Greg Restall on arithmetic

Greg Restall has put a very nice paper online, called ‘Anti-realist classical logic and realist mathematics’. I’ve always been tempted by logicism in the very broadest sense: but Greg’s critics, of course, will say that he’s just smuggled the rabbit into the hat before pulling it out again. Still, his piece is nicely thought-provoking.

Oddly, given his multiple-conclusion logical framework, Greg doesn’t mention Shoesmith and Smiley’s great book in his biblio. (Very regrettably, it had the bad luck to be published in 1978, around when a number of publishers used early computers to print books with what looked like typewritten pages. The first edition of Fogelin’s fine book on Wittgenstein suffered the same fate. And in both cases, I think the repellent and amateurish look of the results was enough to put readers off and stop the books making the impact they deserved at the time. At least Fogelin got a properly printed second edition. But Shoesmith and Smiley has gone out of print, and seems widely forgotten.)

Gödeling along again

I’ve just posted a “maintenance upgrade” of the first 14 chapters of An Introduction to Gödel’s Theorems on the book’s website (and, now I’ve got a bit of time, redone the site from scratch as well so it is a lot neater). As I work through the book again, I’ve not myself yet found anything that needed drastic emergency surgery, though the old Chapter 14 messed up at one point, conflating Frege and Russell. Oops. I’ve managed to delete a few distracting paragraphs and tidy some discussions enough to save four pages so far, which will gladden the publisher’s heart.

About 500 people have now downloaded the book. I guess I don’t really want 500 sets of comments at this stage in the game — no chance of that, though! Still, it would be very good to get a few more than the very small handful I’ve had so far. Comments can be immensely helpful, often in unexpected ways. Keep them coming!

Categories: episode three

Kai von Fintel has just e-mailed, to send a link to the Good Math, Bad Math blog on category theory which is excellent — a series of mini-essays on concepts of category theory with some very helpful introductory explanations of some of the Big Ideas. Worth checking out, so thanks Kai!

I’ve now read quite a lot of Steve Awodey’s new book. Disappointing: or at least, it doesn’t do quite what I was hoping it would do. Awodey’s two papers in Philosophia Mathematica were among the pieces that got me interested in category theory in the first place (see his ‘Structure in mathematics and logic: a categorical perspective’, 1996, and his reply to Hellman, 2004). So I suppose I was hoping for a book that had more of the discursive, explanatory, commentary that Awodey is good at. But there’s very little of that. And although Awodey says in the preface that, if Mac Lane’s book is for mathematicians, his is for ‘everyone else’, in fact Category Theory is still pretty well orientated to maths students (for example, there is a telegraphic proof sketch of Cayley’s Theorem that every group is isomorphic to a permutation group by pp. 11-12!). Of course, there are good things: for just one example it helped me understand better the general idea of limits and colimits. But I wouldn’t really recommend this book to the non-mathematician.

So over the next week or two, I think it’s Goldblatt’s lucid Topoi, and Lawvere and Rosebrugh’s Sets for Mathematics for me.

Slow work …

It’s slow work, going through my Gödel book a couple of chapters at a time checking for typos (down to the level of missing brackets or periods, ‘x’ for ‘y’, etc.), and looking for thinkos, sentences where the prose could be improved, passages where the book drags unnecessarily, paragraphs which could be deleted, cases where what happens in one chapter doesn’t quite tally with what happens in another, and so on and so forth. Thankfully, I’m not finding too much that needs attention yet; but there is more than enough to make the exercise well worthwhile.

It’s slow work too, getting into category theory. As I remember it, it was a lot easier mastering quite a bit of non-linear dynamics (when I was working on what became Explaining Chaos). I suppose that it could just be that I’m getting too old to readily learn new tricks. It could be that category theory’s high level of abstraction makes it more difficult to get your head around. But I rather think that it’s because I then had a whole stack of wonderfully clearly written, well-structured, zestful, example-packed, highly explanatory, dynamics books to lean on, while category theory seems not at all so well served.

But I’ll press on, as the partially understood glimpses I’m getting are intriguing! Being in sight of retirement, with little prospect of promotion, at least has one very enjoyable advantage, which I might as well make the best of: I can cheerfully follow such interests wherever they happen to take me, without getting at all fussed about whether they will ever lead to publications that will “count”.

Categories: episode two

A copy of Steve Awodey’s new Category Theory (Oxford Logic Guides 49) has just been delivered. His preface starts

Why write a new textbook on Category Theory, when we already have Mac Lane’s Categories for the Working Mathematician? Simply put, because Mac Lane’s book is for the working (and aspiring) mathematician. What is needed now … is a book for everyone else.

First impressions look very encouraging, though the book covers quite a lot in under 250 pages, judging from the table of contents, so the pace promises to be pretty speedy. (Grumble to OUP: given this is intended to be a textbook for a relatively wide audience, why exactly is it published only in hardback and at the extremely steep price of £65 pounds/$124.50?)

I’ve just noticed too that Robert Goldblatt’s Topoi: The Categorical Analysis of Logic — which has been out of print for ages — has been republished by Dover (Amazon have it for £14.63/$19.77 which is a bargain for a book twice the length of Awodey’s). I found before that Goldblatt’s book starts pretty gently in a very helpful way, even though it seems to accelerate a bit alarmingly after the first three chapters. Anyway, my plan of action now is to parallel-process Goldblatt, Awodey, and Lawvere/Rosebrugh. Watch this space for further reports!

(Mmm, I hope that one or two greedy booksellers on abebooks.com who have been trying to sell on second-hand copies of Topoi for quite extortionate prices up to $500 have just found themselves stuck with now unwanted stock!)

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).

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