>With rationals you can only approximate.

Approximate relative to what? All actual measurement is implicitly or explicitly approximate such as L = x meters +/- epsilon. There is no infinite precision by which to discount rational measures as "approximate" and thus "invalid" in any way.

>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.

You "buy" all of mathematics which operates on the assumption of "infinite" precision. It is an abstraction necessary to prove theorems and relationships of math. Abstracting from precision isn't a denial that it exists, it is the assumption that I can ignore it or leave it undefined. This is the assumption that distinguishes math from physics/engineering. Mathematicians deal with abstract e or pi but in the real world pi=3.14 if you are tiling your patio and 3.14159265... or whatever is necessary to get to the moon.

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andrewla an hour ago | parent [-]

You are overestimating what real numbers buy you.

pi and e and sqrt(2) are real numbers and not rational, to be sure. But they are computable! Computable just means that they are arbitrarily approximable. "approximate relative to what" is that whatever criteria defines the number. You can't represent the "true" value of a non-rational number in the rationals, but you can prove that the error of an approximation is (rationally) bounded above and below, and you can have another approximation with a tighter bound.

Rational numbers are already infinitely precise relative to other representations -- finite decimals are another representation that is functionally equivalent to the rationals, but even a simple rational like 1/3 does not have a finite decimal value.

You can prove all the interesting theorems with computable numbers and rational/decimal numbers. You don't need the real numbers because you can't name a real number that exists and is not computable, BY DEFINITION! No mathematical construction can define a real number that is not constructible. These numbers are useless and there's no reason to continue even in abstract mathematics to pretend that they are useful because we have the formalisms to ignore them.

	▲	xscott 39 minutes ago \| parent [-]
		I'm on your side for most of what you say. This topic has been interesting to me for years. I've considered going back to school to build on my math degree, specifically because of this topic. However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem. Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.