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
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.
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 MS and KLS. 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
Emeritus Vrije Universiteit Amsterdam;
Otterlo, 26 March 2019
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