MacTutor
A 1968 interview of György Alexits
In 1968 István Kardos interviewed György Alexits. The interview appears on pages 34-53 of the book Scientists face to face (Corvina Kiadó, 1978). We give below a version of the interview.
György Alexits interviewed by István Kardos.
György Alexits was born in Budapest, in 1899. He studied in Graz and at Budapest University. He pursued his studies in mathematics and medicine during the war years when he was in active service. There was a break in his career when, under the 1919 Hungarian Republic of Councils, he was a member of the Socialist Students' Association. However, an exceptional talent and a sense of vocation finally led him back to mathematics. During the Second World War he was a member of the resistance and was interned in the Dachau concentration camp. After the German defeat, he returned home. In 1947-48 he was Under-Secretary of State for Education, from 1948 a university professor. From 1950 he was also general secretary of the Hungarian Academy of Sciences. He is honorary president of the Mathematical Society. His main subject is the orthogonal series and the constructive theory of functions. He has held temporary lectureships in a number of famous universities. His biography of János Bolyai, the famous Hungarian scientist, is well-known as a valuable contribution to the history of science. György Alexits's flat in Városmajor, a suburban district of Budapest, is full of valuable pieces of peasant-baroque furniture, beautiful tapestries and other objects of folk art. He has a broad-shouldered, sportsman-like physique and is a man of forceful temperament. He receives me in shirtsleeves, and in our conversation there is very little of the "academic ".
István Kardos: I have always looked with a certain awe on those who understand mathematics, who, by some miracle, as it seems, know the ins and outs of this crystalline system. How do you see yourself? How does a mathematician think?
György Alexits: Would it not be possible to ask something less difficult? No doubt you find it hard to realise that a mathematician, while working, thinks just like anyone else, only much more precisely as he uses unambiguous concepts only. And it is this unambiguity which is decisive. This is the pith and marrow of mathematics. I would like to give an example. The notice you have all seen in the trams, "Anyone travelling without a ticket or with an invalid ticket must pay extra", works splendidly in everyday life. The message is clear to everyone. However, if this sentence were to be written in the strict form of a mathematical concept, it would probably take half a page, that is, it would be unusable in practice. For a mathematician, on the other hand, the inexact, not entirely unambiguous notice in the tram is unusable. Why? Let's take "Anyone without a ticket". Well, if you please, who is the man who travels without a ticket? Everyone who has no commuters' card and who takes a tram will be without a ticket when the tram starts. Therefore, according to the final words in the notice, he, too, should pay extra. Obviously, that is not the intention. The notice also says that those who travel with an invalid ticket will pay extra. How do I know whether, crumpled in the pocket of a commuters' card carrier, there is or is not an invalid ticket? But of course this is not being questioned. In everyday practice, we all understand what is actually at issue, but a mathematician could not work with such an ambiguous statement and such vaguely exact concepts. For him even the seemingly absurd cases must be taken into consideration.
István Kardos: I believe there are many developments in the field of mathematics today?
György Alexits: Mathematics is a science which, from ancient times to the present day, has scarcely ceased to show development. In the Middle Ages scholars were not permitted very much scientific investigation. For a long time the Church held the view that speculation in the natural sciences was dangerous. But this was not so in the case of mathematics. It is, of course, true that the heliocentric world view which was so hotly challenged in the 16th and 17th centuries was based on mathematical calculations. Fortunately the Church did not realise it at that time. The study of pure mathematics was tolerated, even supported.
István Kardos: One of my mathematics teachers at grammar school used to say that anyone could understand mathematics. Today, however, it is not unusual to find that one mathematician does not understand the other. How can that be?
György Alexits: Well, I have to agree that it may very well come about that two mathematicians, outstanding in their respective fields, do not understand each other. Perhaps I show you a Springer catalogue. This, as you see, mentions a book called "". Frankly, I not only do not understand what it means, but I would not even know that it is about mathematics had it not been listed with the other books on mathematics. One shouldn't be surprised if one mathematician does not understand what the other is talking about. It is so just because - as I have said - mathematics has been developing for two and a half thousand years and during this long time it has developed many ramifications. As a matter of fact, we shouldn't say of a man that he is a mathematician, but that he is a topologist, a theoretician of series, an algebraist, etc., and there are subdivisions even within these fields. Medicine - I mean, the science of medicine - is not so old. Nevertheless, it is obvious that an orthopaedic surgeon and an ophthalmologist will scarcely understand each other when speaking of their own field. It is therefore not difficult to imagine a failure of communication between mathematicians who deal only in abstractions. That is one reason why mathematicians may not understand each other. Of course, those working in the same field do not experience this difficulty. There is another way in which mathematicians may fail to understand each other but it is extremely rare for this to happen. It is when some quite exceptional genius advances an unbelievably bold, even quite radically new range of ideas, ideas never previously entertained. One typical case is that of the Hungarian János Bolyai, one of the creators of non-Euclidean geometry. For two thousand years everyone looked in one particular direction for the solution of a well-known problem of geometry; that is to say, everyone believed that the so-called Axiom of Parallels could be deduced from the remaining Euclidean axioms. But János Bolyai had the idea - revolutionary and daring in his age - that perhaps this was not true, that the truth was exactly contrary to this, that this postulate was independent of the other axioms. This János Bolyai in fact proved according to the expectations of his days. At first, not even his father, Farkas Bolyai, an outstanding Hungarian mathematician, could accept the idea. It was to him such a daring, even absurd theory that he contradicted his son. They argued about it. Farkas thought his son would do well not to pursue the idea. Finally he sent the paper written by his son to Gauss. Gauss, who had also studied this subject and whose thoughts were running along similar lines, was able to understand it. In short, what I am saying is that Farkas Bolyai only began to realise that his son was right when he saw that a great authority like Gauss was ready to support him. There are of course similar cases in the field of mathematics, but they are very infrequent. Usually the failure of understanding between mathematicians is due to the fact that they work in different fields. And then they do not really care even to talk to each other of their special interests. And of course that's not a good thing. It would be good if representatives from the various fields could get together - and there is an attempt to arrange this - but there is such a vast quantity of material it is not at all easy. Today there are about three hundred periodicals which every research worker ought to read. Well, it is a physical impossibility. There would be no time left for thinking. It would take up every minute of their precious time.
István Kardos: How would you advise a talented student interested in mathematics to set about becoming a specialist? Someone with the ability to contribute something new to the field?
György Alexits: There is practically only one way: to have the kind of nature that makes him deal with mathematics all the time. He should want to deal with mathematics. Today this is advanced institutionally. There are, for instance, the special mathematical classes in some grammar schools, competitions, and so on. When I was a child it was a different matter. I had a teacher who - poor thing - had a special talent for discouraging us in our efforts to understand the formulas on the blackboard. And I got discouraged all right. Or rather, I didn't learn anything. I had no idea of the subject as a whole, I didn't even understand what the teacher was talking about during the lessons in mathematics. As a result of this, at the end of the first half-year I got what they called a starred three, meaning, "though I haven't yet abandoned him, if things go on like this he will fail at the end of the year". And it was then that my father and a dose of influenza played their part. I was thoroughly down with flu and had to stay in bed convalescing for quite a long time. And my father said: "This is intolerable. Take your book and learn something. You simply must not fall behind in mathematics!" Sadly, I produced the excellent König-Beke algebra book and began to read. By the second or third page I had realised that there was nothing to learn, everything was clear. So I went on step by step until I came to equations. There were verbal problems which you had to solve by simple equations. I found it amusing. So it dawned on me that here was something worth learning. Then my father didn't interfere any more: he was satisfied that I was learning something. He, in fact, had no notion of mathematics: he was a philologist. He compiled a collection of folksong texts, which was republished with critical notes in Romania on the occasion of the 100th anniversary of his birth.
István Kardos: Do you really think that it was because of your father's remarks when you happened to have influenza that you were guided towards a career in mathematics?
György Alexits: Partly. Of course, you shouldn't take too seriously the enthusiasm of a child on the threshold of adolescence. I was always in mischief, I was a perfect scoundrel. But I was interested in many things. In music, for example. For a long time I had the illusion that I would make a good musician.
István Kardos: How were you encouraged in that idea?
György Alexits: I played the piano from childhood. I was lazy - I did not practise enough - but still I managed to reach a tolerable level. When I was all set to be a musician, my father took me to see Béla Bartók, with whom he was doing a good deal of work on Romanian folk-song texts. And this, of course, influenced me still more towards music rather than mathematics. Many years later I, too, had the honour of working with Béla Bartók on his Romanian folk-song texts; that is, I helped to verify the texts. The way we worked was for Bartók to give me the folk-song text on which I made comments in green pencil for easy recognition; then Bartók went over it with me. He then indicated with a red pencil whether he approved my comments or not. So he arrived at his final selection. By the way, Bartók was a stickler for accuracy. It was not enough for him to take down the music perfectly, he also wanted the text to be faultless. To work with him, just to meet him, was an unforgettable experience.
István Kardos: Did you dream of taking up anything else?
György Alexits: Oh, yes, indeed. My mother would have liked me to be a doctor. Imagine this absurd situation: from March 1917 I was an Austro-Hungarian soldier. As such I could be accepted into any university. To attend, of course, was not possible, because the lectures were not exactly audible from the front: I was admitted to the Medical School, but mathematics, my first love, still kept me enthralled, so I also registered for mathematics. As for the lectures, I didn't go to any. By some miracle, however, in 1919, at the end of four terms, I somehow managed to pass my exams. And, as a matter of fact, it was then that I began to take a more serious interest in mathematics, though I was not clear which branch to take up: at first I thought of studying pure mathematics. It was not possible to continue my studies in Hungary. After the fall of the Republic of Councils it seemed more expedient to take refuge abroad. I went to Graz (Austria) where I had an outstanding teacher. There I settled down in earnest to study theoretical physics (although I have since managed to forget everything about physics). But then Hans Hahn's wonderful book on mathematics came out: Theorie der reellen Funktionen was the title. How it came into my hands, I do not know. But from that moment I could not free myself from this range of subjects. Really, that was how I became a mathematician: I taught myself. In 1922, I was allowed to return home at last, but the only advantage I had was that I was not pestered by the police. I could not sit for the examination for a master's degree at the university because, as a founding member of the Socialist Students' Association, I had been expelled from the university. It was only in 1928 that a ministerial amnesty made it possible for me to sit for my exams after attending two semesters. In the meantime, of course, I also had to live, so I did everything under the sun. I was a music critic, an actuary for the Hungaro-French Insurance Company, the Phönix. I spent a year in Romania, because I couldn't earn a living at home. I translated articles for the Schering Pharmacological Factory, I gave private lessons, in short, I did everything that came my way. At last in 1930, when I had already had a teacher's diploma for a year, I was sent to one of the higher elementary schools of Budapest, as a supply teacher paid by the hour. This meant that I was paid only for the lessons which I actually gave. That is Sundays, holidays, Bird and Arbour Day, were all at my expense.
István Kardos: Did you find it possible in such circumstances to do any research or scientific work?
György Alexits: Don't think that I am urging mathematicians to live in the worst possible circumstances, but it is a fact that a mathematician can do research under rather adverse conditions. There are public libraries where he will find the literature he needs and, after all, his only real tool is his own head. Paper and pencil are the most he requires and they are not essential. On the contrary. Thus, even if circumstances were difficult, and whatever the background situation, mathematics was like a game to me, one I thoroughly enjoyed. In those days I worked on the theory of sets and the theory of real functions, orthogonal series, approximation theory, because that is what I was interested in. Let me show you a book about some small results achieved during those years. It was published by the Hungarian Academy of Sciences. Then Pergamon Press brought out a revised edition in English; later it was translated into Russian and published in Moscow.
István Kardos: These formulas seem rather complicated.
György Alexits: Naturally. Systems like this are difficult to explain exactly. As for the approximation theory, perhaps that is what I could speak most clearly about. The description of the abstract connections of the real world is expressed mostly in functions. But there is a function which, although its existence is already proved, cannot actually be expressed in simple functions. The question is how is it still possible to express such a function as simply as possible. In most cases, by polynomial sequences only, step by step. In the first, second, third, tenth millionth etc. step, I get closer and closer - through the appropriate functions already known - to the one I am looking for. To take infinitely many steps is impossible in practice. What is to be done on such occasions? One has to stop somewhere after the -th step, as we say. Let's presume that I stop after step 500. Who will tell me the extent of the error I have made by omitting infinitely many members and stopping at 500? The expert on the approximation theory has told me in advance that after so many steps the error will be limited to such and such a figure at most. And this is very important. Both the pure mathematician and the applied mathematician will know in advance what deviations must be taken into account if they stop after a certain point.
István Kardos: How can such a theoretical method be applied in practice ?
György Alexits: Today, when we make use of extremely rapid computers, it is of enormous importance to be able to give the margin of error in advance. To know that after a certain number of steps with what exactitude, with what little margin of error, we have approached the perfect solution. This is very important. But, to give a more spectacular example: when the bus garage in Hamzsabégi út was being built, Pál Csillag was asked to do mathematical calculations concerning the stressed-skin structure. Csillag was a first-rate mathematician, but there was a lot to do and he asked me to assist him with the calculations. We first produced figures to show the necessary strength of the structure. Our figures were agreed to by the structural engineers who said that a practical application was possible. Then, coupling these calculations with the appropriate engineering expertise, they constructed the garage. That our calculations were not made in vain was proved by the fact that during the war the garage was hit six or eight times; flying bombs went right through the roof and exploded on the floor; the garage, however, remained erect and the roof did not fall in. There are other such instances too. Of course, we mathematicians don't always envisage all the possible applications of our propositions. But then that's not supposed to be our job.
István Kardos: Do you mean that the practical application of your mathematical calculations comes later?
György Alexits: That is so. The mathematician starts from the idea that he wants to do something beautiful, something important, essential. From the point of view of those who put his calculations to practical use we could perhaps compare the mathematician to a screw manufacturer. He manufactures the screws and doesn't ask in what machines or buildings these screws will be used.
István Kardos: I believe that in 1941 you became a teacher at the Eötvös Grammar School in Reáltanoda utca, then docent. Presumably you then had better conditions for your work?
It was Professor Alexits's wife who replied: That would have been so had it not been for what was taking place around us in the world. It was the fourth of November 1944, on the day when the Margaret Bridge, undermined by the Germans, was blown up. My husband didn't come home at the usual time. My heart misgave me as I knew he was active in the illegal resistance movement. I took his most recent photograph to the police to see if he was listed as missing. The police official went through a great many papers, made several telephone calls, then strongly advised me not to go on looking for him. I came to know only later that my husband had been carried off by the Gestapo to Dachau.
Professor Alexits resumed: After the Liberation I was given a formidable number of jobs. First I became headmaster of a grammar school for girls, then Under-Secretary of State for Education, then, then General Secretary to the Scientific Council. In 1948 I also became a member of the Academy of Sciences and in 1949, when the Scientific Council was dissolved, General Secretary to the Academy of Sciences. I admit that those years gave me little time for mathematics. Still, in the midst of organising, I busied myself with it in secret. Later I lectured on mathematics at the Technical University and, at long last, I was able to find time for mathematical research to make this my main occupation. Right up to last year I was at the Budapest Technical University as well as having a second job in charge of a department at the Research Institute for Mathematics. Now this has become my principal occupation. I have nothing to do apart from research. It is a pity, it has come a little late ...
István Kardos: You are a mathematician with an international reputation and must certainly have travelled a great deal abroad as Hungary's "roving ambassador" for science?
György Alexits: I have been to a good many places. I have given lectures at perhaps thirty universities. My first extended series of lectures - I mean, for a whole term or more - I gave in Argentina in 1960. From 1963 to 1966 I was in the U.S.A. Salt Lake City; there for a year I had the chance to sum up by and large the things I am dealing with, which are of interest to me and which, I hoped, were of interest to them too. My last visit was to Canada where I stayed for three months, a short Canadian term; in fact not even a term. I gave lectures on approximation theory.
István Kardos: How would you assess our progress in the application of mathematical methods, especially in the field of social sciences?
György Alexits: That's a very difficult question to answer. Nowadays there is something of a fashion for mathematics. Nothing wrong with that. But it is no good expecting marvellous applications such as we have seen in the field of physics, for instance. Marx said, a science is truly advanced when it is able to make use of mathematics. Taking this as a yardstick we can say that in some respects the social sciences have made use of mathematics and have been doing so for a long, long time. Take statistics, for example. The use of statistics in the social sciences is well known. It is applied in several spheres; in economics and planning, for example. Then there is the so-called theory of games, which deals with strategies. Such theories have been put to practical use. Here, however, the trouble is very often that the mathematician and the scientist who wishes to apply the mathematics cannot understand one another. Each has something important to say in his own field. But it is as if one spoke in Turkish and the other in Norwegian. It would seem, then, that to achieve what is known as a mathematical model, a fusion of two types of knowledge is necessary: the mathematician should have a certain level of knowledge of the relevant science, while the scientist - or any specialist wanting to make use of mathematics - should have an aptitude for mathematical thinking which ensures the conditions for mutual adaptation.
István Kardos: Exactly. Nowadays there is a good deal of talk about the connection between mathematics and the arts, and even more about the connection between mathematics and music. This is then linked to the fact that there is a renewed interest in pre-classical music. The extraordinarily flourishing cult of Bach is usually mentioned as an example, and also, to a certain extent, the great international interest in Bartók.
György Alexits: Well, it is very hard to comment on that; it is a difficult thought to express in words. The connection between mathematics and music. ... Well, let me begin perhaps with something negative: I do not in the least subscribe to the idea that the application of mathematics, elementary mathematics in particular, to certain compositions must involve a relationship between mathematics and music. That is quite ill-conceived. But there is some connection which it is very hard to express and which I will try to explain as best I can in the following way: both mathematicians and musicians and indeed all true artists strive to express the inner, abstract beauty, harmony and unity of things. Each in his own language. Music is understood by many. Mathematics by a few. This does not mean that a mathematician does not experience through his own creative work the same aesthetic pleasure we all feel when listening to some very beautiful piece of music, or the joy of the composer when it dawns upon him that he has achieved a truly beautiful solution. Certainly, it is a fact that many mathematicians are deeply interested in music. I would not be so bold as to claim that the reverse is also true, though during my conversations with Bartók he said once that he had always been interested in trying to solve the problems in his son's textbooks on mathematics. That is, he liked mathematics.
István Kardos: We have already heard that the methods of mathematical analysis have been used in literature, and even in other arts, for the identification of stylistic traits. But I wonder whether mathematics could ever help to create a work of art, that is, to play a part in the actual composition of a piece of music, for example?
György Alexits: It may, perhaps, help to give a more accurate analysis of the elements of style. This is, of course, necessary. We know that composers, too, have their own techniques for composing music. But I don't believe that mathematics could directly contribute to the value of a composition as we cannot construct a mathematical model of the beauty of a work of art. For that, one would need to know more about the abstract structure of what is aesthetically pleasing. I mean that if someone knows the theory of all the elements of style necessary for the fugue form and, on the basis of this, writes one, it is possible that this fugue will be faultless from the point of view of musical form. But it will not be like a fugue composed by Johann Sebastian Bach. He gives us something more, just as there is something more in the creative work of the mathematician who does not merely apply, in a simple, automatic way, the various rules and methods of mathematics. There is always the plus, the creator's plus. He realises when and what to apply, to what extent. And so it is, of course, in music as well. Someone once queried - I think it was Romain Rolland writing about Beethoven - how it was that a chromatic scale progressing downwards expresses tragedy in Beethoven's music while in other music it is merely trivial. This exemplifies the fact that there is a plus which we cannot express, and for which it is also impossible to construct a mathematical model. Just what this plus consists of, it is difficult to say. Let us perhaps listen to a Bach record, at any rate one passage from "The Well-tempered Clavier". It is beautiful, but can you tell why it is beautiful? And if I made a fugue, conforming to all the rules, why would it not be beautiful? For it certainly would not be beautiful, merely regular. And something else comes into it too, something not so easy to grasp: I mean the creator's genius.
István Kardos: I scarcely need to ask about your hobbies: you have many pictures on your walls, one more beautiful than the other, period furniture, fine old carpets - and music.
György Alexits: I love anything beautiful. You could say that's natural. The love of beauty is seldom restricted to a narrow field, to mathematics only, for instance, certainly not to one branch of it. One should love every kind of beauty: that is to live as a human being. I love beautiful women, of course; I like good coffee - I love to live. And I love people. It sounds so pompous I hardly like saying it, but it really is so. I believe I may safely say that working in the field of mathematics, doing mathematical research is not merely a matter of logic: it also has an emotional element. There is a feeling of pleasure when one has solved some difficulty. That is, I love both the struggle and the joy of ending it. It's very difficult to speak of such things without slipping, all unawares, into platitudes. And I wouldn't like to do that I believe that there are strong links between beauty, truth, the world, nature itself. Our task is to attempt, using our individual talents and opportunities, to reveal these links.
Last Updated March 2026