There’s Something about Gödel, Ch. 9

Chapter 9 of There’s Something about Gödel is about Platonism. There are two tricky issues here — first getting clear about what Gödel’s own views were and how they changed over the decades between 1931 and his late philosophical papers, and then second assessing those views. Berto’s chapter is only sixteen pages long. And five of those are an explanation of Tarski’s theorem on the indefinability of truth. Unsurprisingly then, the rest is too rapid and superficial to get very far. But does it at least start off in the right direction?

Berto writes: “Gödel appears to have believed [that] the Incompleteness Theorem … refutes the idea of mathematics as pure syntax, and validates the metaphysical claim that numbers are real, objective entities in the timeless Platonic sky”. The first half of that is right. Gödel did believe that the Theorem refutes the idea of mathematics as pure syntax: but work is needed if we are to characterize a general notion of “mathematics as syntax” that is wide enough to cover potentially attractive programmatic views but sharp enough to be vulnerable to a crisp refutation (which is probably why there are six drafts of his paper on Carnap left in the Nachlass). But the second half of Berto’s claim about the “timeless Platonic sky” is just crass. It takes real work too to tease out the non-metaphorical content of Gödel’s Platonism — and just ramping up the level of metaphor and talking of timeless Platonic realms (which as far as I can recall, Gödel never does) is no help at all in doing that work, which is left entirely undone here. If to talk about objects in a Platonic sky is to “treat the analogy between the existence of physical objects and the existence of mathematical ones seriously as a literal account of the way things are” (as Michael Potter puts it), then it is highly arguable — and has been argued — that this quite badly misrepresents Gödel.

Berto doesn’t mention that at all, but instead seems rather keen on Rebecca Goldstein’s crude account in her Incompleteness: The Proof and Paradox of Kurt Gödel, and quotes her as giving “a summary interpretation” of a supposed Platonic interpretation of the First Theorem (p. 158). That’s a pretty extraordinary choice, given that Goldstein’s book is frankly awful. It’s not just my view that “the book as a whole is marred by a number of disturbing conceptual and historical errors” — those words are from Feferman’s damning review.

Striking out for ourselves just for a moment, here’s what we establish in proving the First Theorem applied to PA (making the usual assumption about omega-consistency). There’s a primitive recursive relation two-place relation Prf, and a number g, such that for all numbers x, it isn’t the case that Prf(x,g), i.e. ∀x¬Prf(x,g): but PA can’t prove or refute ∀x¬Prf(x,g), where Prf(x,y) formally represents Prf and g is the formal numeral for g. There’s no metaphysically loaded notion of truth involved in stating that theorem, because there is no notion of truth involved, full stop. Of course, since ∀x¬Prf(x,g) expresses that, for all numbers x, it isn’t the case that Prf(x,g), and indeed for all numbers x, it isn’t the case that Prf(x,g), we can say ∀x¬Prf(x,g) is true, and so say that the Theorem shows that there is a truth that can’t be proved (nor, thankfully, disproved) in PA. But this use of the notion of truth is anodyne and basically disquotational, still without metaphysical ooomph. If we start generalizing, and talking about not just PA but suitable axiomatized theories more generally, then we can again say that more generally that for each such theory there will be truths that can proved in that theory. But still, the notion of truth involved remains metaphysically anodyne, and not distinctively platonistic: an anti-realist can be content so far. Which suggests that — without more philosophical input as side premisses — the First Theorem doesn’t have specifically Platonistic implications.

Did Gödel think otherwise? At a second pass, with some further premisses, can we after all draw Platonist morals from the incompleteness phenomenon? Well, a useful theme to pursue would be this. Dummett famously argues that that Gödelian incompleteness is tied up with indefinite extensibility, and taking indefinite extendability seriously should leads us to be anti-realists, specifically intuitionists, about mathematics. The later Gödel, however — particularly in work first published after Dummett’s famous paper — seems to take the extendability of the notion of set, for example, to be a count in favour of his conceptual realism. Where exactly is it, then, that Dummett and Gödel disagree? However, this kind of investigation — which is the sort of thing we need to throw some more light on what constitutes Gödel’s Platonism — takes us a long way from anything that Berto touches on (or even mentions in notes or bibliography).

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One Response to There’s Something about Gödel, Ch. 9

  1. Aldo Antonelli says:

    “The First Theorem doesn’t have specifically Platonistic implications.”

    That should come as no surprise at all, since the First Theorem is about PA and other formalized theories, and as such cannot imply anything about the existence (or non-existence) of mathematical entities. That’s just Craig interpolation: unless you have principles connecting claims about the elementary properties of the natural numbers to claims about the separate existence of mathematical entities, nothing at all about the latter can follow from the former. And any such principles would have to be supported by substantial philosophical argumentation — a possibility about which one would best be advised to be skeptical.

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