Logic keeps you sane

Non-philosophical events have really been rather stressful over the last six weeks or so. While I’m keeping my head above water as far as teaching and seminars are concerned, lots of other things — like sounding off here! — have been pushed right into the background. And I’m going to pressed for time this coming week too. For I’m down to respond to Luca at a Faculty Colloquium this Friday: we’re talking about Graham Priest’s animadversions against the iterative conception of sets, and I need to write a talk. How on earth this will go down with a general audience of non-logicians, heaven alone knows. But then, the whole idea of a general colloquium seems to me mildly daft. Still, the topic is a good one to think about.

Indeed, logic matters have been a lot of fun this last week, and a very welcome distraction to keep me sane. Two of our MPhil students gave very nice and very helpful presentations at the two seminars — one on Chap. 3 of Parsons’s book in the reading group on that, and one on Big Typescript §113, the section on “Ramsey’s Theory of Identity”. None of us ended up very much clearer about what Parsons’s noneliminative structuralism comes to exactly: but I’ll return to that here shortly. Lectures are going pretty enjoyably too (for me, at least!). I’m probably going too slowly talking about Gödel’s theorems to cover everything I really should, but at least people seem to still be on board. And I unthinkingly produced a frisson of mildly scandalized reaction in the first year logic course by saying en passant that, while Wittgenstein might have been a great philosopher — he’d just come up in the context of talking about the idea of a tautology — he seems to have been a bit of a shit as a human being. Still, we do need to encourage some healthy disrespect!

The Gowers’ Companion/Davenport’s Higher Arithmetic

Tim Gowers’s Princeton Companion to Mathematics — which has been available from Amazon USA for a discounted price for a little while — is now available at a 25% discount from Amazon UK. Hooray! I’ve sent off for it and will no doubt be commenting here.

Meanwhile, to get in practice for reading a stack of mathematics (outside the logician’s usual diet, for once), I’m devouring Davenport’s The Higher Arithmetic which has just appeared in the C.U.P. bookshop in its new eighth edition. I first read chunks of this a long time ago as a schoolboy: and it’s old-style mode of presentation — mercifully without Bourbachiste over-formalization — is a delight. I’d of course entirely forgotten most of this stuff: some of it is very pretty!


We’ve now had three seminars on Wittgenstein’s remarks on the foundations of mathematics in the Big Typescript, taking things very slowly. The first week, I talked about Sec. 108 (the contrast between arithmetic and a game). One of our final year undergrads gave an admirable presentation in the second week on Secs 109-11. This week, we battled with Secs 112 and 114 (leaving the discussion on Ramsey and identity in Sec. 113 till next week).

Now, a few pages ago, in Sec. 108, it seemed that it is the use or application of arithmetic that is supposed to distinguish it as mathematics from a mere game. What kind of applicability is in question? “It is mathematics, I should think, when it is used for the transition from one proposition to another.” (Sec. 108, p. 372e). So there, at any rate, Wittgenstein offers the beginnings of a story about applicability. But now, in Sec. 112, we have some distinctly odd remarks about applications. For example, “Arithmetic is its own application.” (p. 382e, repeated p. 385). What does that mean? I think it’s fair to report that we were left baffled.

To be sure, we presumably do want a story about the difference between using an empirical theory to take us from one proposition to another and using arithmetic. And Wittgenstein in effect remarks that if we use arithmetic and get empirically the wrong answer, we don’t blame arithmetic. “It might look as though the mathematical computation entitled us to make a prediction [e.g. about how many apples each a group of people will have, if you divide the pile of twelve apples between four]. But that isn’t so. What justifies us in making this prediction is a hypothesis of physics, which lies outside the calculation. The calculation is only an examination of logical forms, of structures, and of itself can’t yield anything new.” [p. 383e]. But just what does that last sentence mean? And let’s suppose for the sake of argument that, as part of our overall practice, we have the rule that we do not revise arithmetical propositions in the light of empirical results. That doesn’t make it any less the case that arithmetic is being applied to apples or whatever, or make it appropriate to say instead that “arithmetic is its own application”.

Any pointers to helpful discussions in the literature that makes sense of what is going on here will be very gratefully received!

Parsons’s Mathematical Thought: Sec. 35, Intuition of finite sets

Suppose we accept that “it is not necessary to attribute to the agent perception or intuition of a set as a single object” in order to ground arithmetical beliefs. Still, we might wonder whether some such intuition of sets-as-objects might serve to “give an intuitive foundation to theories of finite sets“.

But Parsons finds problems with this suggestion too. One difficulty can be introduced like this. Suppose I perceive the following array:


Then do I ‘intuit’ six dollar signs, a single set of six dollar signs, a set of three elements each a set of two signs, or even a set containing the empty set together with a set of six signs? Which way do I ‘bracket things up’?


The possibilities are many — indeed literally endless, if we are indeed allowed the empty set (and what is our intuition of that?). So it seems that the “intuition” here has to involve some representational ingredient to play the role of the brackets in the various possible bracketings. But then we are losing our grip on any putative analogy between intuition and perception (as Parsons puts it, “in a perceptual situation involving the application of certain concepts, we not expect that a linguistic of other embodiment of the concepts should be perceptually present in that very situation”).

Secondly, note that we can in fact give a theory of those “bracket terms” — putatively for hereditarily finite sets constructed from a given domain D of individuals — which uses a relative substitutional semantics. That is to say, we can start with a first-order language for which D is the domain, add terms for hereditarily finite sets of elements from D, and variables and quantifiers for them, which we then interpret substitutionally relative to D. Parsons spells this out in an Appendix, but the general idea will be familiar to readers of his old paper on ‘Sets and Classes’. And the upshot of this, Parsons says, “is that if we take the relative substitutional semantics as capturing a speaker’s understanding of the language of hereditarily finite sets … then we largely remove the motives for characterizing awareness of such sets as initution”. That’s a significant “if” of course: but we might indeed wonder why we should take elementary talk about finite sets (and sets of those, and so on) to be more committing than the substitutional interpretation allows.

Note that this isn’t to say that we have entirely eliminated a role for intuition. For on the relative substitutional interpretation we still need the idea of sequences of individuals from D. And we might suppose that that notion is grounded in intuition. But even if true, that still falls well short of the original thought that we could need intuitions of sets-as-objects to give a foundation to theories of finite sets.

D960 for a desert island?

Sometimes, in an idle moment, I jot down — be honest, don’t we all? — a list of the eight discs I would select as my Desert Island Discs. Impossibly difficult of course! But one constant choice is the last Schubert piano sonata, D960 (and if I had to save one of the eight from the waves, then this would probably be it). But which recording? …. Well, that’s almost impossible too.

I’ve for a while had Schnabel, Brendel (1972, 1988, 2000), Richter (three recordings of his too — extraordinary, Schubert stretched to the limit), Imogen Cooper (underrated, but very fine), Schiff, Mitsuko Uchida, Perahia, and Kovacevich. Surely I had Kempff too but that seems to have gone walkabout. And very recently I bought the recording by Paul Lewis, which has received much praise. Yes, I agree! — I really should kick this buying habit. But the prospect of another possible great performance was irresistible. Still, Lewis doesn’t quite work ideally for me: I’ll listen more — but it isn’t the one for the desert island. I still think that that has to be one of the Brendel recordings: after listening to others, I always listen to him again with a sense of coming home. Perhaps I love his 1988 recording the most.

Parsons’s Mathematical Thought: Secs 33, 34, Finite sets and intuitions of them

So where have we got to in talking about Parsons’s book? Chapter 6, you’ll recall, is titled “Numbers as objects”. So our questions are: what are the natural numbers, how are they “given” to us, are they objects available to intuition in any good sense? I’ve already discussed Secs 31 and 32, the first two long sections of this chapter.

There then seems to be something of a grinding of the gears between those opening sections and the next one. As we saw, Sec. 32 outlines rather incompletely the (illuminating) project of describing a sequence of increasingly sophisticated but purely arithmetical language games, and considering just what we are committed to at each stage. But Sec. 33 turns to consider the theory of hereditarily finite sets, and considers how a theory of numbers could naturally be implemented as an adjunct to such a theory. I’m not sure just what the relation between these projects is (we get “another perspective on arithmetic”, but what exactly does that mean? — but, looking ahead, I think things will be brought together a bit more in Sec. 36).

Anyway, in Sec. 33 (and an Appendix to the Chapter) Parsons outlines a neat little theory of hereditary finite sets, taking a dyadic operation x + y (intuitively, x U {y}) as primitive alongside the membership relation. The theory proves the axioms of ZF without infinity and foundation. I won’t reproduce it here. In such a theory, we can define a relation x ~ y that holds between the finite sets x and y when they are equinumerous. We can also define a “successor” relation between sets along the following lines: Syx iff (Ez)(z is not in x and y ~ x + z).

Now, as it stands, S is not a functional relation. But we can conservatively add (finite) “cardinal numbers” to our theory by introducing a functor C, using an abstraction axiom Cx = Cy iff x ~ y — so here “numbers are types, where the tokens are sets and the relation ~ is that of being of the same type”. And then we can define a successor function on cardinals in terms of S in the obvious way (and go on to define addition and multiplication too).

So far so good. But quite how far does this take us? We’d expect the next step to be a discussion of just how much arithmetic can be constructed like this. For example, can we cheerfully quantify over these defined cardinals? We don’t get the answer here, however. Which is disappointing. Rather Parsons first considers a variant construction in which we start not with the hierarchical structure of hereditarily finite sets but with a “flatter” structure of finite sequences (I’m not too sure anything much is gained here). And then — in Sec. 34 — he turns to consider whether such a story about grounding an amount of arithmetic in the theory of finite sets/sequences might give us an account of an intuitive grounding for arithmetic, via a story about intuitions of sets.

Well, we can indeed wonder whether we “might reasonably speak of intuition of finite sets under somewhat restricted circumstances” (i.e. where we have the right kinds of objects, the objects are not too separated in space or time, etc.). And Penelope Maddy, for one, has at one stage argued that we can not only intuit but perceive some such sets — see e.g. the set of three eggs left in the box.

But Parsons resists at least Maddy’s one-time line, on familiar — and surely correct — kinds of grounds. For while it may be the case that we, so to speak, take in the eggs in the box as a threesome (as it might be) that fact in itself gives us no reason to suppose that this cognitive achievement involves “seeing” something other than the eggs (plural). As Parsons remarks, “it seems to me that the primary elements of a story [a rival to Maddy’s] would be the capacity to classify what one sees … and to recognize identities and differences” — capacities that could underpin an ability to judge small numerical quantifications at a glance, and “it is not necessary to attribute to attribute to the agent perception or intuition of a set as a single object”. I agree.

Life is too short …

After empty months over the summer, you are suddenly faced with hectic weeks when there is far too much going on by way of seminars and discussion groups. I assume it is much the same in most places; but perhaps the phenomenon is exaggerated here in Cambridge, where ‘Full Term’ is so short and intense.

My coping strategy is to be hyper-selective, on the policy ‘if in doubt, miss it out’. But I do try to keep an eye on what is going on, just in case. I note, for example, that there is a philosophy of mind reading group, and wonder what it is getting up to. Reading next, apparently, Charles Travis’s ‘Reason’s Reach’ European Journal of Philosophy 2007, pp. 225-248. So I take a speculative look. Ye gods. Life is surely far too short to bother with stuff written in such a ghastly pretentious style (this sort of thing would have been straight to the instant reject pile in my editorial days!). If philosophy is worth doing at all — and of course much of it isn’t — then let’s have it in straight-talking prose with absolutely maximal clarity so we can see precisely what the arguments are. To the flames with the rest!


There’s a piece on academic blogging in today’s Times Higher Education. It doesn’t contain any big surprises, but it is mildly interesting — and I do get a few mentions. Fame at last.

I have an entirely enviable walk into the Faculty: through a couple of pleasant late Victorian/Edwardian backstreets, across Midsummer Common (yes, there really are cattle grazing near the centre of Cambridge) and Jesus Green, then perhaps through Trinity Great Court and Neville’s Court and out over the river to the backs; along behind Clare and King’s (here there are white cattle in the meadow, colour-coded to match the Gibbs Building) and then a few hundred yards further to the Raised Faculty Building. I couldn’t wish for much better.

That walk is mostly very quiet: so I can listen to music en route (the Lindsays playing Haydn works really well). And the weather these first few days of term has been stunning: perfect early autumn. Which has all put me in a good mood for the first couple of intro logic lectures which went by with only very minor hiccups — though however many times I do this, the first lecture or so is still surprisingly nerve-wracking. I’ve also given the first class in my Gödel’s Theorems course (depressingly few are taking the math. logic paper again this year: despite our best efforts, a Cambridge tradition seems to be in decline). And the first logic seminar went pretty well. Or at least, I enjoyed it. There were over twenty there, to battle with §108 of the Big Typescript (of course, the numbers won’t last!). I gave the talk which I posted a draft of here, and Michael Potter added some very useful comments. When I’ve got a moment I’ll put together a revised version to take account of some things said in the discussion.

So that’s the first days of term survived. And now, I hope, back to Parsons!

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