Wednesday, January 24, 2007

Gentzen, praise and regret

By popular request, I'm continuing an informal lunchtime Mathematical Logic Reading Group with a number of grad students. This term, the plan is to do a 'slow read' of Gentzen's two great papers on the consistency of arithmetic. But we started today with the lecture he wrote between the two papers, 'The concept of infinity in mathematics'. This is short, very accessible, and gives a great sense of the conceptual problems that Gentzen sees as shaping his work. It is also very clearly sets out the headline news about the structure of his (first) consistency proof and about its supposed finitist/constructivist credentials.

The lecture has its shortcomings -- there's a general murkiness about the notion of a 'constructivist' view of infinity (why should a constructivist view of sets in the sense of the paradox-busting idea of a hiearchy in which sets at higher levels are formed from sets already constructed at lower levels go along with a constructivist rejection of excluded middle at the level of classical analysis?). But still it is wonderful, thought-provoking stuff.

I was moved to try editing the piece on Gentzen on Wikipedia in very modest ways (e.g. adding that he was Hilbert's assistant, which you might have thought was a rather central fact about his intellectual trajectory). But twice my efforts were removed. And I wonder if that was because I'd over-written the claim that he was imprisoned after the war "due to his Nazi loyalties" (I'd put something less specific, but more detailed, i.e. the story as told by Szabo in his introduction to the Collected Papers). Is it true about Gentzen having Nazi sympathies? Regrettably it seems so. Discovering this was really rather depressing, as I've belatedly become a great admirer of Gentzen.

Sunday, January 21, 2007


One of my Christmas presents was Renée Fleming's Homage: The Age of the Diva. It is luxurious, wonderful, singing, with some of the arias quite unknown to me (and I'd have thought to many opera fans). A few of Fleming's interpretations are perhaps a bit over the top -- there is a stunningly sustained note near the end of "Vissi d'arte" which is awesome but ... Though it certainly does make the hairs stand up on the back of your neck, so let's not get too purist!

The CD insert booklet has a little about Fleming's great predecessors in this repertoire. One name stood out for me. For my mother has a copy of Kobbé's Complete Opera Book, given her as a girl: the evocative black and white photographs include a publicity photo of the Moravian soprano Maria Jeritza in her costume as Turandot. That picture made a deep impression on me when I was small! Jeritza was one of Puccini's favourite singers; but I've only ever heard short clips of her voice. I feel an internet search coming on ...

Saturday, January 20, 2007

Size matters

One thing that distracted me for a while was getting myself a new laptop, a 17" MacBook Pro. Simply fantastic, of course. Huge and very bright sharp screen -- I got a matte one -- and fast. Oh, very fast. On my previous 1.5 Ghz 15" PowerMac (a thing of beauty too), it took 28 seconds to typeset the Gödel book. Now the same LaTeX processing takes just over four seconds. And no, of course it isn't that I really need all those saved twenty seconds in a day! But I don't have to think any more about the business of refreshing the preview of the printed page -- you just stop noticing the mechanics, so to speak, of using LaTeX.

I did think long and hard about whether to go for broke and get the big screen version. I'm glad I did. The laptop fits in the same bag I had before, and is not significantly heavier, so I don't notice any real difference in the bother of carting it around. And after all, neither size MacBook Pro is the sort of thing you'd want to tote if you were really doing a lot of travelling (mmmm, the rumours of a really small laptop for 2008 are promising). So get the bigger one if you can: having the extra real estate again helps to make e.g. working with LaTeX (with TeXShop and BibDesk windows open, etc.) quite a lot more pleasant.

I've now just got to write another book to justify the expense ...

Friday, January 19, 2007

Spluttering into life again ...

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 ...