Johan van Benthem, Dick de Jongh In Memoriam Anne Sjerp Troelstra NAW 5/20 nr. 3 september 2019 225

Johan van Benthem Dick de Jongh ILLC ILLC Universiteit van Amsterdam Universiteit van Amsterdam [email protected] [email protected]

In Memoriam Anne Sjerp Troelstra (1939–2019) Creating order in a vast and diverse area

On 7 March 2019 Anne Sjerp Troelstra passed away at the age of 79. Troelstra was emeritus the academic year 1966–1967 there, to- professor at the Institute for Logic, Language and Computation of the University of Am- gether with Olga whom he had married the sterdam. He dedicated much of his career to intuitionistic mathematics and wrote several year before. In this period, Anne sharpened influential books in this area. His former colleagues Johan van Benthem and Dick de Jongh and modified Kreisel’s ideas on choice se- look back on his life and work. quences, the basic notion underlying the real number continuum in an intuitionistic Anne Troelstra was born on 10 August 1939 istic mathematics, a concept that was to perspective. Working together they creat- in Maartensdijk. After obtaining his gym- play an important part in his research in ed formalizations of analysis resulting in a nasium beta diploma at the Lorentz Lyce- the years to come, in many different forms. large article where the typically intuitionis- um in Eindhoven, in 1957, he enrolled as tic concept of a lawless sequence of num- a student of mathematics at the University Stanford bers that successfully evades description of Amsterdam — where eventually his inter- Anne then obtained a scholarship to Stan- by any fixed law, reached its final form. ests converged on intuitionism with Arend ford to visit Georg Kreisel, a leading scholar Though the latter notion is very charac- Heyting as his teacher. Students he was in intuitionism and proof theory, and spent teristic for intuitionistic mathematics as close with include Olga Bakker and E. W. based on free creation of objects, lawless Beth’s students Dick de Jongh and Hans sequences are one of the rare concepts Kamp. With Dick de Jongh, he wrote a pi- in intuitionism not already introduced by oneering paper on intuitionistic proposi- Brouwer himself. tional logic, published only in 1966, that contained the first definition of the central Constructivism notion of a p-morphism between relational In August 1968, Anne played a central role frames, widely used today in intuitionistic in the famous Buffalo Conference on In- and modal logics, as well as the simplest tuitionism and Proof Theory, a meeting of form of the duality between Heyting alge- all important logicians of the day with an bras and relational frames. After obtaining interest in constructivism. He gave a series his master’s degree in 1964, Anne at once of ten lectures there, which turned into his became an assistant professor, according first book, published in 1969, that con- to a custom of the time. It took him just tained the core of his seminal ideas on in- two years from there on to finish his dis- tuitionistic formal systems and their meta- sertation, supervised by Heyting. Besides mathematical investigation. Back home in intuitionism, a main interest of Heyting 1968, he became a lector (associate profes- was geometry and perhaps not accidental- sor) in 1968, and a full professor in 1970. ly, Anne’s PhD thesis was a study of intu- Further early recognition was to follow. In itionistic topology. This subject made him 1976, he became a member of the Dutch aware of the role of continuity in intuition- Anne Troelstra Royal Academy of Sciences. As a full pro- 226 NAW 5/20 nr. 3 september 2019 In Memoriam Anne Sjerp Troelstra Johan van Benthem, Dick de Jongh

mulated within number theory. Church’s Thesis says in this format that, for any true intuitionistic quantifier combination 67xyBx(,y) (the typical sort of statement where constructive mathematics demands a procedure to produce the y’s from the x’s), there exists a recursive function f such that 6xB(,xf()x ). From an informal thesis, Church’s thesis has now become a precise axiom that can be true or false in intuition- istic mathematics. In the Extended Church Thesis the universal quantifier is relativ- ized to antecedents A that have to be spe- cial ‘almost negative formulas’, untouched by realizability: If 67xA((xy)(" Bx,)y ), then 6xA((xB)(" xf,(x))) for some re- cursive function f. This is a typical form of careful metamathematical attention to syntactic fine-structure of mathematical as- sertions that has become standard in the subsequent literature. Photo: School en Universiteit School Photo: Troelstra’s result was the first of its kind, Anne Troelstra in 1985 and it is widely seen as making the force and meaning of realizability clear within fessor, Anne was the successor of Arend and model theory. Perhaps the most prom- the context of intuitionism. The influence Heyting, himself a successor to L. E. J. Brou- inent among the many results obtained by of this work shows in a broad literature wer. Each was in his time the principal con- Anne in this style concern ‘realizability’, analyzing other important notions of real- structivist in the world. a mathematical analysis of constructive izability with similarly structured character- truth introduced by Stephen Kleene in the izations. This study of realizability later de- Realizability 1940s. Its purpose was to explain what is veloped broader mathematical connections The metamathematics of intuitionistic sys- ‘constructive’ about constructive mathe- with category theory and topos theory. An tems was a chaotic jumble of results when matics in terms of recursive functions on interest in this topic remained with Anne Anne entered it. Here he showed his great- natural numbers, then and now a standard for life. In 1998, a chapter on realizabili- est strength: creating order in a vast and paradigm for effective computability. Long ty by his hand came out in the Handbook diverse area. In 1973, the rigorous order viewed as an outside look at intuitionism of Proof Theory. Characteristically, Anne’s was there in his book Metamathematical from the standpoint of classical logic, Anne text had been finished a few years before, Investigation of Intuitionistic Arithmetic took realizability to the heartland of intu- faithfully meeting his deadline, but delays and Analysis. Especially striking are the itionism. Anne’s studies of the topic had by other authors kept him updating, some- clarification of the properties of different already resulted in a long article in the pro- what grumblingly, with all new results in models for constructive mathematics, sev- ceedings of the Second Scandinavian Logic the area. What he published, on this and in eral types of realizability, and various func- Symposium of 1971, and they were now fact on any topic, had to be the complete tional interpretations. The last chapters on expanded in Springer 344. state of the art. special topics were written by Jeff Zucker, Troelstra’s celebrated theorem character- Craig Smorynski and Bill Howard, but the izing realizability states that the formulas Choice sequences lion’s share had been written by Anne, ed- that are provably realizable in Heyting Arith- A next landmark was Anne’s book Choice itor and architect of the whole. The book, metic (intuitionistic natural number theory, Sequences from 1977, exploring the heart- known in the community as ‘Springer 344’, arguably the most basic formal system of land of intuitionistic analysis in mathemat- still functions as a landmark for research- intuitionistic mathematics) are exactly the ical depth and with conceptual finesse. A ers in the subject. ones provable from Heyting’s Arithmetic principal result from this book is Troels- The metamathematical style of investi- plus the so-called Extended Church Thesis. tra’s Elimination Theorem, which says, es- gation in this work represents a significant This result displays some typical features sentially, that any statement quantifying broadening of earlier research. In tandem of the metamathematical style of investiga- over lawless sequences in formalized in- with developing branches of intuitionistic tion. First, the equivalence is formulated in tuitionistic theories is provably equivalent mathematics such as constructive arithme- a proof-theoretic setting. Secondly, infor- to a statement that is only about lawlike tic, analysis, topology or algebra, one now mal claims such as Church’s Thesis saying sequences and objects. This is reminis- also studies the resulting formal theories that every intuitively computable function cent of well-known elimination results in as mathematical objects, using the tools of can be computed by a recursive function, mathematics concerning, say, the reduction logic such as proof theory, recursion theory, or equivalently a Turing machine, are for- from numbers to sets, though the proof Johan van Benthem, Dick de Jongh In Memoriam Anne Sjerp Troelstra NAW 5/20 nr. 3 september 2019 227

in Troelstra’s case is more complex. Lat- proved formats for the then still only par- Natural history er, he went on to prove similar theorems tially understood new paradigm of linear Anne retired in 2000, but not to rest. All his for other types of choice sequences. The logic merging proof theory and computa- life, he had a deep interest in natural history Elimination Theorem clarifies the position tion. Anne also contributed the majority and a wide knowledge of the plants of the of lawless and choice sequences in import- of the chapters in a book with Helmut Netherlands and abroad. His annual linocuts ant formal systems, but it does not make Schwichtenberg called Basic Proof Theory of plants discovered on his travels in Europe the concept of a lawless sequence lose its (1996) that is still a standard resource. were famous. To those on a walk with Anne, value: lawless and other choice sequences listening to him turned what looked like an remain crucial to guarantee a really intu- Students ordinary city street to the untrained eye into itionistic analysis. Troelstra’s result rather An important part of Anne’s life were his a rich landscape of flora, history, and natural means that these concepts can be based PhD students, of whom he supervised wonders. This very year 2019, an article by on a theory that does not contradict stan- seventeen, and with many of whom he him will appear on new species of blackber- dard mathematics. How the concept of maintained a close relationship. His first ries, his special interest. Anne also made a choice sequence remains crucial is shown PhD student Daniel Leivant finished a the- further name for himself as an author on very clearly by Anne in an extensive arti- sis in 1975 on the metamathematics of natural history travel narratives, chronicling cle in 1983 in the Journal of Philosophical intuitionistic arithmetic, and later made the exotic characters and adventures of the Logic. He shows how lawless sequences his career in computer science. Initially a past denied to the average academic of to- and other choice sequences arise by infor- scarce commodity, in the 1980s, the num- day. Tijgers op de Ararat from 2003 is a typi- mal, yet rigorous concept analysis (often ber of PhD students increased, and Anne’s cal sample of this mix of information and hu- called ‘informal rigor’), the main source of students wrote on a broad variety of top- man interest. Characteristically, he managed knowledge in intuitionistic mathematics ics, such as intuitionistic metamathemat- to create order even in this new field. His in general and about choice sequences ics, combinatory algebra, category theory, major Bibliography of Natural History Travel in particular. Martin-Löf type theory, bounded arithme- Narratives was published in 2016. tic, linear logic, and provability logic. Many History of intuitionism and proof theory of these topics reflected the introduction A very special person Other topics pursued in depth by Anne by Anne, often in a close collaboration with Anne will be missed in the first place be- from the 1970s onward are the history of Dirk van Dalen in Utrecht, of new topics cause he will no longer be there to tell intuitionism and its philosophical basis. on the Dutch scene. These students then us what he thinks about a question that Over time, all this work, consolidated in carried the torch further by themselves. For you may have about constructivism. You his monographs and articles, fed into the instance, Ieke Moerdijk became an inter- always knew that you would get a com- magisterial Constructivism in Mathematics, national leader at the interface of topos pletely honest answer from somebody in two volumes co-authored with Dirk van theory, logic and category theory gen- who knew all the issues and had already Dalen, published in 1988, which remains erally, while Jaap van Oosten became a thought much further than you. But it is the standard text on constructivism right worldwide authority on realizability. In the just as much the personal qualities that until today. Netherlands alone, four of Anne’s students will be missed. Anne was a very special, Moving beyond intuitionism proper have become full professors, in mathemat- and to some, occasionally intimidating over the years, Anne broadened his scope ics, computer science, AI and philosophy. person: penetrating, honest, critical, iron- to proof theory in general and wrote two But Anne was a dedicated teacher at all ic, sharp at times, but always open to ar- more books that again created new order academic levels, whose precision, clarity guments and unfailingly supportive of his in developing fields. In 1992, his textbook and scholarship influenced generations of students and colleagues. He will be deeply Lectures on Linear Logic proposed im- students in Amsterdam. missed by all. s

Anne Troelstra in 1988 with Boris Kushner from Moscow (later Pittsburgh) Anne Troelstra in 1998 with Justus Diller from Münster