Why is mathematics so good at describing stuff?

[dahduh]

Dec 14, 2009 18:51:42 GMT vazscep said:

Dec 4, 2009 19:46:38 GMT dahduh said:

I would not say that failure to accurately model the social sciences using mathematics is a failure of mathematics; rather it is a reflection of a lack of good modelling, or that the systems are inherently chaotic and sensitive to the smallest perturbations. Even the latter we can understand perfectly using mathematics, even as it imposes a limitation on what we can do with mathematics.

I look at it this way: had it turned out that the basic systems of physics were chaotic and sensitive to small perturbations, I don't think we'd be asking why mathematics is so wonderful at describing stuff, because we would not have got much use from it. Indeed, we might even wonder: "why isn't the world more usefully described with mathematics?" My point is that we are in this position with many things we want to talk about.

I agree, in principle, that the world could be simulated to arbitrary precision and therefore described mathematically. But no-one's got a chance of doing that, so I wouldn't say this very dismal "in principle" prompts us to ask "why is mathematics so good at describing stuff."

Besides, most of the time I spend understanding the world is time spent not talking in maths.

I guess we got lucky! Or maybe it was just inevitable. For example, take any differential function f(x). To a first approximation for small x, f(x) = a + bx - it's linear! So as long as you don't jiggle things too much, everything is described by nice simple linear functions. So maybe it's not so surprising, living as we do in a cold universe, that lots of things are described very nicely by simple mathematics.

[blowfly]

Dec 14, 2009 21:34:09 GMT dahduh said:

Dec 13, 2009 23:19:23 GMT blowfly said:

I believe it is noteworthy, for a very good reason - the universe is not performing computation in order to work out what to be next, which is why the analogy above doesn't fit. The two boxes experiment isn't a case of two computational systems with the same underlying concepts, it's a case of a computational system and a non-computational system producing exactly the same output.

Now this is something I wish we could really be certain about. You say "the universe is not performing computation"; very well, let's be precise what is meant by 'computation'.

Computation as I understand it requires a) causality, b) a state space, and c) some systematic means of transforming your state space. If we look at the world, we do indeed observe we have all of these things, and more: it appears that our transformations are lossless, so no information is ever gained or lost.

Seems we have different ideas about computation. By computation, I simply mean the formal symbol manipulation that computers and arguably neurology do. This leads to a non-trivial definition of computation, ie. there is actually stuff which isn't doing computation.

I'll happily concede there's a bit of grey area, and eventually we might have to talk in terms of "computation-ish", but we can still comfortably identify the black and the white on either side.

Given this version of computation, to say that an electron "is mathematical" is to say that the electron "is performing computation" is to say it uses computational apparatus to work out what to be next. (ie. whips out its calculator) This is nonsense of course, so I conclude the universe is not mathematical.

So as far as I can see the universe is performing computation, but I have no idea how. The real dilemma as I see it, is that the universe is supposed to be, by definition, 'everything'; so whatever it is that is doing the computation, including the state space and the transformation rules, must be part of that computation. It is a bit like demanding that a Conway game of life, which we know can run an arbitrarily complex computer that can simulate anything, must simulate itself with perfect fidelity in real-time. Now maybe there exists some incredibly tricky configuration with very special redundancies in it that allows it to do just this, but frankly I just find the whole idea confusing.

I'd suggest this self-referential problem is the result of assuming the universe "is" computational (as opposed to "contains isolated instances of computation"). Drop this assumption and the problem disappears.

But pardon the discursion: I disagree, the universe definitely does perform computation, we just don't know how.

Or to put it another way - the universe doesn't "follow rules". Rather, we make formalized descriptions of the universe. Really - it's near impossible to make a coherent philosophy of the universe actually "following rules" - it ultimately implies that electrons are whipping out calculators and running QED formulas in order to determine where to pop up next.

I don't imagine it's quite so complex as this. For a long time there has been a suspicion that the quantum fields of the standard model are simply an effective theory; underneath it all there may be some very simple rules, with the apparent complexity an emergent phenomenon. For example, waves and whirlpools and turbulence are all very interesting phenomena described by a nice continuous differential equation, but this emerges from very simple particle-particle interactions. Maybe at the very simplest level there really are bits being manipulated by simple rules, but again what does the computation I have no idea.

It's an appealing idea. Let's assume it's true - if the simple "rules" we discover are mathematical, then again, the mathematical computation would be entirely in our heads, of our construction. The mathematics describes the physical phenomena, rather than controls it. We are the mathematicians, not the subjects of our observation.

Of course, if the Simulation Argument is accurate, then I'm wrong, and our universe IS mathematical ie. performing computations to work out what to be next.

Cheers,
-blowfly

[Kyrisch]

I think this is so because mathematical statements and rules are actually tautologies. For instance 2+2 = 4 is not a statement of fact but rather that four is defined as the sum of two and two, and that the idea of four -- the actual, abstract idea -- consists in and of itself of a sum of two and two.

Extending this conceptual basis, any additional mathematical "truths" can just as well be reduced to tautology, however complex: for instance, the 'fact' that the square root of two is irrational. The square root of two is just the number which, multiplied by itself, results in two, which hearkens back to the statement 2+2=4. But that the square root of two squared equals two is not being questioned here, rather its identity as rational. However, rationality is again defined to be the quality of ability of representation by the quotient of two integers.

In conclusion, it seems to me that the whole body of math does not include any actual knowledge, just an infrastructure of definition based entirely on the abstract idea of quantity. And as such, since things exist in ways easily described by quantity, math simply follows.