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tdvance · #37 (**permalink**) 07-April-2008, 02:30 AM

Modeling the universe with real numbers a-la classical mechanics: one reason we use real numbers rather than, say, million-digit rationals is you get a lot more complication with a "rational" model, but without any discernible improvement in accuracy.

On the other hand, you get a tremendous (in terms of what can be measured) increase in accuracy by switching to Quantum Mechanics. However, instead of infinite-precision reals, we now have---complex-valued wave functions! Not only do we go from finite precision to infinite precision, we go from 3 dimensions for a particle in space, 3 more for momentum, but we actually have an infinite number (worse, a continuum of) dimensions--a complex-valued wave function is essentially a vector of complex numbers of dimensionality equal to the number of real numbers. One might be able to come up with a Rube-Goldberg model that avoids the use of wave functions and complex numbers, but I doubt you'll get an improvement in accuracy for the increase in complexity.

The upshot is, mathematics wins again in the race to understand the universe. Concepts of mathematics that were discovered by pure mathematicians turned out to fit reality very well. General Relativity is another example--differential geometry and tensor algebra, and some other things, come together to make the most accurate large-scale theory there is. If Einstein hadn't had differential geometry available to him, it is very likely he would either have stopped at Special Relativity, or at best, come up with a patchwork Rube-Goldberg theory for how matter behaves in various situations--no gravity field, constant field, constant rotation, constant acceleration, rotation+acceleration, rotation+gravity field, and so on.