As you might remember, I’m supposed to be writing a review of Alan Weir’s *Truth through Proof*. I started blogging here about the book some time ago, intermittently discussing the first couple of chapters at length, and then I’m afraid I stopped. That was partly through pressure of other things. But also I got to a point where I was finding it quite difficult to be sure I was getting the picture. (A seminar discussion at Birkbeck was comforting, as it revealed that I’m not the only one to find Weir’s exposition of his “neo-formalism” rather opaque.)

Ok: the review is now overdue, and I don’t have the time or energy to carry on blogging in such detail. (Who knew retirement could be so busy?) But let me try in this post to give the headline news about the content of Weir’s neo-formalism as first sketched in Chapter 3. Reading the chapter again, I think I’m following the plot better. But rather than try to summarize his own presentation, let me start by telling a story, one that I hope works up to a toy model of the sort of position which Weir wants to develop. The story involves developing three games, and each game we develop in three steps.

*The abacus game* Step one. We teach children to play with an abacus — let’s imagine one with seven horizontal wires, six with nine beads on them, with the last wire having just a single bead. In the game, the inital state of the abacus has the beads in the first four rows each distributed into two (possibly empty) groups, a left one and a right one, and the beads on the final three rows are all shuffled to the right. Then allowed moves are made until a position is reached where all the beads on the top four rows are shuffled to the right, and there’s some new distribution on the bottom three rows. In fact the rules are such that the game tracks the addition of two numbers under 99 (represented in the initial state of the abacus by units and then tens of one number on the top two rows, then units and tens of the other number) with the result represented by the state of the final three rows at the end of the game. But the children don’t know this: they are taught allowed moves, which they apply (not by counting but) by pattern-recognition.

Step two. We now teach the children another step, augmenting the abacus game. Initially they learn to write in a ledger e.g. the symbols ’74 + 46 = 120′ when they have just got from a certain initial configuration (the configuration that we would describe as follows: the beads on the first ‘units’ row separated four to the left, five to the right; on the second ‘tens’ row, seven to the left, two to the right; etc.) to a certain end configuration (on the first four rows, the beads all to the right; the same on the next row represent the null ‘units’ in the conclusion; then two beads on the left of the next row, and the final row the single bead is on the left). Note, however, it isn’t enough just to have ended up with the appropriate configuration of beads: for a legitimate entry in the ledger, the configuration must have been achieved by correct play. A child’s entry on the ledger can be challenged by getting them to repeat the shuffling and showing a misplay, and a challenge met by a correct re-play.

Step three. The children now get rewards when they write down (what we would regard as) a ‘correct’ addition, and there are disincentives for incorrect ones. Moreover, they quickly learn that the goodies will be forthcoming whether or not the ‘additions’ are cued to actually playing with the abacus (behind a screen, perhaps) — it is only the written ‘additions’ in the ledger which are inspected. They learn that if an ‘addition’ is challenged, the challenge can met by actually going through the abacus game and getting the result cued to the addition, and the challenge is lost if they get a different result.

Now, in the original abacus game (at the G level for short), the tokens the children are moving around — the beads on wires — have no significance. But, at the level of the written tokens, where the children are giving a kind-of-commentary (call this the C level for short), the tokens in the language game can be thought of as having a certain significance, in that there are now correctness conditions for the issuing of a token ’74 + 46 = 120′.

However, although the correctness condition for issuing ’74 + 46 = 120′ is that you can get from a certain initial state of the abacus to a certain final state via moves according to the rules of the abacus game, it would arguably be over-interpreting to suppose that this is what the ‘equation’ *means* for the childish player, who after all need have no reflective grasp of e.g. the concept of a rule-of-the-game, and indeed no descriptive concepts for initial and final states either. If we tie the idea of having content to the idea of having correctness conditions in the language game, then the token ’74 + 46 = 120′ can be said to have content: but the content falls short of the explicit thought that the correctness conditions obtain.

In sum, this gives us a toy model for a language-game in which (i) there are correctness conditions for the issuing of ‘equation’-like tokens, (ii) these correctness conditions for tokens are given in terms of the availability of moves in an abacus-shuffling game, but (iii) it would be over-interpreting to suppose that the players, in issuing such an ‘equation’-like token *mean* that the correctness conditions obtain.

Ok, so far let’s suppose, so good. Now let’s move on to imagine another scenario, again with a G-level formal game which involves shuffling items around, and with a second C-level of linguistic tokens keyed to the availability of moves at the G-level.

*The school-room addition sums game* This time, take the G-level to be decimal arithmetic, as taught to children as a package of routines. So at step one they write down, as it might be, ’74’ above ’46’, draw a horizontal line underneath, and then — following what they are taught as an uninterpreted syntactic game — they ‘add the units’, carry one, ‘add the tens’ and write down ‘120’.

After being taught these symbol-shuffling routines (so far empty of significance), the children are then taught a new step, and learn to write down ’74 + 46 = 120′ after their addition routine is executed. And then thirdly, as before ,the game is expanded so that they are allowed to write down such a token even if they haven’t done the routine, so long as they could respond to a challenge by ‘doing the sum’.

Here, unlike the abacus cases where we had beads-on-a-wire at one level and symbols to play with at another, the same token ’74’ can appear at the G-level as what an uninterpreted symbol-shuffling game operates on, and at the C-level as part of a language game keyed to the G-level game. In the first case it is empty of significance, in the second case part of a move with content. For as before, the thought goes, (i) there are correctness conditions for the issuing of ‘equation’-like tokens at the C-level, (ii) these correctness conditions are to be given in terms of the availability of moves in a formal symbol-shuffling game, but (iii) it would again be over-interpreting to suppose that the players, in issuing such an ‘equation’-like token, *mean* that the correctness conditions obtain. So the content of the token ’74 + 46 = 120′, such as it is, falls short of the explicit thought that you can get from ’74’ and ’46’ to ‘120’ by legitimate moves in the adding game. Still, even if ’74 + 46 = 120′ doesn’t explicitly represent a fact about the formal game, its correctness conditions might be said to be, in an obvious sense, formal.

Now, a smallish tweak takes us to …

*The DA game* Again, we are dealing at the G-level with decimal arithmetic again, but we this time imagine decimal arithmetic presented as a formal quantifier-free system of equations, with axioms and rules of infererence, giving a formal theory which we’ll call DA (on the model of our old friend PA). Again we go through three steps. First we imagine the neophyte learning to play with DA — and being taught by the ‘direct method’ to recognize legimate DA manipulations by pattern-recognition and training, not by explicit instruction that mentions ‘axioms’ and ‘rules’ etc. This time the items being shuffled in the formal game are equation-like, but as yet — at the G-level — they have no content, any more than the arrangement of beads in the abacus game.

At the next step, however, the game is expanded: a player is taught to enter one of those equation-like tokens on a ledger if they are produced at the end of a DA game. Then, thirdly, the practice is expanded to allow a player to write down such a token even if they haven’t done the DA ‘derivation’ routine, so long as they could respond to a challenge by ‘doing the proof’.

So now the very same kind of equation-like token ’74 + 46 = 120′ gets into the story twice over. Firstly, a token can appear in the G-level DA symbol-shuffling game, which is again as empty of content as bead-arrangements in the abacus game. And second, a token can appear again at the C-level in a kind-of-commentary on the DA game, making a move in a language game which has the correctness condition that you can derive that sort of token inside DA.

As before, then, at the C-level, (i) there are correctness conditions for the issuing of equation-like tokens, (ii) these correctness conditions are given in terms of the availability of moves in a formal game, but (iii) it would be over-interpreting to suppose that the players, in issuing such an equation-like token *mean* that the correctness conditions obtain.

*Neo-formalism* Now here, at last, comes the punch. The neo-formalist claims that the content of *our* arithmetical claims is, or is like, that of the tokens at the C-level cued to the DA game. So: unlike the classic formalist who avers that arithmetic is an empty game with signs, the neo-formalist allows that arithmetical claims do have content. But he’s a formalist because the correctness conditions for such claims are given in formal terms, in terms of moves in a formal game. However, the claim goes, the content is not as rich as the thought that the correctness conditions obtain. So it would be wrong to say, as a cruder formalist might, that an arithmetical claim is *about* the formal game facts about which supply the correctness conditions.

That, then, is the basic story about arithmetic in introductory form. And similarly, it is hoped, for other claims in other areas of mathematics. But does this sort of account work? Watch this space!

Alan WeirThanks Peter for that excellent account. I like the abacus example and the nice three-step way of setting out my position for arithmetic.

As regards the ‘rather opaque’ exposition, that’s of course not for me but for readers to judge. I feel my eleven page introduction setting out the general position and structure of the book is written in fairly clear, straightforward prose. But I don’t for a minute think that reading it would give one more than a superficial idea of what the book is about or how the objections to the neoformalist position are met; otherwise I wouldn’t have written the book but just published a short article! I did talk to some of the participants at the Birkbeck seminar afterwards, and it was clear they hadn’t read the book, just the one chapter which had been set. (Nor would I have expected them to, being very busy folk in the middle of the term with a lot of other stuff on their agenda.) So I think the test for opacity should really only apply to those who have struggled through to the death.

Peter SmithI’m glad you think the abacus model hits off one of the key ideas: feel free to snaffle it for the second edition :-)

And we’ll let others judge questions of opacity. (Tho to be fair to the Birkbeck seminar, we/they were

supposedto have read your intro pages 6 to 11, plus some of my blog posts giving the headline news about Chs 1–2, before tackling the pivotal Ch.3. So while others hadn’t struggled through to the death, they weren’t just ripping the chapter out of all context!)Alan WeirA second edition — chance would be a fine thing Peter! Actually, I’ll probably need the sales from one as the maintenance on the Lear jet I bought with the sales from the first edition is pretty costly (airfix glue isn’t as cheap as it was).