Categories: episode one

I said I’d post occasional progress reports on getting to grips with Category Theory, to pass on recommendations about what I’ve found helpful, etc. (Perhaps I should explain that — when trying to get into a new chunk of maths or math. logic — I find it works best to dive in and read a number of books quite quickly, skipping and skimming when the mood takes me, rather than plod very carefully through one key text. Also, though I know a few bits and pieces, I thought I should start over again from the very beginning.)

Two frequent recommendations for entry points are Lawvere and Schanuel’s Conceptual Mathematics (CUP, 1997), and Lawvere and Rosebrugh’s Sets for Mathematicians (CUP, 2003). But I can only say, give the first book a miss. It’s not very good: the authors seem to have no clear idea about their intended audience, so they veer between the irritatingly condescending/extremely laboured on the one hand, and sudden jumps in difficulty/sheer opaqueness on the other. For just one example, it is very difficult to believe that someone who needs a noddy explanation of why proving if p, q establishes if not-q, not-p is going to make anything at all of the sudden excursus on Gödel and Tarski at p. 307. But worse, you can get to the end of the book with no clear idea about what it is supposed to have achieved, or why it might matter. The second book is so far proving also a bit uneven but much better.

I’m dipping into various expository/philosophical articles as I go along. One I’ve just come across which I found prokoving and illuminating is Barry Mazur’s ‘When is one thing equal to some other thing?‘. Recommended.

Getting categorical

For a bit of light relief from matters Gödelian, I’m hoping to spend the next couple of months getting more to grips with category theory (well, and why not? — there are world-class category theorists just down the road at CMS, Martin Hyland and Peter Johnstone for a start, and it might be fun to be able to sit at the back of the category theory seminar and have some sense of what is going on). So I’ve gathered a somewhat daunting stack of books, and am plunging in … I’ll report progress!

Meanwhile, over 250 people have downloaded the Gödel book, and I’ve already had some very useful comments. In particular, Toby Ord has quite rightly taken me to task for getting a bit overexuberant in saying of Section 33.5 that it gives a proof that the Church-Turing Thesis entails the First Incompleteness Theorem. What the section does, in fact, is take the Kleene Normal Form Theorem and deduce incompleteness, assuming CTT along the way. But like any appeal to CTT in proving a formal result, that’s a labour-saving device that is dispensable — and if it weren’t, we’d be able construct a related counter-example to CTT (as I’d already pointed out in Section 28.7). So really, I guess I should have said, less dramatically, that KNFT entails incompleteness. Still, it’s a lovely argument if you don’t know it: and the point remains that in thinking about CTT, and proving recursiveness is equivalent to Turing computability as a step in its support (and that equivalence yields KNFT very easily), then we get incompleteness almost immediately — and that’s surely a nice surprise!

Gödel at long last

Back from Tuscany, with — at long last — a complete draft of my book on Gödel’s Theorems; if you are interested, do download a copy of the PDF, for all comments/suggestions will be very, very gratefully welcomed (I’d rather hear about gruesome mistakes now while there is a chance to change things!). I’ve just sent the PDF off to the publishers for a final review: it is late and over the originally contracted length, so fingers crossed. But I’ve already cut out an amount of stuff, and I don’t see how to cut out more without spoiling things.

I was staying at my daughter’s house at Certano near Siena. Sadly that was last time I’ll be there as they are moving. I’ll greatly miss the view from the kitchen table where I often wrote.

It’s a strange feeling ‘finishing’ a book — scare quotes, because I’ll have to do an index and tidy some of the typography and read for typos and thinkos and respond to comments: it won’t be finally gone for weeks. But there comes a point with any book where, although you know you must be able to improve it, you basically have to let it go. Which is both a relief and an anxiety.

Thank goodness that’s over

Examining over for another year. Thank goodness. I don’t particularly mind the process of marking tripos papers itself (though there is that inevitably huge and always rather dispiriting gap between what you tried to put across and what comes back in the generality of scripts). But having to run the show has its tense moments. But justice, of course, was perfectly done to everyone, as we all retained our immutable grasp of the Platonic form of the first-class script and marked accordingly.

Off to London on Thursday to the annual meeting of the Analysis committee. It seems a very long time ago that I was editor, and I can’t really recall why it then struck me as such a bright idea to spend twelve years at so time-consuming a job. But the journal continues to flourish, which is good. And there was time for a quick detour to visit my favourite Nereids nearby. So now, back — at last — to full-time Gödel!

"The best and most general version"

I’m still half-buried under tripos marking, but the end is in sight. And I’m Chair of the Examining Board for Parts IB and II this year, which is also not exactly an anxiety-free job. But between times, I’m trying to reorganize and finish the chapters of my book on the Second Theorem. I’m stuck for the moment wondering what exactly to say about “the best and most general version of the unprovability of consistency in the same system” which Gödel so briefly alludes to in the first part of his 1972a note (which repeats a footnote from 1967). Feferman in his editor’s introduction explains things by bringing to bear Jeroslow’s 1973 result. But it isn’t entirely clear to me that this rather esoteric result must be what Gödel had in mind.

Meanwhile, a nerdy footnote. There’s an even better new version out of NoteBook just out (if you are a Mac user, it really is just indispensable, and the academic price is absurdly low).

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!

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