KGFM

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.)

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.

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.

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!]

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.

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!