SC285: Does Math Exist? - Cool Video

Here's an interesting video produced by PBS Idea Channel.

When I first got into philosophy and heard about this thing called philosophy of mathematics and began to get a sense that there were questions like, "Do numbers exist?" I was pretty puzzled by the whole thing.  And I'm still pretty baffled by most of it.  This video does a nice job of introducing one of the basic questions.

"How does our mind connect with the external world around us?" is a very, very old question in philosophy.  When I hear about anti-realist views, I'm struck by the thought that they seem to be worried by the idea that the fit between our minds and the world seems to be almost too good.  (How one determines how well our minds fit with the world is an entirely different question that I won't take up.)  When I say they're worried, I don't mean that they consider the close fit between mind and world to be a bad thing.  But they do see the close fit between mind and world as something of a puzzle to be solved.  What explains the close fit between our minds and the world?  Their answer: it must be that our minds actually shape the world in important ways.  To put it less precisely, if a bit more provocatively, the reason that our minds and the world seem to fit so well must be that what we encounter in the world is actually coming from our minds.  Or, in a sense, it's all in the mind.

The fact that our mathematical concepts find such ready application in the world and seem to apply so universally--that seems to push these anti-realists to conclude that we must be imposing that sort of order.  But why not go the other direction?  Why not think that the reason we see math everywhere is because, in an important sense, the math really is there?  Is it because scientists and philosophers have just accepted that the only things that could possibly be there are material things; and since math clearly isn't material it couldn't possibly be there?  Or has the (possibility of the) existence of immaterial objects been decisively refuted?

Connecting mathematical realism with faith seems an odd move.  Is anti-realism supposed to be more scientific?  Does the thesis that mathematical concepts are produced by the human mind make mathematics more scientific?

Is there another reason for not thinking that the math is really out there?