What Would It Mean for Math To Be Empirical?

Over at Obscure and Confused Ideas, Greg discusses possibilities for arithmetic to be empirical.

But what would it mean for arithmetic (or geometry) to be empirical?

I proposed this distinction:

a)Mathematical description can be applied to a given system. But we empirically find out that the mathematical truths which follow from the description are not true in that system. The example would be that we can describe something as ‘two’ (i.e. the system IS system of two things), but through empirical research we find out that there are no “one and one more thing” (of course while still there are two things).

b)Mathematical description can’t be applied to a given system. For example we can’t use simple arithmetics to track truths about number of rabbits in some room, if we just add one to a sum for a rabbit that enters the room, and subtract one if a rabbit goes out of it. That because rabbits can be born and die within the room too. However this doesn’t mean that arithmetic is wrong, or that it is not true that 1+1=2.

I think only in cases like (a) we can speak of arithmetic or geometry to be empirical.

But as a commenter Drake there pointed (btw, check out his cool MySpace page. But wait, first finish reading this post!):

The criterion for a state of affairs S’ falling under a mathematical description D is that D “holds” in all respects relevant to S. If it doesn’t, some other description D* is required. Conversely, to see that
D is not the right description for S, we have to see that D in some relevant respect doesn’t hold.

So, it is not clear that cases like (a) are intelligible at all. And I agree. To relate to the example – what would it mean that there are two things, if there are no one and one more thing?

BTW, I don’t use 1+1=2, as I think the formalism might hide the basic analytic truth that whenever we have two, we have one and one more thing, or the other way around. This is not saying that “two” means nothing but “one and one more thing”; but that when we have two things we can both think of them as a pair (“two”), or think of them as separate (“one and one more”). But as this is true just in virtue of the meanings of the terms, it is analytical truth.

Also, further interesting note would be, if we e.g. find something like (a) (yes, I’m saying it is unintelligible, but just for sake of argument let’s say it is) , e.g. a system where we have two things, but we have one and one more and still one more thing, would that mean that we should now change our math books, and put in there that 2=1+1+1?  What we will do with systems where we have two things we have one and one more thing? Will this show that arithmetic is inconsistent or something? I think, again, what we would have is a case of (b).

Two further cases that might be interesting:

A. Theory of Relativity

Can we say that the theory of relativity is case of (a)? I think not. What it shows, I think, is just that Euclidean geometry is not applicable to things in movement/which apply forces (gravity) to each other. However, what is put in place of Euclidean geometry is not a geometry that we have “constructed” from empirical research. It is as a priori (and non empirical) as Euclidean geometry is. What was empirical was the figuring out which of those mathematical descriptions fit the universe.

B. Unavailable permutations of particles in QM

Both ‘classical’ and ‘quantal’ objects of the same kind (e.g. electrons) can be regarded as indistinguishable in the sense of possessing the same intrinsic properties, such as rest mass, charge,
spin etc…That a permutation of the particles is counted as giving a different arrangement in classical statistical mechanics implies that, although they are indistinguishable, such particles can be regarded as
individuals…If such permutations are not counted in quantum statistics, it follows that quantal particles cannot be regarded as individuals … In other words, quantal objects are very different from most everyday objects in that they are ‘non-individuals’ in some sense. (SEP)

Is this the case of (a), do we have here a case where we have two particles, but not one and one more? This might be closest something can come to (a), but one can ask having in mind Drake’s comment, why do we think that there are two things after all? If we don’t think that there is one and one more thing, shouldn’t the conclusion be that there are not two particles either? I’m inclined to this second conclusion.

Anyway, any further idea of what would it mean for mathematic to be empirical?