Formalism versus Platonism

Abraham Robinson once mentioned that the situation in number theory seems to favor a Platonist outlook, whereas the situation in set theory seems to favor a Formalist outlook.

The context of Robinson's remarks on number theory and set theory is provided by the (then) recent result by Paul Cohen who introduced the method of forcing to show that the Continuum Hypothesis is independent of ZF. Cohen's method preserves ℕ and introduces new entities only at the level of P(ℕ) (equivalently, the reals), so it sheds no light on the philosophical/ontological status of ℕ itself. This is what Robinson means when he refers to "number theory". According to Robinson, Cohen's work demolished any idea of a "preordained" real number line (in spite of its touted properties such as categoricity, unicity, etc.), therefore favoring a Formalist approach in set theory.

The question remains whether similar independence results can be obtained for purely arithmetic statements, such as the twin prime conjecture. This is currently unknown.

What is the role of Goedel incompleteness in all this? To fix ideas, let's take the sentence Con(PA). As we know following Goedel incompleteness II, this cannot be proved in PA (though there is some interesting recent related work by Artemov; see below). This seems to be evidence against Platonism. However, Con(PA) can be proved in ZF. Many mathematicians consider this fact to be significant evidence in favor of Con(PA). So that would seem to overcome this particular objection against Platonism. Of course, the Continuum Hypothesis is famously indederminate (in a number of senses that have been discussed in the literature). However, CH is not a purely arithmetic sentence. Therefore CH provides evidence against Platonism in set theory but not in Arithmetic.

It needs to be emphasized that there is a difference between consistency of ZF and Platonism about ZF. Robinson thought it reasonable to work as though ZF were consistent.

The situation with Con(PA) and its negation is not symmetric. One can have a nonstandard model of arithmetic where there is a proof of the negation of Con(PA), but such a proof would have to have nonstandard length. This means that as far as metalanguage integers n are concerned, there will never be a proof of length n establishing the negation of Con(P). In this sense, the logicians assume that it is reasonable to assume that PA is consistent, in the sense that practically speaking a concrete (and therefore metalanguage-integer-length) proof of inconsistency will never be found.

It is fine to be sceptical about the consistency of PA; in fact, Edward Nelson was seriously searching a proof of inconsistency of PA. Of course, much of modern mathematics is based on the informal assumption that PA is consistent (and that ZF is consistent).

Again, working under such an assumption is different from adopting Platonist views about mathematical entities. Note that the eminent scholar Sergei Artemov recently pointed out that there is an important distinction between Con(PA) and Con^S(PA) (the latter being Hilbert's view). In fact, Artemov proved Con^S(PA) within PA itself; see the discussion here. So this arguably settles the issue as far as the consistency of PA is concerned.