Logic Matters

I’ve just put another full draft of the Gödel book online. There’s an amount of work still to be done (in particular, I’ve yet to take account of some comments I’ve had on the last chapter); but things are progressing towards a terminus. One task now, while I’m thinking about more substantive things as well, is to decide what kind of index or indexes to do. What a thrill.

I’ve been distracting myself a bit by dropping into the newsgroup sci.logic and posting in some of the threads. Which probably shows that I need to get a life. Yet I can’t help finding it irritating (indeed offensive) when loud-mouthed ignorant blockheads are allowed to dominate a public forum which can occasionally can be very useful. So I’ve been diving in and trying to help sort things out here and there.

Two enthusiastic book recommendations, one logic-related, one not. First, I didn’t know Piergorgio Odifreddi’s Classical Recursion Theory before, but I’ve been quickly working through it and it is excellent — I very much like the style and pace. In a way, I’m quite glad that I didn’t read it before or I might have been tempted to follow some of his nice modes of presentation. As it turns out, the treatment in my book is nicely complementary at various points.

The other, my late-night book at the moment, is Alan Bennett’s Untold Stories. Perhaps I should take the words at the end of his Introduction, which he quotes from his own play The History Boys, as the motto for the Gödel book: “Pass it on. Just pass it on.” For that’s exactly what I’m trying to do …

A revised version of Chapters 1 to 26 is now on-line. Next, the two chapters which probably need the most work/thought, so off to the Moore library again to minimize distractions.

I’ve been wrestling with that loose and baggy chapter which needed carving into two. Annoying that what should have been a simple task took so long, but I think I’ve cracked it. So the latest version of what are now the first 23 chapters of the book can be downloaded from the website.

Greg Restall has a link to a terrific Australian radio talk on ‘they’ as a singular pronoun. Which reminds me I’d better check through my book that I haven’t, for example, called all logicians ‘him’.

[Added] Damn. Heaven knows why, but I still find after all these years using computers that when I waste a tree and print out hard copy I immediately spot glaring errors that I hadn’t noticed on-screen. Very odd. So there’s now a better version of those first 23 chapters on-line.

… but very good news for me! CUP have just agreed definitely to publish my Gödel book, having had a very positive “clearance review” from an obviously Very Wise Reader with Impeccable Logical Judgement. It just remains to continue working away improving it here and there, and taking account of the very assorted but very useful comments I’m getting (including from the Very Wise Reader). The next major task: to disassemble the rather rambling Chapter 22 and put it back together as two chapters.

Meanwhile, the project of getting on top of category theory will inevitably have to be put on the back burner for a while.

Jacob Plotkin has e-mailed to point out a stupid mistake just five pages into the previous version(s) of the Gödel book. I’d given a really, really bad reason for saying (what is true) that the incompleteness of arithmetic entails the incompleteness of the theory of rational fields. Of course, it’s not enough for Gödelian incompleteness to carry over to a theory T that T can define 0, 1, 2 and so on (and knows about adding and multiplying these). T has got to be able to define a predicate Nat applying to the numbers so that T can replicate numerical quantifications. Only that way can T count as e.g. embracing Robinson arithmetic, and hence be incomplete-if-consistent for Gödelian reasons. Well, Q, the standard theory of rational fields, can define a suitable predicate Nat, so incompleteness does apply. But that is not at all obvious. In fact, defining Nat in Q was part of the work that Julia Robinson got her PhD for (see her JSL 1949 paper).

Now I kind of knew all that, but it didn’t stop me writing something that completely ignored it. And it stayed in a number of successive drafts. These kinds of cognitive glitches — these failures in joined up thinking — are very odd, and maddeningly annoying when you succumb to them. (I once saw my wife, who’d spent some of the afternoon making a large pot of chicken stock, very attentively pouring the stock away down the sink, carefully saving the boiled bones and vegetables …)

Another psychological oddity: why do I often find it easier to read in my favourite noisy Italian café (Savino’s, since you ask) with Italian radio blaring and a lot of comings and goings, than in a quiet library which can be all too conducive to sleepiness. I don’t think it is just the supply of espresso. Presumably it is something to do with the noise and bustle keeping part your brain alert for signs of danger and threat in the background, and so stopping processing from shutting down …

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!

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”.