Two Questions Regarding Hume’s Account of Relations of Ideas

I’m not sure Hume deals very specifically with the idea of number, beyond to say that it is a relation of ideas. He has a very odd, obscure argument, borrowed from the mathematician Malezieu, in Treatise 1.2.2 that seems to assume that numbers presuppose that lines are composed of finite indivisible points; but, as I said, it’s an odd and obscure argument, so there are probably several ways to take it. He does, however, have a very important and extensive discussion of equality in Treatise 1.2.4 (see also the summary in Treatise 1.3.1), which was an important influence on Frege’s attempt to come up with an account of number.

It’s also at the beginning of Treatise 1.3.1 that Hume addresses something like your second point. Some relations of ideas can be changed without changing the relations of ideas; some, however, depend entirely on the ideas themselves, and therefore cannot change as long as the ideas remain the same.

I should have been slightly more precise. Hume is very explicit that number (or ‘proportion in quantity’ as he also calls it) is a relation of ideas. But we can have ideas of particular relations of ideas, which are copies of the original ideas in these relations, and these are our ideas of numbers.

That’s one way to read him (and the way Frege himself did read Hume, I take it). But as I said, Hume’s remarks on number are rather scattered and not deeply developed. He spends much, much more time discussing equality than he does discussing number; but it is reasonable to think that the two fit closely together, in which case something like the broadly Fregean reading, which you are suggesting, is right. I’m not wholly convinced that Hume’s notions are as well-formed as the Fregean reading requires, but it’s certainly a powerful reading, one that’s plausible in many ways.

(Hume seems to think of number as relatively unproblematic, and so he doesn’t discuss it at length. The reason he discusses equality at length is that he thinks there are severe problems with the notion of geometrical equality that don’t exist with arithmetical equality. So he has to trace out the purported problem and his suggested solution.)

Part of it may just be that the arithmetic side wouldn’t have been seen as obviously rationalist-favoring at the time, whereas a number of features of geometry (its various infinities) were already a pretty obvious issue that empiricists had to deal with. And you can accept the necessity of arithmetical and algebraic truths without running afoul of the copy principle, which is really the whole of Hume’s empiricism; as I think you are suggesting in your comment.