I did slightly lose the will to blog, partly through overwork, partly through the blog being spammed, partly through other distractions. But the mood takes me again, so let’s see how things go …

The Gödel book hasn’t left my desk yet, and won’t for another month or so. I’m still waiting to hear back from the CUP proof reader; and meanwhile I’m working through 126(!) comments sent over the holidays with immense kindness by Richard Zach. Thanks Richard! I’ve had to take the book off-line now that it is going through the press, at the reasonable insistence of CUP: but making it available on the web certainly made the whole business of writing it so much more enjoyable, and has made the result much, much better. (More enjoyable, because of encouraging feedback along the way: it can be a pretty lonely and sometimes dispiriting business writing a long book. Nice comments kept me going. And the book might still have its many flaws, but it has certainly been much improved by comments from literally dozens of people. I’m immensely grateful.)

I’m beginning to wonder what to do next. The Preface to the Introduction to Gödel’s Theorems promises a sequel, Incompleteness after Gödel, but that’s going to be a long job, getting really on top of and then organizing the materials. I promised this term to talk about Gentzen’s proof of the completeness of arithmetic to a group of grad students, and have found there isn’t anything treating quite what I had in mind to cover at the level I wanted to cover it. So a tempting smaller project suggests itself, and I’ve even a title: Ordinals, Cuts and Consistency. But we’ll see: watch this space …

Indexing a book is no fun. None at all. Yet it can be so annoying when a book has a perfunctory index that I feel compelled to try to do the job decently.

You can only work on so many pages at a stretch. Distractions are needed. One that works for me is opening up sci.logic in Google groups every so often and sounding off. Hence a rash of posts battering away at some of the dafter stuff there. Who knows if anyone appreciates it … but it keeps me amused, and is one of the more harmless ways of wasting time on the web.

Passing the CUP bookshop, I couldn’t resist picking up a copy of Béla Bollobás’ brand new The Art of Mathematics: Coffee Time in Memphis — no less than 157 mathematical problems and their solutions, problems (he says) of the kind that would have delighted Littlewood and Erdös.

Here’s the very first in the book. A lion and a Christian in a closed circular Roman arena have equal maximum speeds. Must the Christian in the end be caught by the lion? Here’s another later one: Is an infinite family of nested subsets of a countable set necessarily countable. (These are fun, because it is difficult to shake off the temptation to say that the answer in each case is “yes” when it is “no”.) Hours of amusement to be had here.

But I suspect that there are two kinds of mathematicians, puzzle-setters and puzzle-solvers on the one hand, and (for want of a better word) more philosophically minded mathematics who want to see the deep interconnections between Big Ideas. Erdös vs. Gödel, perhaps. Once upon a time I used to be fairly good at the puzzling; but these days, it’s Gödel for me.

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 …