Volume 33 in the AMS/LMS History of Mathematics series, Logic’s Lost Genius: The Life of Gerhard Gentzen, by Eckart Menzler-Trott, is a very serious contribution to the indicated discipline, dealing with the life and times of the founder of structural proof theory. Gerhard Gentzen died very young, in his middle 30s, starved to death in 1945 in a Czechoslovakian prisoner-of-war camp. The description of his arrest in Prague, his subsequent imprisonment, and then his horrible death, make for quite harrowing reading in this meticulously researched work. But given the attention to detail paid to his subject by Menzler-Trott, it cannot be otherwise, and we are necessarily brought face-to-face with the horrors of the Nazi period in Germany and Western Europe. And Gentzen was right in the middle of it all.
Born in Greifswald, Prussia, into an attorney’s family, Gentzen proceeded swiftly along the path laid out for him in the direction of a professional, specifically an academic, career. Starting at the University of Greifswald (the oldest in Prussia), Gentzen moved to Göttingen at the instigation of Hellmuth Kneser. At Göttingen, he busied himself “fitting out his radio, playing chess, going to the movies and to [the] theatre,” but he also “attended Hilbert’s lectures on Set Theory… [and] learned everything necessary about algebraic numbers, sets, logical entanglements, the paradoxes of Russell and Zermelo, the paradox about the class of all ordinals, Dedekind’s construction of number theory, permissible and impermissible inferences, the logic of tertium non datur , and the basic concepts of mathematical logic.”
Can one imagine a better preparation for a logician in those days when giants walked the earth, especially around Göttingen? But Gentzen presently moved to Munich, and Menzler-Trott presents the following somewhat melodramatic rationale (be that as it may): “Gentzen was an independent thinker, and independent thinkers don’t think much of descriptions they find but haven’t thought through themselves nor of impressions they might receive on the basis of what they heard or read in the thoughts of others. Gentzen remained deeply protestant. His motto could have been: Everything I believe and know comes from within and if it must be, I will write it myself.” Nonetheless, after Munich and a semester in Berlin, Gentzen returned to Göttingen and stayed, starting his research, properly so-called, in logic.
Of course, this last interval of Gentzen’s university days saw the appearance of Gödel’s first papers in logic, causing an unparalleled upheaval in the mathematical world, and in particular in its centre, Hilbert’s Göttingen. Gentzen apparently caught fire: independent of Gödel, Gentzen, crediting Bernays as his collaborator, showed that the consistency of axiomatic number theory reduces to the consistency of intuitionistic arithmetic. He went on to write his PhD dissertation under Hermann Weyl in 1933, on the subject of logical deduction. His thesis work had been done under Bernays, but, following the rubrics in place at German universities, the actual promotion had to be performed by a full professor, to wit, Weyl.
Two days after Gentzen’s promotion to PhD, together with other Jewish academics such as Landau, Neugebauer, Lewy, and Hertz, Bernays lost his license to teach (venia legendi) and deeper and deeper darkness descended on Göttingen and Germany as the Nazi terror was unleashed. Presently, “[o]n 5 November 1933, following the advice of someone yet unidentified, Gentzen joined the (university) SA as Sturmmann (storm trooper) perhaps… not to endanger his university or possible teaching career.” This naturally raises the question of Gentzen’s motives behind an act that, from our historical vantage point, surely requires explanation. And Menzler-Trott attempts to do so: “One can ascertain in Gentzen a certain passive, almost phlegmatic trait in all things which did not concern mathematics. Gentzen even based his career on the advice of Hasse [and] Rohrbach.” (Of course Hasse’s troubles along similar lines are well-known.)
In any event, the last decade of Gentzen’s life saw him first as an academic in Nazi Germany, then as a soldier in the Wehrmacht up to his discharge in 1942 in a “state of nervous exhaustion.” Subsequently, “[a]fter a short recovery in a sanatorium his condition improved quickly and he began his logical research again — commuting between Putbus, Liegnitz and Sigmaringen.” Gentzen’s mother and sister lived in Liegnitz, and his mother’s sisters lived in Sigmaringen, a condition doubtless figuring in critically as regards his recovery. But in the fall of 1943, Rohrbach, who had it in his power to do so, appointed him to a post as an unpaid lectured in Prague. It was on 7 May 1945 that Gentzen was arrested in Prague. He died on 4 August, only three months later.
So much for this paltry sketch of the abundant biographical material found in Logic’s Lost Genius: The Life of Gerhard Gentzen. The mathematical logic is taken care of with equal meticulousness and accordingly the work serves two audiences very well, historians (of mathematics) and logicians (with a historical bent). It is a very serious work and is likely not suited for light reading, particularly given the tragic and lugubrious features of Gentzen’s life and times. But it should be of great value to specialists.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.