Well, math works by definition through deductive logic. If the question were, 'Why is math such an effective tool for modeling and solving real world problems" the answer is that those problems are not inherently physical; they are just abstractions built on assumptions of form and function, and these assumptions, if properly expressed in a formal grammar, can form a basis for manipulation. Again one might ask, 'Why do valid logical deductions work?' but we immediately discover the tautology; 'valid' and 'working' too are merely definitions.

Hmmmm.... but if something's a tautology, shouldn't it be true in every conceivable situation?

Here's a thought experiment. What follows sort of assumes a deterministic universe but I think it could apply to a less deterministic one too. Say I have a rock, and when I drop it, it falls. I just tried it right now so I know it really does happen. Let's say the state of the system includes the position/velocity/etc of each atom making up the rock. Number every possible state s1, s2 ... sN. And say f(sk) is the successor to state sk.

So that's a lot of states, way more than I could understand, but there's a pattern, that the rock falls. And I can reliably see this. If I want to come up with a model for the rock's behavior, I need some kind of a pattern. But I don't think that that is possible in general.

Let's make a new system with states r1 ... rN based on a renumbering of the states s1 ... sN, so that rk = P(sk) for a permutation P. Then the successor g(rk) = P( f( P~1(rk))). I think this system is just as valid as the first one, aside from violating the laws of physics. It's perfectly describable. But I think you'll agree that for most choices of P, I won't be able to find a recognizable pattern and hence there is no hope for using math in a useful way here.

About r1...rN being an unrecognizable pattern, I suppose that just means that our brains didn't evolve to analyze structures like that one. Certain patterns exist in the world, and our brains have developed to deal with those sorts of patterns as well as similar ones. Eventually, we draw these patterns out of our unconscious and write them down in a neat formal language. Sometimes, after writing it down, we can refine our sense of pattern, leading to a deeper understanding of the universe and structure in general.

If mathematics indeed models the world so well, I attribute its success to selection bias.

I would really like to see some empirical demonstrations of Goedel's Theorems, but I'm not even sure how they'd be manifested, unfortunately.

"If mathematics indeed models the world so well, I attribute its success to selection bias."
I suspect you are right, but I do not know how to find out. I'm in school for engineering right now, and do a lot of hand problems analyzing circuits and such. After a long night of homework those few situations where the math comes out friendly feel so much _nicer_. If I had a choice I'd do more of _those_. And so I am subject massively to selection bias, to the point I don't even know how I'd get out of it.

Nature, of course, has no difficulty carrying through in physical form all the problems presented to it, with no regard for the math. Nature and I see the world differently.

hartford27th Aug 2009, 8:38 AM edit deleteKMFE27th Aug 2009, 5:02 PM edit deleteHere's a thought experiment. What follows sort of assumes a deterministic universe but I think it could apply to a less deterministic one too. Say I have a rock, and when I drop it, it falls. I just tried it right now so I know it really does happen. Let's say the state of the system includes the position/velocity/etc of each atom making up the rock. Number every possible state s1, s2 ... sN. And say f(sk) is the successor to state sk.

So that's a lot of states, way more than I could understand, but there's a pattern, that the rock falls. And I can reliably see this. If I want to come up with a model for the rock's behavior, I need some kind of a pattern. But I don't think that that is possible in general.

Let's make a new system with states r1 ... rN based on a renumbering of the states s1 ... sN, so that rk = P(sk) for a permutation P. Then the successor g(rk) = P( f( P~1(rk))). I think this system is just as valid as the first one, aside from violating the laws of physics. It's perfectly describable. But I think you'll agree that for most choices of P, I won't be able to find a recognizable pattern and hence there is no hope for using math in a useful way here.

Okuno Zankoku6th Apr 2011, 8:43 AM edit deleteIf mathematics indeed models the world so well, I attribute its success to selection bias.

I would really like to see some empirical demonstrations of Goedel's Theorems, but I'm not even sure how they'd be manifested, unfortunately.

KMFE14th Apr 2011, 8:14 AM edit deleteI suspect you are right, but I do not know how to find out. I'm in school for engineering right now, and do a lot of hand problems analyzing circuits and such. After a long night of homework those few situations where the math comes out friendly feel so much _nicer_. If I had a choice I'd do more of _those_. And so I am subject massively to selection bias, to the point I don't even know how I'd get out of it.

Nature, of course, has no difficulty carrying through in physical form all the problems presented to it, with no regard for the math. Nature and I see the world differently.