In TTP 11, I emphasized that Weir’s position interweaves two separable strands. One strand I called “formalism about arithmetical correctness”: at a first approximation, what makes an arithmetical claim correct is something about what can be done in some formal game(s) played with uninterpreted tokens. The other strand proposes, as I put it, that the content of arithmetical claims is not “transparently representational”. So far, in these blog posts, I’ve been talking mostly about the way the second strand gets developed. It has been Hamlet with only brief appearances of the Prince of Denmark.
Weir’s story about content is intended to serve as a kind of protective wrapping around the formalist core (so he can say e.g. that although arithmetical claims have formalist correctness conditions we aren’t actually talking about synactic whatnots when we make common-or-garden arithmetical claims — thus avoiding at least some incredulous stares). But there is no getting away from it: when the wraps are off, the story about those correctness conditions is indeed very starkly formalistic. What makes arithmetical claims correct, when they are correct, is facts about plays with concrete tokens in some rule-governed practice of token-shuffling (actual plays, or practically possible ones).
Well, here I am, making arithmetical claims. These are supposedly made true by facts about concrete moves in some formal practice. Which formal practice? Look again at the toy models I offered in TTP 10. There was a (1) a game with an abacus, with facts about this making true tokens like ‘68 + 57 = 125’ (whose content is tied to such facts, but non-representationally). Then there was (2) something like school-room practice where we write down “long additions” etc. on our slates, with facts about what can be written down in this practice making true equiform tokens ‘68 + 57 = 125’ (which therefore have a different content from before). Then there was (3) a formal proof-system for quantifier-free arithmetic, and tokens such as ‘68 + 57 = 125’ are now tied to facts about what can be derived inside the formal system. So when I say ‘68 + 57 = 125’ which formal game is my utterance tied to (one of these or another)? What content does my utterance have? What’s to choose?
Weir’s response is: don’t choose.
The neo-formalist position is pluralist rather than relativist. The truth-value of ‘68 + 57 = 125’ is not relative to a formal system. Rather there is a plurality of systems, and the sentence expresses different senses in each, whilst it is made true (or false) in the context of a given one iff it is provable therein. (p. 108)
Which is the line he has to take (irrespective of the details of his story about non-representational content). Take the child brought up “bilingually” to play the abacus game and comment on that, and play the school-room game and comment on that. By mishap the child could come to believe 68 + 57 = 125 in the comment mode in one context and disbelieve in the other: so the contents had better be different.
OK: two children taught two different games and two different commenting practices will mean something by equiform comments. Yet it would be mighty odd, wouldn’t it, to say that two real children taught real arithmetic by different methods mean something different by ‘68 + 57 = 125’? The natural thing to say, most of us will think, is that if the kids end up as practical arithmeticians counting the world in the same ways, using addition when they want to put together the counts of two different piles to give a count of their combination, then whether they get to ‘68 + 57 = 125’ by abacus, school-room sum, doing a formal proof (heaven help us!) or using a calculator, they mean the same. For the sense of what they say is essentially grounded in their applied practice, in how they use arithmetic in the world.
From this natural perspective, Weir’s story begins to look upside down. He first talks of unapplied formal games (as it might be with an abacus, or writing down sums as in the school room, or operating with an uninterpreted proof system); assigns various contents to ’68 + 57 = 125′ treated as comments on moves in those various as-yet-unapplied games. And only afterwards, with the various informational contents fixed by the liaisons to the uninterpreted formal games, does Weir talk about applying the ‘arithmetic’ to the world. But at least as far as arithmetic is concerned, this looks topsy turvy: it’s the embedding in the applied practice of counting and adding and multiplying that gives sense to arithmetic; claims, properly so called. (Cf. Wittgenstein in the Big Typescript discussing what makes arithmetic different from a game.)
Later (p. 218) Weir modifies his pluralism. He says two speaker S and S* using e.g. ‘68 + 57 = 125’ “express the same sense” if there is an “admissible mapping” between theie linguistic practices L and L*. The details don’t matter (which is good, because they are not clear). The trouble is that I don’t see what entitles Weir to play fast and loose with the notion of sense like this. He’s spent chapters earlier on the in book trying to give us a story about sense or “informational content” that allows him to distingish the thin sense of ‘68 + 57 = 125’ from the rich sense of a statement of the correctness conditions of the claim in terms of moves in a formal game. He can do this because he cuts sense fine. And his criterion of difference in sense is that you can believe 68 + 57 = 125 (as a comment on the game) without being able to frame the concepts necessary to state the metaphysical correctness conditions. But then, by the same Fregean criterion of difference, the bilingual L/L* speaker like the abacus user who also does sums on paper could still express two different senses by different uses of ‘68 + 57 = 125’. Weir seems, later, just to be changing the sense of “sense”.
So I think Weir is landed with the radical pluralism he cheerily embraced earier, and his later attempt to soften it is a mis-step by his own lights. Many will count this as a strike against neo-formalism.
To be continued