So at long last, it’s back to discussing Alan Weir’s Truth Through Proof (henceforth, TTP). And apologies to Alan, and anyone else, who has been eagerly waiting for further instalments.
Let’s quickly, in this post, review where we’ve got to (cutting-and-pasting a few snippets from previous posts which you’ve now forgotten!). In his short Introduction, Weir sketches out the ground he wants to occupy. He wants to say that, as a mathematical claim, it is true that are an infinite number of primes. And this common-or-garden mathematical truth isn’t to be reconstrued in some fictionalist, structuralist or other way. However, he wants to say, a mathematical claim is one thing, and a claim about how things are in the world is another thing. Speaking mathematically, there are an infinite number of primes; but there is also a good sense in which THERE ARE NO primes at all.
How is the gap here to be opened up? Not by construing talk of what really EXISTS as a special level of ontological talk, distinct from other talk that aims to represent the world. Rather, the small caps just signal that straightforwardly representational discourse is in play, and the key idea is going to be that mathematical discourse (like e.g. moral discourse) plays a non-representational role — if you like, mathematics makes moves in a different language game.
If Weir is going to be able to develop this line, we’ll need to hear more in general about styles of discourse, representational vs non-representational. It’s the business, inter alia, of Chap. 2 to provide some of this background. And in the next posts I’ll start discussing this chapter. But some semantic groundwork, and some terminology, has already been provided in Chap. 1.
Suppose we aim for a systematic story about how sentences of a certain class get to convey the messages that they do: take, for example, sentences involving a demonstrative ‘that’. The systematic story will, perhaps, use a notion like salience, so for example the story tells us that ‘that man is clever’ expresses a message which is true when the most salient man in the context is clever. Now, for this to be part of a semantic theory that is suitably explanantory of speech-behaviour, speakers will have to reveal appropriate sensitivity to what we theorists would call considerations of salience. But note: those we are interpreting needn’t themselves have the concept of salience. So the explanatory account given in our theoretical story doesn’t supply a synonym for ‘that man is clever’: we need to distinguish the literal content of the demonstrative sentence as speakers understand it (what is shared by literal translation, for example) from what we might call the explanatory conditions as delivered by our systematic semantic theory.
It is a familiar and not-too-contentious point that such a distinction needs to be made, and made not just in the case of the semantics of demonstratives. Somewhat unhappily, I’d say, Weir has chosen different terminology to mark it: in particular, he talks not of ‘explanatory (truth)-conditions’ — which indeed was his initially preferred term — but of metaphysical content. And he says “metaphysical content specifies what makes true and makes false a sentence in a circumstance”. This talk of truth-making might suggest that the business of metaphysical content is to specify truth-makers in the sense favoured by some metaphysicians. But not so! Weir in fact is quite sceptical about truth-makers, so understood. Hence we mustn’t read more into Weir’s terminology than he really intends to put into it: to repeat, so-called metaphysical content is just a specification of the situation where an utterance of the sentence in question would be correct or appropriate or disquotationally-true.
The thought is going to be then that, when it comes to giving the ‘metaphysical content’ of mathematical claims, the story about what makes a mathematical sentence correct or appropriate or disquotationally-true doesn’t mention mathematical entities of a platonist kind but runs on quite different lines. But how? Back in his Introduction, Weir says “The mode of assertion of [mathematical claims] … is formal, not representational”. And what does this mean? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is rendered correct by “the existence of proofs of strings which express the infinitude of the primes”. Hence Weir’s “neo-formalism”. Our task is going to be that of making sense of this surprising claim, and evaluating it.
Now read on …