**Karl Sigmund**attended the Lycée Français de Vienne, a school founded in 1946. It moved in 1954 to the park of the Palais Clam-Gallas. Although it had a number of French pupils, at the time when Karl attended the vast majority of the pupils were Austrian. He makes this interesting comment in [3]:-

He graduated with his Baccalaureate in 1963 and began his university education at the Institute of Mathematics of the University of Vienna.... the Vienna of my youth was an intellectual wasteland, the "town without Jews," as novelist Hugo Bettauer had predicted in the1920s.

Sigmund undertook research advised by Leopold Schmetterer and was awarded his doctorate in 1968 for his thesis *Über Verteilungsmaße von Maßfolgen auf lokalkompakten Gruppen* Ⓣ. He published a paper with the same title in 1969 which gave conditions for the existence of the distribution measure of a sequence of normalised measures. After completing his doctorate, Sigmund went first as a postdoctoral worker for a year to the University of Manchester in England where he spent the academic year 1968-69. There he had useful discussions with the probabilist Kalyanapuram Rangachari Parthasarathy (born 1936) who was Professor of Statistics at Manchester in 1968-70. Sigmund then spent a second postdoctoral year, 1969-70, at the Institut des Hautes Études, Bures sur Yvette, France. There he benefitted from discussions with Stephen Smale who spent the autumn of 1969-70 at the Institut des Hautes Études.

Sigmund's second paper, published in 1970, was *Generic Properties of Invariant Measures for Axiom A-Diffeomorphisms*. In this he states (we have given the names in full):-

At the time he was writing this paper Rufus Bowen was a research student working with Stephen Smale. When Sigmund published this 1970 paper he gave his address as Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel. He spent a third postdoctoral year at the Hebrew University where he had many useful discussions with Benjamin Weiss who was an American who had been appointed to the faculty of the Hebrew University in 1967.It is a pleasure to thank David Chillingworth, Kalyanapuram Rangachari Parthasarathy, Stephen Smale and especially Rufus Bowen for helpful suggestions.

Sigmund spent two further years undertaking postdoctoral studies both back in Austria, the academic year 1971-72 being spent at the University of Vienna and 1972-73 also spent in Vienna at the Austrian Academy of Sciences. In 1973 he was appointed Professor at the Institute of Mathematical Statistics in Göttingen, Germany where he spent one year before moving back to Vienna when became a full professor at the Institute for Mathematics, University of Vienna. In the same year, 1974, he married the historian and author Anna Maria; they have one son Willi.

Throughout the various years leading to his appointment as a full professor in Vienna, Sigmund had continued to work on ergodic theory, measure theory, topological dynamics and dynamical systems. For example, he published *Topological Dynamics of Transformations Induced on the Space of Probability Measures* in 1975, a paper written in collaboration with Walter Bauer. In some ways one can see the culmination of his work in these areas as the book *Ergodic Theory on Compact Spaces* (1976), jointly authored with Manfred Denker and Christian Grillenberger. Bill Parry, in a review, states:-

A major change in the direction of Sigmund's research occurred around 1977 after discussions with the theoretical biochemist Peter Schuster and the mathematical biologist Josef Hofbauer. He explained [4]:-... current ergodic theorists will find the book indispensable.

Sigmund and Hofbauer began collaborating in 1978 and the result of this collaboration was the book... the first topics that interested me were in the tradition of Ludwig Boltzmann on statistical mechanics - mass action kinetics and physical problems. Very soon, I found extremely interesting problems involving ensembles of beings in population biology, whether it was population ecology, population genetics, or even animal behaviour, it always concerns large populations of interacting agents. In biology, particularly, interesting questions arose because these interacting agents could change their behaviour, in contrast to what physical entities are doing.

*The Theory of Evolution and Dynamical Systems*(1988). For extracts from reviews of this book, see THIS LINK.

There were other influences on him as he explained in [4]:-

Let us quote Martin Nowak's description of how their cooperation worked [5]:-I started applying mathematical models to population ecology and genetics, and just when I gave my very first talk - on Robert Axelrod's book, 'The Evolution of Cooperation' - I noticed a student sitting in the front row who was starting to glow. His eyes were getting larger and larger and I found I was talking only to him. This was my first introduction to Martin Nowak, who has a magnetic personality. I hijacked him from his former thesis advisor, and he wrote his thesis on the iterated prisoner's dilemma with me. He was not the only brilliant student I had, but he was the one who was the most mind-opening for me. He then went as a post doc to Bob May in Oxford, and then he went to Princeton, and then to Harvard ... .

Sigmund discussed the prisoner's dilemma and similar topics in his booksOnce a year the theoretical chemist Peter Schuster used to take his students from the University of Vienna to a small house in the Austrian mountains. During the day we skied, of course, but in the evening the emphasis was on science. I was a first-year PhD student looking for a project. The mathematician Karl Sigmund was there and gave a talk on what was a new topic for him: the prisoner's dilemma. At the end of the talk I asked a question, and the next day we travelled back to Vienna, endlessly debating this game. In subsequent days, I visited Karl's office and we started to do calculations. We had become prisoners of the dilemma. We often met in coffee houses, the genius loci of past glory. Here Kurt Gödel announced his incompleteness theorem, Ludwig Boltzmann worked on entropy, and Ludwig Wittgenstein challenged the Vienna circle. Or we walked in the Vienna forest, visiting a meadow called 'Himmel'(Heaven), where a sign noted that here Sigmund Freud first understood the nature of dreams. Within a year, we had conceived an evolutionary description of probabilistic strategies in the prisoner's dilemma struggling for cooperation by natural selection ...

*Evolutionary Games and Population Dynamics*(1998),

*The Calculus of Selfishness*(2010), and

*Games of Life: Explorations in Ecology, Evolution, and Behaviour*(2017). For information on these books and extracts from reviews, see THIS LINK.

Let us return to giving some details of Sigmund's career. In addition to his professorship at the University of Vienna, he became a part-time worker at the International Institute for Applied Systems Analysis in Laxenburg, Austria, in January 1984. Laxenburg is only around 20 km south of Vienna. At this Institute he worked in the Evolution and Ecology Program. His work includes game-theoretical modelling of animal behaviour, mathematical models in ecology and population dynamics, and stochastic and deterministic treatment of immunological processes. Sigmund was head of the Institute of Mathematics at The University of Vienna in 1983-85. He was also much involved with the Austrian Mathematical Society, being its vice-president in 1995-97 and president 1997-2001.

With such a range of interests, it is perhaps not entirely surprising that Sigmund should become interested in the history of science. His book *Exact Thinking in Demented Times: the Vienna Circle and the Epic Quest for the Foundations of Science* (2017) has been described as "serious and first-rate history - written like a novel. [It is] a masterpiece." When asked how he became interested in the Vienna Circle, Sigmund replied [3]:-

[In the recent [October 2018] interview Sigmund explains his journey through the different areas on which he had done research and where his latest ideas are leading [4]:-My interest]was sparked by Ludwig Wittgenstein, a philosopher of mathematics and language who fascinated me when I was a schoolboy. I got over that crush, but I still cannot make up my mind about his work or about him as a person - and when I found that the Vienna Circle had exactly the same problem with Wittgenstein, I instantly felt drawn to them. Today the members of the circle are pigeonholed as "logical empiricists," but that label does not do justice to their diversity, their internal and external fights, their turbocharged philosophical environment, and their dramatic individual fates. They were right in the middle of an amazing philosophical firework, one that sent Ernst Mach, Ludwig Boltzmann, Karl Popper, and Wittgenstein soaring into the sky.

Let us now look at some of the honours given to Sigmund. He gave the plenary lectureRight now I'm in the middle of a big change in gear. I've been thinking for the last four or five years about the history and philosophy of science, including the Vienna Circle, the influence of Albert Einstein, etc. Now, I'm getting back to evolutionary game theory, the theory of evolution of cooperation and the social contract, and how the social contract can be subverted by corruption. That's what interests me most currently. Of course, that is not a new story. I believe it explains a lot of what I see happening in my field and in related fields. The ideas that survive are the ideas that are fruitful in the sense of quickly producing a lot of publications, and that's not necessarily correlated with these ideas being important to advancing science.

*The Population Dynamics of Conflict and Cooperation*at the International Congress of Mathematicians held in Berlin in August 1998. He was elected a member of the Austrian Academy of Science in 1999, and a member of the German Academy of Science (Leopoldina) in 2003. He delivered the Gauss lecture at the German Mathematical Society held in Würzburg in 2003. He was declared Austrian of the year (research) for 2006.

On Monday, 9 April 2007, the Department of Applied Mathematics at the Illinois Institute of Technology in Chicago hosted the inaugural Karl Menger lecture by Karl Sigmund on "Menger, Games, and Morals."

In 2010 Sigmund was elected a member of the European Academy of Science and, in the same year, awarded an honorary doctorate by the University of Helsinki. He received the Preis der Stadt Wien für Naturwissenschaften Ⓣ, also in 2010. He was awarded the Würdigungspreis für Wissenschaften durch das Land Niederösterreich Ⓣ in 2011 and, in the same year, the Blaise Pascal Medal for Mathematics of the European Academy of Science.

He was invited to give the Pacific Institute for Mathematical Sciences Distinguished Colloquium at the University of British Columbia on 13 April 2012. He delivered the lecture *Sanctions on the Commons: Social Learning and the Social Contract*. He gave the following Abstract:-

Also in 2012, Sigmund received the Isaacs' Award of the International Society on Dynamic Games. On Tuesday, 4 June 2013, the University of Vienna presented the "UNIVIE Teaching Award 2013" for outstanding teaching achievements to Karl Sigmund. A Symposium in honour of Sigmund was held at the University of Vienna in October 2015. He received the Science Book of the Year Prize from the Austrian Federal Ministry of Science, Research and Economy in 2016 for his book on the Vienna Circle.Evolutionary game theory helps to investigate the role of incentives in promoting cooperative behaviour in joint enterprises. In particular, this lecture deals with the surprising effects of optional participation in collaborative enterprises. Coercion works better for voluntary rather than compulsory collaboration. A social contract need not be based on rational deliberation or the command of a higher authority. It can emerge spontaneously through social learning of individuals guided by no more than their myopic self-interest.

**Article by:** *J J O'Connor* and *E F Robertson*