--- Autoplectic <autoplectic at gmail.com> wrote:
> Part of the difficulty resides in the notion of what
> we mean by a
> solution, or as Davis put it: "What kind of answer
> will you accept?"
...
> Yacht designers must also wrestle with these
> legendarily difficult
> equations. Over lunch, Davis told a story about
> yacht racing. He had
> recently talked to an applied mathematician who
> helped design a yacht
> that won the America's Cup. This yachtsman couldn't
> have cared less if
> the Navier-Stokes equations were solved; what
> mattered to him was
> that, practically speaking, he could model the
> equations on his
> computer and predict how water would flow around his
> hull. "Proofs,"
> said Davis, "are just one of the tools that
> mathematicians now use."
He's being a little wooly about what he's calling mathematics. Pure mathematics typically concerns itself with logical proofs, and judging the evaluation of complex ones is way beyond my skills (and probably the author's, too). But the example above is nothing new. It's been standard practice in science and engineering (and hence applied mathematics) for at least 150 years to work out a "pure" equation as far as you can take it, and if you get to point where you can't evaluate it any more, try to make it solvable by dropping out parts that you think are going to make a negligible contribution to the final answer. Knowing how to do this is a key skill in physics. It also tends to make pure mathematicians blanch. The result isn't 100% correct, but that doesn't matter in the context since you don't have 100% precision in anything else when dealing with the physical world, either.
Andy
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