Peter Smith's Blog, page 126

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-naturalist 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 them as well. So, to be honest, I got more out of sitting quietly at home looking at the drawings 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), you 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.

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!]

The Pavel Haas Quartet again

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.)

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.

Vermeer in Cambridge

We went today to the Fitzwilliam Museum to see for the first time (but definitely not the last) Vermeer's Women: Secrets and Silence. A rather wonderful exhibition, astonishingly gathering four Vermeers in the same room, with another twenty-eight pictures from the Dutch 'Golden Age', mostly small intimate pictures of women at home. It's just the right size of exhibition to take in without feeling overwhelmed. A delight, affecting, and all quite free too. If you are in reach of Cambridge, do see it.

As years go by, I seem to find more and more (quite untutored) pleasure in looking at pictures. A quiet philosophical voice sometimes wonders why: can any reader of this blog recommend something insightful to read on why we can find the old masters so affecting?

Heck's Frege's Theorem — and KGFM, 11–14

When I was in London for the Tennenbaum Workshop, I picked up a copy of Richard Heck's very recent Frege's Theorem, which collects together eleven of his papers — with some changes and some postscripts — together with a 39 page introductory 'Overview'. I've quickly read the overview which is immensely helpful, as you'd predict, and it is terrific to have the previously  very widely scattered papers in one place. Even if you aren't a great fan of the neo-logicist project, you'll want to know just how much Frege achieved, and where the pressure points are, technical and conceptual. You won't get a better guide than Heck. So this collection (the sort of thing that tends to add up to quite a bit more than the sum of its parts) is just great to have, and I really look forward to (re)reading it all.

So there you are — proof positive that I'm not always a cantankerous reader/reviewer! But I'm afraid that I'm again not going to be so friendly about the next four instalments of Kurt Gödel: Foundations of Mathematics.

Next up is another piece like Svozil's that ranges widely over notions of incompleteness in mathematics and science, though at least John Barrow writes very clearly in his 'Gödel and physics'. He aims at accessibility, but it is all slightly slapdash (from irritating little things like trying to define syntactic consistency using the notion of truth to bigger things like quite mis-stating how a Turing machine is used to decide 'undecidable' questions in Mark Hogarth's now famous construction). So despite the comparative readability, this piece can't really be recommended to beginners.

The twelfth paper is by Denys Turner, a theologian, on 'Gödel, Thomas Aquinas, and the unknowability of God'. The author himself thinks that any analogies between Gödel and the tradition of 'negative theology' are pretty tenuous, and says "I simply do not know whether the superficial parallel is genuinely illuminating". Well, it isn't. Skip this.

The following paper is a really surprising disappointment. I much admire Piergiorgio Odifreddi's Classic Recursion Theory which seems a paradigm of how to write such a book: the exposition is wonderfully clear, but what really makes the book stand out are the historical/conceptual asides about what lies behind the technical developments. I'd have predicted, then, that Odifreddi could have interesting things to say how Gödel's logical work can be seen as in some way shaped by or encouraged by philosophical ideas. But no: we get less than five pretty superficial pages. Strange.

Finally in this batch, the fourteenth paper — Petr Hájek writing on 'Gödel's Ontological Proof and Its Variants' — may, for all I know, be quite outstanding. Enthusiasts for exploring that strange 'proof' will want to read the paper, I'm sure. But I've never caught that particular bug: so I frankly confess I've just no way of telling how much insightful novelty this is here. Sorry!

OK: that's taken me over 300 pages through KGFM, and so far — Feferman apart — I've not been enthused. But there's Hilary Putnam, Harvey Friedman and Hugh Woodin among those yet to come. So I still live in hope!

 

Tennenbaum's Theorem Workshop

Sean Walsh organized a one-day workshop on the philosophical significance of Tennenbaum's Theorem on Saturday. It kicked off with me presenting a short piece that Tim Button and I have forthcoming in Philosophia Mathematica: here's a preprint of our paper.

But for a quicker read, my overheads give the headline idea — that's there no implication about how we grasp the standard model to be got out of the elegant but non-trivial Tennenbaum's Theorem that you can't get out of the very easy theorem that every model of PA where every element has a finite number of predecessors is isomorphic to the standard model. Tennenbaum's Theorem has no extra oomph against the Skolemite sceptic. Indeed, appealing to either model theoretic result just doesn't touch the sceptic's worries. (The talk timed nicely, and having Tim there to help fend questions made giving it a lot more fun!)

The current temporal parts of Walter Dean and Leon Horsten were agreed, contra earlier parts, that Tennenbaum's Theorem cuts no ice against the model-theoretic sceptic (I wasn't so clear where Paula Quinlon now stands). But I think all three other speakers in different ways wanted to squeeze something philosophical out of Tennenbaum's Theorem. If/when published pieces emerge, I'll say why I wasn't so convinced. But a fun occasion (as such closely-focused workshops tend to be).