I get the appeal, it's a minimalism thing. The thinking mind makes up patterns and checks them against each other, there is no ultimate reality, blah blah blah. Besides, a mathematical ground truth would be a kind of transcendence, in that it's prior to human thought, which makes it uncomfortably close to ideas of God.
Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.
I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?
The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.
An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.
Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.
At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.
I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?
The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.
An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.
Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.
At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.