[fasterik]

>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?

I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbounded operators, without which we wouldn't be able to define probability (the Born rule) or time evolution (the operator exponential e^-iHt).

Can you formalize quantum mechanics using computable numbers? I don't actually know, but let's say yes. I assume it's more work with more edge cases, so I would ask the same question: what do you get for the trouble of building a formalism around computable numbers?

[btilly]

Bad example. You can do all of this with constructivism. Any constructable Cauchy sequence converges to a constructable member of the space.

What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing could have.

[fasterik]

Having constructible Cauchy sequences doesn't guarantee that we can construct unbounded operators. I'm no expert, but the little searching I've done suggests this is an open research question.

I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.

Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.