Peter Smith's Blog, page 129

TTP, 11. Disentangling neo-formalism

The introductory sketch in the last post reveals at least this much about Weir's neo-formalism: it is the marriage of two independent lines of thought.

One idea — call it "formalism about arithmetical correctness" — is that, at a first approximation, what makes an arithmetical claim correct (and we'll stick for the moment to the example of arithmetic) is something about what can be done in a some formal game(s) played with uninterpreted tokens.

To introduce the other idea, let's first say that the content of a claim C is transparently representational when what we grasp in grasping C is just its correctness condition according to our best explanatory metaphysical-cum-semantic theory. Thus, plausibly, in grasping "cats are mammals" we grasp what best theory surely will say is its correctness condition, i.e. just that cats are mammals. But it isn't like this, according to Weir, for arithmetical claims. Here, the suggestion goes, the content of a claim, as grasped by an ordinarily competent speaker, falls short of the thought that its correctness conditions are satisfied (which the formalist takes to mean: falls short of a thought about what can be done in the relevant formal game). Here, then, the content is not transparently representational; so call this the idea that arithmetic has  "NTR content" for short

Both ideas need to be embroidered in various ways to give us a decently worked through position, and Weir offers his versions, about which more anon. But the first point to make is that these plainly are two quite different basic ideas and their further developments will be very largely independently of each other. You could buy Weir's line on one without buying his story about the other.

For example, if you are a modal structuralist, who gives a quite different account of the correctness conditions for arithmetical assertions, you might — very reasonably — also   want to say that the content of the ordinary arithmetician's claims doesn't involve concepts of possible worlds or whatever. In other words, although no formalist, you might very tempted to embrace some story about how arithmetic content is not transparently representational (again holding that what is grasped in our ordinary understanding of an arithmetical claim falls short of grasping what makes the claim correct, according to your preferred story). So you might be interested to see whether you could borrow themes from Weir's story about NTR content. Alternatively, of course, you could embrace something like Weir's formalism about correctness while not liking the way he spins his account of NTR content.

So we'd better clearly disentangle the two themes. We'll turn to "formalism about arithmetical correctness" in later posts. Here let's begin to consider how Weir handles the idea of NTR content.

Weir's first two chapters are supposed to have softened us up for the viability of this idea. For recall, he has given three examples of cases where, it seems, the idea of NTR content looks appealing. First, and least controversially, there is the case of assertions involving demonstratives: "that animal is a mammal". Here the story about correctness (in such a case, correctness is plain truth) will talk about the most salient animal given contextual indications (pointings etc.) But what the ordinary conversationalist grasps surely isn't a thought involving the concept of salience, etc. Second, and much more controversially, there's the case of assertions supposedly inviting a Blackburnian projectivist treatment, e.g. moral claims. Here the story about correctness (and perhaps correctness can be thought of a matter of truth again, if we are sufficiently minimalist about truth) will talk about appropriateness of attitudes: but the content of a moral claim is not a thought about human attitudes. Third, Weir considered claims made in elaborating a fiction: "Holmes lived less than five miles from the Houses of Parliament". Here correctness ('truth in the fiction', perhaps) is keyed to what experienced readers would, on reflective consideration, judge must belong to an elaboration of the Holmes stories if everything is to make good enough overall sense. Again the content of the claim about Holmes, on the lips of a casual conversationalist, is not plausibly to be said to involve ideas about the coherence of a fictional corpus.

OK. But let's note again, as I've noted before, that these are three very different stories about instances of NTR content. We might say that, in the demonstrative case, the content — though not fully transparently representational — is still partially representational: the claim "that animal is a mammal", in context, aims to represent the world as it is. But a projectivist will say that moral judgements, by contrast, are in a different game from representation, they get their content from the practical business of encouraging and coordinating attitudes. As for fictional claims, they aren't representational either, but are articulating a make-belief.

So: a claim can fail to be (fully) transparently representational because it is representational but part of the representation has to be supplied by context; it can fail because it actually isn't primarily in the game of representation at all; or it can be fail because it is only pretending to represent. But that's not the end of it. Here's another sort of case which Weir doesn't mention. On a plausible metaphysical view, it is correct to say of something that it is green just if it is disposed, in normal viewing conditions for things of the relevant kind, to produce a certain characteristic response in normal viewers. But again, it seems wrong to say that the ordinary speaker, in grasping the content of "grass is green" is grasping a thought about dispositions or normal viewers. (To use a favourite style of argument of Weir's, Alan can believe that grass is green without believing that grass is disposed, in normal viewing conditions, etc. etc.). So, "grass is green" also has NTR content. But it isn't that "grass is green" is non-representational or is only pretending to represent: rather, it represents, but in a foggy way (that doesn't transparently yield correctness conditions). It seems apt to echo here Leibniz's talk of "confused ideas".

The question, then, for Weir is this. Let's grant him the idea that (in our terminology) some claims are not transparently representational: their sense peels apart from their explanatory correctness conditions. But the divorce can arise in various ways. Weir himself distinguishes three model cases; we've just added a fourth. So which of these models, if any, is the appropriate one when it comes to elucidating the idea that arithmetical claims have NTR content?

To be continued …

TTP, 10. Neo-formalism introduced

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).

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!

TTP, 10. Neo-formalism

As you mightl remember, I'm supposed to be writing a review of Alan Weir's Truth through Proof. I started blogging here about his 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 in the book 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 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).

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!

Good news on 2nd edn of IGT

Good news. I've a contract to do a 2nd edition of my Gödel book. Not quite under the terms I'd have ideally liked, like another 50 pages, a couple of years or more to do it, and of course a humungous slice of the royalties. In fact the press want it next year, and I've only about another 20 pages to play with. But the discipline will be good for me, and some will think that 380 pages will be quite enough, thank you, for what's called an 'Introduction'. So … down to work.

Or at least, it's back to Gödel after I've written the overdue review of Alan Weir's book. Watch this space for, belatedly, a few more thoughts on that. Meanwhile, if you've bright ideas about how to improve the Gödel book, do let me know!

Rediscovering reading …

One of the joys of retirement, I'm discovering, is that there seems to be a lot more time for reading. I mean reading that isn't either logic/philosophy on the one hand, or wind-down late-night novels on the other. Of course, this probably says a lot more about me than about how much extra time I've got (as I seem to be doing a lot of philosophical things about which more anon). For I guess it is really a matter of now giving myself permission to read stuff that isn't work-related during the day.

So what have I read the last couple of weeks? I finished Sarah Bakewell's How to Live: A Life of Montaigne in one question and twenty attempts at an answer, which is just terrific, has been rightly much praised, and gets you back to reading the man himself with even more pleasure. I also read Susan Richards Lost and Found in Russia: Encounters in a Deep Heartland, though I found that pretty disappointing. I've read books about Russia on and off since my student leftie days, this promised much, and again it was warmly praised by some reviewers. But Richards's writing somehow gives very little concrete sense of place, and not much either for the changing feel of everyday life in the turmoil of post-Soviet Russia remote from Moscow.

But what I enjoyed most — though the end is wrenching — was Tony Judt's remarkable The Memory Chalet, written (or rather dictated) as he was dying from motor neuron disease. The writing is simply wonderful, his recollections of times and places so evocative for someone of much the same generation, and his sharp observations on our times very much after my own heart. I loved this book (and the mind behind it). Read it.

The Belcea Quartet again

I've mentioned here before my ever-growing admiration for the Belcea Quartet. Last week at a late night Prom concert they played the Schubert Quintet (with Valentin Erben as the second cello) with their usual intensity. You can hear it here for another few days. You have to forgive the inter-movement sprinklings of applause: this is a rather wonderful performance, and probably heard better on the radio than it would have been in the vastness of the Albert Hall. Listen — and then get their even better CD performance.

What's so great about Kuratowski pairs?

I'm currently writing about countable order types — kinds of orderings of the naturals — and for various reasons want to do as much as I can without explicit talk about sets. But I obviously need to talk about ordered pairs of numbers so I can talk about "adding" and "multiplying" order types. So the trick I adopt is to code up the pair m, n by using the code  2m3n as 'pair numbers' (I also use the same powers-of-primes trick to code up finite sequences.)

It's been put to me that the fact I have resort to cheap tricks like this simply shows that there is something really a bit perverse about trying to do without sets. But not so, and perhaps it is worth explaining why (though I guess the following line of thought ought to be familiar — so don't expect novelties).

Why should handling ordered pairs of natural numbers be regarded as somehow inferior to other, albeit more familiar, devices? It might be said that (i) a single pair-number is really neither ordered nor a twosome; (ii) while a number m is a  is one of the pair m, n, a number can't be a one of (or a member of) a pair-number; and  in any case (iii) our construction was pretty arbitrary (we could equally well have used, e.g., 17m16n to code the pair).

Which is all true. But of course we can lay exactly analogous complaints against e.g. the  Kuratowski definition that we all know and love, which treats the ordered pair (m, n) as the set {{m}, {m, n}}. For (i) that set is not intrinsically ordered (it is a set!), nor is it always two-membered (consider the case where m = n). (ii) Even when it is a twosome, its members are not the members of the pair: in standard set theories, m cannot be a member of {{m}, {m, n}}. And (iii) the construction again involves pretty arbitrary choices: thus {{n}, {m, n}} or {{{m}}, {{m, n}}} etc., etc.,  would have done just as well. Thus far, then, our coding of pairs of numbers using pair-numbers involves  no worse a trick than coding them using Kuratowski's gadget.

To pursue the point further, suppose you are working in standard ZF (for example).  Pure ZF knows only about pure sets. So to get natural numbers into the story — and hence to get Kuratowski pair-sets of natural numbers — you have to choose some omega-sequence of sets to implement the numbers, (or 'stand proxy' for them, 'simulate' them, 'play the role' of numbers, or even 'define' then — whatever your favourite way of describing the situation is). But someone who has opted to treat natural numbers within set theory by selecting some convenient sets to implement them is hardly in a position to complain about our opting to treat ordered pairs in arithmetic by choosing some convenient numbers to implement them. Suppose, alternatively, that you are working in ZFU, a set theory with urelements (so that at the base level of the iterative hierarchy you have not just the empty set but sets whose members are non-sets). You can then allow natural numbers as sui generis urelements, and then form  Kuratowski pair-sets of these elements. But given the prior commitment to numbers as urelements and the availability of the numerical coding device, you don't need the extra commitment to sets to do the job of representing pairs of numbers. So why shoulder the extra ontological load?

You might respond that the Kuratowski trick at least has the virtue of being an all-purpose way of getting pairs of anything, while you can only use the powers-of-primes trick for coding pairs of numbers. But that's like saying that you can use sledgehammers to crack all sorts of things, while you can only use nutcrackers for nuts; true, but not to the point if it happens to be nuts you currently want to crack.

Putting the point more abstractly, suppose X are some objects and Y are some objects (maybe the same, maybe different). Then we'll say a pairing scheme for X with Y comprises some 'pair objects' O and a surjective two place function pr from X and Y to O, such that there are functions fst from O to X, and snd from O to Y which 'recover' the first and second items in a pair object in the obvious way. Set things up right, and pairing schemes are unique up to unique isomorphism. Then the 'pair numbers' trick and Kuratowski trick (with their obvious respective functions pr, fst, snd) yield equally good pairing schemes.

This is a common situation: we specify conditions on some mathematical gadgets, and then show that we have pinned them down uniquely up to unique isomorphism. And doing this, i.e. fixing gadgets  up to unique structure-preserving bijection, arguably gives us all that we will normally need for mathematical purposes.

To take the case of natural numbers as the trite example, we might reasonably suppose that for mathematical purposes one omega-sequence is as good as another for implementing the numbers. True, to continue with the example, we will only be concerned as number-theorists with what holds true of numbers irrespective of the implementation. So, following Dedekind, perhaps we might insist that we should go on to abstract from the various 'concrete' implementations within richer frameworks (implementations which give numbers unwanted extraneous properties), and say that numbers themselves are distinguished as having no properties other than those they get in virtue of their place in the structure shared by different more concrete implementations. Well say that if you like: it doesn't matter one way or the other for most purposes.

And my present point is that we can be equally insouciant about what pairs of numbers 'really' are. My preferred scheme for pairing numbers and the more familiar set-theoretic scheme are just as good as each other, in that both 'define' pairs equally well for mathematical purposes.  But if you do remain minded to go on in a Dedekinian spirit to say that, still, neither our pair-numbers nor the Kuratowski sets are really pairs (because both schemes treat pairs as objects with too much irrelevant structure), and we ought really to abstract away from these concrete implementations, then so be it. But that certainly doesn't spoil the symmetry and make my way of treating pairs of numbers any worse that the familiar standard way.

Thank heavens that's over …

Examining, I mean. For the last time ever. And, after a long-drawn-out and rather depressing experience marking tripos, at least I finished on a high note, viva-ing a particularly excellent M.Phil. thesis.

Now it is back to clearing my office. Into the bin with lecture notes from courses twenty-five years ago! Out with old overheads and handouts! Onto the come-and-help-yourselves shelves in another room for a lot of never-read/never-to-be-read books!

But now it is getting harder. I'm slowing down, and it is all getting more discombobulating.

In some cases it is a matter of regretfully having to acknowledge that — being realistic — I am never going to have a year or so to really get my head again round X or Y. I'd love to really get to the point where I was back on top of the state of play in the philosophy of quantum mechanics (say); but it is never going to happen — or at least, it's never going to happen if I am to have half a chance of finishing some logicky projects. So that whole area will have to remain a closed book, or rather a small pile of closed books. A cheering reminder of faded hopes, eh?

Then there are the books to which I still feel an odd attachment and find difficult to let go for no reason I can easily articulate. Irrational, as I've not read them for decades, and I'm surrounded by Cambridge libraries. For instance, I've just found myself rereading some of  Cornford's Unwritten Philosophy, which I must have bought in 1967, and not had occasion to read much since. I'm sure it is all very creaky: ancient philosophy has come such a very long way since when Cornford was writing (the essays date from the thirties and forties). I've long since lost touch, and my Greek has quite disappeared. And yet, and yet … The charm of his writing still weaves its magic. No; this I think I will keep, just for a bit longer.

Back to the pile for sorting …

Cambridge news: congratulations/two jobs

First, the very warmest of congratulations to my colleague Michael Potter, who will become a Professor in the Faculty of Philosophy from October 1st. And about time too.

Second, two temporary teaching posts have been advertised for October. They are primarily in Ethics and History of Modern Philosophy. But the Faculty has a pretty desperate need for logic teaching given my retirement, so I imagine that an ethicist or historian who can step up to the plate and help out teaching some of our first-year formal logic course too might get a warm reception!

Teaching philosophy, what to do?

So here I am, the last teaching done, and an office to clear out, and books to dispose of. What do I keep?

By one of those odd coincidences, I was thumbing through Richard Gale's The Language of Time wondering whether I'd ever want to look at it again, just as a tweet appeared onscreen announcing Craig Callender's brand new Oxford Handbook of Philosophy of Time.

Gale's title pretty much gives away its date. Few now would baptise like that a book substantially about metaphysical questions. But back in 1968 this excellent book was the best available. Terrific stuff. Read it, and maybe a dozen significant articles or chunks from McTaggart, Broad, and others (collected in one or two handy readers of the time), and you were up to speed on the state of the art in perhaps a couple of weeks work at the outside.

Callender's handbook weighs in at almost three times the number of pages, and much more than that in terms of numbers of words. And — if it is anything like the other excellent Oxford Handbooks — the many articles will only aim to give a glimmer of where we've got to in the philosophy of time, and will each have daunting bibliographies of must-read papers, and books. How long now to get up to speed, to the point of being able to launch out on research even in one corner of the philosophy of time? Six months?

We could of course multiply such examples across philosophy. A case I've mentioned before here, the theory of descriptions. What was there to read in 1968? Remember, Donnellan's paper had just been published. A student could still pretty easily read everything worth reading in two or three days. And now?

The explosion of publications in philosophy over my time — good publications, pieces you'd want to take note of — has been staggering. It has changed what it is like to work in philosophy in all kinds of ways, changed therefore what it is like to be a "professional" philosopher, to the point where the demands of the discipline are increasingly at odds with that yearning to make connections and see big patterns that gets many of us into philosophy in the first place (remembering again Sellars: "The aim of philosophy," he wrote, "is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term"). But I wonder how much of this great change we have really come to terms with.

More specifically, how much has all this been taken aboard in our teaching practices? Of course, over the years, we (meaning, at least, we in the UK) have had to do some hard thinking about teaching, but it has driven by how to cope with a great worsening of staff-student ratios (at least in most places), how to jump through the hoops of "quality" assessments, how to cope with a broader ability band.

Oddly, though, philosophers don't seem to have done much thinking about how to cope with the changing nature of the discipline itself. How should our teaching respond? We carry on much as before, don't we? No doubt our lectures have nicer overheads — where were they in 1968? — and we give better handouts, and so on and so forth. But have we been thinking about how best to structure or teaching, given the changes in the very practice of philosophy itself? I suspect not. Or have I been missing something?