Year: 2011

Going, gone

The main road west from Cambridge used to go down the main street of the market town of St. Neots. But there has long since been a bypass, and it is quite a while since I’ve turned off to take the old route. But I wanted a coffee, so today I stopped in the town and went to a scruffy and run-down branch of Caffè Nero on the large market square.

Their espresso is best passed over in silence, but that’s probably only to be expected. What I hadn’t really bargained for was just how depressing the view out to the square is now. Even on a bright autumn morning, it looked as scruffy and run down as the coffee shop. This was never a very wealthy place: but there was once some small domestic grace to the surrounding mostly nineteenth century buildings. But now many of them are quite disfigured with the gross shop-fronts of cheap stores, and others look unkempt. There’s a particularly vile effort by the HBSC bank, which gives a special meaning to “private affluence and public squalor” — only an institution with utter contempt for its customers and their community could plonk such a frontage onto a main street. Where once even small-town branches of banks were solid imposing edifices in miniature, with hints of the classical orders here and a vaulted ceiling there, now they are seem to take pride in having all the visual class of a here-today, gone-tomorrow betting shop. How appropriate.

And the square itself (like so many other urban spaces in England) seems to have been repaved on the cheap, with the kind of gimcrack blockwork that always seems, a few years in, to settle into random waves of undulation. The bleakly open space cries out for more trees to surround it, and inviting wooden seats. But no, on non-market days it is just the inevitable carpark.

Next to coffee shop, still on the square, a horrible looking cafe is plastered outside with pictures of greasy food. I walk a little further down the road before driving on. It is a visual mess. Even Marks and Spencer manages a particularly inappropriate shout of a shop-front, as sad-looking charity shops cringe nearby. Could anyone feel proud or even fond of this street as it now is?

A couple of hundred yards away there are lovely water-meadows by the bridge over the river, and fancy residential developments. On the outskirts of town the other side, as the road leaves towards Cambridge, there is a lot more quite expensive-looking new housing (though heaven knows how it will seem a few years hence). But the town centre itself is in a sorry state. “Most things are never meant,” wrote Larkin when he foresaw something of this in ‘Going, going’. And we — I mean my generation, for it is we who were in charge — surely didn’t mean this, for the hearts of old country towns like St. Neots (or the larger next town,  Bedford) to become such shabby, ugly, run-down places. But it has happened apace, all over the country, and on our watch.

KGFM 20, 21: Woodin on the transfinite, Wigderson on P vs NP

And so, finally, to the last two papers in KGFM. I can be brief, though the papers aren’t. The first is Hugh Woodin’s ‘The Transfinite Universe’. This inevitably mentions Gödel’s constructible universe L a few times, but otherwise the connection to the ostensible theme of this volume is frankly pretty tenuous. And for those who can’t already tell their Reinhardt cardinals from the supercompacts, I imagine this will be far too breathless a tour at too stratospheric a level to be at all useful. Set-theory enthusiasts will want to read this paper, Woodin being who he is, but this seems to be very much for a minority audience.

By contrast, the last paper does make a real effort both to elucidate what is going on in one corner of modern mathematics for a wider audience, and to connect it to Gödel. Avi Wigderson writes on computational complexity, the P ≠ NP conjecture and Gödel’s now well-known letter to von Neumann in 1956. This paper no doubt will be tougher for many than the author intends: but if you already know just a bit about P vs NP, this paper should be accessible and will show just how prescient Gödel’s insights here were. Which isn’t a bad note to end the volume on.

So how should I sum up these posts on KGFM? Life is short, and books are far too many. Readers, then, should be rather grateful when a reviewer can say “(mostly) don’t bother”. Do look at Feferman’s nice paper preprinted here. If you want to know about Gödel’s cosmological model (and already know a bit of relativity theory) then read Rindler’s paper. If you know just a little about computational complexity then try Wigderson’s piece for the Gödel connection. And perhaps I was earlier a bit harsh on Juliette Kennedy’s paper — it is on my short list of things to look at again before writing an official review for Phil. Math. But overall, this is indeed a pretty disappointing collection.

KGFM 19: Cohen’s interactions with Gödel

The next paper in KGFM is a short talk by the late Paul Cohen, ‘My Interaction with Kurt Gödel: The Man and His Work’. The title is full of promise, but there seems relatively little new here. For Cohen had previously written with great lucidity a quite fascinating paper ‘The Discovery of Forcing‘ and he already touches there on his interactions with Gödel:

A rumor had circulated, very well known in all circles of logicians, that Gödel had actually partially solved the [independence] problem, specifically as I heard it, for AC and only for the theory of types (years later, after my own proof of the independence of CH, AC, etc., I asked Gödel directly about this and he confirmed that he had found such a method, specifically contradicted the idea that type theory was involved, but would tell me absolutely nothing of what he had done). … It seems that from 1941 to 1946 he devoted himself to attempts to prove the independence [of AC and CH]. In 1967 in a letter he wrote that he had indeed obtained some results in 1942 but could only reconstruct the proof of the independence of the axiom of constructibility, not that of AC, and in type theory (contradicting what he had told me in 1966).

In this present paper, Cohen can shed no more real light on this unclear situation. But still,  what he writes is perhaps interesting enough to quote. So, Cohen first repeats again the basic story, though with a comment that chimes with other accounts of Gödel’s philosophical disposition:

I visited Princeton again for several months and had many meetings with Gödel. I brought up the question of whether, as rumor had it, he had proved the independence of the axiom of choice. He replied that he had, evidently by a method related to my own, but he gave me no precise idea or explanation of why his method evidently failed to succeed with the CH. His main interest seemed to lie in discussing the truth or falsity of these questions, not merely their undecidability. He struck me as having an almost unshakable belief in this realist position that I found difficult to share. His ideas were grounded in a deep philosophical belief as to what the human mind could achieve.

And then at the end of the talk, Cohen sums up his assessment like this:

Did Gödel have unpublished methods for the CH? This is a tantalizing question. Let me state some incontrovertible facts. First, much effort was spent analysing Gödel’s notes and papers, and no idea has emerged about what kinds of methods he might have used. Second, I did ask him point-blank whether he had proved the independence of CH, and he said no, but that he had had success with the axiom of choice. I asked him what his methods were, and he said only that they resembled my own; he seemed extremely reluctant to give any further information.

My conclusion is that Gödel did not complete any serious work on this topic that he thought was correct. In our discussions, the word model almost never occurred. Therefore I assume that he was looking for a syntactical analysis that was in the spirit of his definition of constructibility. His total lack of interest in a model-theoretic approach quite astounded me. Thus, when I mentioned to him my discovery of the minimal model also found by John Shepherdson, he indicated that this was clear and, indirectly, that he knew of it. However, he did not mention the implication that no purely inner model could be found. Given that I also believe he was strongly wedded to the syntactical approach, this would have been of great interest. My conclusion, perhaps uncharitable, is that he totally ignored questions of models and was perhaps only subconsciously aware of the minimal model.

That hints at an interesting diagnosis of Gödel’s failure to prove the independence results he wanted.

The Book Problem

Hello. My name is Peter and I am a bookaholic …

Well, perhaps it isn’t quite as bad as that. But I’ve certainly bought far too many books over the years. Forty-five years as a grad student and a lecturer, maybe acquiring forty or more work-related books of one kind or another a year (research, “keeping up”, books for teaching, books outside my interests that colleagues recommend, passing fads …). It’s pretty easy to do. Especially if you have something of a butterfly mind. That easily tots up to some 1800 philosophy and logic books. OK, OK, round that up to 2000. Ridiculous, I know. (Though not quite so mad as it might seem, having spent a long time in places without the stella library facilities of Cambridge.)

Chez Logic Matters (sort of ...)

Retiring and losing office space means there is now a serious Book Problem (ok, we’re certainly talking a First World problem here: bear with me). I’ve already given away a third. But now at home we want to do some more re-organization, which will mean losing quite a bit of bookshelving. So lots more must go. Dammit, the house is for us, not the books. One hears tell of retiring academics who have built an extension at home for their library or converted a garage into a book store. But that way madness lies (not to mention considerable expense). And anyway, what would keeping thirty-year-old one-quarter-read philosophy books actually be for? Am I going to get down to reading them now? In almost every case, of course not!

“A little library, growing larger every year, is an honourable part of a man’s history. It is a man’s duty to have books. A library is not a luxury, but one of the necessaries of life.” Yes. But let “little” be the operative word!

Or so I now tell myself. Still it was — at the beginning — not exactly painless to let old friends go, or relinquish books that I’d never got that friendly with but always meant to, or give away those reproachful books that I ought to have read, and all the rest. After all, there goes my philosophical past, or at any rate the past I would have wanted to have (and similar rather depressing thoughts).

But I think I’ve now got a grip. It’s a question of stopping looking backwards and instead thinking, realistically, about what I might want to think about seriously over the coming few years, and then aiming to cut right down to (a still generous) working library around and about that. So instead of daunting shelves of books reminding me about what I’m not going to do, there’ll be a much smaller and more cheering collection of books to encourage me in what I might really want to do. The power of positive thinking, eh?

At least, that’s the plan. I’ll let you know how it goes.

KGFM 17, 18: Kohlenbach and Friedman

Next up in Kurt Gödel and the Foundations of Mathematics is Ulrich Kohlenbach, writing on ‘Gödel’s Functional Interpretation and Its Use in Current Mathematics’. This rachets up the technical level radically, and will be pretty inaccessible to most readers (certainly, to most philosophers). The author has done significant work in this area: but as an effort towards making this available and/or explaining its importance to a slightly wider readership than researchers in one corner of proof theory, this over-brisk paper surely quite misses the mark. (I guess enthusiasts who want to know more about recent developments will just have to go for the long haul and try Kohlenbach’s 2008 book on Applied Proof Theory, but that too is very hard going.)

Then, for the eighteenth paper, we have Harvey Friedman, aiming to discuss a ‘sample of research progjects that are suggested by some of Gödel’s most famous contributions’ — a prospectus that immediately alerts the reader to the likelihood that the paper will cover too much too fast. The piece has the remarkably self-regarding title ‘My Forty Years on His Shoulders’ and ends with the usual Friedmanesque announcements of results about the equivalence of the provability-in-various-arithmetics of certain combinatorial claims with the consistency of certain set theories with large cardinals. The style and content will be very familiar to readers of the FOM list, and probably pretty baffling to others.
One place where Friedman’s paper goes a bit slower is in discussing the Second Incompleteness Theorem, and there are intimations by the author that he has found a neater, more insightful way of developing the result. But with his customary academic incivility, Friedman doesn’t bother to explain this in accordance with the normal standards of exchange between colleagues, but refers to online unpublications … where things remain equally unexplained. This is, to put it mildly, irritating: and I know I’m not the only person who has long since lost patience with this mode of proceding. Humphhhh!

KGFM 16: Penrose on minds and computers

Stewart Shapiro has had two shots at exploring the troubles with Lucas/Penrose-style arguments, first in his well-known paper ‘Incompleteness, Mechanism and Optimism’ Bull. Symb. Logic (1998), and then — expanding his treatment of Penrose’s efforts in Shadows of the Mind (1994) — in ‘Mechanism, Truth, and Penrose’s New Argument’ Jnl. of Philosophical Logic (2003). As you’d predict, Shapiro’s discussions are eminently lucid and very sharp; and his treatment of the Penrose argument in particular is extraordinarily patient and constructive, trying to get something out of the argument, and finding some interesting lines (though nothing that gives Penrose what he wants). He concludes with a

challenge to the anti-mechanist to articulate the new Penrose argument in a way that blocks the Gödel–Kreisel–Benacerraf ploy [i.e. the move of saying that perhaps we can be simulated by a computer but if so we can’t, with mathematical certainty, know which] but does not invoke unrestricted truth and knowability predicates [as apparently, but problematically, required by the Penrose argument, when the wraps are off].

If you don’t know the papers, they are terrific. And Shapiro’s insightful exploration surely has become the necessary starting point for any subsequent discussion here.

It is disappointing to have to report, then, that Penrose’s contribution to KGFM is written as if Shapiro had never made the effort to try to sort things out.

Well, that isn’t quite true: there’s a footnote which has a reference to Shapiro 2003. But otherwise, as far as I can see, Penrose just gives a (too brief to be useful) thumbnail sketch of his 1994 argument, and doesn’t address at all the technical problems that Shapiro explores. In so far as he does respond to critics, Penrose just offers some rather thin remarks about the sort of worries concerning idealization and vagueness that we noted that Putnam rehearses. But of course, the interesting thing about Shapiro’s discussion is that, for the sake of the argument, he gives the game to Penrose on those matters, allows Penrose’s anti-mechanist argument at least to get to the starting point, but then still finds trouble. Lots of trouble. And there’s nothing in Penrose’s paper here which offers any reponses. So I can’t say that this is a useful contribution to the debate on the impact of Gödelian arguments.

KGFM 15: Putnam on minds and computers

In his 1967 paper, ‘God, the Devil, and Gödel’, Paul Benacerraf famously gives a nice argument, going via Gödel’s Second Theorem, that proves that either my mathematical knowledge can’t be simulated by some computing machine (there is no particular Turing machine which enumerates what I know), or if it can be then I don’t know which machine does the trick. Benacerraf’s argument is perhaps not ideally presented, so for a crisper, streamlined, version see my Gödel book, §28.6: but the idea should be familiar.

Of course, how interesting you think this result is will depend on just how seriously you take the notion that there might such a determinate body of truths as my mathematical knowledge. For one thing, any real-world mathematician makes mistakes: what I know will be a subset of what I think I know, and I won’t in fact know which subset (so it’s no surprise if I wouldn’t recognize which Turing machine enumerates my actual knowledge). OK, it will be replied that the Benacerraf argument is supposed to apply to my idealized knowledge, prescinding from mistakes in performance etc. But how is that story supposed to work? And even if we can make the idea fly, and can sensibly idealize away from common-or-garden error, isn’t it going to be vague at the margins what I count as a proof? So isn’t it still going to be irredeemably vague what belongs to my idealized mathematical knowledge? If so, the question of simulating it with the crisply determinate output of a Turing doesn’t arise.

Similar worries about idealizing mathematicians and the vagueness of the informal notion of proof will beset other attempts to get sharp anti-computationalist conclusions about the mind from Gödelian considerations. And in the his quite brief paper, ‘The Gödel Theorem and Human Nature’, Hilary Putnam brings such worries to bear against Penrose in particular. Rather than pick holes again in the details of Penrose’s arguments (which have been chewed over enough in the literature, by Putnam among many others), he now stresses that the whole enterprise is misguided. “The very notion of an ideal mathematician is too problematic” to enable us to set up a contrast between what a suitably idealized version of us can do and what a naturalistically kosher mechanism can do. The complaint is quite a familiar one, but perhaps none the worse for that.

But interestingly, for all his worries about the pointfulness of such tricksy arguments, Putnam does return to explore a relation of Benacerraf’s argument, spelt out this time in terms of the notion of justified belief rather than knowledge.

The target is a (surely implausible!) Chomskian hypothesis to the effect that we have a ‘scientific faculty’ such that this faculty — in idealized form — can be simulated by some particular Turing machine T. In other words, (C) T enumerates (a coded version of) every true sentence of the form ‘we are justified in accepting p on evidence e‘. Then Putnam has an argument that either (C) isn’t true, or if it is we aren’t justified in believing it (I can’t have a justified belief about which machine does the simulation trick).

Oddly, however, Putnam doesn’t mention the analogous Benacerraf argument at all, so — if you are interested in this sort of thing — you’ll need to do your own “compare and constrast” exercise. And as with his predecessor’s argument, Putnam’s too isn’t ideally well presented and a bit of work needs to be done. Perhaps I’ll return to the exercise in a later posting, if it proves fun enough.

Or then again, perhaps I won’t … For in any case, the more interesting tack is to return to Penrose and ask whether he or a defender can sidestep the sort of general worry that Putnam has about arguments with a Lucas/Penrose flavour. Well, the next paper in KGFM is another shot by Penrose himself. So let’s turn to that.

Leonardo at the National Gallery

Bother. All pre-bookable tickets for the Leonardo Da Vinci exhibition at the National Gallery are sold out from now to the end of the show in February. It would have been good to go a second time. But at least we got to see the exhibition once.

Even though they are admitting fewer people per hour than in previous blockbuster exhibitions (didn’t I read that was so?), it was still somewhat uncomfortably crowded. Which made it quite difficult to get up-close and personal with the fifty or so drawings in the show (and somehow the bustle doesn’t put you in quite the right mood for them either). The light has to be low for the drawings too. So, to be honest, I got more out of sitting quietly at home looking at them as reproduced in the stunningly well-produced exhibition catalogue, which we bought in advance of our visit, even if it was good to see the originals.

So what really made it worth jostling through the crowds were the paintings. In particular, there’s the (first time ever) chance to see the two versions of the The Virgin of the Rocks in the same room — and the earlier version is surely here much better displayed and lit than I remember it in the Louvre. Stunning. And then there’s the portrait of the young Cecilia Gallerani, The Lady with an Ermine. Somehow — although the image is reproduced on the posters which are now all over London and has suddenly become very familiar — seeing the painting itself was very affecting, and just by itself made the trip worthwhile.

Those of you out there who already have tickets booked (or are up for queuing for the restricted number of tickets available on the day) still have a wonderful treat in store. And for others, you really could do a lot worse than buying yourself the bargain catalogue for Christmas as a consolation.

But ok, that’s quite enough culture for now. Back to grumpy logic-chopping in my next posts …

The Pavel Haas Quartet again [updated]

Driving back and forth to the town dump with a load of garden waste, I unexpectedly caught on the radio  — with an innocent ear — a concert performance of (most of) the first Rasumovsky quartet. It was stunningly good. I thought it sounded like a Czech — or at least middle European — quartet, and to be young too. And (having recently bought their Dvorak and Prokofiev CDs) I wondered if it was the Haas Quartet. Well, indeed it was. One of the very best performances I’ve ever heard. You can listen to it here for the next week. (But if you miss that chance, you’ll get an idea of how good they are from this film of them playing the last movement from the third Rasumovsky, also courtesy of the BBC.)

[Added later. You can also, for the next few days, listen to them playing the Schubert Quartettsatz and “Death and the Maiden”. More great stuff. Thanks, BBC!]

Three philosophy jobs in Cambridge

Three lectureships have been advertised, and the details are here. No, this isn’t the wildly overdue expansion of the Cambridge philosophy faculty from an establishment of twelve (as it has been for thirty years). I’ve already retired, Jane Heal retires at the end of this academic year, and — as I understand it — the remaining post is an early filling of the post which falls vacant when Raymond Geuss retires shortly. Still, it does mean that in these very troubled times, the faculty shouldn’t be shrinking in the near future: so well done to those who have ensured that this is so!

I’ve not been privy to the debates about the hoped-for future shape of the faculty (and quite right too, I suppose — the retired should keep out of such things). But the faculty anyway has a good tendency to sit a bit loose to plans, and appoint the smartest people who apply. The advert certainly doesn’t rule out one of the posts being a logic-minded replacement, for which in any case there’s a particularly desperate teaching need. Great. For as we all know, logic matters.

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