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.


eckartmenzler said...

Gentzen by no means had any sympathy to the Nazis.
I wrote his scientific biography (a rewritten version of my german edition, Birkhauser 2001):
Eckart Menzler-Trott:
Logic's Lost Genius - The life of Gerhard Gentzen.
Translated from the German by Craig Smorynski and Edward Griffor.
With appendices and two essays by Craig Smorynski and Jan von Plato.
American Mathematical Society: Providence, Rhode Island (USA)
and the London Mathematical Society: London (U.K.)
Series: History of Mathematics, vol. 33.
ISBN-10: 0-8218-3550-5
ISBN-13: 978-0-8218-3550-0
October 2007. 464 pages. Hardcover. US $ (price not yet fixed).
Here you will find alle three lectures of Gentzen (Szabo has just two).
Why are biographies read but not mentioned in the history of modern mathematics?

Peter Smith said...

Many thanks for that. I'm really pleased to hear that your biography has been translated, and I look forward a lot to reading it (for, regrettably, I don't read German).

Alasdair Urquhart said...

Did Gentzen have "sympathy for the Nazis"?
Well, the facts as recounted in the recent
translation of the Menzler-Trott biography are as follows:

1. Gentzen joined the SA in November 1933,
though he was by no means compelled to do so (p. 52);

2. There are two photographs of Gentzen wearing the party insignia on his lapel, one in Paris as an official representative of German philosophers (p. 54);

3. He joined the Nazi party in 1937 (p. 52);

4. He swore the oath of loyalty to Adolf Hitler in April 1939 as part of his academic appointment (p. 119);

5. He apparently carried out computations relating to the V2 rocket project under a contract from the SS (p. 238).

What conclusions should we draw from this? My own guess is that Gentzen was simply a "good German" who never thought critically about the nature of the regime to which he swore loyalty. In other words, his attitude was both naive and opportunistic.

I'd compare him to Heisenberg in this. I don't think he had "sympathy for the Nazis" in the way that Teichmuller did (a Nazi fanatic who believed totally in Nazi ideals). He was just a "good German", that's all.