Aequat. Math. 89 (2015), 1–16 c Springer Basel 2015  0001-9054/15/010001-16 published online March 10, 2015 DOI 10.1007/s00010-015-0343-5

Aequationes Mathematicae

An interview with J´anos Acz´el ´n Daro ´ czy Zolta

This interview was first made in the year 2000 and was finalised in 2003. It was published in the journal Debreceni Szemle in 2004 (volume 3, issue 12, pp. 465–480). This is to express the interviewer’s deep respect for his master on the occasion of his 90th birthday. After living in Hungary for 40 years, and about 35 years abroad, probably it is time to stop and make an evaluation. Your life encompasses important periods in history. Today most people in Hungary have no idea what it was like in the 1920’s when you were born. What memories do you have of your childhood and family? I was born in 1924. Before the war my father had worked for a company producing chemical and medical products, which after World War I moved to Romania. After long hesitation he decided to start his own business so he stayed in Hungary. He sold medical equipment produced by various companies. My parents also ran a pension in B´ alv´ any street to make some extra income. Then we moved to Vilmos cs´asz´ar street—called Bajcsy-Zsilinszky street today. In our house, the so called Ulrich house, there was an elementary school, and that’s what I attended in the first 4 years. I was accepted before I was 6. Right before going to high school I went to an English language school, that’s where I first started learning the English language and had to do all the subjects in English. From that time on my parents were insistent on my learning English but not French, because whenever they wanted to leave me out of the discussion they used French between them. So I teamed up with one of my classmates and we took private classes from a teacher of French.

The preparation of this interview has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651.

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Then I attended Berzsenyi Grammar School (which was later united with Bolyai Grammar School). Life in the grammar school was really enjoyable, even though you had to strive for good results at the GCSE exams, particulary because of the Numerus Clausus. I had several good teachers and it was enough to speed up in the year before the exams, so other than that there was no stress throughout the high school years, which you so often hear about today. I took my GCSE exams in 1943. In autumn 1943 you began your studies at P´ azm´ any P´eter University of Arts and Sciences. How did you get into that university and why did you choose mathematics? There were many talented young boys in my class, and they soon found their fields of interest, so by the time we were in our sixth year (it was an 8 year grammar school), all my classmates or at least my friends already knew exactly what they wanted to be, even if later they changed their minds. Not me. Then once our maths teacher came to class with our corrected test papers and said that nobody could write it properly but J´ anos Acz´el. Then I thought, hmm, I would become a mathematician. And so I did. I started reading maths books and solving problems. Probably, otherwise I would have chosen the University of Engineering. Finally I was accepted without any problem to P´ azm´any P´eter University and studied maths and physics. Back then there were no compulsory courses or credits, you only had to attend for 4 years. But everybody chose teacher training courses as well, to make sure they could make ends meet, and there were some (but not too many) compulsory courses. Who are the professors you remember from those years? Lip´ ot Fej´er, of course. I had met him earlier through my uncle. Our lecturer in analysis was P´ al Sz´ asz. The analytical geometry lecture was given by professor B´ela Ker´ekj´art´ o. P´ al Sz´ asz did a very good job of teaching us the fundamentals but Ker´ekj´art´ o was different. Although his lectures were held with the charm of a French charmeur, they were often not clear enough and he also made mistakes that he took very lightly. Jen˝ o Sz´ep was his assistant professor and he gave the seminars, but when he was called up, his substitute was Istv´an F´ ary. If he was late for class we lightheartedly left and did not wait for him. Experimental physics was taught by Istv´ an Ryb´ ar and after the war theoretical physics was taught by K´ aroly Novob´ atzky, who was really very good. We had to invest a great amount of energy into studying physics, as we majored in mathematics and physics. We also had the later Nobel laureate Gy¨orgy B´ek´essy as a lecturer in experimental physics, but we had more contact with the assistant professor Zolt´an L´ aszl´o. We took chemistry courses from Grob. So you had a pleasant time at grammar school and in the first year of university, while there was a terrible war going on in Europe. Then on 19th March 1944 Hungary practically lost its independence. This period, the “shoah”, must have been hard for you.

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Although we were certainly aware of the war going on and were worried about the outcome, it was only after 19th March that we realised how terrible it was. Soon also the university closed. For 15th May I was called up for labour service, and I worked near Miskolc. At the narrow-gauge railway station there was the wood-mill and wood-hole. It was not very hard work. Then on 5th October I escaped: I was hiding at one of the workers’ home, since we thought it was all over but the Hungarian Nazi arrow-cross troopers came. I returned to Budapest and from there I went to forced labour again, to Fert˝ or´ akos this time. We had to dig trenches. The real hardship was the cold, and we had to sleep in barns. The work of those who were building tank defenses was exhausting, that of the trench diggers was not so bad as they were hard to supervise: you could walk along the trenches whenever an SA foreman came. J´ anos Horv´ ath (university professor at Maryland) was also there and we had a maths textbook “Real Analysis” by P´ al Veress on us. (That reminds me: P´al Veress also taught me in the first year.) We even dared to read and discuss that book there in the camp. It happened once that an SA foreman came perpendicular to the trenches and we noticed him in the last minute only. We dropped the book and started digging, but he saw the book and asked what it was. J´anos Horv´ ath had enough presence of mind to say it was Scheisspapier that is toilet paper. That was alright, you need paper for that. After the war we told the story to Lip´ ot Fej´er, who was angry with P´ al Veress and so he answered that after all it was not a big lie, actually. Then we moved on. On our way from Fert˝ or´ akos to Mauthausen a retreating Waffen SS troop started hitting us with the butt of their rifles. I still have the mark of a hit on my head. We were taken to Mauthausen then to a smaller camp in Gunskirchen. Fortunately we were soon liberated by the Americans, but it took a long time to get home with many of my comrades. Because of the war the academic year 44–45 lasted well into the summer. Even so, I was a little late for signing up. Later, I made up for this lost semester, so I could finish university as if I had attended continuously. What happened to your parents and your sister? My parents were in the ghetto in Budapest and my sister was hiding. She was my only sibling, considerably older than me, actually, my step sister, we have the same mother. Most of our more distant relatives were killed in 1944– 1945. But by the time I got home, my parents were already back in our old home, or at least part of it, as the flats were divided. After the war you continued your studies and lost no time, like you said. Was it then that your famous friendship with J´ anos Horv´ ath and the others, the so called “big five”, was born? “The big five” started very early. The youth organization MADISZ organized seminars for us as there were few mathematics seminars at the university. ´ Finally there were only five of us left. Akos Cs´asz´ar, J´ anos Horv´ ath, Istv´ an Ga´ al, L´ aszl´o Fuchs and me. But there is another reason why these lectures

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were important for me: that’s where I met a girl called Zsuzsa Kende, whom I married later. Our group of five gave lectures to ourselves only, but with great intensity. In the golden days it was 4 hours a week. Our exclusive company had an open meeting once a year, where others could come as well. My big brothers think it was Lip´ ot Fej´er who dubbed us the “big five”. (The victorious powers were called the “big five”: the USA, England, France, the Soviet Union and China—that’s where the name comes from.) At the beginning of your mathematical work you had a strong connection to Lip´ ot Fej´er, whose problem of mean values was a challenge not only for you but for the others in the big five as well. Was this the first functional equation you had come across or had you seen problems involving functional equations? No, it wasn’t. P´ al Veress had an article about the theory of mean values. But Fej´er said, and rightfully so, that both Veress’ book and this article are a bit like a badly adjusted microscope, in the sense that the picture you get is blurry. The original problem of Fej´er was to generalise the property of exponential functions that at the arithmetic mean of the abscissas the function is the geometric mean of the two ordinates. That’s how I came to quasi-arithmetic means. Maybe there was something written about them in Veress’ article as well. I cannot remember clearly now, but anyway it was then that I learnt about the classical Kolmogorov–Nagumo characterization of quasi-arithmetic means. There could have been some other sources. Frigyes Riesz was in Budapest at the time and gave lectures on inequalities and I had read the book by Hardy–Littlewood,1 which contains quasi-arithmetic means, inequalities and their characterization. You say that you learnt about quasi-arithmetic means from P´ al Veress’ article. P´ al Veress was familiar with the characterization given by Kolmogorov, Nagumo and de Finetti, probably he had read the book by Hardy–Littlewood, since it was published in 1934. The Kolmogorov–Nagumo characterization is based on the axiom that if you substitute the first k variables by their mean, the mean value will not change. This had two drawbacks. On the one hand it did not characterize a mean value but a sequence of mean values. Today I would say it was not a functional equation but a sequence of functional equations that characterized this sequence of mean values. On the other hand it was not fulfilled by weighted quasi-arithmetic means. I realised that the so-called bisymmetry equation, which has accompanied me in the theory of functional equations, once again became important, as it solved both problems by characterizing quasi-arithmetic means for a fixed number of variables and it is also satisfied by weighted means. My first paper was published in 1947, and already in 1948 the detailed paper was also published, and it also contained the general solution, more general 1

The book Inequalities written by Godfrey Harold (G. H.) Hardy, John Edensor Littlewood and Gy¨ orgy P´ olya. (The Editors.)

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than weighted quasi-arithmetic means, that is quasi-linear means. I also studied convex functions: I received a lot of help from Frigyes Riesz. I generalized the Jensen inequality, so that it had two different, not necessarily arithmetic means and then using this I examined how a Hardy–Littlewood inequality changes and also examined the specific property of convex functions that the difference quotients increase. I told Riesz about this and he gave me three weeks in his lectures, where I could publicly speak about these results. You can learn a lot this way and I am very grateful to him for this. And did you know back then that the book of Hardy–Littlewood had a third author, P´ olya, who was of Hungarian origin? Yes, I did know that. The book by Gy¨ orgy P´ olya and G´ abor Szeg˝o was great experience as well. We went through the problems, and Lip´ ot Fej´er was making jokes about the pronunciation of Szeg˝ o’s name abroad. Considering that you were practically giving lectures to yourself, when did you get your degree? The other members of the “big five” were all a year ahead of me. We had to write a thesis in maths and one in physics, so I decided that I would choose variational calculus as my topic in maths and in physics it would be variational principles of physics, and I wanted to hand in the same thesis for both. But then I started investigating convex functions and P´ al Sz´ asz asked me why I did not write my thesis on this field. L´ aszl´o Fuchs recommended me to write about convex functions in physics, which I did not accomplish. So when my mates wrote their thesis, I also wrote mine and asked at the dean’s office if I could hand it in, once it was finished. They took my thesis in saying that it probably would not be evaluated before I started my last year. Fej´er and Riesz read my thesis along with the other ones and they wrote their review. Then I started getting emboldened and asked if I could take the exam as well. They said I could take the examination but I would not get my degree. It was wise to pay attention to what Fej´er was talking to you about as most likely he asked you about that in the exam. Once he asked me if I had heard about the Gaussian medium arithmetico-geometricum. Alright, I thought and I learnt it very well, but he did ask something else. So you received a doctoral degree even before you graduated? That’s right. Well, this doctoral exam preceded the qualifying exam as well. My question at the qualifying exam was about the pendulum. I described the small swings and then added that the larger swings could be described by an elliptic integral, which can be approximated with the use of the Gaussian medium arithmetico-geometricum. So that effort was not wasted either. I was wondering why I could not graduate together with my friends. So I went to the dean’s secretary again and asked if I could finish earlier. He said that I could hand in a petition (back then you could write a petition for anything), but the next time there was a meeting and decision could be reached was the following autumn. He suggested instead that I have my index signed by

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all the lecturers, whose course I had attended, and he would have it signed by the dean. So that’s what happened. I got my doctoral degree in 1947 and soon I passed my pedagogy exams as well. I had graduated, but I had no job. I cannot quite recall the political events of the year of the change, but I can very well remember what happened to me. I applied for the position of a quality controller, so I had to take a written test. One of the questions was: what is the next element of the following sequence: 1, 4, 9,. . . I wrote that it could be any number. Of course I was not accepted for the job, but they were nice enough to say that I was overqualified for the job. Then I became a statistician for a trade union. Never before had I dealt with statistics, but I took it. The head of the union summoned me once. He wanted some statistics. I asked him what he wanted to show. He answered statistics. So I said, ‘Yes, but what do you need it for, what do you want to show?’ Finally he told me and I gave him the statistics he wanted. He doubled my salary right on the spot. So when I was offered the position of an associate lecturer in Szeged in 1948, I took it for half the salary I was making. It is well-known that in Hungary the university of Szeged had the most reputation in mathematics; after moving the university of Kolozsv´ ar to Szeged following the Trianon Treaty, there were excellent professors there. How do you remember the time you spent in Szeged? Those were good times. L´aszl´o Kalm´ ar was my boss and I had colleagues like B´ela Sz˝okefalvi-Nagy, L´ aszl´o R´edei and also Gyula Sz˝ okefalvi-Nagy, but he could only give lectures and classes in his home. To earn some extra money I also took up a job in a secondary school, situated in the building which is the Bolyai Institute today. Gy¨ orgy Alexits, who held a good position there and whom I had known from my secondary school years, as I had attended the lectures he gave at the open university and had also regularly discussed mathematics with him, arranged for me to have all my lessons from 8 to 10 in the morning, though I had a full time position. After that I went to the maths institute in Ady Endre street. Zsuzsi, my wife, usually left for university in the morning, so walked together to the square in front of my school. Once as we were kissing each other good-bye we noticed that my class was standing at the window clapping eagerly. It was a very good class, I enjoyed teaching there. Were you already married? We got married in Budapest before we moved to Szeged. Zsuzsi was still a student at university. When she wrote her preliminary exam she was already pregnant with Kati, our first daughter. You must have been inspired by the spirit of Szeged, the atmosphere of the Bolyai Institute, its excellent library and the long standing traditions. Riesz had already moved to Budapest, Ker´ekj´ art´ o had passed. But you left Szeged for some reason. How did it happen? The atmosphere in Szeged was very good, it was great talking to the various people. The hot tempered Kalm´ ar had great theories about maths education.

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I was impressed by the calm B´ela Sz˝okefalvi-Nagy. Zsuzsi thought that the bit strange R´edei was the best lecturer. We had a good relationship with K´ aroly Tandori, who was another assistant lecturer. It was a pleasant period in our life. Why I moved to Miskolc? In 1950 I was asked or rather ordered to go to the new university in Miskolc and I did not mind it at all. I was promoted from an assistant lecturer to an associate professor and head of department. I often say that my only boss in my career was Kalm´ ar (actually the trade union representative as well), after that I was head of department or in Canada I had no real boss either. To be honest, in Miskolc there was no tradition in mathematics or a good library, or colleagues with strong mathematical expertise. Still it was there that you started to build a circle of students, representing a unique mathematical discipline in Hungary, and you were unbelievably successful. I invited Mikl´ os Hossz´ u to Miskolc from Budapest. Endre Vincze was a graduate there, he was my student. The assistant lecturers and senior lecturers were good: Iv´ an Raisz, Ern˝ o Gesztelyi, Zolt´ an Szarka. I gave lectures and seminars to the members of the department and they did, too. We discussed a whole chapter of P´ olya and Szeg˝ o’s book. I still have some of my notes from that time. The level of the students at the university continuously decreased even during the 3 years I spent there. One of the first groups was very good, they were students who had wanted to study before but had not had the chance, but after that the students got weaker. In one of the exam periods I was really very much upset about their studies and diligence, but the student union did not like my reaction at all. I went to one of their meetings where they were discussing how many plans they had. I simply asked if they wanted to study as well. But all in all that time in my life was not bad at all either. I know that during your time in Miskolc you already started organizing the research on functional equations in Hungary as well as making international contacts. You organized mini conferences revealing your talent in pulling together researchers of the topic both in Hungary and abroad. We organized conferences but there were no foreign participants. Our good friend Alfr´ed R´enyi had the habit of leaving a poem in the toilet whenever he visited us. I can recall part of one of these poems: every month a conference or two, he deserves some reverence, that’s true. I organized conferences on the methodology and applications of mathematics, not particularly on functional equations, but we had seminars on this field. The first foreign mathematician I had intensive correspondence with was Jan Mikusi´ nski, then with the Danish mathematician Børge Jessen, which I did not remember clearly, but his autobiographer found our letters. I must have seen his name in Hardy, Littlewood and P´ olya’s book and wrote to him. With Mikusi´ nski I also worked on a common research topic, so I always wanted to travel to Poland. By the time I managed to go for a visit, probably in 1954, he was working on something different. I met Stanislaw Gola¸b in Krak´ ow and

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he knew about functional equations. He was studying geometric objects, where functional equations are applied. I think it was in 1957, during my second visit, that we decided to write a book together. It is known that in Lemberg (or Lw´ ow) Banach and his colleagues founded the Scottish Book, in which they presented problems. The society of Polish mathematicians moved to Wroclaw after the war and they started a new Scottish Book. I have seen that J´ anos Acz´el also has a problem in that. I don’t think I would have mentioned it if you had not brought it up. I think it was in 1954 when Jan Mikusi´ nski, Czeslaw Ryll-Nardzewski, Bronislaw Knaster and I were working on some related problems. One of them was autodistributivity. Ryll-Nardzewski gave the symmetric solution, which was also a quasi-arithmetic mean, just like for bisymmetry, assuming symmetry and reflexivity. So the common solution implies the equivalence of the cases. Knaster said this could be done directly, but Mikusi´ nski had his doubts. Then Karol Borsuk suggested that if a professor thought something was possible, but it raised doubts, then he would go ahead and prove it. And Borsuk did publish a paper about this. The new Scottish Book was usually given to the visiting professors. They also started giving prizes, well before Paul Erd˝ os did, but it was a bottle of wine, a piece of ham or a goose and not sums of money. We had our discussions in English and French while I was writing my problem, so half of it was in English and half in French. I added a remark at the end, that the answer need not be in two languages. Last year you were nominated doctor honoris cause at the University of Miskolc, and they highly acclaimed your pioneer work at the university. Why did you leave Miskolc? This nomination was a very pleasant surprise for me because I left Miskolc with some bitter feelings. I would rather not discuss the details, but there was often a bad relationship among the colleagues, especially the two maths departments had a lot of disagreement. I moved to Debrecen, the Ministry assigned me there, and it was a good move, because in Miskolc I would not have had a chance to do so much scientific research, considering the number of students and the number of lectures. When I learnt about the move I went to Debrecen to have a look round. The people there did not know that I was coming, so they were surprised. I had a good relationship with Tibor Szele, Ott´ o Varga as well as B´ela Gyires. I spent 13 years in Debrecen. That’s a very long time. You moved to Debrecen at an interesting time, as it was getting close to 1956. I think you spent that year in Debrecen. I was a freshman then and I did not know you. I only took your course in analysis in 1957. But I taught your brother, who was a member of a talented physics group. I wrote a paper based on the lectures I gave them. 1956 was less turbulent in Debrecen than in Budapest, especially because the Soviet military base was close. I remember that on 4th of November it took some time for us to

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realise that soviet troops had entered or re-entered the country. When we realised what was going on we wanted to buy bread, but there was none. So I went to a confectioners’ to buy some cake and later I enjoyed beginning my story to my friends in Budapest by saying “as I was queuing up for cake at a confectioners’...” They had very different memories of that day. After the war, the German occupation and the “shoah” we had great hopes in the new system and disillusionment came gradually. Most of the mathematicians were members of the Socialist Party before 1956 and were not after that. We had to persuade each other to enter the Party because it was not advantageous for the Institute if it had no party members. You just started to realise what was going on and were getting disappointed. There were prisoners working at the new university in Miskolc. Like you said a lot of people were very much looking forward to the changes in Hungary. Did you have information about what was going on in the country who these prisoners were and what was happening in the villages? I think after 1956 you were quite well informed politically. You just slowly came to accept things you could not believe at first. Many had relatives in the countryside and talked about requisitions and kulak lists. Even jokes had a role in making people aware of the reality. It was obvious by 1954 when Imre Nagy became prime minister. Nikita Khrushchev’s speech also spread fast. Imre Nagy’s speech reminds me of a story. A couple we knew told us that the husband was listening to his speech when the wife entered and told him off for listening to America’s Voice so loudly. You grew up in Budapest, so why did you choose to live in the country instead of the capital, where you had better connections? Many of your friends and acquaintances like Gy¨ orgy Alexits and Istv´ an Feny˝ o must have held good positions. You may very well have found a job in Budapest. Your work in Hungary, however, is definitely connected to the countryside and that’s where your disciples were from. First I moved to the countryside because I was offered a job there, but then I got to like that way of life and I did not feel excluded. I often went to Budapest and met my friends there. If I could not get something in the library in Miskolc (which was poor) or in Debrecen (which was good), then I could look in Budapest. I did not miss Budapest really. We went to the theatre there or the opera and then came back and we enjoyed our everyday life. The atmosphere was friendlier in a smaller group of colleagues. I really enjoyed working at the Maths Institute. When you moved to Debrecen the journal Publicationes Mathematicae had already been launched. I think it was during this time that the departments were formed in the institute, the department of analysis was founded when you got there as head of department. You lived in a very nice neighbourhood, near the university. That was the time when you were most intensively working on your book that was published by the Swiss Birkh¨ auser and a publishing house

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in the German Democratic Republic. As far as I know the one published in the GDR was sold in the socialist countries and Birkh¨ auser’s in the non-socialist countries, but it was the same book, wasn’t it? You seem to have had a fantastic bibliography. You found papers that were certainly not available in Debrecen, some of them not even in Hungary. One example would be Sutˆ o’s article from 1914. Where did you get the information from? I know you had a card file system, before the time of the computer, and you put down the particulars of the papers and a short extract on these cards. Would you talk about the “Acz´el work routine”? When writing the book on functional equations it was really the bibliography that was the hardest. I was working simultaneously on this book and the one I was writing with Gola¸b and which was published in 1960. Probably I found the papers you find surprising in review journals like Mathematical Reviews, Zentralblatt and Fortschritte der Mathematik. The others I knew from papers that quoted them in details. You are right about the card file system. I did not only use it when there were no computers but even today. It is more convenient because if you see a paper somewhere a piece of paper is always at hand. The English publication contained even more references. Since then other bibliographies have been published. For a while I continued collecting the data, then Zsuzsi, then we worked on it together and then left it to others. Now it is Luigi Paganoni and Gian Luigi Forti who are working on it. In the 1960 edition and then the extended but still manageable 1966 edition I aimed at including all the papers and books on functional equations and also referring to them at some place. Many people liked it that in the bibliography you could find what page was referred to. It would be impossible to do this today. That’s how much the theory and application of functional equations have developed. Surprisingly I still get letters or e-mails saying that they have come across a functional equation or another and it is not included in the book, which was published 34 years ago. You attended an English language grammar school in Budapest and you learnt French and Italian. Then you wrote your first two books about geometric objects in German as well as the one titled “Vorlesungen u ¨ber Funktionalgleichungen und ihre Anwendungen”. You speak fluent German even today. Although I was taught German from elementary school it was in the family that I learnt it. My grandmother, who was born in Vienna, stayed with us and she couldn’t speak Hungarian properly, so we usually used German in the family. Except when we wanted to exclude her from the conversation, then we spoke in Hungarian. I used to speak as well as my grandmother, but I have forgotten a lot, and to my astonishment I have developed an accent. Can you talk some more about the years you spent in Debrecen? I found great pleasure in the excellent students I had, some of whom continued studying the topic and I became their doctoral supervisor. Mikl´ os Hossz´ u and Endre Vincze came over from Miskolc. You received your doctoral degree

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in 1962 under my supervision. I also supervised L´ aszl´o Losonczi and Albert Balogh. The times changed and you could travel abroad. I visited Poland again and in 1960 I went to both Germanies. My schedule was so busy that I could not accept the last minute invitation from Karlsruhe. However Zsuzsi suggested giving a lecture at the railway station. It was not that weird, as I later learnt, since the Maths Institute in Bern is practically in the same building as the railway station. I made a lot of connections. I found out that in social and behavioural sciences they also used functional equations. The question of bisymmetry was first raised by Wolfgang Eichhorn and then R. Duncan Luce, who I am still in contact with.2 You were still in Hungary when you organized the first conference in Oberwolfach, which was the beginning of a series of international symposia. In 1961 I was invited to a conference in geometry in Oberwolfach—I was then dealing with geometry as well. It took so long to get the permissions and visa that I arrived 2 days before the conference was over. I was so tired that I had to be woken up to give my talk. Then the head of the institute, Barner and other mathematician suggested that there should be a conference on functional equations as well in Oberwolfach. Of course I agreed with the suggestion and it soon became obvious that the committee would consist of Alexander Ostrowski, Otto Haupt and me. In 1962 the International Congress of Mathematicians was held in Stockholm and we invited researchers from there, too. However, the number of participants was still very low, there were about a dozen. For example Einar Hille, Jack Todd and Olga Taussky were among them. It was successful, so we organized a meeting almost every year after that. In the beginning it was in Oberwolfach, then in Waterloo and its vicinity when I was living there, then in other countries like Italy and later all over Europe, in the US and Canada. We had two conferences in S´ arospatak with several foreign participants, and then a third one in the fall of 1964 in Debrecen. Meanwhile the Bolyai Society had a conference on differential, integral and functional equations, where a strong team in functional equations participated. Some time you decided to continue you career outside Hungary. L´aszl´o Fuchs, who also left the country a few years later, agrees with me that if we had been left alone then we would have stayed in Hungary. I was to go to Florida in autumn 1963. My permission to travel and the visa (for which you could not apply individually but only through the department of foreign relations) arrived 2 months later. I am a precise man, and I wrote a letter to the Ministry that the 10-month holiday they permitted was beginning right then. I had been invited to a longer session, a six-week conference at Stanford University. There I received a telegram asking why I had not returned to Hungary yet. Then I contacted the Hungarian Embassy in Washington, 2

R. Duncan Luce passed away in 2012. (The Editors.)

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where I was told that the problem had been settled and my holiday ended in September. It was a narrow escape from getting disciplinary measures. My saving grace was that it was all in writing. But I was banned from travelling abroad for a few years. It would not have been that bad as I had already travelled a lot, but I had been invited to give a plenary talk at the congress of the Mathematical Society of the GDR, and it was very embarrassing with the colleagues. Finally my talk was read to the participants. Then came the notice of the delegation to China. Within the framework of an exchange program of the Academies I was supposed to have gone earlier, but because of the bad relationship with China it had been postponed. In autumn 1964 Khrushchev fell and the situation changed. Someone at the Academy thought that the country’s good relationship with China was dependent on whether I went and gave a talk on functional equations or not. I was notified through the dean ´ ad Haraszty. I wrote back and informed him that I had been banned from Arp´ travelling. For a few months I felt like a king in Debrecen and Hungary: if something was not the way I wanted I simply said I wouldn’t go to China. In the end I did. Finally we had had enough of the inconveniences, especially Zsuzsi. In this exceptional position I said I had travelled and worked a lot in the US and China and I needed a rest, so I would like to go to Austria for Christmas. We could go, because I had to apply for a passport at the university, Zsuzsi had to go to the town council and the kids to the local party representative. These applications should somehow have come together, but they did not, so all of us could go. We spent some time in Austria, then Johann Pfanzagl invited me to K¨oln. Then at the next conference on functional equations Michel A. McKiernan asked me if I wanted to go to Canada. Soon the invitation arrived and I started teaching there in September 1965. Until my retirement, more precisely my emeritus, I had been working and teaching there except for my sabbaticals and short journeys. If I am not mistaken, you were first invited by the University of Massachusetts. Did you have a definite position when you left Hungary? There were promises. I had an invitation to Amherst but back then I was too right wing for Hungary but too left wing for America, so it was hanging in the air. That’s why I welcomed the offer from Waterloo. Schweizer could go to Amherst, because when they invited me, I asked them to invite him as well. In the end he got there and I did not. The hesitation about the journey to the US took so long that when I was offered the position in Waterloo I decided if I got an immigration visa to Canada before the beginning of the academic year, then I would go. I received a special permit from the minister of immigration. Let’s talk a little bit about your family. You had two daughters when you left the country. How did they take the changes? Zsuzsi wanted to come, though she had her doubts every now and then. As for my daughters, Kati told me in Vienna that she wanted to live there. When

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we told her that we would stay she was very happy. Then we moved to K¨ oln. We were surprised to see that after a few weeks they wanted to go to school, to a German school. We had a nice garden, they could have watched television, but they chose the school. Kati finally accepted it when she went shopping for a blouse once. When she asked for a blouse in the shop, they asked what kind she wanted, short sleeved, long sleeved, elegant or more casual. It was a great experience for her that there was such a wide choice. Waterloo, Canada is still your home. You have had three outstanding disciples there: John Baker, Mark Taylor and Che Tat Ng. The university in Waterloo was relatively new. The research council was very happy to see me there, so I received quite a huge grant. I had several doctoral students, but the ones mentioned are really the most outstanding ones. I became a Canadian citizen in 1971. I often went to conferences and we also went on holiday a lot around the world. The conferences in functional equations and information theory continued. I also met you more often, and a place where we were together gave its name to an information measure. It is well known that you are a great traveller, but I think Paul Erd˝ os cannot be surpassed in that. You must have made a lot of friendships and found new collaborators during your journeys. You have the special skill of collecting people around you to solve problems. I enjoy solving problems with others. One of my papers was almost a world record with its five authors. I strived to find methods that are not only valid for a particular functional equation but a whole class. I never thought that it is the applications that one should do mathematics for but you should always be aware of them. There is a heated debate in Hungary about what is science and what not. Within mathematics there is considerable tension between the advocates of pure mathematics, who do not understand the applications and applied mathematicians do not see why others do not appreciate their work. The citation index and the impact factor, which are there to measure achievement do not ease the tension either. Do you think the performance of a mathematician can be described by a function of any variables? I do not underestimate applications, like I have said and I believe that we must be aware of them. The popular attitude they have in America that only what is applicable is valuable (and the sooner the better) ignores the fact that mathematics is applied in layers. For example mathematical logic may have an application in algebra, algebra may find an application in analysis, analysis in probability theory and statistics, which again may be applied in physics, chemistry, theoretical engineering and then finally we are getting to practical applications. But if this process is broken at any stage, the whole system breaks down. Being after fast applications is like saying let’s keep the beautiful flowers in the garden but get rid of the ugly roots. The impact factor seems to be an Eastern European myth. In North America or in Western

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Europe nobody seems to worry about the impact factor of the journal they publish their paper in. It does not matter where it is published the quality of the paper is what matters. When applying for a grant you can include how often your results are cited and the impact factor, but I am not sure if it matters, after all. Today even among scientists there is a fight for good positions. We have recently heard that there were two vacancies at a not so big American university. There were 140 applicants for one and 180 for the other. Obviously there is a need for filtering. In Hungary we see that publication is an existential need. The results do not mature, young researchers are urged to publish often and in great numbers. Achievement is constantly measured. This is far from being desirable. In the last decades also in North America it has been difficult especially right after graduation to find a good job. On the other hand those 140 or 180 or sometimes 300 applicants send their resume to 200 universities. A few years ago mathematical societies started suggesting that they should not send the same application to every university but they should tailor it to the particular university. The number of publications has increased indeed. There are more mathematicians and thus more good mathematicians with more good results. The Canadian system, and I think it has been adopted in the US, too, has effective protection against exaggerated publication and attempts to publish a dissertation in ten parts. In grant applications you can include anything you have written, but you can attach only 5 papers. So the applicant would want them to be valuable ones. There is also an alarming tendency of plagiarism. It can even take the form of putting down a result you hear in a talk and quickly publish it as your own. Let’s talk a little bit about rewards. I was given the title “distinguished professor” in 1969. At a conference on functional equations in Spain I received a Cajul Medal from the scientific council. In 1990 I was elected honorary doctor at the university of Karlsruhe, and in economics, not in mathematics. In 1971 I became member of the Royal Society of Canada, then in 1991, that of the Hungarian Academy of Sciences. I have also been an honorary doctor at Graz University since 1995, Katowice since 1996 and Miskolc since 1999.3 You have invited many Hungarian mathematicians to the university in Waterloo. There is a recent story about L´ aszl´ o Kalm´ ar, who made a 90 min contribution at the faculty meeting explaining how he thought the investments should be made. Then a regulation was passed that no visiting professor can attend the faculty meeting. The story is a little bit exaggerated but essentially it is true. L´ aszl´o Kalm´ ar—whom I had invited—expressed his opinion, maybe not about investments but in connection with the curriculum, for an academic year when 3

J´ anos Acz´ el has been an honorary doctor at Debrecen University since 2003. (The Editors.)

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he would not even be there. Then—not at that particular meeting, because it would have been rude, but at the next one—they changed the regulations so that only full time professors could attend the meetings, so visiting professors were excluded. In Canada faculty meetings are short, so I gave up smoking. They usually started at half past three and it needed a voting to continue it after five o’clock. I did not need a cigar like at the extremely long meetings in Miskolc or in Debrecen. It was surprising that they were so short as there are more people present at the meetings: all the assistant and associate professors, not only full professors. The schedule is tough so people do not get to talk about their hobby horses. How could you follow the development of Hungarian mathematicians? In the beginning it was through colleagues I invited because I had known them and thought we could work together. Later through correspondence, papers, proofs I knew about your disciples, for instance, what they were investigating and if I found that interesting or thought they were good and some research experience would be useful, I invited them. Can you say something about the education at Waterloo University? Earlier Waterloo University was unique in Canada and among a few in the world because of its cooperative system, where students attend university for 4 months and then work in their profession for 4 months. Mathematicians work at computer or insurance companies. Students could see where they could get a job after graduation. Companies and firms already knew them. Neither the company nor the student had any future obligations. By now many universities have introduced this system. Usually there are few printed notes but there is a wide selection of textbooks, even graduate textbooks. It’s amazing how poor their knowledge is when they finish secondary school and go to university. It is true all over North America. The university education, however, works miracles, because by the time they become graduate students, they excel. Some of them quit but not many. Sometimes it also happens in the co-op system that the company wants to keep the student even before they graduate. What are you most proud of ? I am most proud of my disciples and their disciples and those who learnt from me but not at the university. I also learnt from them when we were working together. I have many colleagues, such as mathematical psychologists, who stand their ground among mathematicians in applied sciences. In particular in the field of behavioural sciences there are many well qualified experts, who with some help became creative mathematicians. I am very proud of giving that little help to some of them. Do you have strong connections to the Hungarian culture and language? We would not have left Hungary had they left us alone. Hungary is essentially a pleasant country. J´ anos Horv´ ath (professor in Maryland) once said that there are cities where you can live well. Paris, Vienna and he also mentioned

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Budapest, if only they would let us, and I think Debrecen is also one of these. Probably I would not include Miskolc, but maybe Szeged. I have had a love of the Hungarian language, literature, poetry and music since my childhood. I do not see any reason why it should not stay like that. How comfortable do you feel about North American culture? Andr´ as Pr´ekopa once said: “You can find the most beautiful and the most dreadful things in America. The extremes exist side by side...” He was right when he said that. It is more valid for the US than Canada, though. Canada has its drawback, but it has the advantage of mixing the American and the European culture (especially the British and the French cultures, but because of the immigrants the entire European (and Asian) culture). It does not expect assimilation as much as the US has done until very recently. Financial and social differences are not that striking. However, it has only been so in the US for the last few decades. 30 years ago it was very difficult to tell how much wealth the person you were meeting had if it was some public place. They wore the same clothes, they spoke the same way, would go to the same pub or bar. There are shocking differences today. Most rich people are not happy with their yacht, they want a ship. The unemployment rate is not very high, it has gone down lately, but there are huge differences in salaries and with the technological development the differences will keep growing. Besides, I see two astonishing things in the USA: one is that it is the only developed industrial country where there is no social and health insurance, the other is that it is the only country where there is capital punishment. But I do not know the USA very well, although I spent 3 or 4 months in California in the past few years.

Zolt´ an Dar´ oczy Institute of Mathematics, University of Debrecen Egyetem t´ er 1. Debrecen, 4032 Hungary e-mail: [email protected]