I have always had two interpretational questions about Hume’s account of relations of ideas. These issues come up in my into class all the time and I am constantly foiled in my attempt to locate an exact answer either in Hume’s corpus or in the secondary literature. Maybe someone else knows where I should look…

The first question is about Hume’s account of our ideas of numbers. Locke is very clear that the ideas of numbers are what he calls modes. We start with our simple idea of a unity and then form the complex ideas of 2, 3, 4, 5…etc by combing this idea with itself. So the idea of the number three is a complex idea composed of three of the simple ideas of a unity (‘III’). Does Hume accept this account of our ideas of numbers? Or does he have some other account of them? I somehow started to think that he had a set-theoretic accoount but I may be turning him into a logical positivist…

The second question is about the possibility of change in relations of ideas. If the mathematical truths are simply definitional truths defined in such a way as to exclude contradictions then it seems that it should be possible for us to change these definitional relations. Is this what Hume actually thinks or am I turning him into Quine? If he doesn’t think this, then how do the relations get set? And what makes it the case that they can’t be changed?

Does anyone know where in Hume’s work he is more explicit about these issues than he is in the Enquiry? Or the name of a good secondary source that addresses these issues?

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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.

Hi Brandon, thanks for the tips (I thought you might know where I should look!)…I am going to look up those passages this weekend when I get a chance…one quick question though. Did you mean to say that the ideas of numbers are relations of ideas for Hume? I took it that the ideas of numbers had to be either complex ideas (like Locke) or simple ideas and that it was mathematical truths like 5*10=50 which were relations of ideas…

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.

Ah, thanks Brandon, that’s very helpful!

So, I take it, that Hume is anticipating a move like the one that Frege makes later? We start with a definition of number (the number of x’s is equal to the number of y’s iff we can put the x’s and y’s in a one to one correspondance) and then we form our ideas of particular numbers by coping the relata in these relations of ideas? So the idea of ‘2’ comes from the relation of ideas that has ‘x-y, x-y’ as its relata? Is that the idea?

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.)

Thanks again Brandon, this is very helpful!

I don’t know why Hume disavowed the Treatise…it seems to me superior to the Enquiry. Maybe, like Stephen Colbert, he let the market decide which was the best đź™‚

But all of this does leave one wondering how this fits with his empiricism. One gets the feeling that he thinks denying that numbers have an independant ontological status is enough, but then he seems to consent to the usual rationalist dogma that the arithmetic/algebraic truths are necessary…

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.

Hey, this seems to be an old thread but since studying hume I’ve got a question I’m uncertain about. He seems to state that 2 2=4 is a tautology and therefore it is the vsame as saying that 4=4 following this thought you must surely say that 1.5 2.5 is a definition of four as well and so on. If this is held to be the case then that would mean 4 has an infinitely large number of possible definitions which, considering we cannot have a clear concept of infitesimal amounts but a concept of whatcthey represent, would imply we cannot make sense of numbers at all. I think this is stated by a philosopher but I cant remember who.