Was Gödel right?

A coincidence. Rereading John Dawson’s Gödel biography Logical Dilemmas, I’ve just got to the point where Dawson recounts how Gödel thought he’d discovered an inconsistency in the American Constitution, which would allow a dictatorship to arise (pp. 179-80). And then the same day I come across Elizabeth Drew’s recent article in the New York Review of Books explaining some of the ways in which the Bush White House has grabbed powers to itself and undermined the constitutional settlement between the three branches of government. Perhaps Gödel’s anxieties were well founded.

Gödeling along

I’m still working away on my draft book on the incompleteness theorems, in between the delights of marking tripos papers. I’ve just uploaded a new near-final(?) version of Chapters 1 to 22 — the first two hunded pages — to www.godelbook.net; all comments are still most welcome. Don’t all rush at once …

It’s that time of year again …

… when I’m buried in tripos marking. Distractions between marking sessions are necessary. So I’ve just finished reading Margaret Atwood’s The Penelopiad which made a wonderful diversion from the usual mixed bag of metaphysics scripts.

Incidentally, the white smoke has at long last gone up from the consistory chapel window, and the Knightbridge Professorship has been offered to X. But unlike popes, who don’t get to negotiate their terms, potential professors do. So we’ll have to wait and see if X indeed arrives. [Later breaking news, 11 June: X = Quassim Cassam, who is indeed coming to Cambridge from Oxford via UCL for January 2007.]

Libraries should be circular?

When I was in Aberystwyth, I had a decent sized room in the Hugh Owen Building which is halfway up Penglais, with panoramic views over Cardigan Bay. In Sheffield, I had a huge room on the 12th floor of the Arts Tower — and while the daytime urban view wasn’t exactly a delight, on winter evenings the transformation into a glittering landscape of lights was magical. These days I have a very small room in the Faculty, with a window into the grad. centre and otherwise tiny windows too high to look out of, which isn’t as bad as it sounds, but equally isn’t very enticing.

So I work a lot in the Moore Library. It took me a while to really “get it”, but now it strikes me as in many ways a quite splendid building, and I love being there. The reading tables run around the perimeter, so you are looking out to trees and to the modern buildings of the rest of CMS; even when the library is busy, you can only really see a few people either side of you because of the curve of the building and the book shelves which are arranged as along the spokes of a wheel. And while the bookstacks in the UL seemingly run off to infinity (so you can feel lost in a Borgesian nightmare), there is a sense that here the readers are surrounding the mathematical knowledge shelved behind them. There is a rather calming feel to the place, which draws me back especially when things aren’t going well with my book. So I should get down there now …

Tired of ontology?

It requires a certain kind of philosophical temperament — which I seem to lack — to get worked up by the question “But do numbers really exist?” and excitedly debate whether to be a fictionalist or a modal structuralist or some other -ist. As younger colleagues gambol around cheerfully chattering about these things, wondering whether to be hermeneutic or revolutionary, I find myself sitting on the side-lines, slightly grumpily muttering under my breath ‘And who cares?’.

To exaggerate a bit, I guess there’s a basic divide here between two camps. One camp is primarily interested in analytical metaphysics, or epistemology, or the philosophy of language, and sees mathematics as a test case for their preferred Quinean naturalist line (or whatever). The other camp is puzzled by some internal features of the practice of real mathematics and would like to have a story to tell about them.

Well, if you’re tired of playing the ontology game with the first camp, then there’s actually quite a bit of fun to be had in the second camp, and maybe more prospect of making some real progress. In the broadest brush terms, here are just a few of the questions that bug me (leaving aside Gödelian matters):

  1. How should we develop/improve/augment/replace Lakatos’s model of how mathematics develops in his Proofs and Refutations?
  2. What makes a mathematical proof illuminating/explanatory? (And what are we to make of unsurveyable computer proofs?)
  3. Is there a single conceptual grounding for the standard axioms of set theory? (And what are we to make of the standing of various large cardinal axioms?)
  4. What is the significance of the reverse mathematics project? (Is it just a technical “accident” that RCA_0 is used a base theory in that project? Can some kind of conceptual grounding be given for that theory? Would it be more principled to pursue Feferman’s predicative project?)
  5. Is there any sense in which category theory provides new foundations/suggests a new philosophical understanding for mathematics?

There’s even a possibility that your local friendly mathematicians might be interested in talking about such things!

Laws of nature

I’m giving just four second-year lectures on the philosophy of science this term, revisiting Lakatos (I’m a long-time fan). Last year I talked instead about laws of nature; rather to my surprise I found myself taking exactly the opposite line from that I used to take in supervisions, and warmed to a wild Humean subjectivism. Re-reading the notes from the lectures on laws they seemed at least provoking enough to be worth handing out again to this year’s class. I don’t promise that I believe any of this stuff: I was just interested to see if you can play the game the Humean way.

Ancestral logic

One of the many things I want to do once I’ve got my Gödel book finished is to slowly trawl through the first twenty years (say) of JSL to see what what our ancestors knew and we’ve forgotten.

I was put in mind of this project again by finding that John Myhill in JSL 1952 (‘A derivation of number theory from ancestral theory’) already had answers to some questions that came up in re-writing a section of the book last week.

As is entirely familiar, we can define the ancestral of a relation R using second-order ideas: but it doesn’t follow from that that the idea of the ancestral is essentially second-order (as if the child who cottons on to the idea of someone’s being one of her ancestors has to understand the idea of arbitrary sets of people etc.) Which in fact is another old point made by e.g. R. M. Martin in JSL 1949 in his ‘Note on nominalism and recursive functions’. So there is some interest in considering what we get if we extend first-order logic with a primitive logical operator that forms the ancestral of a relation.

It’s pretty obvious that the semantic consequence relation for such an ‘ancestral logic’ won’t be compact, so the logic isn’t axiomatizable. But we can still ask whether there is a natural partial axiomatization (compare the way we consider natural partial axiomatizations of second-order logic). And Myhill gives us one. Suppose R* is the ancestral of R, and H(F, R) is the first-order sentence which says that F is hereditary down an R-chain, i.e. AxAy((Fx & Rxy) –> Fy). Then, putting it in terms of rules, Myhill’s formal system comes to this:

  • From Rab infer R*ab
  • From R*ab, Rbc infer R*ac
  • From H(F, R) infer H(F, R*)

where the last rule is equivalent to the elimination rule

  • From R*ab infer (Fa & H(F, R)) –> Fb

which is an generalized induction schema. Myhill shows that these rules added to some simple axioms for ordered pairs give us first-order Peano Arithmetic. But do they give us more?

Suppose PA* is first-order PA plus the ancestral operator plus the axiom

  • Ax(x = 0 v S*0x)

i.e. every number is zero or a successor of zero. Then — if we treat the ancestral operator as a logical constant with a fixed interpretation — this is a categorical theory whose only model is the intended one (up to isomorphism). But while semantically strong it is deductively weak. It is conservative over PA. To see this note that we can define in PA a proxy for R*ab by using a beta-function to handle the idea of a finite sequence of values that form an R-chain, and then Myhill’s rules and the new axiom apply to this proxy too. And hence any proof in PA* can be mirrored by a proof in plain PA using this proxy. (Thanks to Andreas Blass and Aatu Koskensilta for that proof idea.)

So the situation is interesting. Arguably, PA doesn’t reflect everything we understand in understanding school-room arithmetic: we pick up the idea that the numbers are the successors of zero and nothing else. In other words, we pick up the idea that the numbers all stand to zero in the ancestral of the successor relation. So arguably something like PA* does better at reflecting our elementary understanding of arithmetic. Yet this theory’s extra content does nothing for us by way of giving us extra proofs of pure arithmetic sentences. Which is in harmony with Dan Isaacson’s conjecture that if we are to give a rationally compelling proof of any true sentence of basic arithmetic which is independent of PA, then we will need to appeal to ideas that go beyond those which are constitutive of our understanding of basic arithmetic.

Lighten up, Ludwig

Went to the one-day Tractatus workshop here in Cambridge (the last in a series that has mostly taken place in Stirling). I was there in my role as the village sceptic.

Julian Dodd and Michael Morris kicked off with a joint talk on Making sense of nonsense. What are we to make of the fact that Wittgenstein officially seems to think of his claims in the Tractatus as nonsense (yet in the Preface he says ‘the truth of the thoughts communicated here seems to me to be unassailable and definitive’)? One line is that claims of the Tractatus communicate truths that can be shown but not said (the ‘truth-in-nonsense’ view). Another line is that actually not all the claims are non-sensical (the ‘not-all-nonsense’ view). Julian and Michael think there is a third way. All of the claims of the Tractatus are nonsense and they don’t communicate any genuine truths indirectly either (the ‘no-truths-at-all-view’); the prefatory remark is just another bit of philosophical nonsense.

This was interestingly done, though they also seem to want to suggest that the movement of thought in Tractatus, read their way, naturally leads to its mystical conclusion. I just don’t see that. Somewhere in the middle of the 6.somethings, sensible readers of the Tractatus can perfectly well think “Oh come off it, Ludwig, lighten up!”. The mystical guff about feeling the “world as a limited whole” is no more an upshot of what’s gone before than would be, say, something like Lichtenberg’s wryly amused attitude to the scattered occasions of his life.

Next up, Fraser MacBride and Peter Sullivan talked about Ramsey, Wittgenstein, and in particular the argument about complex universals. Peter hinted at, but didn’t in this talk really explore, an interesting thought. Ask: ‘How much do the principles of logic reveal about the nature of things/the constitution of facts?’ It seems Frege answers “a great deal” (logic reveals the deep distinctions between objects, properties, properties of properties etc.) while Ramsey answers “next to nothing” (e.g there isn’t a deep object/universal distinction reflected in the logical subject/predicate form). Peter suggested that there is a lot in the Tractatus that comes from Frege and a lot that feeds into Ramsey’s position. Which suggests that Wittgenstein’s position might be an incoherent mixture.

Finally, Mike Beaney talked about the chronology of the interchanges between Frege and Wittgenstein. And Michael Potter talked more specifically about when W. might have learnt from F. to distinguish sharply complexes and facts (early, according to Michael).

Which was all mildly instructive, though the discussions sometimes became bogglingly theological, in the way that Wittgenstein-fests can do. It was occasionally like listening to rounds of ‘Mornington Crescent’ without the jokes (and no, I’m not going to explain!).

I leave it too long …

I leave it too long between visits to the Fitzwilliam. But since the really excellent new café started up, I’ve been going rather more often. Take a book, read over a coffee, then take a break to look at just a few pictures (that is surely much the best way to “do” a gallery). I was very struck again today by The Holy Family by the seemingly rather unregarded Sassoferrato. I just wish I was clearer in my mind about how an unbeliever should regard religious art, without double-think or sentimentality.

Universities on the cheap

Reading the Guardian isn’t always good for my blood pressure. Today there is an article under the name of Tony Blair, no less, saying how important it is that “we maintain and improve the high reputation of higher education in Britain” (note, it is the reputation that has to be improved). But not, of course, because education might be a good in itself; no, it’s because we want to sell the product and make the most money possible out of overseas students.

But I wonder who is going to staff these high reputation universities? Some of our brightest and best might enjoy a year of graduate study; but even here in Cambridge they seem increasingly reluctant then to launch into a PhD. And who can blame them? It could be six years more before they are in the running for a permanent academic job. Getting one is a very chancy business (since employment numbers are kept down by ludicrous staff-student ratios). And the pay is then dreadful, at least compared with what they might hope to get elsewhere. Oh well …

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