[nwhitehead]

> But then, is any mathematical abstraction real? I guess it's all beside the point.

I actually think the reality of mathematical abstractions is hugely important because of... computer programs! In a real way, programs are the embodiment of mathematics. I want my programs to work so I need the underlying math to work as well.

That's why I'm a constructivist. I reject the law of the excluded middle because proofs that use it don't translate into real programs; they translate into programs that ask an omnipotent oracle to decide which branch to take. Constructive proofs translate into working programs.

It also ties into philosophy. I am a skeptic, so when someone tells me either A or not A must be true even if we can never know which one, I ask for proof or justification of that fact. The justifications that I get are remarkably similar to logical "proofs" that god exists, and just as fallacious. This isn't to say that there couldn't be an ultimate truth about A or a god, just that it is not logically necessary.

[mtdewcmu]

Interesting. I'm not all that up on philosophy, and I had to look up constructivism and the law of the excluded middle. Computers can only deal directly with discrete math, so that eliminates quite a bit of mathematics. Moreover, despite being regarded as Turing machines, you could in reality count all the states that a computer could be in, by counting the bits of memory, and you will find that it's a finite number. So computers are, in fact, only finite state machines. It's often overlooked that what makes a Turing machine a Turing machine is the infinitely-long tape. So computers occupy a fairly small sliver of the infinite universe of math.

Philosophically, my worldview is like that of science: the way to know something is by making observations and formulating and testing hypotheses. What we can observe is limited, and hypotheses are only models that are tested by successive approximations. I'm not sure I understand what you mean that that statement isn't required to have an ultimate truth. Presumably a statement like that has an answer, but the fact that something is knowable in principle doesn't mean that there's any way to get the answer. For instance, the Hubble telescope can see far away galaxies that we can never get an up close look at. The question of whether there's life somewhere else in the universe must have an answer, but we can't know what's in those galaxies; even with a better telescope, we'd be seeing what they looked like a billion years ago. Many things will never be known.