**Karl Aubert**'s parents were the headmaster Karl Fearnley Aubert (1879-1945) and Solveig Laake (1895-1970). Karl was one of his parents' two children, having an older brother, Vilhelm Aubert (1922-1988), who became a leading sociologist and professor in the University of Oslo. The family were descended from Benoni Aubert (1768-1832), a Norwegian military officer who also did important work on cartography. Benoni Aubert was the father of Ludvig Caesar Martin Aubert (1807-1887), whose son Vilhelm Henrik Julius Adelsteen Aubert (1847-1911) was the father of Karl Fearnley Aubert, the father of the subject of this biography.

Aubert, the subject of this biography, was educated in Oslo. We note that because of Danish influence the city was known as Christiania or Kristiania from the 17th century to 1925. In that year, one year after Aubert was born, it reverted to its original Norwegian name. In April 1940 German troops invaded Norway and on 1 May the Norwegians surrendered. At this time Aubert was a fifteen year old schoolboy while his brother Vilhelm was just completing his schooling and preparing to enter the University of Oslo. The following years were difficult with the Nazis attempting to convert the university to the Nazi philosophy. When this was unsuccessful, the Germans closed down the University of Oslo in November 1943.

Aubert's education was disrupted by the war but, with the help of others, he overcame the difficulties in a remarkable way for he began publishing mathematics papers in 1944 while still a teenager. The first was *Summation of some series of binomial coefficients by means of Cauchy's integral formula* (1944) followed by *Remark on the middle binomial coefficient* (Norwegian) (1944), *Summation of some series of binomial coefficients on the basis of Cauchy's integral formula* (Norwegian) (1945), *A group-theoretical remark of E Hoff-Hansen concerning certain expressions in the quantification theory* (1947), and *On making precise and generalizing the concept of relation* (Norwegian) (1948). In this last mentioned paper Aubert claims that most mathematicians have only vague ideas concerning the general notion of relation, in spite of the fundamental importance of this notion. His aim in this paper is to try to define this notion in as precise and general a way as possible.

He received a cand.real. degree (equivalent to a Ph.D.) from the University of Oslo in 1951. Although based at the University of Oslo where he held a fellowship, Aubert made several long-term study trips Paris. For example Paulo Ribenboim writes (see [1]):-

In 1957, Aubert was awarded the degree of dr. ès sciences from the University of Paris for his thesisIn1954, Karl Egil Aubert was in Paris, very much interested in valuation theory and participating in Krasner's seminar. He came to Bonn to get personally acquainted with Krull and Lorenzen, leaders in valuation and ideal theory. This is when we met for the first time - as I was myself working with Krull. We became good friends at once and stayed so until his untimely death.

*Contributions à la théorie des idéaux et à la théorie des valuations*Ⓣ. He became a senior lecturer at the University of Oslo in 1960 and professor of mathematics there in 1962. He held this position until he became a senior fellow at the Norwegian Research Council for Science and the Humanities (NAVF) in 1990. However, he was also affiliated with other universities both before and after his appointment in Oslo: among other things, he took part in developing the mathematical research community in Tromsø. From 1958 to 1960, he was a research fellow at the Institute for Advanced Study in Princeton and he was later a visiting professor at the University of Washington in Seattle and at Tufts University in Boston.

Paulo Ribenboim writes about his friendship with Aubert and tells of both his trips to Norway and Aubert's trips to the United States (see [1]):-

Much of the material in following paragraphs are taken from [5]. After his initial work, which was in several different branches of mathematics, Aubert's research was almost exclusively in algebra. He was particularly interested in the general concept of an ideal, which can be traced back to the German mathematician Eduard Kummer's "ideale Zahlen" introduced in his attempts to solve Fermat's Last Theorem the 1800s, but which plays a key role in modern algebra.As we had many common interests, at various occasions, I went to Norway and he came to Kingston. It was Karl Egil who showed me the Vigeland sculptures and Munch's museum; both left me an everlasting impression. I will always remember how proud he felt showing Abel's statue in the King's Palace gardens. ... It was Karl Egil who first made me taste whale's meat - however I wouldn't recommend it to a friend. Karl Egil was particularly fond of visiting Kingston, where he would work with Izzy Fleischer, our common friend. Moreover, Kingston was for him a base point from where he would start the merry-go-round of visits to friends' universities, but also - and this was not less important - to the Canadian Rockies. Karl Egil was very keen on his work on x-ideals and rightly thought that he completed in satisfactory manner the work initiated by Prüfer, Krull, Lorenzen and Jaffard. I also believe that these are his most substantial papers.

Aubert is, as Ribenboim states in the above quote, famous for introducing the term "*x*-ideal", a concept which appears in his doctoral thesis, and this made it possible to combine a number of previous results into one common theory. He developed this theory further, and gave a comprehensive and complete representation of it in the article *Theory of x-ideals* (1962), which has been a standard reference for anyone interested in the subject. The theory of ideals continued to be his main interest, and he contributed to it with several important papers, such as *Additive ideal systems *(1971), *Localisation dans les systèmes d'ideaux* Ⓣ (1971) and *Divisors of finite character *(1983).

Although Aubert's own research essentially concentrated on pure algebra, he was also concerned with the interplay between algebra and other fields of mathematics, and this became the subject of a new seminar in contemporary analysis that he started when he was given a fellowship at the University of Oslo. The seminar was an informal working group of students and university teachers who joined forces to read up on all the new research in mathematics that had not reached Oslo's small academic community during the years of war and isolation. There was no formal hierarchy, and they all worked together with the fervour and enthusiasm that comes with the feeling of breaking new ground. Despite the group not having a formal leader, Karl Egil Aubert was its central figure. He was the one to set the path of the seminar, using his broad perspective on mathematics and his contacts with the international research communities, and his commitment and enthusiasm for the project drove everyone forward.

Throughout the years, Aubert supervised a number of students. He was also eager to help younger mathematicians go abroad to study with leading experts. Thanks to his good contacts at French and American universities, he contributed to a whole generation of Norwegian mathematicians receiving much of their education at some of the world's best educational institutions.

The most important aspect of Aubert's research was the basic ideas, the simple and general concepts, that he introduced. Nevertheless, he could also hold advanced lectures and participate in seminars on current topics far outside his own specialty, as when he was part of organising a major international seminar on number theory held in Oslo in 1987 on the occasion of Atle Selberg's 70th birthday. Aubert was also a driving force behind the meetings in the series "Ski and mathematics" at Gausdal Hotel, where teachers and students from universities and colleges from all over the country gathered during the first week of the new year for outdoor activities and inspiring lectures.

Aubert was also ready to become involved in issues beyond academia when he thought it was necessary, as when he worked to strengthen women's representation at the University. This led to the appointment of four female professors at the University of Oslo during the mid-1980s. Their appointment was made on a strictly academic basis however, and he was indignant at the idea of increasing the proportion of women by lowering standards. He used to say: "But they're just as good, and we would be making them a disservice by appointing them as second-rate professors using second-rate criteria."

Karl Egil Aubert also addressed other important university questions. He was the impetus to the work that led to a new and better 'examen philosophicum', the common compulsory first-semester course for all incoming students, with the goal of "creating a common basis for academic practices based on rationality and reasoning."

For Aubert, the requirement of intellectual honesty was a moral concern. He could never accept an easy and superficial argument, even when it would support a good cause and corroborate his own views. This moral indignation over an argument that he considered as insufficient, was expressed for example in his strong and continuing involvement in the spy case against Arne Treholt. He did not accept that one could fail to prove the existence of specific violations of specific laws for every instance and instead settle for "an overall assessment".

Karl Egil Aubert perceived mathematical science as being part of general culture. His interests were varied and he participated eagerly in debates on philosophy and social issues. Although he could be harsh against those who did not follow his demands for stringent and honest argumentation, he had a deep respect for the truly talented who were masters of their craft and could made edifices soar. Let us now look at the debate that Aubert carried on with Herbert Simon.

Herbert Alexander Simon (1916-2001), was an influential social scientist and professor at Carnegie Mellon University for many years. He was awarded the Nobel Prize for Economics in 1978. In the early 1980s, Aubert criticised Simon's irrelevant use of mathematics beginning a fascinating debate in which they responded to each other in a series of papers. Aubert also criticised the fact that, in his paper *Bandwagon and Underdog effects in election predictions* (1954), Simon used the Brouwer fixed point theorem for a proof when the Intermediate Value Theorem would suffice. Aubert writes in the Abstract of his paper [2]:-

Herbert Simon replied to Aubert's criticisms. to which Aubert responded with the paperIn his well-known paper from1954, Herbert A Simon sets out to demonstrate that it is possible, in principle, to make public predictions within the social sciences that will be confirmed by events. However, Simon's proof by means of the Brouwer fixed point theorem not only rests on an illegitimate use of continuous variables, it is also founded on the questionable assumption that facts - even on the level of possibilities - can be established by purely mathematical means. The 'proof' also appears redundant since we already know from past experience that, for instance, the confirmation of a public election prediction is possible.

*Mathematical modelling of election predictions: Comments to Simon's reply*(1983). Aubert's abstract is as follows:-

Aubert discusses his ideas on mathematical modelling in general, and gives an overview of the Aubert-Simon debate in [3]. We present two short extracts from this interesting paper. For Aubert's introduction to this paper, see THIS LINK.Herbert A Simon's reply to my criticism of his1954paper is not to the point. He fails to respond to some of my arguments and misconceives others. One of his misconceptions is that any mathematical deduction from empirical premises which are formulated mathematically will necessarily lead to empirically valid conclusions. This claim is particularly unwarranted in Simon's case since his mathematical premise, the continuity of the reaction function, is empirically meaningless.

For Aubert's discussion of Zeno's 'arrow' paradox, see THIS LINK.

Given Aubert's interest in the philosophy of mathematics and also in its history, it was therefore no coincidence that he would play a crucial role in shaping the course "The development of mathematics and its uniqueness," intended specifically for prospective teachers in the subject. His goal was to provide them with the necessary background, not primarily new knowledge and technical skills, but insight and understanding, so that they could pass on some of the enthusiasm and joy about mathematics that he himself felt.

During the last 10 years of his life, Aubert was strongly committed to stimulate interest in mathematics among the very young, and his efforts were essential for the Abel competition to become an annual competition in schools across the country. He was the chairman of the jury, welcomed the finalists in Oslo and went through the questions with them after the competition.

Aubert's personal qualities were of paramount importance for the role he played in Norway's academic environment. He was open and outgoing, and had many friends as well as an impressive international network.

He was also very good at languages and a great admirer of French culture and mathematics. He was also fond of nature, went for long skiing treks and enjoyed mountaineering. He climbed several major mountains in the Alps, America and Norway with good friends and with his brother Vilhelm, whom he remained close to his entire life.

Aubert's primary legacy is his role in building up the academic community. For younger colleagues and students, he was the wise and experienced advisor and the beloved teacher who always kept his office door open. For the older ones, he was also the instigator who brought in new research from abroad during the 1950s. There can be no doubt that he built much of the foundation that the good mathematical academic community there is today in Norway rests upon.

His outstanding contributions led to him receiving many honours. At the University of Oslo, he held several substantial honorary positions and he was widely used as an expert witness both at home and abroad. He was the Norwegian editor for the respected journal *Acta Mathematica*, published in Stockholm, and he sat on the board of the Mittag-Leffler Institute there for many years. He was a member of the Norwegian Academy of Science and Letters, the Royal Norwegian Society of Sciences and Letters in Trondheim and the Finnish Academy of Science and Letters. He never married.

Let us end this biography by quoting a comment from Ane Maria Dohl, the author of [5]. She writes:-

I just learned that Aubert was one of my father's mathematics professors at Oslo University during the1960s. Apparently he was a charming man who was also a very handsome bachelor, and he "charmed many of the female students." In any case he seems to have been very well liked by the students!

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

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