Logic’s Lost Genius

It has to be said that Eckart Menzler-Trott’s Logic’s Lost Genius: The Life of Gerhard Gentzen is a strange work.

For those who haven’t seen it, a word about the structure. The main part (pp. 1-283) of this large-format, small-print, book is notionally a biography of Gentzen, but as I’ll note in a moment there are very long digressions. Then there are four appendices. The first (pp. 285-292) is a note by Craig Smorynski (who is also the main translator from the original German version of the book) on an elementary but neat proof in geometry re-discovered by the school-boy Gentzen. Next (pp. 293-343) there is a long essay by Smorynski on Hilbert’s Programme. The final appendix (pp. 369-405) is another essay, almost as long, by Jan von Plato, called “From Hilbert’s Programme to Gentzen’s Programme”. Lastly, rather oddly sandwiched in between those last two essays, Menzler-Trott includes three lectures by Gentzen himself. These are

1. The Concept of Infinity in Mathematics (item #6 in Szabo’s Collected Papers),
2. The Concept of Infinity and the Consistency of Mathematics,
3. The Current Situation in Research in the Foundations of Mathematics (item #7 in Szabo).

1 and 3 are newly retranslated. 2 is just over two sides long (and is little more than a summary of 1).

I’ll comment in later posts on the essays by Smorynski and von Plato. But what about the main biography?

Well, as I said, this really is rather strange. For a start, one very long chapter (pp. 141-232, ‘The Fight over “German Logic” from 1940 to 1945: A Battle between Amateurs’) concerns Nazi attitudes about the “decadence” of mathematics supposedly due to Hilbert. This tells us just a bit about how Gentzen’s work was viewed in some quarters. But most of the discussion is only distantly relevant (pages and pages go by without Gentzen being even mentioned). In fact, however interesting this all will be for those researching on the politicization of academic life under the Nazi regime, it tells us almost nothing about Gentzen’s intellectual development.

And the other chapters, which really are on Gentzen, are oddly written (and I’m not talking about the translation into an English replete with far too many sentences no native speaker would use). Rather the text too often reads like unprocessed working notes, stringing together remarks on intellectual events, or on unrelated family affairs, with excerpts from Gentzen’s letter and reviews. For a particularly staccato example, on p. 94 we read [and yes, these are consecutive mini-paragraphs]:

In December 1937 Gentzen informed at least Paul Bernays that he had carried out his consistency proof in a simpler and more thorough form.

Since December 1937, Gentzen’s sister, Waltraut, and her husband lived in Liegnitz/Niederschlesien (today: Lignice, Poland).

On 3 January 1938 Bernays wrote from Besenrain Str. 30 in Zürich that he had finished §11 of the foundations book for two weeks, but it was not yet typeset: “As soon as the copy is made, I will send it to you.”

In Zentralblatt für Mathematik 17 (1938), p. 242, there appeared: [and Menzler-Trott then reproduces Gentzen’s review of Barkley Rosser’s 1937 JSL paper ‘Gödel theorems for non-constructive logics’.]

So no one can call this a gripping, well-structured, story!

But stylistic complaints (and the very long aside on Nazi attitudes) apart, is the biography at least illuminating? The answer, I’m afraid, is “not very”, at least not if you are looking for an account of Gentzen’s intellectual trajectory. And this is because — unlike Dawson on Gödel or the Fefermans on Tarski — Menzler-Trott doesn’t just engage enough with Gentzen’s logic. For example, if you don’t already know about his work on natural deduction and sequent calculi, you’ll hardly get any sense of what Gentzen was up to here and why it matters. (Ok, Gentzen’s work is going to be addressed more directly in the two long appendices by Smorinski and von Plato: but it does feel as if Menzler-Trott narrative has a large hole at the centre.)

Still, there is a lot of detail about Gentzen’s milieu, about whom he met when, about who influenced him, and whom he was corresponding with. And this scene-setting is interesting enough. Moreover, quite a few letters are quoted, which have some interest — though note that there is little by way of e.g. novel informal exposition here. Hence you won’t get a new understanding of Gentzen’s results from them. But you’ll at least come away with a better sense of the external shape of the context in which his papers were written. You’ll have to wait until the appendices, however, to learn more about the internal dynamic of the ideas there.

I should add, though, that the last main chapter, on Gentzen’s death in prison, contains moving accounts from fellow prisoners: do read this chapter even if you don’t read the rest of the book.