1. First up, spotted in the CUP bookshop and snapped up, is the just-published Proofs and Computations by Helmut Schwichtenberg and Stanley S. Wainer. A mere 450 action-packed pages, this looks as if it should be an instant classic, a welcome filling of a gap in the literature on the interactions between proof theory and computability theory.
2. Arnie Koslow told me about Lloyd Humberstone’s The Connectives which has been been out a couple of months and somehow I’d missed seeing. This one weighs in at some 1500 pages (which makes the price rather remarkably cheap). Again, on a quick browse it looks daunting but amazing.
3. Very differently, I spotted an announcement a couple of days ago by Michael Gabbay of the publication of the first instalment of a translation of Hilbert and Bernays (or rather a bilingual text, German and English on facing pages). This only gets to p. 44 of the German text (over fifty pages of the book reprint a long essay by Wilfried Sieg on Hilbert’s proof theory). But it is again very inexpensive, and being German-less I certainly wish the project well. So I’ll be sending off for a copy, and will report back here.

So there you are: how can we resist? (Any suggestions for other recent books on logic matters that I might have missed?)

Time then to pause to draw breath for Christmas after a busy/distracting time. But then what’s next, logically speaking?

Well, I plan to blog here in the new year about two more books that I’ve been asked to review together — Leon Horsten’s The Tarskian Turn: Deflationism and Axiomatic Truth and Volker Halbach’s Axiomatic Theories of Truth. Not that I claim any special expertise about their topic: but then both books are written for a reader like me — a philosopher/logician interested in theories of truth, who wants to get a handle on recent some formal developments (to which the respective authors have been notable contributors). Should be very interesting. And since both books have been out for few months, I hope some readers of the blog will be able to chip in helpfully in comments!

But mostly, it will have to be back to Gödel. At the moment, I’m re-writing the opening chapters of the book for the second edition, and I think very much improving them — about which I have mixed feelings! On the one hand, it’s very good to feel the effort of doing a second edition is going to be worth while, but on the other hand, I’m a bit downcast to see how very far from ideal those chapters previously were. Sigh. Anyway, when I’ve got the first tranche of chapters more to my liking, I’ll post them here for comments.

And I’ve one or two other plans too … So watch this space!

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.

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.

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.

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.

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.

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!

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!]