I'm reading Benacerraf's What numbers could not be, where he provides the following argument against platonistic account of numbers.

  1. 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.
  2. 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 = {{∅}}, ....
  3. 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.
  4. 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?

  • QM and arithmetic are very dissimilar. QM is a black box, we have not determined the 'right' description of what's inside yet, but perhaps we might. Arithmetic is not a black box, it is a conception we use. We know exactly what's inside, "conditions which were both necessary and sufficient for any correct account", as he puts it. Yet, with sets "we have two... each of which satisfies what appear to be necessary and sufficient conditions". Either one of them has platonically 'right' 3, but its 'distinction' has nothing to do with the 3 we use, or 3 can't be a set. That's the dilemma.
    – Conifold
    Apr 30 at 12:50
  • What we do not yet know about physics may well tell us which interpretation of QM is really the case, if any. On the other hand, nothing we do not yet know about arithmetic (some deep theorems or whatnot) can tell us which set 3 'really' is. Thus, set-theoretic platonism about numbers is either irrelevant to our arithmetic, or false.
    – Conifold
    Apr 30 at 13:00
  • @Conifold Thanks for the comments, but I don't think QM and arithmetics are "very" dissimilar here. Imagine a case where scientists made all the possible empirical observations about QM, yet there are still different interpretations of QM standing. Such case seems identical to the current coexistence of several set theoretic interpretations of arithmetic. Then wouldn't your conclusion imply that realism about quantum theory is either irrelevant to our (empirical) quantum theory, or false?
    – Dimen
    Apr 30 at 16:26
  • @Dimen In physics, we deal only with approximate models, so it is very possible that 2 approximations with differing metaphysical implications both produce reasonably accurate models. However in the case of mathematical logic, numbers and sets are very well defined, not as approximations, but as objects obeying an exhaustive collection of axioms and deduction rules. When dealing with approximations, there is the possibility of finding yet another approximation that will work better. But that is not the case for numbers and sets.
    – Frank
    Apr 30 at 17:38
  • 1
    I don't see that the argument is circular, but it seems to assume that for numbers to exist as abstract objects, they have to be reducible to something else. Also, the claim that any theory would have the same problems he points out for sets has already been shown to be false by neo-Fregeans. Apr 30 at 23:19


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