Year: 2011

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 and Rindler 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).

Santa’s singleton

Here’s a question which I’m sure bugs all my logical readers. Modern mathematics standardly recognises partial functions which can take something as input but deliver nothing as output (like the reciprocal function which isn’t defined for zero). Do we also need to allow for co-partial functions which can take nothing as input but deliver something as output? Exciting eh?

Well, perhaps not so very thrilling. But for what it is worth, here’s a version of a paper in dialogue form given to the Serious Metaphysics Group here in Cambridge this evening. Not exactly the talk as delivered (with help from Rob Trueman!): the discussion has led me to cut out a long passage which was getting too involved by half and to tinker elsewhere a small amount.

One comment: if/when I get round to further rewriting this a bit, I’d drop the ‘Santa’ example, as I don’t want any noise coming in from issues about fiction. It would be better to use e.g. ‘Vulcan’ instead.

[Added later] Hmmm. The argument about partial-functions-by-stipulation certainly won’t do as it stands …

KGFM 9, 10: Gödelian cosmology, Rindler and Svozil

The next piece is ‘Gödel, Einstein, Mach, Gamow, and Lanczos: Gödel’s Remarkable Excursion into Cosmology’ by Wolfgang Rindler.

Rindler’s books on Relativity are real classics of exposition, so I was hoping for good things from this paper. I wasn’t disappointed. As Rindler says, Gödel famously “invented a model universe that was consistent with general relativity but that nevertheless exhibited two startlingly disturbing features: bulk rotation (but with respect to what, as there is no absolute space in general relativity?) and travel routes into the past (enabling one to witness or even preventone’s own birth?)”. If you want to know what Gödel’s cosmological model looks like, and have a smidgin of knowledge about relativity theory, then this paper is a great place to start. There’s no philosophical discussion though about worries concerning the very idea of closed time loops: but that’s no complaint — the paper does beautifully what it does set out to do. Recommended!

The tenth paper — grouped with Rindler’s in a subsection called ‘Gödelian Cosmology’ — is Karl Svocil’s ‘Physical Unknowlables’. But this piece in fact doesn’t even mention Gödel’s model universe, but rambles about indeterminism, ‘intrinsic self-referential observers’, unpredictability, busy beavers, deterministic chaos, quantum issues, complementarity, and lots more. Hopelessly unfocused, I’d say. Not recommended!

[That finishes the first part of KGFM. There will now be a gap for ten days or so before I can return to the second part, as I’ve promised to give two different talks next week and need to work on them!]

Stefan Collini on the very idea of a university.

I much admire Stefan Collini’s writing on the current situation in UK universities (see here, for example). He has a book forthcoming next year What are Universities For? which should be a major event at least for academics struggling to find principled ways of thinking about and reacting to the battering. In the meantime, you might be interested in his recent Cambridge lecture on the very idea of a university, now online.

KGFM 7, 8: Computers and computation, Papadimitriou and Copeland

Looking at the postings on KGFM, I’ve been pretty negative so far. Sorry! OK, Macintyre’s paper is indeed a tour de force but is for a pretty specialized reader. Otherwise I can only really recommend Feferman’s paper. Am I being captious? Well, collections like this one do tend to be very mixed blessings, don’t they? Blockbuster conferences invite the great and good who perhaps don’t always have much new left to say, and in any case interpret their briefs in very different ways, at different levels of sophistication; and the resulting edited volumes then bung more or less everything in with little editorial control (printing papers that wouldn’t make the cut in top journals). So you get collections like this one.

And now I fear I’m going to be pretty negative about (most of) the next two papers as well. I really am getting cranky in my old age. Sigh. But Christos H. Papadimitriou writes briefly on ‘Computation and Intractability’. He touches on Gödel’s 1956 letter to von Neumann and his prefiguring of something like the question whether P=NP which has been extensively discussed elsewhere (and there is nothing new here). And he adverts to a result about the intractability of finding Nash equilibria which is proved by a method of arithmetization inspired by Gödel: but you won’t learn how or why from this paper.

Next up is a much longer paper by Jack Copeland ‘From the Entscheidungsproblem to the Personal Computer – and Beyond’. Most of this is a story about the development of computing devices from Babbage to the Ferranti Mark I (complete with photos): interesting if you like that kind of thing, but utterly misplaced in this volume. But randomly tacked on is a final section which is germane: so after Feferman, you can start reading again here, with Copeland’s ‘Epilogue’, which is indeed worth looking at.

The issue here is Gödel’s 1970 note which attributes the view that “mental procedures cannot go beyond mechanical procedures” to Turing. Copeland responds not by worrying about Gödel’s anti-mechanism but with evidence that Turing shared it. He cites passages where Turing criticises what he calls an “extreme Hilbertian” view and  writes of mathematical intuition delivering judgements that go beyond this or that particular formal system. In fact,

Turing’s view … appears to have been that mathematicians achieve progressive approximations to truth via a nonmechanical process involving intuition. This picture, in which minds devise and adopt successive, increasingly powerful mechanical formalisms in their quest for truth, is consonant with Gödel’s view that “mind, in its use, is not static, but constantly developing.” These two great founders of the study of computability were perhaps not quite as philosophically distant on the mind-machine issue as Gödel supposed.

Copeland’s evidence seems rather convincing.

KGFM 4, 5, 6: More history, from Sigmund, Kennedy, and Feferman

I should have explained that Kurt Gödel and the Foundations of Mathematics is divided into three main parts, ‘Historical Context’, ‘A Wider Vision: the Interdisciplinary, Philosophical and Theological Implications of Gödel’s Work’, and ‘New Frontiers: Beyond Gödel’s Work in Mathematics and Symbolic Logic’. There are no less than seven papers in the first part left to talk about, and I’m not really the best person to comment on their historical content. But on we go.

Next, then, is Karl Sigmund on “Gödel’s troubled relationship with the University of Vienna based on material from the archives as well as on private letters”. The bald outlines will be familiar e.g. from Dawson’s biography, but there’s a lot of new detail. This is a nicely readable paper about the facts of the case, but I don’t think that there’s anything here that sheds new light on Gödel’s intellectual progress.

The fifth paper is ‘Gödel’s Thesis: an Appreciation’ by Juliette Kennedy. This concentrates on the philosophical Introduction of Gödel’s 1929 thesis on completeness (the material which wasn’t included in the 1930 published version), and in particular on his remarks about whether consistency implies existence. So part of Kennedy’s paper is in effect an elaboration of what Dreben and van Heijenoort say on p. 49 of their intro in the Collected Works when they point out that Gödel’s remarks are a bit misleading. She goes on also to urge that that there is evidence that Gödel was already, in 1929, thinking about incompleteness for theories (also not really a new suggestion). Unless I’m missing something — and Kennedy isn’t ideally clear — this also doesn’t really add much to our understanding. [That’s too brisk — see the Comments.]

The following paper is a typically lucid piece by Solomon Feferman, on the Bernays/Gödel correspondence, and as always is well worth reading if you are interested in the history here. Feferman has already introduced the correspondence in the Collected Works: but this piece concentrates at greater length on the theme of ‘Gödel on Finitism, Constructivity, and Hilbert’s Program’. As he writes,

There are two main questions, both difficult: first, were Gödel’s views on the nature of finitism stable over time, or did they evolve or vacillate in some way? Second, how do Gödel’s concerns with the finitist and constructive consistency programs cohere with the Platonistic philosophy of mathematics that he supposedly held from his student days?

Re the first question, Feferman wrote in CW of Gödel’s ‘unsettled’ views on the upper bound of finitary reasoning. Tait has dissented. But in fact,

Tait says that the ascription of unsettled views to Gödel in the correspondence and later articles “is accurate only of his view of Hilbert’s finitism, and the instability centers around his view of whether or not there is or could be a precise analysis of what is ‘intuitive”’ (Tait, 2005, 94). So, if taken with that qualification, my ascription of unsettled views to Gödel is not mistaken. As to Gödel’s own conception of finitism, I think the evidence offered by Tait for its stability is quite slim …

On the second question, of why Gödel should have devoted so much time to thinking about a project with which he was deeply out of sympathy, Feferman writes

Let me venture a psychological explanation … : Gödel simply found it galling all through his life that he never received the recognition from Hilbert that he deserved. How could he get satisfaction? Well, just as (in the words of Bernays) “it became Hilbert’s goal to do battle with Kronecker with his own weapon of finiteness,” so it became Gödel’s goal to do battle with Hilbert with his own weapon of the consistency program. When engaged in that, he would have to do so –  as he did – with all seriousness. This explanation resonates with the view of the significance of Hilbert for Gödel advanced in chapter 3 of Takeuti (2003) who concludes that … [Gödel’s] “academic career was molded by the goal of exceeding Hilbert.”

Sounds plausible to me!

Touched by greatness

A very moving concert last night. In the small Peterhouse Theatre (a lovely space for intimate music), Menahem Pressler played Beethoven’s A-flat major sonata, Op. 110, Debussy’s Estampes, and then Schubert’s last piano sonata D. 960. He talked touchingly at the beginning of the evening, and this was evidently music that meant a great deal to him. Pressler’s playing now is not the most technically secure, but his desire to communicate with his audience is undimmed.  Pavel Haas Quartet play DvorakThe Schubert in particular was very affecting: in the second movement, the poignancy of an old man now 87 playing the searing music of a young man facing early death was almost too much to take. We will remember the occasion a long time.

It can be irritating though — can’t it? — to hear tell of great concerts that you’ve now missed (and couldn’t have got to anyhow). So let me mention something else which is quite wonderful in a different way, the Pavel Haas Quartet’s Dvořák disk. Hardly a discovery by me! — it’s the recently announced new Gramophone Recording of the Year. But it really is astonishing. I’ve always thought the old performance of the “American” quartet by the Hollywood Quartet was in a league of its own (one generation from the shtetl, is it fanciful to hear the tug of a vanished Europe in their playing?). But this new recording from the young Czech quartet is at least as great. I’m bowled over.

KGFM 2, 3: Kreisel and Grattan-Guinness

The second paper in the collection is a seven-page ramble by Georg Kreisel, followed by twenty pages of mostly opaque endnotes. This reads in many places like a cruel parody of the later Kreisel’s oracular/allusive style. I lost patience very quickly, and got almost nothing from this. What were the editors doing, printing this paper as it is? (certainly no kindness to the author).

Something that struck me though, from the footnotes. Kreisel “saw a good deal of Bernays, who liked to remember Hilbert  …. According to Bernays … Hilbert was asked (before his stroke) if his claims for the ideal of consistency should be taken literally. In his (then) usual style, he laughed and quipped that the claims served only to attract the attention of mathematicians to the potential of proof theory” (pp. 42–43). And Kreisel goes on to say something about Hilbert wanting use consistency proofs to bypass “then popular (dramatized) foundational problems and get on with the job of doing mathematics”. Which chimes with Curtis Franks’s ‘naturalistic’ reading of Hilbert, which I discussed here.

The book’s next contribution couldn’t be more of a contrast, at least in terms of crisp clarity. Ivor Grattan-Guinness is his usual lucid and historically learned self when writing quite briefly about ‘The reception of Gödel’s 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s’. But in a different way I also got rather little out this paper. There are some interesting little anecdotes (e.g. Saunders Mac Lane studied under Bernays in Hilbert’s Göttingen in 1931 to 1933 — but writes that that he was not made aware of Gödel’s result). But the general theme that logicians got to know about incompleteness early (with some surprising little delays), and the word spread among the wider mathematical community much more slowly could hardly be said to be excitingly unexpected. Grattan-Guinness has J. R. Newman as a hero populariser (and indeed, I think I first heard of Gödel from his wonderful four-volume collection The World of Mathematics) — and Bourbaki is something of a anti-hero for not taking logic seriously. But, as they say, what’s new?

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