BOOK REV I EW

Logic’s Lost GeniusandGentzen’s CentenaryA Dual Review by Jeremy Avigad

Logic’s Lost Genius: The Life of Gerhard GentzenEckart Menzler-TrottAmerican Mathematical Societyand London Mathematical Society, 2007, US$97, xxii+440pagesISBN: 978-0-8218-3550-0Gentzen’s Centenary: The Quest for Consistencyedited by Reinhard Kahle and Michael RathjenSpringer, Cham, x+561 pagesISBN: 978-3-3191-0103-3

Among the mathematical disciplines, logic may have thedubious distinction of being the one that always seemsto belong somewhere else. As a study of the fundamentalprinciples of reasoning, the subject has a philosophi-cal side and touches on psychology, cognitive science,and linguistics. Because understanding the principles ofreasoning is a prerequisite to mechanizing them, logicis also fundamental to a number of branches of com-puter science, including artificial intelligence, automatedreasoning, database theory, and formal verification. Ithas, in addition, given rise to subjects that are fields ofmathematics in their own right, such as model theory, settheory, and computability theory. But even these are blacksheep among the mathematical disciplines, with distinctsubject matter and methods. The situation calls to mindthe words of Georges Simenon, the prolific Belgian authorand creator of the fictional detective Jules Maigret, whotold Life magazine in 1958, “I am at home everywhere,and nowhere. I am never a stranger and I never quitebelong.” If we use his self-assessment to characterizemathematical logic instead, the description is apt.

One of the things that gives the field this peculiarcharacter is its focus on language. Every subject has itsbasic objects of study, and those of logic include terms,

Jeremy Avigad is professor of philosophy and mathematicalsciences at Carnegie Mellon University. His e-mail address isavigad@cmu.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1451

expressions, formulas, axioms, and proofs. Logic was noteven viewed as a part of mathematics until the middleof the nineteenth century, when George Boole’s landmark1854 treatise, The Laws of Thought, took the issue ofviewing propositions as mathematical objects head on.Since the time of Aristotle, mathematics was commonlydescribed as the science of quantity, encompassing botharithmetic, as the study of discrete quantities, and geom-etry, as the study of continuous ones. But Boole observedthat propositions obey algebraic laws similar to thoseobeyed by number systems, thus making room for thestudy of “signs” and their laws within mathematics. Putsimply, we can calculate with propositions just as wecalculate with numbers.

Making sequences of symbols the subject of mathe-matical study was the cornerstone of metamathematics,the program by which David Hilbert hoped to securethe consistency of modern mathematical methods. By

George Gentzen, as pictured on the cover of Logic’sLost Genius: The Life of Gerhard Gentzen, by EckartMenzler-Trott.

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mathematizing things like formulas and proofs, Hilberthoped to prove, using mathematical methods—in fact,using only a secure, “finitistic” body of mathematicalmethods—that no contradiction would arise from thenew forms of reasoning. Hilbert gave a mature pre-sentation of his Beweistheorie, or proof theory, in theearly 1920s. Ironically, the representation of formulasand proofs as mathematical objects is an importantcomponent of Gödel’s incompleteness theorems as well.Published in 1931, these showed that Hilbert’s programcould not succeed, in the following sense: No consistentsystem of mathematical reasoning that is strong enoughto establish basic facts of arithmetic can prove its ownconsistency, let alone that of any larger system thatincludes it.

Nonetheless, proof theory is alive and well today. Itsfocus has expanded from Hilbert’s program, narrowlyconstrued, to a more general study of proofs and theirproperties. For Hilbert, as for Gödel, a proof was a se-quence of formulas, each formula of which is either anaxiom or follows from previous formulas by one of thestipulated rules of inference. Perhaps these qualify asmathematical objects, but they are the kinds of mathe-matical objects that only a logician could love. If thereis one person who should be credited with developing amathematical theory of proof worthy of the name, it isundoubtedly Gerhard Gentzen, whose work is now fun-damental to those parts of mathematics and computerscience that aim to study the notion of proof in rigorousmathematical terms.

Gentzen was born in Greifswald, in the northeasternpart of Germany, in 1909. He began his studies at theUniversity of Greifswald in 1928, but after two semestershe transferred to Göttingen, where he attended Hilbert’slectures on set theory and was introduced to foundationalissues. He spent a semester visiting Munich and anothervisiting Berlin, and then returned to Göttingen in 1931. Helearned about Hilbert’s program and Gödel’s results fromPaul Bernays, whose collaboration with Hilbert ultimately

Gentzen’s workis fundamentalto the study of

proof inrigorous

mathematicalterms.

culminated in the Grund-lagen der Mathematik, thetwo volumes of which werepublished in 1934 and1939.

Hilbert and his studentshad studied classical first-order arithmetic as animportant example of aclassical theory whose con-sistency onewould hope toprove by finitistic methods.In 1930 Arendt Heytingpresented a formal axiom-atization of intuitionisticfirst-order arithmetic in accordance with the principlesof L. E. J. Brouwer. In 1932 Gentzen showed that theformer could be interpreted in the latter, using an explicittranslation that is now known as the double-negationtranslation. This showed, in particular, that the consis-tency of classical arithmetic can be derived from the

consistency of intuitionistic arithmetic using finitisticmeans. Gentzen withdrew his submission, however, whenhe learned that Gödel himself had obtained the sameresult, by essentially the same method. The translationis often referred to as the Gödel-Gentzen translation, inhonor of both.

David Hilbert hoped toestablish a provablyconsistent foundation formathematics.

The reduction of classi-cal arithmetic to intuition-istic arithmetic seems tohave encouraged Gentzento think about the possi-bility of obtaining finitisticconsistency proofs of eitherof these theories (and henceboth). The second incom-pleteness theorem impliesthat such a proof could notbe carried out within thetheories themselves, so thechallenge was to find suit-ably strong principles thatcould be argued to conformto the vague criterion of be-ing finitistic. In order to doso, he needed a notion of deduction that was mathemati-cally clean and robust enough tomake it possible to studyformal derivations in combinatorial terms. This was thesubject of Gentzen’s 1934 dissertation.

In fact, Gentzendeveloped two fundamentally differentproof systems. The first, known as natural deduction, isdesigned to closely model the logical structure of aninformal mathematical argument. Formulas in first-orderlogic are built up from basic components using logicalconnectives such “𝐴 and 𝐵,” “𝐴 or 𝐵,” “if 𝐴, then 𝐵,”and “not 𝐴,” as well as the quantifiers “for every 𝑥,𝐴” and “there exists an 𝑥 such that 𝐴.” For each ofthese constructs, natural deduction provides one or moreintroduction rules, which allow one to establish assertionsof that form. For example, to prove “𝐴 and 𝐵,” you prove𝐴, and you prove 𝐵. To prove “if 𝐴, then 𝐵,” you assume𝐴 and, using that assumption, prove 𝐵. Natural deductionalso provides elimination rules, which are the rules thatenable one to make use of the corresponding assertions.For example, from “𝐴 and 𝐵,” one can conclude 𝐴, andone can conclude 𝐵. The elimination rule for “𝐴 or 𝐵” isthe familiar proof by cases: if you know that 𝐴 or 𝐵 holds,you can establish a consequence 𝐶 by showing that itfollows from each.

Gentzen also designed a system known as the sequentcalculus, with a similar symmetric pairing of rules and,for some purposes, better metamathematical properties.In both cases, Gentzen considered reductions, steps thatcan be used to simplify proofs and avoid unnecessarydetours. TheHauptsatz (main theorem) of his dissertation,now also known as the cut elimination theorem, is aseminal and powerful tool in proof theory. It shows thatappropriate reductions in the sequent calculus can beused to transform any proof into one in a suitable normalform, in which every derived formula is justified, in asense, from the bottom up.

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While Gentzen completed this work, the Nazi party wason the rise, and the political and academic environmentin Germany was deteriorating. Göttingen was particularlyhard hit. Hermann Weyl, who had assumed Hilbert’s chairin 1930, fled to theUnited States in 1933with hiswife, whowas Jewish, and joined the Institute for Advanced Study.Emmy Noether fled similarly to Bryn Mawr. Bernays, whohad become Gentzen’s advisor, was summarily dismissedfrom his post in 1933 because of his Jewish ancestry.

In 1935 Gentzen submitted an article to the Math-ematische Annalen describing a consistency proof forarithmetic. He began by extending the system of naturaldeduction to intuitionistic arithmetic, adding a rule toencapsulate proof by induction. The strategy was to showthat proofs in the system can be reduced to ones innormal form, since from the description of the normalforms, it was then immediate that no such proof couldconclude in a contradiction. A copy of the paper was sentto Weyl, who felt that in “the immediate future it shouldplay the rôle of the standard work on the foundationsof mathematics.” The paper was also discussed by Gödeland Bernays on a boat to New York in 1935, though wedo not know the content of their discussions. Apparentlythe paper met with criticism at the Annalen, and a letterthat Gentzen wrote to Bernays suggests that the problemwas that it was not sufficiently clear that the reductionprocedure he described would always terminate. Gentzenrewrote the argument entirely, adapting it from naturaldeduction to a sequent calculus for classical arithmetic.This time he assigned to each proof an ordinal less thanan ordinal known as 𝜀0 in such a way that with each reduc-tion the associated ordinal decreases. Since, by definition,there is no infinite descending sequence of ordinals, everyreduction sequence necessarily terminates.

The revisedproofwaspublished in theAnnalen in 1936,but Gentzen’s original proof is independently interesting,and one can show that the associated reduction proceduredoes in fact terminate. This version was published inEnglish translation in 1969 and in the original Germanin 1974. Jan von Plato has provided a detailed history ofthese results, as discussed below.

It is perhaps a signof themarginal role that syntaxplaysin most branches of mathematics that Gentzen’s name isgenerally unfamiliar outside proof theory, but he is, today,considered to be a seminal figure in computer science. Nat-ural deduction and the sequent calculus are fundamentalto automated reasoning, where normal form theoremsplay a key role in reducing the space that algorithms haveto search to find an axiomatic derivation. The connec-tions between deduction and computation that Gentzenforeshadowed play an important role in the study of func-tional programming languages, where syntactic typingjudgments are used to help ensure that programs meettheir specifications. His ideas have been generalized tostronger logics, including second-order and higher-orderlogic, as well as modal logics and other special-purposelogics designed for reasoning about computation andcomputational processes. Although Gentzen’s structuralanalyses were really byproducts of his work on the con-sistency problem, they are fundamental to contemporary

Kurt Gödel provedHilbert’s plan hopeless.

proof theory. Sequents, rules,and normal forms are tothat community of researcherswhat functions, operators, andderivatives are to the analyst,and nowwe can think of formaldeduction only in those terms.

The first book under review,Logic’s Lost Genius, is a biog-raphy of Gentzen that wasoriginally published in Ger-man with the title GentzensProblem. The work was thenrevised by the author, EckartMenzler-Trott, and translatedto English by Craig Smoryński and Edward Griffor. Theappendices include a history of Hilbert’s program bySmoryński, translations of three expository lectures de-livered by Gentzen in 1935 and 1936, and a friendly andinformative overview of Gentzen’s work by von Plato.

Readers willenjoy followingthe backstageexchangesbetween

Gentzen andmany importantfigures in earlymathematical

logic.

Menzler-Trott does agood job of conveyingthe intellectual milieuof German foundationalresearch. Readers willenjoy following the back-stage exchanges betweenGentzen and many im-portant figures in earlymathematical logic, in-cluding Weyl, Bernays,Gödel, Heyting, AlonzoChurch, and Wilhelm Ack-ermann. But the greaterand more moving dramais the utter collapse ofthe German intellectualenvironment as a resultof the Nazi rise to power.It is painful to read first-hand reports of German

science being purged of Jewish involvement by Nazi ad-ministrators and Nazi sympathizers within the academiccommunity. AndhowdidGentzen respond?Disgracefully,in fact: in 1933 he voluntarily joined the paramilitary wingof the Nazi party militia, known as the Sturmabteilung,or SA. It is jarring to see Gentzen end letters to hisold mentor in Greifswald, Martin Kneser, with a hearty“heil Hilter!” For those of us who admire Gentzen’s workand contributions to logic, this raises knotty questionsas to the extent to which we can admire someone’sscientific contributions while isolating them from lesscommendable, and even deplorable, aspects of their lives.

But Menzler-Trott’s portrayal of Gentzen is sympa-thetic. The sense we get from the stories of Gentzen’syouth and from the letters he wrote to colleagues isthat of someone bright, affable, creative, and polite, andabove all devoted to his work. There is little to suggestthat Gentzen had any interest in politics; joining the SAcomes across as a pathetic attempt to keep himself in

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good graces with the establishment while maximizing hischances of securing financial support to continue withhis research. In Menzler-Trott’s assessment, Gentzen had“a certain passive, almost phlegmatic trait in all thingswhich did not concern mathematics.”

Indeed, he seems at times to have had no real sense ofwhat was going on around him. In April of 1934 he wrotean upbeat letter to Bernays in which he complained of thedifficulty of obtaining a teaching position and casuallymentioned that he had joined the SA, “as has beenurgently advised from various quarters.” After relatinghis progress on consistency proofs, he asked, cheerfully,“Are you coming again to Göttingen for the summersemester?” It is almost as though he had forgotten thatBernays had been stripped of his license to teach andwas trying to convince himself that nothing had reallychanged.

To be sure, being generally clueless and fixated onresearch is not an excuse for failing to take a standagainst the Nazi atrocities and the indignities that weresuffered by his colleagues. But I suspect that manyreaders of the Notices will find Gentzen’s naiveté andpreoccupation to be disarmingly familiar. We see thesetraits in ourselves and in our colleagues, and Gentzen’sstory challenges us to wonder whether we would havedone better and, indeed, whether there are things wecould be doing better in the present day.

In any case, Gentzen’s attempts to withdraw into hiswork did not succeed, and he did not lead a happyor easy life. In 1939 he was called to military serviceas a radio operator inside Germany. In 1942 he washospitalized in a state of nervous exhaustion and thenreleased from military service. He was then assigned toteach mathematics at the German Charles University inPrague and chose to remain there after the Third Reichfell. As part of a general backlash against Germans inPrague, he was sent to a prison camp, where he died ofmalnutrition in August 1945. He was thirty-five years old.

Logic’s Lost Genius is not easy reading. Its primarypurpose is to serve as a historical record rather than anentertaining narrative, and Menzler-Trott accumulated alitany of names, places, dates, and events throughoutthe course of his prodigious research effort. Wheneverpossible, he lets source documents speak for themselves,and as a result many facts and opinions are conveyedthrough letters, scholarly reviews, firsthand narratives,government reports, and academic assessments. A longchapter details the Nazi transformation of logic andfoundational research from 1940 to 1945, which had thegoal of establishing racial purity and a proper “German”mathematics. Telling this story cannot have been an easytask for Menzler-Trott, who often lets his personal voicerise over the dry assemblage of facts in order to expresshis anger, frustration, and disgust. His admiration andrespect for Gentzen is apparent throughout.

The second item under review, Gentzen’s Centenary,is a very different work. It comprises a collection ofessays that, taken together, provide a broad appraisal

Paul Bernays, Gentzen’sadvisor, was summarilydismissed from his post in1933 because of his Jewishancestry.

of Gentzen’s mathe-matical legacy on theoccasion of the 100-year anniversary of hisbirth. Although a few ofthe articles are targetedat logicians and prooftheorists, most of thearticles are written witha general mathematicalaudience in mind.

The book is di-vided into four parts.The first, titled “Reflec-tions,” offers historicalandphilosophical viewsof Gentzen’s work. Rein-hard Kahle considersthe meaning and im-portance of Gentzen’sconsistency proofs andexplores modern varia-tions on Hilbert’s pro-gram.MichaelDetlefsenargues that Gentzen’s formalist position was less far-reaching than Hilbert’s: whereas Hilbert felt that afinitistic consistency proof was sufficient to groundabstract mathematical methods, Gentzen held that aproper grounding of abstract mathematics would haveto ascribe content to mathematical abstractions as well.Anton Setzer proposes a broader approach to Hilbert’sprogram that combines non-mathematical and philosoph-ical validation of basic principles with metamathematicalstudy.

The second part, titled “Gentzen’s Consistency Proofs,”focuses on those. Wilfried Buchholz provides an analy-sis and presentation of Gentzen’s original consistencyproof using modern terminology and notation. Jan vonPlato relies on archival work to describe the evolution ofGentzen’s consistency proof from his original submissionto the final result, and shows that the original versionwas equally prescient in introducing themes that were tobecome central to proof theory. Whereas Gentzen used aclassical sequent calculus in the final version of the con-sistency proof, the one that used the notation for 𝜀0, DagPrawitz presents a Gentzen-style normalization proof fora formulation of arithmetic in intuitionistic natural de-duction, and Annika Siders presents a similar consistencyproof for a system based on an intuitionistic sequentcalculus. William Tait explains the constructive principlesthat can be used to ground Gentzen’s original consistencyproof, and Michael Rathjen gives a clean mathematicalanalysis of Goodstein’s theorem, an interesting numbertheoretic result which, essentially as a result of Gentzen’sanalysis, can be shown to be independent of first-orderarithmetic.

The third part, titled “Results,” presents technicalresults that round out Gentzen’s work. Sam Buss studiescut elimination procedures in terms of time and spacecomplexity, Fernando Ferreira discusses a consistency

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proof for second-order arithmetic due to Clifford Spector,Herman Ruge Jervell explains how properties of ordinalsare established in first-order arithmetic, and WolframPohlers surveys some of the methods of contemporaryordinal analysis. Only the final part, “Developments,” isaimed primarily at proof theorists. It includes substantialand interesting results by leading researchers in the area.The section includes a paper by Grigori Mints, a belovedand important figure in proof theory, who passed awayjust as the collection was about to go to press. The volumeis movingly and appropriately dedicated to his memory.

Logic’s Lost Genius and Gentzen’s Centenary comple-ment each other well, offering informative overviews ofGentzen’s accomplishments and the intellectual and po-litical environment in which they emerged. Contemporarytextbooks provide sufficient evidence of the importanceof Gentzen’s work to modern logic. But the foundationalbackground duly reminds us that the overarching goal ofresearch in proof theory is a better understanding of the

principles of reasoning and what it means to do mathe-matics. And the dark historical narrative reminds us thatresearch doesn’t take place in a vacuum; even the purestof mathematicians have to contend with the exigenciesof their social, political, and institutional environments,and it is important to face them deliberately rather thanaccidentally.

AcknowledgmentsI am grateful to Paolo Mancosu for advice and correc-tions.

Photo CreditsPhoto of George Gentzen provided courtesy of Eckart

Menzler via Erna Scholz.Photo of David Hilbert provided courtesy of Gerald L.

Alexanderson.Photo of Kurt Gödel is courtesy of the Archives of the

Institute for Advanced Study, Princeton, NJ.Photoof Paul Bernays is courtesyof ETH-BibliothekZürich,

Bildarchiv, photographer unknown, Portr_00025, Pub-lic Domain Mark.

Jeremy Avigad

ABOUT THE AUTHOR

Jeremy Avigad’s research interests include mathematical logic, interac-tive theorem proving, and the his-tory and philosophy of mathematics.

ABOUT THE AUTHOR

Jeremy Avigad’s research interests include mathematical logic, inter-

active theorem proving, and the history and philosophy of mathe-

matics.

Jeremy Avigad

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