The delay in getting round to talking about the next couple of papers in Charles Parsons’s Philosophy of Mathematics in the Twentieth Century signals, I’m afraid, a certain waning enthusiasm. I’m still hopeful that the essays in Part II of the book, on his contemporaries, will prove more exciting, but I found the two pieces on Gödel that end Part I of the collection rather frustrating. Though perhaps the problem is more with Gödel as a philosopher rather than with Parsons. Gödel’s more general remarks — e.g. about platonism, or the role of intuition in mathematical knowledge — too often seem un-worked-out (to put it charitably), and Parsons is too careful a philosopher to leap to giving readings which attribute to Gödel a more developed (and hence more interesting) position than the texts really warrant. So we are left, after reading Parsons’s discussions, not much clearer about what Gödel’s position comes to than we were before.
We’ll look at Parsons’s familiar piece on Gödel’s Platonism in the next post. But first comes ‘Quine and Gödel on Analyticity’, written for a 1995 volume of essays on Quine, but in fact concentrating on Gödel.
The initial theme is that there are some similarities between what Quine and Gödel say about analyticity and mathematics that arise from their shared opposition to the views they attribute to Carnap. Both argue against the view that arithmetic, say, can be held to be empty of content (in Tractarian spirit) by noting that in developing that view, the contentual truth of arithmetic must be already be presupposed in arguing that arithmetic, as a syntactic system, has the properties required of it. (In his Postscript to the reprinted paper, Parsons revisits the question of the force of Gödel’s arguments against the real Carnap.)
But Gödel of course goes on to make a claim that Quine would strongly resist: namely that, while e.g. arithmetic is not analytic in the sense of vacuously-true-by-definition, it is analytic in the sense that (as Parsons puts it) “mathematical truths are true by virtue of the relations of the concepts denoted or expressed by their predicates”.
The trouble is that on the face of it Gödel doesn’t have anything much by way of a theory of concepts, or any very helpful way of explicating his talk of “perceiving” a relationship between concepts except from some opaque remarks about intuition. Or so it will seem to the inexpert reader of the relevant passages in Gödel. And Parsons, as our expert reader, doesn’t seem to find much to help us out here, but rather confirms first impressions that Gödel lacks a serious theory. Which isn’t to say that Gödel must be barking up the wrong tree. Indeed, in the Postscript, Parsons goes as far as to say that “a view something like Gödel’s on this point [that mathematics is in a sense analytic] has always seemed to be the default position”, being something that developed out of his experience as a mathematician, and — which Parsons implies — should chime with the experience of other mathematicians. And I’m tempted to agree with Parsons here. But he too doesn’t do much to hint at how we might develop that view. Which leaves the reader — or at least, leaves this reader — not much further forward.
1 thought on “Parsons #5: Gödel on analyticity”
One way to develop this “default position”, without appealing to Gödel’s realism about concepts, would be to cling on the remarks on pp. 141-2, in which Gödel appeals to considerations intrinsic to mathematics in order to justify a given axiom (e.g. the axiom’s fruitfulness). Indeed, I found a bit surprising here that Parson’s didn’t mention Penelope Maddy’s work in this connection (even if in the postcript), though perhaps this is understandable given the aim of the essay.