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A friend of mine who is quite an aggressive Nominalist told me the other day:

"Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are consistent, but any consistent framework will do."

I'm curious as to what other examples of "consistent frameworks" he is talking about. Perhaps there are alternative definitions of addition? Of subtraction, or multiplication?

Different things you can do to numbers that have a "mapping" (isomorphic) to real-world, physical phenomenon? (Such as mathematical addition isomorphically mapping to, say, putting together two groups of apples, or differentiation of position mapping to velocity)

Are there any examples of this?

There might not be; this is somewhat of a vague question and it might end up being very stupid. But I'm interested in what other completely different frameworks of "math" might exist that are meaningful in describing/modeling physical events.

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    $\begingroup$ This is a philosophical question, not a mathematical one. At least your tags should reflect this. $\endgroup$ – Robin Chapman Oct 25 '10 at 7:08
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Philosophers of a nominalistic/constructivist bent often come out with stuff like this, asserting that maths, physics or whatever is some sort of arbitrary, contingent system which could have been totally different if history had turned out differently. What these philosophers are not very good at doing is proposing credible alternative systems. Of course, the burden of proposing alternative "constructive frameworks" should be on your chum. Only he can say what he has in mind (if anything), not us.

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