think my reaction to Parsons’s writing is in fact that unique). But these ideas are certainly worth wrestling with: so I’m battling on!”

One thing I didn’t comment on before was Parsons’s motivation for pushing the notion of intuition and intuitive knowledge. “Intuition that,” he says, “becomes a persuasive idea when one reflects on the obviousness of elementary truths of arithmetic. Two alternative views have had influential advocates in this century: conventionalism … and a form of empiricism according to which mathematics is continuous with science, and the axioms of mathematics have a status similar to high-level theoretical hypotheses.” Carnapian(?) conventionalism is, Parsons seems to think, a non-starter: and Quinean empiricism “seems subject to the objection that it leave unaccounted for precisely the obviousness of elementary mathematics.” An appeal to some kind of intuition offers the needed account.

But I’m not sure that the Quinean should be abashed by that quick jab. For the respect in which the axioms of mathematics are claimed to have a status similar to high-level theoretical hypotheses is in their remoteness from the observational periphery, in their central organizational roles in a regimentation of our web of belief by logical/confirmational connections. That kind of shared status is surely quite compatible with the second-nature “obviousness” that accrues to simple arithmetic — for some of us! — due to intense childhood drilling and daily use. Logical position in the web, a Quinean would surely say, and degree of entrenched obviousness something else.