One thing that that has caused me great puzzlement (in the past few years), is the question of whether math tells us anything 'substantive'. I want to suggest that our intuitive notion of "substantiveness" combines two distinct notions, which come apart in this case.
- mathematical truths DONT rule out any physically or even metaphysically possible states of the world. (This is just another way of putting the truism that mathematical truths are necessary, hence compatible with every metaphysically possible world. I like putting things this way, because it doesn't suggest that necessary mathematical truths arise from something (mathematical objects?) causally blocking any person that tries to being both more than three feet long and less than two feet long)
- mathematical truths DO combine with our background beliefs to lead us to form expectations we wouldn't have formed otherwise (e,g. about the results of future counting procedures, about the programs)
Presumably you admit that these are at least nominally different properties. But you might still wonder *how* these two things could come apart. How could knowing any proposition be useful, if this proposition didn't rule out any possible states of the world? Here's what I think the answer to that is in a nutshell:
Some mathematical facts (i.e. facts which are derivable from math and logic alone) which are useful because they tell us that whenever one description of the world holds, then so does another (e.g. anything that accelerates from standstill at this rate for this amount of time travels that distance, anything that's less than two feet long isn't three feed long.)
And here's the answer in more detail.