I'm reading Benacerraf's What numbers could not be, where he provides the following argument against platonistic account of numbers.
- The only criteria we can ask for in searching the correct account of numbers is that (a) it should include definitions of numbers and operations by which the laws of arithmetics can be derived, and (b) it should explain the application of numbers, notably counting.
- There are many set theoretic accounts of numbers that satisfy (a) and (b), but have conflicting nature. For example, defining 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, ... yields 0 ∈ 2, which is not true with 0 = ∅, 1 = {∅}, 2 = {{∅}}, ....
- As there is no reason to favour one account over another, the correct set theoretic account of numbers does not exist. Thus, numbers are not sets.
- Indeed, numbers cannot be platonistic, since any account of numbers in terms of mathematical objects would encounter the same problem.
However, I find this argument fallacious. In physics, the only criterion we can ask for in searching the correct theory is that it should be empirically adequate. However, we have several metaphysically conflicting interpretations of quantum theory with equivalent empirical consequences. Yet this does not compel us to concede that no correct interpretation of quantum theory exists.
Likewise, a platonist about numbers can reasonably hold that although there are several arithmetically equivalent reduction of numbers, there is only one metaphysically correct reduction. So it seems that Benacerraf's argument against platonism can be rejected by holding platonism, thus circular. Have I misunderstood his argument?
