Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics

Perhaps the most turbulent period in modern mathematical history is the interval from the formulation of set theory by Georg Cantor to the delineation of what is now the status quo in mathematical logic, namely the work of Kurt Gödel, which dealt the coup de grâce to Hilbert’s dream of proving mathematics’ consistency through finitist methods. In judo there is a saying about the great judo competitor Masahiko Kimura (often considered to be the greatest of all time): Kimura no mae ni Kimura naku, Kimura no ato ni Kimura nashi. It means, “Before Kimura there was no Kimura, after Kimura there is no Kimura.” To say that before Cantor there was no Cantor is perhaps even more on target, given that his set theory burst on the scene without any warning or foreshadowing (except perhaps in the private meditations of, say, Dedekind), and wrought an unprecedented revolution. But what about Cantors after Cantor? Certainly there are great set theorists one can list, including slayers of dragons that appeared shortly after scholars doing battle for or against Cantor released them from their caves; Paul Cohen comes to mind right away, for example, with his proof of the independence of the continuum hypothesis, done using the method of forcing. But unlike in judo, in mathematics it’s not about man-to-man contests, so the question is ill-posed. It is worth noting, of course, that after Cantor no one else launched a revolution in the area of set theory on a par with the revolution in mathematics that coincided with the subject’s genesis: Cantor no ato ni Cantor nashi.

Cohen’s great result belongs to set theory, to be sure: there are models of set theory (based on the Zermelo-Fraenkel axioms, i.e. including them) with the continuum hypothesis added, as well as models with its negation added. So, the hypothesis is “independent,” as the logicians say. Here we encounter one of the most profound evolutions in mathematics: the wedding of set theory to logic. The early paradoxes (e.g. Russell’s, about the set of all sets that are not members of themselves — the liar’s paradox come back to haunt us after a full-body make-over) unavoidably introduced this element, and presently the focus shifted to what could be said about the objects Cantor discovered, given the nature of the language(s) in which we might try to say these things.

The area of mathematical logic has a pedigree all its own, of course, and we meet, e.g., George Boole and Gottlob Frege along the way. But after Cantor, mathematico-logical questions took on a particular urgency as David Hilbert sought to safeguard Cantor’s paradise from the demonic attacks on the part of his enemies, notoriously starting with Leopold Kronecker, but also including the aforementioned Bertrand Russell. Even Henri Poincaré was skeptical, and Hilbert accordingly waged war: his famous Paris Problems included three which dealt with set theory and logic. The first Paris Problem was nothing less than the continuum hypothesis question, settled by Cohen in the aforementioned manner; the second asked about the consistency or arithmetic; and the tenth asked for an algorithm whereby to decide on the solvability of any Diophantine equation. Yuri Matiyasevich took care of the tenth problem: no such algorithm exists. The fate of the second problem is particularly interesting in light of our present considerations: Gödel showed that arithmetic itself cannot yield a proof of its consistency, but it was Gerhard Gentzen who showed that if the ordinal \(\varepsilon_0\) is well-founded, we get the desired consistency. Again, we have the marvelous interplay between set theory proper (the theory of ordinal numbers) and logic (the notion of well-founded-ness).

So, yes, Gentzen appears in our story as a very major player, notwithstanding his short life (1909–1945). But before we get to that, and to the book under review, one more remark about the repercussions of Cantor’s revolution. After Hilbert’s attempts to restore a certain status quo were altogether thwarted by Gödel, the discipline which we now characterize as mathematical logic matured very quickly into something rather different from what its precursors had presented in the form of such works as Frege’s Begriffschrift: after Gödel the foci began to include questions of consistency, independence, well-foundedness, computability, and so on. And we encounter some familiar figures, such as Alonzo Church, Alan Turing, John von Neumann, the earlier Paul Cohen and Yuri Matiyasevich, and of course Gerhard Gentzen.

Getting, at last, to the present book, by Jan von Plato (than which no better name for a logician can be thought, unless it were “von Aristotle”), it sports a fabulous “Part I,” spanning over 60 pages, in which not only Gentzen’s life is presented with great attention to detail, but a great deal of excellent information is given about the time in which all this revolutionary work took place. Note that Gödel’s first devastating result dates to 1931; at this time Gentzen was in all likelihood taking a course from von Neumann in which the latter broke the cataclysmic news rather dramatically. Here is what von Plato says (p.9), quoting Hempel who was an eyewitness:

I was taking a course with von Neumann [in Berlin] which dealt with Hilbert’s attempt to prove the consistency of classical mathematics … [I]n the middle of the course von Neumann came in one day and announced that he had just received a paper from … Gödel who showed that the objectives which Hilbert had in mind … could not be achieved at all. Von Neumann … devoted the rest of the course to the presentation of Gödel’s results. The finding evoked an enormous excitement …

A monumental understatement! Von Plato continues (p.11):

During the fall of 1931, Gentzen worked on … especially the translation of arithmetic into pure predicate logic … [which] was the way in which Gentzen in his thesis [1933] was able to reduce the consistency of arithmetic without the full induction principle to the cut elimination theorem of pure predicate logic.

The game was afoot:

In the second part of 1935 [Gentzen] was nominated assistant to David Hilbert … [but] was able to concentrate on his own research. Even [earlier] … he had finished his original proof of the consistency of arithmetic, then changed it around the turn of the year 1935–1936 into the well-known proof based on the consistency of transfinite induction [cf. the earlier remark involving the well-foundedness of \(\varepsilon_0\)].

Von Plato goes on to note that after his phenomenal success of 1936 Gentzen was hit with a very serious bout of depression, requiring treatment, but recovered and regrouped so that by 1939 he was able to present his Habilitationsschrift, readying him for a full-fledged academic career. At this point, however, the dark and deepening shadow of Nazism had spread over Europe and World War II erupted in full force. In §3 of Part I, von Plato conveys the ambiguous nature of Gentzen’s position toward the Hitler regime as we learn that Gentzen had joined the SA as early as 1933, but in 1935 the Nazi teachers’ union reported that he had “had contacts with someone in Jerusalem, clearly Abraham Fraenkel, and thereby ‘had shown his loyalty to the Chosen People.’” Additionally Gentzen’s doctoral advisor at Göttingen had been Paul Bernays, who, as a Jew, had fallen victim to the Nazis’ horrific racial laws and accordingly lived exiled from Göttingen in his native Switzerland.

Gentzen met up with Bernays again in Paris in 1937 and, as von Plato says, “it turned out to be the last time Gentzen and Bernays met, but the two continued their correspondence until the war.” And then: “For Gentzen it meant contacts with the expelled Jewish professor … and obvious risk to his [G’s] own conditions in Göttingen. Bernays in turn must have had some understanding for Gentzen’s decisions … Gentzen gives the impression of having been in practice blind towards the Nazi regime …” Although there is a great deal more in the book under review, suffice it for now to mention that Gentzen entered military service in 1939 in Braunschweig (near Göttingen), suffered a nervous breakdown in 1941, “was called to teach at the German University in Prague [p.15]” and, together with remaining German university professors, was put under arrest there by the Russians in May 1945. He died a POW in 1945, of malnutrition.

Now about the cellar mentioned in the book’s title. The answer to this question adds a layer of intrigue and mystery to the story. It turns out that Gentzen was a prolific note-taker in regard to his researches, generally using a shorthand writing system taught to German grade school children in his youth. Here is von Plato again (cf. p.3 ff.):

Gerhard Gentzen died on August 4, 1945, in a prison in Prague. His fellow prisoners were professors at the local German university, and there are accounts of his last days and how he was, rendered weak by lack of food, still pondering over the consistency problem of analysis … [There was allegedly] a mythical suitcase … filled with papers of a near-proof … Nothing was found … in Prague. In Göttingen, instead, there were preliminary studies for published work … yet again, nothing … More than thirty years later, in 1984, two slim folders of stenographic notes by Gentzen surfaced as if by a miracle, … It appears from a note in [one of the folders] that Gentzen had left the papers in a summer place … on the island of Rügen in the Baltic Sea close to his hometown … This took place in the summer of 1944 …

And then:

Save for the pages that were preserved in the two folders, the Gentzen manuscripts got lost in some forgotten cellar, in the attic of the Göttingen mathematics department to be subsequently discarded, and the rest burnt.

Against this backdrop, Jan von Plato set himself the task of seeking out and fleshing out what Gentzen noted down in the skeletons and sketches in the indicated “two slim folders,” so that what we are dealing with in the book under review is, so to speak, high level mathematico-logical archaeology. The second and third parts of von Plato’s book contain his careful analysis of this hidden work by Gentzen, its interplay with and connections to what Gentzen published in his brief career, and a good deal of autonomous scholarship, i.e. a combination of detective work, critical analysis, and the difficult business of doing mathematical logic properly so-called. It’s an amazing achievement on von Plato’s part.

It is unfortunately the case that there is no royal road to mathematical logic, given its particular character, notation, and style: it is very hard to get into it at a non-trivial level without some sort of boot-camp in which formal languages are covered in some detail. It has always been so: just consider the pages of Russell-Whitehead: even when talking about ordinary arithmetic, logicians seem to speak an alien tongue to the rest of us mathematicians.

So, before going at this extremely fascinating book, unless you have a background in mathematical logic already, say, at the level of Enderton’s Mathematical Introduction to Logic plus Shoenfield’s Mathematical Logic, prepare by reading these books or their equivalents (my own experience in mathematical logic goes back to UCLA in the 1970s). It would also be good to have more than passing acquaintance with Gödel’s proof as per, say, the book by Nagel and Newman by that name. Furthermore, given Gentzen’s and, accordingly von Plato’s, concern with intuitionism, it would also be wise to garner some experience in that area, as per, say, Heyting’s Intuitionism, an Introduction. In other words, von Plato’s book is not for the mathematical masses, but only for those initiated into the secrets of mathematical logic, if only its first few circles.

The table of contents of the book, specifically Part III, provides at a glance the themes Gentzen dealt with in the indicated notes and what von Plato expands on to such a dramatic degree. It is indeed tantalizing, and doubly so given that it’s largely about essentially unknown work by the enigmatic, ambiguous, and largely hidden Gerhard Gentzen, aptly characterized as “logic’s lost genius” in another book reviewed here. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.