傳播數學知識．促進數學教育

Interview with Prof. Peter Lax

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewer:** Tai-Ping Liu(TPL)

**Interviewee:** Peter Lax(PL)

**Date:** January 3rd, 2002

**Venue:** Institute of Mathematics, Academia Sinica

Professor Peter Lax was born on 1 May, 1926 in Budapest, Hungary. Several well-known mathematical terms are named after him, including the Lax Pairs for completely integrable systems, Lax Entropy Condition for hyperbolic conservation laws, Lax equivalence theorem for numerical schemes, and Lax-Milgram Theorem in functional analysis. He is a member of the National Academy of Sciences, was awarded the National Medal of Science in 1986, the Wolf Prize in 1987, the Abel Prize in 2005.

TPL: Peter, first, welcome and thank you for agreeing to this conversation. Let’s starts with early on in your career. As we know, Hungary has produced disproportionately many great mathematicians.

PL: That’s absolutely true. Hungary has produced Bolyai, Fejér, the Riesz brothers, Haar, Polya, Szegö, von Neumann, Erdös, Turán, Rényi, Lovász and many more. I once asked Marcel Riesz what was the reason for that. He thought the example of János Bolyai, who invented non-Euclidean geometry, was the inspiration. So in Hungary being a mathematician is very honorable. I became interested in mathematics at 12. I learned a lot from my uncle Albert Kornfeld who was a good mathematician; as a high school student, he won an important competition; he became an engineer. Later I was professionally tutored by a wonderful woman mathematician. Her name is Rózsa Péter; she was a logician, and wrote the first book on recursive functions; she was a brilliant pedagogue. The first material we studied was Rademacher-Toeplitz’ book, titled in English, “The Enjoyment of Mathematics; ” in German it was called Von Zahlen und Figuren, about numbers and figures. Its short chapters, five, six pages, are ideal for learning mathematics. Even today it’s a book I’d recommend to young persons interested in mathematics. In Hungary problem solving was an important part of mathematical education. Each year there was a contest for high school seniors; I participated unofficially in these contests, and did very well, better than the seniors. So when I came to the United States in December, ‘41…

TPL: How old were you then?

PL: About fifteen and a half.

PL: König, who was in charge of the examinations, wrote a letter to von Neumann, to Szegö, and to Szász, asking them to look after me; my mentor Rózsa Péter also wrote to von Neumann. Very soon after we arrived, von Neumann came to visit me, to advise me about my future. He was extremely kind to me then and also later.

TPL: What’s the age difference you have with von Neumann?

PL: Oh… von Neumann was born in 1903 and I was born in 1926.

TPL: I know von Neumann is one of your heroes and of course a hero to the rest of us. So maybe we can digress a little bit and talk about von Neumann.

PL: The next time I met him was in 1945. When I turned 18 I was drafted into the United States Army. After basic training and six months of engineering study at Texas A&M, I was transferred to Los Alamos, the center of the nuclear bomb project. That was like being in science fiction; von Neumann was a frequent visitor there; he made extremely important contributions to the project. He came as a consultant but he always gave a general math lecture. He was at the Institute of Advanced Study, but he spent a lot of his time at war projects. He was tremendously in demand, everybody wants to speak with him, not only mathematicians, but also physicists, and chemists, because he could understand things very quickly and then give sound advice.

TPL: He was simply very powerful.

PL: He had a very powerful mind. I never met anyone else who had such a powerful mind. Closest perhaps is Jack Schwarz. Paul Erdös was also very friendly and helpful to me. He gave me some problems, and discussed mathematics with me.

TPL: That was when you were still in Hungary.

PL: No. I was here. He was already here. In Hungary I was helped by Paul Turán, who was a very good friend of Erdös and himself was an excellent mathematician. There was a tradition in Hungary to help young people.

TPL: You mentioned that the Hungarians like problem solving. I know Polya has a book about it.

PL: Polya and Szegö have a book of problems and Polya has a book on problem solving… ‘How to Solve It’. Polya was interested in the psychology of mathematical invention. Szegö thought mathematicians should only do mathematics. So he had the informal agreement with Polya that until age 65 Polya will do mathematics. And after that, it turned out that Polya had another 30 years left to think about psychology and the teaching of mathematics. Polya was idolized by high school teachers; he was willing to talk to them and they were willing to listen.

TPL: When you came to US, you studied in high school or…

PL: Yes, I finished high school just in one year. I went to Stuyvesant High School, which is a very good high school that specializes in math and science; many distinguished people came from Stuyvesant, Jack Schwarz, Paul Cohen, Harold Widom and many others, and physicists too.

TPL: Somehow I have an impression that you didn’t finish, you didn’t go through complete high school.

PL: I didn’t go through the complete high school; at age 16 and half, I went to college.

TPL: And that’s NYU.

PL: That’s NYU. Szegö recommended to my father that he entrusted me to Courant, because Courant was very good with young people.

TPL: In fact, you didn’t even go through the complete undergraduate course either.

PL: I knew most of the undergraduate in math already in high school, and then, after a year of college I went to the army, so then I went to school at Texas A&M. And I spent two Summers at Stanford, and at Los Alamos I took some classes.

TPL: With, for example, von Neumann?

PL: No, von Neumann didn’t give a course. Fermi gave a course on nuclear physics, and Edward Teller gave a course on whatever he wanted. I took a course on Classical Mechanics. Anyway then I came back, and in a year I finished the college. But I didn’t have an undergraduate experience.

TPL: The education system wasn’t made for you. I see, so since then, Courant has been your home.

PL: That has been my home.

TPL: Could we turn to Courant, as a person?

PL: Oh, certainly. Before I left, I finished an essay about Courant. It’s really amazing how much Courant has accomplished, building institutions and exerting influence, in spite of being essentially an outsider, both in Germany and perhaps even more so in America. It is very interesting how his psychological makeup helped him to accomplish so much. He had very great enthusiasm, no prejudices of any kind, optimism in the face of what seemed like insurmountable obstacles. He had the ability to gain people's loyalty.

TPL: He was a very charming person.

PL: He was very charming, and very kind. His manner was ironic; when you were talking to him and asked him a question, he figured out what you expected him to say, then said something else. He was interested in putting people off balance, just to see how they would react.

TPL: Different people paint very different picture of him. Of course you had a very long association with him.

PL: Yes. He was a father figure for me. He and my father were very good friends. My father was his physician.

TPL: Then there is Friedrichs.

PL: Yes, Friedrichs was a wonderful influence on me.

TPL: In Constance Reid’s book on Courant, it says that Friedrichs could immediately sit down to work, and Courant would have to get up, walk around, and then finally he settle down to work, but when Courant settled down he can work for even longer period than Friedrichs.

PL: (laugh)I don’t know about that. You know, Cathleen Morawetz edited the Shock Wave book which was very successful; its influence is still felt today. The first task she had as editor was to reconcile the intuitive style of Courant with the great demand for precision that Friedrichs had; it went back and forth very often.

TPL: Friedrichs had admiration for Courant.

PL: Oh, yes, he had complete trust in Courant's judgment, and in Courant's willingness to look out for the people in his group and do the best for us that is possible under the circumstances. Courant was very flexible; the phrase “under the given circumstances” is a very important one.

TPL: The explosively growing US science at that period needed such personality.

PL: That’s right. The explosive growth started in the 30’s when the refugees from Europe started to arrive, mainly from Germany, but it also from Hungary and France, Poland, from all over Europe. It was a difficult task to find positions for all these people, but the American community worked very hard, Veblen particularly was extremely helpful. The level of American mathematics rose to an unprecedented high level.

TPL: MacLane at one time listed the immigrant scientists, and it looks like Who’s Who in science.

PL: Oh, yes, MacLane had also been very helpful.

TPL: He was American.

PL: He was an American. He has studied in Göttingen, and I think he lived in Courant’s house as a student.

TPL: Friedrichs is so different from Courant. At that time, you talk to which one more?

PL: I think, mathematically, I was influenced more by Friedrichs. Just finished a big book on Functional Analysis, which I originally learn it from Friedrichs, although already learned some of it from Courant and Hilbert.

TPL: I understand that the second volume of Courant-Hilbert was written mostly by Friedrichs.

PL: I think that is mostly the seventh chapter, on the variational approach to boundary problem for partial differential equations; it was not included in the English translation of Volumes II of Courant & Hilbert because the subject has grown so much that Courant want to make a separate book out of it. By then he was too old to do it; but I had many discussions with him, he wanted to know my opinion how he should do it. He often asked for opinions but he didn’t necessarily take people’s advice. I told him he should introduce Hilbert space, some abstract theorems, and show how it works in concrete cases. He said, no, he won’t do that; I asked him why not, and he said that when you introduce Hilbert space, you start working on problems of Hilbert space, instead of problem of mathematics. There is something to that.

TPL: This reminds me of something you told me about Friedrichs, who said that he liked abstract mathematics so much but didn’t dare work on it.

PL: Yes, like a potential alcoholic who dares not take an even single drink. I’ve quoted another thing of Friedrichs in the foreword to my book; Friedrichs said it’s easy to write a book if you’re willing to put into it everything you know about the subject. For him a book was a work of art; what was included, what was excluded, all contributed to its shape.

TPL: There was a Chinese Taoist who said that, what to do and what not to do.

PL: Or in a painting you look at the positive form but also the negative form.

TPL: In a Chinese painting, a lot of time, you have complete empty space.

PL: Chinese, and I guess also Japanese prints, has very bold compositions; when westerners first discovered them, they were startled.

TPL: A painting by Titian or Leonardo fills the whole canvas.

PL: It dare not leave so much empty space.

TPL: To change the subject, before you go back to Courant Institute later, many people, when talked about Peter Lax, said that you are noted by your choice of problem, your scientific taste, which is unique and admirable. You know, I am one of your many admirers. (Thank you). So, can you comment on that thing? What are you trying to achieve in mathematics? How do you choose a problem?

PL: I always choose something that interests me. The German mathematician, Schottky, when he reached 80, was interviewed by the press and asked what is the secret of his accomplishment. He said, “But gentlemen, when one thinks about mathematics for 50 years, one must think of something.” Hilbert, on a similar occasion, when asked about the secret of his great success, replied without hesitation, “I owe my success to my very bad memory.” He had to reconstruct everything.

TPL: So, you mean, he always thinks through problems anew.

PL: Something like that. For instance, Fredholm came out with integral equation which is analogous to linear algebra, only linear algebra in infinite dimensional space. Then Hilbert thought of it anew, but he thought of infinite dimensional Euclidean space, and so he developed the theory in that direction, which is more fruitful. Although Fredholm contribution was very important. I keep in touch not quite so much by reading, but by talking and listening to my friends; if something seems interesting then I think about it. Kruskal & Zabusky’s discovery of solitons started me thinking about it. What made me interested shock waves was von Neumann's idea of shock capturing. He didn't have it quite right, but I realized it had to do with conservation formally. There are psychologists of mathematics inventions; Polya was one. Hadamard wrote a little book on the psychology of mathematical invention, and a friend of mine, the psychologist Vera John-Steiner, recently wrote a book on the psychology of collaboration. When János Bolyai, at age 21, created non-Euclidean geometry, he wrote to his father, “Out of nothing, I have created a new world.” He was a very romantic person. But it seems that any discovery creates something new out of nothing, and that contradicts conservation laws. When it comes to collaboration between two people, then one can see the roots of discovery because each person has some incomplete ideas which then they fit together. I did quite a bit of collaborating, mostly importantly with Ralph Phillips, for something like 25 years; it was a very happy time for me, I enjoyed that tremendously. But creations by individuals rely, instead of one collaborator, on observing what others do. It's not something that you even remember, some of it is subconscious, but that's somehow what I believe to be the secret of creative work. It is within the mathematical culture. You move in mathematical culture like fish swimming in water. You are not alone. Even a lone genius isn't alone.

TPL: So you emphasize communication, talking to people is very important.

PL: Yes, yes.

TPL: Now, could we go back to the previous subject, which is your life in Courant Institute?

PL: It was altogether a happy one.

TPL: There are many preeminent mathematicians produced there, going through there.

PL: Oh, certainly. Many came as students, Clifford Gardner, Harold Grad, Joe Keller, Louis Nirenberg, Martin Kruskal, Cathleen Morawetz, and many others all came as students. And many of them stay on.

TPL: These people you mention, they are all very different?

PL: They are all very different.

TPL: Do you feel like to mention some of these people, tell us some...

PL: Louis Nirenberg came from Canada.

TPL: He is one of your oldest friends.

PL: He is one of my oldest friends. We joined the faculty at the same time and retired at the same time. We have only one joint paper but we have discussed everything between us. So the influence is much stronger. He got to Courant Institute because in Montreal he knew Courant's daughter-in-law Sarah Courant, who noticed this brilliant young man and wrote to her father-in-law. When Courant looked at Louis Nirenberg's record at McGill, the leading University in Montreal, he said, “This man has only A's, what's wrong with him?”

TPL: Turns out he was ok.

PL: Turns out he was ok. Louis was very much influenced, at first by Stoker, but really by Friedrichs. He had quite an interest in geometry which lasts to this day. His latest work was in geometry with Yan Yan Li, and from Friedrichs he learned the theory and tool for partial differential equations. The story about Cathleen is that her father was a mathematician, John Synge. Courant and Synge met at a math meeting and started talking about their families. Courant's older daughter Gertrude has just gotten her degree in biology, and in the beginning of your career it's always difficult to choose a path; Synge's daughter has just got a master's degree, and didn't quite know what to do. Courant said: “ Well, there's nothing you can do for my daughter, but I think I can do something for your daughter.” and he did. Courant was very encouraging to women, very early on. For instance, when right after Cathleen got her degree and was hired as a research associate, Courant made arrangement that she could work at home. I don't think there's any other institution that would agree to such an arrangement, but Courant had faith, and didn't give a damn about formal rules.

TPL: He had a long term, bigger goal than following the rule.

PL: Courant was also very nice to Anneli, my wife, also encouraged her. Anneli was Courant’s PhD student; he gave her a very good topic, and she wrote a wonderful dissertation. It is still quoted today.

TPL: Is this about degenerate hyperbolic equation?

PL: Yes. She was the first one to show that certain conditions were not only sufficient but necessary.

TPL: Where and when did you first meet Anneli?

PL: Oh, that’s a very romantic story.

TPL: OK. Can we share with that?

PL: Yes, I met with her in a graduate course on complex variables.

TPL: You don’t remember who taught it.

PL: Yes. I remember very well; it was a man whom I don’t want to name because he was incompetent. He copied out by hand Courant’s lecture and read from them. That was all right, but when somebody asked a question he became confused. After a while he noticed that I always had the right answer. He called on me more and more. That intrigued Anneli, especially when she found out that I was only 17.

TPL: It was very romantic. So, you have to thank that teacher.

TPL: Harold Grad…When I was visiting Courant in 1976-77, he looked like a tough guy.

PL: Yes. Right, right. He came from a poor family. And you have to be tough to survive. He had a brother who became a mathematician and ended as an administrator. He was really tough. He became president of Brooklyn Poly, and he was so tough that the faculty threw him out.

TPL: That’s too tough.

TPL: Kruskal had his degree in Courant Institute?

PL: Yes, he was a student of Courant himself. He wrote a very interesting dissertation about given two minimal surfaces, you can always join them by some thin surface.

TPL: We still have time before your tennis match. Kruskal is a quite interesting person.

PL: Very interesting person. You know, he is one of three brothers, all of whom are very good mathematicians. His older brother, Bill, was one of the leading statisticians in University of Chicago. And his younger brother, Joel, is a brilliant combinatorist. They had a very remarkable mother.

TPL: You have witnessed Mathematics for fifty years by now. Have you seen some changes in its culture?

PL: Sure. When I started out pure mathematics was king in United States; applied things were looked down upon. That has completely changed today.

TPL: I remembered reading in a book on Courant a sentence by Courant about pure and applied saying that you have struck a balance.

PL: You have to strike a balance. That is absolutely right.

TPL: No, he said that YOU have struck a balance.

PL: Everybody strikes their own balance.

TPL: But he liked your balance.

PL: I am very pleased.

TPL: But what do you view about pure, applied?

PL: You know, I got into applied mathematics early. After I got my degree, I took a job for a year at Los Alamos. The lab has just started using computers; numerical solutions of shocks were a big problem. It was fascinating. Also, it helps to be in a non-academic atmosphere, where the problems are the problems of the institution, and where you work with and talk to physicists and chemists. Ever since that time, I tried to send my students for a limited time to Los Alamos and to industrial labs. It’s a very good experience. You can’t quite get in a academic environment.

TPL: But in academia, we can’t reproduce such an environment. You feel that we should not try to reproduce such an environment? Is it not a role that academia should play? For example, in NYU, there is no engineering school.

PL: We have no engineering school. We lost it. That’s a rather big blow. There is now a big neuroscience institute, and there is a lot of collaboration going on with them.

TPL: And to speculate on the future of applied mathematics?

PL: I refuse. Certainly biology will be a very fertile field, many aspects of it.

TPL: I remember you said that mathematics is central to science?

PL: It’s the language of physics and becoming the language of chemistry. It lies at the heart of computing. Whatever the future for applied mathematics, computing will be a central tool. As von Neumann said “it is computing that will give hints of what is going on”; that has happened a number of times.

TPL: You feel that undergraduate math major should take some course in computing?

PL: Oh, Certainly. Even pure mathematics uses computing. And it’s fun because it’s an active thing. After you’ve been in school for so many years, it’s good to break out of the passivity and do computing. Today, we have so many wonderful tools. Matlab, Maple, Matematica.

TPL: They can have fun with relative ease.

PL: They can have a lot of fun. They really can find out something for themselves. At a time when they are not quite ripe for research, it could help them get into research.

TPL: In the Courant Institute you have a substantial computation program now, and even early on.

PL: That’s right. We have some brilliant computational scientists, Marsha Berger, Leslie Greengard, Olaf Widlund. If I look back fifty years ago, we knew nothing. Certainly, there was no computing. There was no theories of partial differential equations, at least not much. Second order elliptic, hyperbolic, parabolic, a little bit of the wave equation, three dimension, a little bit of fluid mechanics. There were few texts on partial differential equations; Courant and Hilbert, a book by Webster and Riemann-Weber. There were very few texts altogether when I was a student. In algebra, there was Van der Warden; in real analysis, there was nothing. To learn measure theory, Lebesgue theory, you had to go to early monographs, or to Saks’ Theory of the Integral.

TPL: You are maybe the only one who work on every aspect of PDE.

PL: I haven’t done much work on parabolic equations.

TPL: But the Lax-Milgram is sort of the basis of existence of solutions of elliptic and parabolic equations.

PL: The history of that is this. Art Milgram was a very good topologist who spent a year at Courant. He wanted to do some analysis; so we became friends. In fact, we used to play tennis. He asked me to give him a problem to work on, that is the best way to get into a field. I told him that for the self-adjoint problem there are variational methods. But we don’t know how to handle non-self-adjoint problems. He came up with what’s called the Lax-Milgram lemma.

TPL: PDE theory has developed so much these 50 years, half of a century. Maybe you can mention what were the most exciting things for you.

PL: The most exciting mathematical moment for me was when the discovery of differential topology by John Milnor. Milnor was able to prove that in 7 dimensions, there are spheres which are homeomorphic but not diffeomorphic. That was a tremendous surprise. The discovery of solitons made a big impression on me.

TPL: You have introduced Lax pairs into soliton theory and you have continued to work on solitons. But you haven’t followed all the other developments.

PL: I did the zero dispersion limit for KdV with Levermore. True, when I work on the subject and I turn to something else I have a tendency not to follow it up. Just laziness. It’s very hard to learn new mathematics. I remember once I was working with Friedrichs on something and found an important reference, which was in Russian. Friedrichs volunteered to read it; he said he knows a couple of hundred words of Russian and can do it. I was a little bit worried, that it will cause tremendous difficulty because it’s in Russian? He said: “ That it is in Russian is nothing; it’s the mathematics that presents the greatest difficulty to reading it”. For most people, it is very difficult to read mathematics. I know of only one exception, possibly two. I think von Neumann had no difficulty, nor did Jack Schwartz. When Schwartz was appointed to the faculty, he had secretly decided that he will teach all the courses that are listed in our Bulletin. When he finished, he switched to computer science. I asked him how he did it; there must have been some subject that he didn’t know. He said that the summer before, he would take out a few textbooks on the subjects and read them. I don’t know anyone else who could do that.

TPL: You are writing a book on functional analysis.

PL: I finished it, including the page proof. It’s done.

TPL: You have written other books with Ralph Phillips.

PL: I’ve written two books with Ralph Phillips. I included some of my works with Phillips in a chapter in my book on functional analysis and in the process I cleaned it up a little bit; I think Ralph would have been pleased. And our other book on automorphic functions, we wrote that prematurely. We wrote it before we have solved the problem; it’s an unsatisfactory book. But later on, we did very well. I think if Ralph were alive, he and I would rewrite it. But I’m not up to it. The experts in the field have ignored our contribution, except for Peter Sarnak, who has continued to work with Ralph and has contributed a lot.

TPL: Now that you don’t have to teach- or do you still teach?

PL: In the fall of 2000 I taught a course for seniors on linear algebra. I’ve written a book on it. Last spring, February, March, I taught a mini-course at Berkeley. And this coming May, I will teach for three weeks a mini-course in Budapest. Then, in the Fall of this year, I was invited to spend a month at Stanford. I may give a mini-course. I hope you will be there.

TPL: What’s the topic at Stanford?

PL: I haven’t quite decided. At Berkeley I’ve talked about the zero dispersion limit for KdV. I could also talk about some of the things that Phillips and I did. Or something in harmonic analysis.

TPL: So you will continue teaching and be busy?

PL: Yes. Now that my book is finished, I have to decide on some major project, and I don’t know. I would love to learn about random matrices. They have something to do with everything, including the zero dispersion limit. There are very good notes by Percy Deift that I can recommend.

TPL: Can I ask you a general question? What do you advise on the young PhD or undergraduate students interested in mathematical sciences?

PL: Oh, they should keep their eyes open, that what’s I can say.

TPL: Ok. Another general question. You have known so many mathematicians; who are your heroes?

PL: Among the older generation, Friedrichs was a hero of mine. von Neumann I place above all others. Charles Loewner was a hero of mine, as was Gabor Szego. Among the younger people that I know well, I admire Martin Kruskal, Louis Nirenberg, Peter Sarnak, Charlie Peskin and Charlie Fefferman. Wonderful mathematicians.

TPL: What was so good about Friedrichs?

PL: Let me tell you a story. The Courant Institute nominated Friedrichs for a National Medal of Science and I was sort of the manager. The first thing I wanted was the endorsement of the American Mathematical Society. I called them and I was told that there was actually a committee devoted to it; the chair is the latest recipient of Medal of Science, Paul Cohen. I was glad to hear that; he was an old friend of mine. I was coming to Stanford anyway. So I talked to him. He wanted to know what difficult problem Friedrichs has solved. I said, that the nature of his work is different; for instance, he introduced the concept of the Friedrichs extension that made the theory of self-adjoint operators usable. Paul would have none of it, so I went around him and got the endorsement. In fact, at one point I was put on the Medal Committee; then it was rather easy, because usually when a mathematician’s name comes up and his accomplishments are described; they mean nothing to the Committee, which consists of scientists. They say, OK, mathematics get one medal. They knew Friedrichs because of his contributions to fluid dynamics. So that year mathematics got two medals. When it was all done, I congratulated Friedrichs. Friedrichs said, “You know, I never solved a difficult problem”.

TPL: (laughs) Okay.

PL: Another person I admire very much is Tosio Kato.

TPL: He is a very quiet person.

PL: Very quiet person. But I did read what he wrote. He brought functional analysis to bear on more problems than anybody else. He started his life as a physicist.

TPL: You give us the impression that your life with mathematics has been good.

PL: Yes. I have enjoyed it tremendously; I’m enjoying it still. In fact, I sent emails to my friends telling them I have steady the diet of Chinese food and mathematics.

TPL: Well, you should come regularly to this part of the world.

PL: This is my third visit. My first visit was in 1980; it was a very critical time for Chinese mathematics. In fact, everything in China has been ravaged by the Cultural Revolution. It wasn’t clear whether it could be repaired. To obtain graduate training students were sent abroad. It wasn’t even clear that China could offer undergraduate training. I have tremendous respect for the outstanding people who saw it through. But now of course it’s quite different. At the second Chinese International Congress, I was really deeply impressed how high the level mathematics in China is. In fact, I think that in a short while it won’t even make sense to maintain the tradition of special Chinese International Congress, although it’s a nice tradition. As long as it was important to build it up, it was a good thing. Now, it’s just a historical thing.

TPL: It was a historical period.

PL: The period is ending.

TPL: Now China is a part of the global mathematical community.

TPL: Indeed, when a culture has vitality, it ought to be global. Under the Chinese Tang dynasty, it was very global. You have Persian astronomers, Hindu mathematicians, Arabian mathematicians, all in the Tang dynasty Capital. This is no such thing as Chinese mathematics.

PL: You know, Arabic culture at the time of Caliphate was very high. They contributed mathematics and astronomy. Perhaps it was at its height, the strongest school of science in the world. Europe was sunk in the Medieval dark age.

TPL: Tang was 6-9th century.

PL: It was even before the fall of Caliphate.

TPL: Maybe that’s why Tang could import mathematicians from that part of the world.

PL: How much contact did Tang have with Europe?

TPL: Oh, a lot, definitely, more than what we know of. They used the Silk Road. They sent a Tang monk, called Xuan Zhung (玄莊), to India. If you can get to India, from there it is easy. Before that, the high mountains between deserts were considered too hard to pass. And sea, of course also.

PL: Did the Chinese have ships in order to make such a long journey?

TPL: Sung dynasty had big ships. Quan-Zhow (泉州) and Alexandria during the Sung dynasty were the two biggest sea ports in the world. And then China became isolated and began to decay. And so we now go a full circle. We now have this International Congress of Chinese mathematicians. As you pointed out, there is really no need for it today.

PL: No need for it, but it’s a nice tradition. It’s nice to see how many Chinese occupying important positions everywhere in the world.

TPL: Yeah, that’s nice word of encouragement. So maybe we make this as your concluding remark.

PL: Yes.

TPL: Thank you again. And come back soon.

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.