Does one have to become a Platonist to refuse to be a Platonist?

Question

I believe the answer is no, but Scott Aaronson on his blog just gave in interesting argument to the contrary. This is in connection with the now famous paper Undecidability of the Spectral Gap, and what it would mean for a Hamiltonian describing an actual physical system to be mathematically undecidable.

"This is a case, in my opinion, where you should have the courage of your intuitions! :) Imagine, for example, that God assured us that Goldbach’s Conjecture was independent of ZF. Even then, I would say: Goldbach’s conjecture is true in the real world, and what I mean by that, is that every even integer greater than 2 really is a sum of two primes. (Indeed, I would know that, since if there were a counterexample, then it wouldn’t be independent of ZF)... suppose you reject this viewpoint as “Platonism,” and you say: no, for me there’s no fact of the matter about anything until it’s proven or disproven in ZF. In that case, I reply: why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF, or about whether ZF itself is consistent? You seem caught in an infinite regress, where the only way out is to admit that, while you might or might not have any intuition about what transfinite sets are that’s conceptually prior to axioms, at any rate, you do have such an intuition about the positive integers".

I believe "infinite regress" is not the right word, but what Scott seems to mean is that proof constructions in ZF (or other formal theory) are essentially equivalent to arithmetic. Let's grant that God is not a deceiver. Scott's ZF fanatic appears to be in a predicament. She already accepted God's assurance that there is a fact of the matter concerning provability of the Goldbach’s Conjecture. Is she now compelled to accept that there is a fact of the matter concerning its truth as well? Is her only way out to doubt God?

Answer 1

EDIT 30 May 2018

I was reminded of this question when I was reading one of Aaronson's papers over the weekend. I think I now understand the argument Aaronson is making here.

Aaronson catches the ZFer on shaky ground when he notes:

In that case, I reply: why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF, or about whether ZF itself is consistent?

The problem is that independence (and consistency) results are results from mathematical logic about ZF rather than results in ZF. Therefore, if the ZFer is going to only accept results proven in ZF then how can she even think in meta-theoretic terms of independence and consistency.

Thus, the ZFer is forced to either accept the realist argument that the independence of GC implies its truth, or she must accept that God's divine declaration is meaningless.

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(Original Answer)

I am having a hard time pinning down Aaronson's argument, both philosophically and logically.

Aaronson begins by arguing that the independence of GC from ZF would imply its real world truth. This is Platonism in the sense that it assumes that GC is either true or false (in the real world), amongst other things.

He then turns around and rebuts his antagonist's objection to his Platonism with a non-Platonist argument when he says "why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF". He is rejecting LEM as applied to ZF, saying there are statements S in ZF such that "S is ZF-provable or S is not ZF-provable" is not true.

I do not see the ZF fanatic as facing a predicament. I see Aaronson facing a predicament. How can he defend his Platonist argument with a non-Platonist argument.

I have probably misunderstood both the question and Aaronson's argument. My mathematical knowledge is not very deep. I am also not entirely comfortable with his claim that our intuition concerning integers is somehow stronger than our intuition concerning transfinite sets, but that's another question.