# Anne Troelstra (1939-2019)

Sadly, we have to inform you that Anne Troelstra, after a short illness, suddenly died on March 7th, aged 79. A world-renowned eminent researcher, a supportive colleague, a teacher who trained many students to become important scholars, is no longer with us. Beginning in 1957 as a mathematics student, he remained at the University of Amsterdam all his life, except for a number of visiting professorships. He rose quickly to take his place as the successor of Brouwer and Heyting. As a full professor from 1970, he was the universally recognized authority on intuitionism and constructivism in general, leaving behind a number of books that will remain landmarks for many years to come. Retiring in 2000, he made a further name for himself as an author on natural history travel narratives. Anne was regularly seen at the ILLC until this year. His impressive personality, always intensely occupied with his present interests, will be greatly missed.

Anne Sjerp Troelstra (August 10, 1939–March 7, 2019)

Anne Troelstra was born on August 10th, 1939, in Maartensdijk. In 1957, he enrolled as a student of mathematics at the University of Amsterdam – and eventually his interests converged on intuitionism with Arend Heyting as his advisor. Students he was close to include Olga Bakker and E.W. Beth’s students Dick de Jongh and Hans Kamp. With Dick de Jongh, he even wrote a pioneering paper on intuitionistic propositional logic, published in 1966, that contained the first definition of the central notion of a p-morphism, as well as the simplest form of the duality between Heyting algebras and relational frames. After obtaining his master’s degree in 1964, Anne at once became an assistant professor, according to a custom of the time. It took him just two years from there to finish his dissertation, supervised by Heyting. Besides intuitionism, a main interest of Heyting was geometry and perhaps not accidentally Anne’s PhD thesis was a study of intuitionistic topology. This subject made him aware of the role of continuity in intuitionistic mathematics, a concept that was to play an important part in his research in the years to come, in many different forms.

Anne then obtained a scholarship to Stanford to visit Georg Kreisel, and spent the academic year 1966-7 there, with Olga whom he had married the year before. Anne sharpened and modified Kreisel’s ideas on choice sequences, and together they created formalizations of analysis resulting in a large article where the typically intuitionistic concept of a lawless sequence of numbers that successfully evades description by any fixed law, reached its final form. In August 1968, Anne played a central role in the famous Buffalo Conference on Intuitionism and Proof Theory, a meeting of all important logicians of the day with an interest in constructivism. He gave a series of ten lectures there, which turned into his first book, published in 1969, that contained the core of his seminal ideas on intuitionistic formal systems and their meta-mathematical investigation. Back home in 1968, he became a lector (associate professor) in 1968, and a full professor in 1970. Further early recognition was to follow. In 1976, he became a member of the Dutch Royal Academy of Sciences.

The meta-mathematics of intuitionistic systems was a chaotic jumble of results when Anne entered it. Here he showed his greatest strength: creating order in a vast and diverse area. In 1973, the order was there in his book Metamathe-matical Investigation of Intuitionistic Arithmetic and Analysis. Especially striking are the clarification of the properties of different models, various types of realizability, and functional interpretations. The last chapters on special topics were written by Jeff Zucker, Craig Smorynski, and Bill Howard, but the lion’s share had been written by Anne, editor and architect of the whole. This book, known in the community as ‘Springer 344’ still functions as a landmark for serious researchers in the subject. Over time, this work developed into the much larger Constructivism in Mathematics, in two volumes co-authored with Dirk van Dalen, published in 1988, the standard text on constructivism right until today.

Of course, Anne also published in depth on special topics. A central notion in the study of intuitionistic formal systems is realizability, introduced by Stephen Kleene in the 1940’s. Anne’s thorough studies of the subject resulted in a long article in the proceedings of the Second Scandinavian Logic Symposium of 1971. The interest remained with him for life. In 1998, a chapter on realizability by his hand came out in the Handbook of Proof Theory. Characteristically, Anne’s text had been finished a few years before, faithfully meeting his deadline, but delays by other authors kept him updating, somewhat grumblingly, with all new results in the area. What he published had to be the complete state of the art.

Other topics pursued in depth by Anne, through the 70s, 80s and 90s are the history of intuitionism and the philosophical basis of the theory of choice sequences. In his important 1977 book Choice Sequences, he proved lawless sequences to be essentially just a figure of speech by an elimination theorem, showing how statements about lawless sequences can be expressed in a theory containing only lawlike sequences. But he did stress that the notion of lawless sequence still serves a purpose as a clear notion derived by informal rigor.

Moving beyond intuitionism proper over the years, Anne broadened his scope to proof theory in general and wrote two more books which again created new order in diverse fields. In 1992, a textbook Lectures on Linear Logic came out proposing improved formats for a then still only partially understood new paradigm. He contributed the majority of the chapters in a book with Helmut Schwichtenberg called Basic Proof Theory, 1996, that is still a standard resource.

An important part of Anne’s life were his PhD students, of whom he supervised 17, and with many of whom he maintained a close relationship. His first PhD student Daniel Leivant finished a thesis in 1975 on the meta-mathematics of intuitionistic arithmetic, and later made his career in computer science. Initially a scarce commodity, in the 1980s, the number of PhD students increased, and Anne’s students wrote on a broad variety of topics, such as intuitionistic meta-mathematics, combinatory algebra, category theory, Martin-Löf type theory, bounded arithmetic, linear logic, and provability logic. Many of these topics reflected the introduction by Anne, often in a close collaboration with Dirk van Dalen in Utrecht, of new topics on the Dutch scene. These students then carried the torch further by themselves. For instance, Ieke Moerdijk became an international leader at the interface in topos theory and logic and category theory generally, while Jaap van Oosten became a worldwide authority on realizability. In the Netherlands alone, four of Anne’s students have become full professors, in mathematics, computer science, AI and philosophy. But Anne was an dedicated teacher at all academic levels, whose precision, clarity and scholarship influenced generations of students in Amsterdam.

Anne retired in 2000, but not to rest. All his life, he had a deep interest in natural history and a wide knowledge of the plants of the Netherlands and abroad. His annual linocuts of plants discovered on his travels in Europe were famous. To those on a walk while listening to him, what looked like an ordinary city street to the untrained eye would turn into a rich landscape of flora, history, and natural wonders. This very year 2019, an article by him will appear on new species of blackberries, his special interest. Anne also made a further name for himself as an author on natural history travel narratives, chronicling the exotic characters and adventures of the past denied to the average academic of today. His major Bibliography of Natural History Narratives was published in 2016.

Anne will be missed in the first place because he will no longer be there to tell us what he thinks about a question that you may have about constructivism. You always knew that you would get a completely honest answer from somebody who knew all the issues and had already thought much further than you. But it is just as much the personal qualities that will be missed. Anne was a very special, and to some, occasionally intimidating person: penetrating, honest, critical, ironic, sharp at times, but always open to arguments and unfailingly supportive of his students and colleagues. He will be deeply missed by all.

Our thoughts go out to his wife Olga and to his daughters Willemien and Ine.

*Johan van Benthem*

*Dick de Jongh*

Prof. Troelstra's archive is available at the the *Noord-Hollands Archief.* The index of the archive is filed as X-2003-01 in ILLC's Technical Notes Series. A list of publications and his CV are available here and here.

When Anne and Olga came to California in 1966 a lifelong friendship began, eventually bringing them and their teenage children to visit Yiannis and me and our teenage children in Greece. We admired Willemien and Ine's beautiful travel journals, learned from Olga the restorative power of tea with bread and good Dutch cheese, and were amazed by the family's botanical erudition. Anne taught me to recognize Phlomis fruticosa, Malva sylvestris and other common Greek wildflowers. In later years he and Olga visited us for botanical adventures and good fellowship, and we visited them in Muiderberg, where Anne guided us on forest walks and longer excursions of which vivid memories remain. We exchanged books and handcraft, cards and photos, looking forward to our next meeting.

At conferences, and in papers, books and correspondence, I have learned more about intuitionistic logic, arithmetic and analysis from Anne than from anyone else. He was a firm but kind critic, encouraging even modest advances by others, always giving fair credit. He took seriously his position as Brouwer's and Heyting's successor at the University of Amsterdam and once confessed to feeling exhausted by the responsibility to understand every new development concerning intuitionism. At retirement he donated much of his own mathematical library to the University of Athens, where he was a visiting lecturer for the graduate program in logic. His generous spirit has enriched innumerable lives. We miss Anne's presence but his essence lives on in books and papers and letters and in all our memories.

*Joan Rand Moschovakis*

**Anne Troelstra**

Around 1977, I was a master's student under Dirk van Dalen and Henk Barendregt in Utrecht. We students attended the `baby seminar' under supervision of Jan Willem Klop. That year's baby seminar was concerned with a book that was simply called 344. We students found the book somewhat hard to study, since it gave many details but little motivation. Nevertheless, we learned a lot from the book. The book was written by Anne Troelstra.

From 344, I learned, among many other things, about Heyting Arithmetic and Kleene Realizability. In my career, I returned again and again to Anne's presentation and I used the things I learned in my work. For example, Anne characterized the theory of Kleene realizability as Heyting Arithmetic plus an extended version of Church's Thesis. My paper on the Completeness Principle can be viewed as an answer to the question: what happens if we do the same thing Anne did when Kleene Realizability is replaced by the provability translation?

I met Anne for the first time in 1977/78 when he visited Dirk van Dalen at the Mathematical Institute in Utrecht (now the Freudenthal Building). Even if Anne looked, at first sight, somewhat stern, he turned out to be friendly and accessible. He was always prepared to answer questions and to discuss problems. At a certain point, I started calling Anne `Anne'. Dirk van Dalen noted this and asked `Did professor Troelstra specifically invite you to call him `Anne'?' No, but it felt appropriate. After that I avoided, for some time, any form of address for Anne and then returned to `Anne'. There was never any sign that Anne disapproved of `Anne'.

Anne did not care much for worldly fame. For, him the main motivation was understanding a thing in all possible detail. This was a defining property of his personality. We see it both in his work in logic and in his work on botanical travel stories.

I did not see Anne very often in the last years, but, now and then, we attended the same meeting. It was always good seeing him and talking to him. It is hard to understand that now this is not possible anymore.

*Albert Visser*

In my formative years around 1975 in the logic school of Dirk van Dalen and Henk Barendregt, at the Mathematical Institute, Boedapestlaan 6 in Utrecht, I was happily sharing a room with Albert Visser, adjacent to the rooms of Dirk and Henk and at the

same fourth floor corridor as the room of our friend Jeff Zucker. We had a well-attended Intercity Colloquium, bi-weekly taking place alternatingly at the institutes in Amsterdam and Utrecht. Anne Troelstra, Dick de Jongh, Roel de Vrijer were regular participants,

together with visitors in those years such as Walter van Stigt, David Isles and Craig Smorynski, and many more short or long term visiting friends and colleagues from abroad. As a junior Ph.D-student under supervision of Dirk and Henk my (too) difficult assignment was to assist a group of newcoming students, including Albert, in digesting and mastering various chapters of Anne’s famous Springer LNM 344, shortly known as ‘344’.

The encounter with 344 did not influence me as deeply as would have been desirable, but in my subsequent development to a theoretical computer scientist I did greatly profit from the treatment in 344 of important recursion theory theorems such as the ones of Myhill-Shepherdson and Kreisel-Lacombe-Schoenfield. They were instrumental in a later paper (1982) by Jan Bergstra and me about parametrized data types, continuing Jan’s well-known series of papers together with John Tucker from Swansea, analyzing the theory of abstract data types.

I used to drive our Utrecht group, Dirk, Henk, Jeff, Albert in my small car to the Amsterdam sessions of the Intercity seminar, and dually, drive the Amsterdam participants, Anne, Dick, Roel after the Utrecht sessions back to Utrecht Central Station. I remember Anne’s stimulating comments, when I had made good progress with my PhD-thesis, about term rewriting systems and lambda calculus. I remember Anne as a true scholar and a gentleman. He was an example, a role model for junior logicians and computer scientists. At the event of his emeritate, I thanked him for his continual inspiration. I remember Anne’s facial expression, somewhat amused and ironic.

Later, the past fifteen years, I encountered Anne many times in the meetings of the Section Mathematics of the Academy. His life and work will be for many of us, in logic and computer science, an ever-lasting inspiration and enrichment.

*Jan Willem Klop*

I first saw Anne in action when he taught "Analysis II", the major stumbling block for beginning mathematics students. It was whispered in our group that he was very clever, having become an Associate Professor at a very early age, but it would be saying too much that the field of Analysis came sparklingly alive. What did come alive was my image of Anne, he looked very much then like he looked all through his life: serious, sharp, erudite, and with his technical subjects at his fingertips. Later on, I took his all courses on intuitionism, even though I chose the path of model theory eventually, having tasted the sinful delights of getting mathematical results without constructivist proofs.

Anne, characteristically, never held my choice against me,
nor did he object to my broader activism in philosophy, linguistics, computation, and even further areas over time. Anne was clearly a mathematical logician from the heartland, but he saw pursuing wider frontiers of logic as good for the whole field, rather than as a threat to established rank and dogma.

I owe a lot to Anne. He helped me at a crucial stage of my dissertation project in 1976, he had confidence in me when I was appointed as his collega proximus in 1986 (an honor that I am still vividly aware of after all these years), and in the formative time of the ILLC, he was quietly but persistently supportive, even though governance and organizational activism were not among his favorites. For many years, our offices were side by side, and our contacts happened daily. That does not mean Anne never criticized his next-door neighbor, sometimes with the aid of a list of points on his whiteboard, but always for good reasons: and in his turn, he was always open to arguments, and able to change his mind.

Familiarity may breed contempt, as is said, but it can also breed respect. Over the years, I came to appreciate Anne's qualities as a researcher and as a person more and more.
I also admired his starting a new life
after retirement, rather than following the inertia that people call 'still going strong in one's field'.
With Anne gone, my world in Amsterdam looks reduced: it has lost a dimension.

*Johan van Benthem*

Arriving in September 1961 in Amsterdam for a master study with E.W. Beth I soon found myself in contact with one of Heyting's students: Anne Troelstra. Since our subjects were close we participated in a number of the same classes. He was meticulous, neat, always ahead of deadlines, everything I was not. After a while, I had to recognize he was very clever as well. We shared a love of intuitionistic logic, and looked at the basics together. Soon we found the 14 equivalence classes of formulas with only p, q and implication. More seriously, we delved into my subject, the theory of the models later called Kripke models, and established some important results. He stayed here for a PhD with Heyting, I left for one in the U.S. When I returned to take up a position at the UvA, he had already a solid reputation. It was always a safe feeling that he was there. When I had gone back again to the U.S. for two years of teaching, and on wanting to return found it was less easy than I had expected to find a position again in the Netherlands he was able to ease me back into the UvA. We both worked in intuitionism but from different angles. When we discussed issues in that area he often surprised me, suddenly showing insights I didn't suspect. This remained so even when his main interest had shifted from logic to natural history. Life will be different now that he is no longer there.

*Dick de Jongh*

I first met Anne Troelstra at the Mathematisches Forschungsinstitut Oberwolfach. It was at the time when he just had finished his marvellous lecture notes volume entitled "Metamathematical Investigations of Intuitionistic Arithmetic and Analysis". This impressive piece of work quickly became the standard source of knowledge for at least a generation of mathematics students with an interest in the logical foundations of constructive mathematics. In fact, at the time it was the most-read volume of the whole series of Springer Lecture Notes in Mathematics at the Mathematical Institute Library of LMU Munich. Later it was extended to the almost encyclopedic two-volume book "Constructivism in Mathematics", which he wrote together with Dirk van Dalen, and later again I had the great pleasure to be his coauthor in the book "Basic Proof Theory", with appeared in two editions around the turn of the century. For many years we also cooperated in organizing the regular workshop on Mathematical Logic at the Mathematisches Forschungsinstitut Oberwolfach.

In all these years he strongly impressed me by the clarity and originality of his mathematical work, and also his ability to relate it to the vast literature of his field. He was an absolutely honest person, who always insisted to give other researchers the deserved credit for their work. Apart from mathematics he had many other interests, which he pursued with similar quality and endurance as his mathematical work.

I will very much miss him as a dear colleague and friend.

*Helmut Schwichtenberg*

One of the great pleasures of the academic life is that occasionally you meet (and become friends with) people who look different, talk strangely and generally have nothing in common with you except math---in this case; and so it was that Joan and I met and became friends with Anne and Olga back in the sixties, when we were all very young. Joan had a lot to talk about with Anne, of course, but I, too, always enjoyed the scientific exchanges with him: he was a tolerant intuitionist, and I had started in constructive mathematics, reading Heyting's little book in '57 or '58, probably before Anne. (I lost my constructive faith in Grad School but like many lapsed Catholics, I never lost my respect for the faithful or lingering guilt for my dropping out.)

There have been many trips over the years---some with children---by the Troelstras to Greece and by us to the Netherlands, most recently three years ago. Many happy memories, mostly of talk about plants.

A very vivid one (some years back) is of a walk in Parnitha, the tallest of the four mountains that surround Athens. It was chilly (Spring or Fall most likely) and Anne was in his element, latin names of species pouring out of him; until he turned silent and said softly that he had seen more species that morning than exist in Holland---the single, greatest praise of the Greek countryside I ever heard.

We will miss him.

*Yiannis Moschovakis*

Professor Troelstra has been one of my main scientific mentors and supporters when I was young and needed support the most. He not only invited me to my first Oberwolfach Meeting in 1990 but also to my first talk abroad (the Intercity Logic Seminar between Amsterdam and Utrecht) again in 1990.

He was co-referee for both my master and my PhD theses and no scientific work had a greater influence on me than his Springer LNM 344 "Metamathematical Investigation of Intuitionistic Arithmetic and Analysis" from 1973 which is only book I had to buy twice since the first copy disintegrated due to its intensive use. There was a time when I remembered the page numbers on which certain facts were presented in this book.

He examplified for me the ideal scholar. I will always remember him in the highest regards and will be forever grateful to him.

*Ulrich Kohlenbach*

In the last quarter of the 20th century, Anne Troelstra, with my assistance, organized an Amsterdam-Münster logic contact between our two institutions: Every second year, a few Amsterdam logicians under Anne's leadership would visit the logic institute at Münster, some of them as well as some Münster logicians would present their recent research, and every other year, it would be the other way around. For many young logicians from Münster -possibly also from Amsterdam- this was their first encounter with the international logic scene, and it certainly influenced and improved the work on intuitionism and constructive mathematics at the Münster institute considerably.

This contact was also a basis for a deep friendship between the Troelstra and the Diller family, resulting in a number of visits to their respective homes at Muiderberg and Münster. We, the Dillers at Münster, are deeply moved by the sudden and unexpected death of Anne, and our feelings of sympathy are particularly with Olga.

*Justus Diller*

It must have been in the second year of my studying maths, in the spring of 1983, when I first met Professor Troelstra when I attended a logic seminar. We read Dana Scott's Lectures on a Mathematical Theory of Computation, and I was immediately tasked with presenting Chapter 3.

Half a year later, I was student-assistent under him. I helped with bibliographical research for the Omega-bibliography, of which Troelstra and Diller compiled the part on Constructivism.

Troelstra could be rather rigid when it came to social conventions. Dutch, like many other continental European languages, has a polite/formal form and a familiar form (different ways to say "you"). In those days, it was still customary to address a student in the polite form, whereas for a colleague one would of course use the familiar form. When I did my bibliographical chores, I was a colleague and was talked to in the familiar form. When, after a discussion about this work, I asked him whether I could ask a question about his course in lambda calculus, which I was following, he switched at once to the polite form.

I accompanied him on trips to Muenster in Germany. Once a year, a delegation of teachers, postdocs, PhD students and undergraduates went for a short weekend-seminar to the department of Diller and Pohlers (and once a year, the Muensteraner reciprocated with a visit to Amsterdam). Once when we passed a sculpture of a nymph-like person, her naked torso rising out of a rough substance, he commented: "the artist probably thought: if I reveal more of her body, it can only lead to disappointment". He had definitely a sense of humour.

Maybe once a year, he invited the whole logic group to his place for dinner. At the end of the evening he would put a large crate, full with books, on the floor, and everyone was invited to take what (s)he liked.

In the fall of 1986 I had been working on my master thesis, but my assistentship had ended and I had several jobs, so the thesis work had been put on hold. Troelstra was not in the habit of telling you what he thought of your work, and naturally I assumed I would do my master's some time and then find a position outside academia. But I was in for a surprise, when early february 1987 he was on the phone. "You should finish your thesis and get your master diploma, for you can start by March 1 as PhD student." It was not a question. At the ceremony for my master diploma (my parents were present) he called me a "rough diamond", which was sort of a compliment, I suppose. The emphasis was on "rough".

As a PhD student I of course got to know Troelstra better as a mathematician. I did my first steps in topos theory, following work by Martin Hyland. Although he had not worked in this field, he always held it in some kind of timorous esteem, although he stressed the importance of applications. Abstraction had its limits. He had lots of criticisms, mainly on my style of writing ("too succinct", "too terse"); later I realized that he taught me a lot during this period. Reading my manuscripts was sometimes a strain for him, and he complained that supervising me would shorten his life by a year. I came to appreciate how he could "feel" a result, even if he was not familiar with the formal details.

Again, he was not going to give you any compliments so after 3 years of work on my thesis I was just dead certain that it had been a total failure. I would leave academia without a thesis. But again there was a surprise: "you should do some research in the library for an overview of the field. And gather your results, for it is about time to start writing up".

One aspect of doing a PhD with Troelstra was that, starting right after the ceremony, one should no longer use the polite form with him and call him "Anne". This took some getting used to, but I managed eventually.

In the northern Dutch province of Friesland (which is not where Anne was born or grew up, but is, I presume, where his ancestors came from) the name "Anne" is a common boys' name. A funny story was told at the memorial conference which Ieke Moerdijk, Harold Schellinx and I organized to celebrate Anne's 60th birthday: when he first met Dana Scott, he told him that he had thought Dana was a girl. He probably didn't reflect on how his own name might sound to an English speaker.

I am grateful to have been able to study under a scholar of Anne's calibre. His staunch honesty will remain a beacon for me the rest of my life.

*Jaap van Oosten*