Generally I take Ed to mean that a mathematical knowledge is (1) not about a mere "matter of fact" because our intellectual grasp of a mathematical truth is of something universal, not just of particular objects of sensory experience, (2) nor about a mere "relation of ideas" because (a) a mathematical proposition is an objective truth that tells us something about external, mind-independent reality, not just about how our "ideas" are related to one another, and (b) Hume's own account of "ideas" conflates intellect with imagination anyway and so gets that part hopelessly wrong.

It's the second of these that puzzles you, because Hume himself did expressly regard mathematical knowledge as about "relations of ideas." But when Ed says that mathematical knowledge is notoriously hard to account for in this way, he isn't denying that Hume made this claim, just that it can be made good in a way that preserves mathematical knowledge as both objective and universal. He doesn't think Hume can protect such knowledge from his own Fork by replying, "Nay, not so; for mathematical Knowledge concerneth Relations, which do obtain, betwixt pale Copies of ſingular ſense Impreſsions."

Last edited by Scott (8/21/2015 6:37 pm)

The criticism that I've heard of Russell and Whitehead is this: that they tried to show that mathematics was about, not just relations of ideas, but relations of purely logical ideas and nothing else.  In particular, I've seen R&W criticised for trying to assimulate the idea of infinity to logic.  Critics argue that the idea of infinity (as used in mathematics) is irreducibly mathematical and can't be reduced to pure logic.

Hume could evade this criticism, because, for him, the ideas in "relations of ideas" aren't restricted to logical ideas.

Last edited by Tyrrell McAllister (8/21/2015 6:57 pm)

But if Hume takes mathematical truths to be about relations between ideas according to his version of ideas, they're relations between particulars only; his account of ideas as "copies of singular sense impressions" doesn't leave room for universality. "Two plus two equal four," if true, could for him be true only of particular mental images. Nor would it be true of any objective extra-mental reality, although it might be true of certain exclusively mental events/entities.

At any rate, those are the reasons I take to be lurking behind Ed's remarks on Hume's Fork and mathematics. (I agree in principle that any such account of mathematical truth is problematic and unworkable, but I'm not sufficiently expert on Hume to know whether he genuinely defended such an account to the end of his days.)

Last edited by Scott (8/21/2015 7:10 pm)

Hume would have to accept that math isn't "true about extra-mental reality", because it is about ideas, which exist only in minds.  But he could still say that math is objective, in the same sense that there are objective truths about minds, even though these truths, trivially, aren't about extra-mental reality.

He has a problem with accounting how truths about "my idea of 2" could have anything to do with truths about "your idea of 2".  (When my idea of squaring is combined with my idea of 2 in a certain way, I get my idea of 4.  But this is supposed to reflect a truth about anyone's ideas, not just mine.)  But this is a general problem. (In what sense is your idea of gold "the same as" my idea of gold?)  It's not obviously a special problem for mathematical ideas.

Last edited by Tyrrell McAllister (8/21/2015 7:18 pm)

Tyrrell McAllister wrote:

He has a problem with accounting how truths about "my idea of 2" could have anything to do with truths about "your idea of 2".  (When my idea of squaring is combined with my idea of 2 in a certain way, I get my idea of 4.  But this supposed to reflect a truth about anyone's ideas, not just mine.)  But this is a general problem. (In what sense is your idea of gold "the same as" my idea of gold?)  It's not obviously a special problem for mathematical ideas.

That reminds me of Frege's criticism to the old psychologists, that they were unable to properly account for mathematical error.

Speaking of Frege, wouldn't Hume's view make necessary mathematical truths contingent on psychological or other contingent truths about minds?

Edit: I was typing when your latest post went up. In In the Light of Logic, Solomon Feferman writes about how foundations of mathematics is suffering from a sort of hangover after becoming drunk on grand reductionist schemes like the old logicism but, Feferman argues, mathematics doesn't need such extreme schemes as its foundations.

Tyrrell McAllister wrote:

Gödel's Theorems are a problem if you want to say that math is just about the logical consequences of a particular fixed set of axioms.  Mathematics proves to be "too rich" to be captured by any such set.  Can't the logicist just say that logic itself is similarly "too rich"?

I think so, yes. Godel's theorems were a problem for some very specific projects that people are still hypnotized by only.

Last edited by John West (8/21/2015 8:45 pm)