BOOKREVIEW Logic’s Lost Genius and Gentzen’s Centenary A Dual Review by Jeremy Avigad expressions, formulas, axioms, and proofs. Logic was not Logic’s Lost Genius: The Life of Gerhard Gentzen even viewed as a part of mathematics until the middle Eckart Menzler-Trott of the nineteenth century, when George Boole’s landmark American Mathematical Society and London Mathematical Society, 2007, US$97, xxii+440 1854 treatise, The Laws of Thought, took the issue of pages viewing propositions as mathematical objects head on. ISBN: 978-0-8218-3550-0 Since the time of Aristotle, mathematics was commonly described as the science of quantity, encompassing both Gentzen’s Centenary: The Quest for Consistency arithmetic, as the study of discrete quantities, and geom- edited by Reinhard Kahle and Michael Rathjen etry, as the study of continuous ones. But Boole observed Springer, Cham, x+561 pages that propositions obey algebraic laws similar to those ISBN: 978-3-3191-0103-3 obeyed by number systems, thus making room for the study of “signs” and their laws within mathematics. Put Among the mathematical disciplines, logic may have the simply, we can calculate with propositions just as we dubious distinction of being the one that always seems calculate with numbers. to belong somewhere else. As a study of the fundamental Making sequences of symbols the subject of mathe- principles of reasoning, the subject has a philosophi- matical study was the cornerstone of metamathematics, cal side and touches on psychology, cognitive science, the program by which David Hilbert hoped to secure and linguistics. Because understanding the principles of the consistency of modern mathematical methods. By reasoning is a prerequisite to mechanizing them, logic is also fundamental to a number of branches of com- puter science, including artificial intelligence, automated reasoning, database theory, and formal verification. It has, in addition, given rise to subjects that are fields of mathematics in their own right, such as model theory, set theory, and computability theory. But even these are black sheep among the mathematical disciplines, with distinct subject matter and methods. The situation calls to mind the words of Georges Simenon, the prolific Belgian author and creator of the fictional detective Jules Maigret, who told Life magazine in 1958, “I am at home everywhere, and nowhere. I am never a stranger and I never quite belong.” If we use his self-assessment to characterize mathematical logic instead, the description is apt. One of the things that gives the field this peculiar character is its focus on language. Every subject has its basic objects of study, and those of logic include terms, Jeremy Avigad is professor of philosophy and mathematical sciences at Carnegie Mellon University. His e-mail address is [email protected]. George Gentzen, as pictured on the cover of Logic’s For permission to reprint this article, please contact: Lost Genius: The Life of Gerhard Gentzen, by Eckart [email protected]. Menzler-Trott. DOI: http://dx.doi.org/10.1090/noti1451 1288 Notices of the AMS Volume 63, Number 11 mathematizing things like formulas and proofs, Hilbert consistency of intuitionistic arithmetic using finitistic hoped to prove, using mathematical methods—in fact, means. Gentzen withdrew his submission, however, when using only a secure, “finitistic” body of mathematical he learned that Gödel himself had obtained the same methods—that no contradiction would arise from the result, by essentially the same method. The translation new forms of reasoning. Hilbert gave a mature pre- is often referred to as the Gödel-Gentzen translation, in sentation of his Beweistheorie, or proof theory, in the honor of both. early 1920s. Ironically, the representation of formulas The reduction of classi- and proofs as mathematical objects is an important cal arithmetic to intuition- component of Gödel’s incompleteness theorems as well. istic arithmetic seems to Published in 1931, these showed that Hilbert’s program have encouraged Gentzen could not succeed, in the following sense: No consistent to think about the possi- system of mathematical reasoning that is strong enough bility of obtaining finitistic to establish basic facts of arithmetic can prove its own consistency proofs of either consistency, let alone that of any larger system that of these theories (and hence includes it. both). The second incom- Nonetheless, proof theory is alive and well today. Its pleteness theorem implies focus has expanded from Hilbert’s program, narrowly that such a proof could not construed, to a more general study of proofs and their be carried out within the properties. For Hilbert, as for Gödel, a proof was a se- theories themselves, so the quence of formulas, each formula of which is either an challenge was to find suit- David Hilbert hoped to axiom or follows from previous formulas by one of the ably strong principles that establish a provably stipulated rules of inference. Perhaps these qualify as could be argued to conform consistent foundation for mathematical objects, but they are the kinds of mathe- to the vague criterion of be- mathematics. matical objects that only a logician could love. If there ing finitistic. In order to do is one person who should be credited with developing a so, he needed a notion of deduction that was mathemati- mathematical theory of proof worthy of the name, it is cally clean and robust enough to make it possible to study undoubtedly Gerhard Gentzen, whose work is now fun- formal derivations in combinatorial terms. This was the damental to those parts of mathematics and computer subject of Gentzen’s 1934 dissertation. science that aim to study the notion of proof in rigorous In fact, Gentzen developed two fundamentally different mathematical terms. proof systems. The first, known as natural deduction, is Gentzen was born in Greifswald, in the northeastern designed to closely model the logical structure of an part of Germany, in 1909. He began his studies at the informal mathematical argument. Formulas in first-order University of Greifswald in 1928, but after two semesters logic are built up from basic components using logical he transferred to Göttingen, where he attended Hilbert’s connectives such “퐴 and 퐵,” “퐴 or 퐵,” “if 퐴, then 퐵,” lectures on set theory and was introduced to foundational and “not 퐴,” as well as the quantifiers “for every 푥, issues. He spent a semester visiting Munich and another 퐴” and “there exists an 푥 such that 퐴.” For each of visiting Berlin, and then returned to Göttingen in 1931. He these constructs, natural deduction provides one or more learned about Hilbert’s program and Gödel’s results from introduction rules, which allow one to establish assertions Paul Bernays, whose collaboration with Hilbert ultimately of that form. For example, to prove “퐴 and 퐵,” you prove culminated in the Grund- 퐴, and you prove 퐵. To prove “if 퐴, then 퐵,” you assume lagen der Mathematik, the 퐴 and, using that assumption, prove 퐵. Natural deduction two volumes of which were Gentzen’s work also provides elimination rules, which are the rules that published in 1934 and enable one to make use of the corresponding assertions. 1939. is fundamental For example, from “퐴 and 퐵,” one can conclude 퐴, and Hilbert and his students to the study of one can conclude 퐵. The elimination rule for “퐴 or 퐵” is had studied classical first- the familiar proof by cases: if you know that 퐴 or 퐵 holds, order arithmetic as an proof in you can establish a consequence 퐶 by showing that it important example of a follows from each. classical theory whose con- rigorous Gentzen also designed a system known as the sequent sistency one would hope to calculus, with a similar symmetric pairing of rules and, prove by finitistic methods. mathematical for some purposes, better metamathematical properties. In 1930 Arendt Heyting In both cases, Gentzen considered reductions, steps that presented a formal axiom- terms. can be used to simplify proofs and avoid unnecessary atization of intuitionistic detours. The Hauptsatz (main theorem) of his dissertation, first-order arithmetic in accordance with the principles now also known as the cut elimination theorem, is a of L. E. J. Brouwer. In 1932 Gentzen showed that the seminal and powerful tool in proof theory. It shows that former could be interpreted in the latter, using an explicit appropriate reductions in the sequent calculus can be translation that is now known as the double-negation used to transform any proof into one in a suitable normal translation. This showed, in particular, that the consis- form, in which every derived formula is justified, in a tency of classical arithmetic can be derived from the sense, from the bottom up. December 2016 Notices of the AMS 1289 While Gentzen completed this work, the Nazi party was proof theory. Sequents, rules, on the rise, and the political and academic environment and normal forms are to in Germany was deteriorating. Göttingen was particularly that community of researchers hard hit. Hermann Weyl, who had assumed Hilbert’s chair what functions, operators, and in 1930, fled to the United States in 1933 with his wife, who derivatives are to the analyst, was Jewish, and joined the Institute for Advanced Study. and now we can think of formal Emmy Noether fled similarly to Bryn Mawr. Bernays, who deduction only in those terms. had become Gentzen’s advisor, was summarily dismissed The first book under review, from his post in 1933 because of his Jewish ancestry. Logic’s Lost Genius, is a biog- In 1935 Gentzen submitted an article to the Math- raphy of Gentzen that was ematische Annalen describing a consistency proof for originally published in Ger- arithmetic. He began by extending the system of natural man with the title Gentzens deduction to intuitionistic arithmetic, adding a rule to Problem.