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A man with no home and no job, Paul Erdös (1913–1996) was the most prolific mathematician who ever lived. Erdös inspired generations of mathematicians throughout the world with his insightful approach and wry humor.
N is a Number: A Portrait of Paul Erdös follows him for four years through four countries, presenting his mathematical quest and its personal and philosophical dimensions. Animated sequences illustrate the kinds of mathematical problems Erdös pursued.
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N is a Number A Portrait of Paul Erds TRANSCRIPTS, SUBTITLES AND CREDITS (CARD OVER MUSIC) I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me. -- Sir Isaac Newton INSERT LOWER THIRD CAMBRIDGE, ENGLAND June 1991 NARRATOR Cambridge Universitys honorary doctorate is one of the worlds most prestigious awards. In 1991 the eight recipients of the degree included Mary Robinson, the President of Ireland, Sir Alec Guinness, German novelist Stefan Heym, Nobel prize winner Pierre Gilles de Gennes...and Paul Erds, a 78-year-old Hungarian mathematician. With more published papers and collaborators than anyone before him, Erds is the most prolific mathematician who has ever lived. (Main titles over procession wide shot) N is a Number A Portrait of Paul Erds A documentary film by George Paul Csicsery NARRATOR To Erds, mathematics is the key to the transcendental truth underlying all of reality. Solving a problem is a triumph over the unknown. Erds has no home and no job. For more than half a century, he has traveled constantly, surviving on fees from lectures and appearances. RONALD GRAHAM (twirling pennies onto a plate) Here, you want to do it? RONALD GRAHAM 2 It's actually not quite circular. PAUL ERDS What is that? What drives this? 3 RONALD GRAHAM It just does it by itself. PAUL ERDS No. That I don't believe. RONALD GRAHAM You want to try it? PAUL ERDS Can you give it to me? RONALD GRAHAM Well, I can't really pick it up. That's the problem. PAUL ERDS Try. RONALD GRAHAM It's not that simple. PAUL ERDS Start it again. RONALD GRAHAM You have to come over. RONALD GRAHAM (v.o.) Paul Erds is perhaps the most remarkable mathematician alive today. INSERT LOWER THIRD RONALD GRAHAM Director, Mathematical Research AT&T Bell Laboratories RONALD GRAHAM He has started so many fields which have blossomed and interacted and have become very important in mathematics and even in computer science. VERA SS He has about thirteen-hundred papers. INSERT LOWER THIRD VERA SS Mathematical Institute Hungarian Academy of Sciences 4 VERA SS I don't know exactly how many, and he doesn't know. And I think there is nobody on the world who knows exactly how many. JOEL SPENCER (v.o.) The Erds Number is a favorite among mathematicians. Erds has an Erds Number of zero. If you write a paper with Erds, then you have an Erds Number of one. INSERT LOWER THIRD JOEL SPENCER Courant Institute (NYU) JOEL SPENCER And I think there are between one and two hundred people that have an Erds Number of one right now. If you write a paper with somebody who has written a paper with Erds, then you have an Erds Number of two. And if you write a paper with someone who's written a paper with someone who's written a paper with someone who's written a paper with Erds, then you have an Erds Number of three. Pretty much everybody has an Erds Number of three or less. RONALD GRAHAM (v.o.) I have the best collection of Paul's reprints outside of Hungary. I made a special effort to get every single one. And some are in journals that don't exist anymore. Some are in obscure Hungarian journals or South American journals, and Paul doesn't even know where some of his papers have appeared. PAUL ERDS You know this joke about the Erds Number? And now there's a new definition: If I have k joint papers with somebody, then his Erds Number is one over k. And Hajnal and Srkzi are leading. but among the women, Vera is clearly leading. Her Erds Number is less than onetenth. NARRATOR Vera Ss has taught mathematics in Hungary since the nineteen-fifties. She was married to the late Paul Turn, a renowned mathematician and one of Erds's closest friends. PAUL ERDS ...among the even numbers, if you only consider the even numbers. VERA SS The even numbers... VERA SS If you ask a question, he will remember what is in one of his papers he wrote in 38 and he will remember what was the problem he discussed with somebody in 47. And if we start to work on a problem and we stop in the middle and we meet two years later, he will remember where we 5 stopped, what we proved, what was the problem. Well, which is fantastic because he works simultaneously, I dont know, maybe with fifty people or more, or hundred. And he remembers with whom on what problem he works. INSERT LOWER THIRD J.W.S. CASSELS, F.R.S. Trinity College University of Cambridge J.W.S. CASSELS Just like the bumblebee goes from flower to flower carrying its loads of pollen, so Paul goes from mathematical center to mathematical center with his problems and his information, thereby being an agent of mathematical cross-fertilization. I dont want to interrupt this process any longer and so I call on him to give his talk. (applause) 6 PAUL ERDS I introduce myself as a joke like this. (writes on blackboard) Now it takes a little bit of explaining. PGOM means Poor Great Old Man (laughter) and the first LD means Living Dead. You get that title when you are 60 (laughter). A D, the second means Archological Discovery, when you are 65. Actually, I remember when I was 65, a friend of mine who can draw quite well sent me a picture of my birth, that they found me and said, Hey, another archological discovery." And this means Legally Dead when you are 70. (laughter) Well, now they asked me, What will you say when you are 75? About five years ago I said always as a joke, maybe I wont have to meet this emergency. But by the time I reached 74, I realized that probably I cant escape, so I found the word CD, which means Counts Dead. So let me stop this nonsense and talk about mathematics. NARRATOR Erds was born in Budapest in 1913 amidst tragic circumstances. Just days before his birth, his only two sisters died of scarlet fever. His mother never really recovered from the loss. World War One took Erdss father to the front where he was captured by the Russians. MRTA SVD The father was an active school teacher, mathematics teacher. The mother was also a trained mathematics teacher who taught for awhile, but after the war, in 1919, as a lot of Jewish teachers, she was sacked. INSERT LOWER THIRD MRTA SVD Australian Mathematical Society MRTA SVD Or most of them werenot Pauls father, who was prisoner of war. NARRATOR With his father in a distant prison camp and his mother teaching away from home, Erds was left alone with a German governess. He learned how to count the days until his mothers school holidays would bring her home. Nobody remembers how he became a child prodigy. PAUL ERDS ... but I could multiply three, four digit numbers in my head and I amused myself by asking people how old they are and told them how many seconds they lived and such things. And once I calculated how far the sun is by... because my mother told me how long it would take to a train to get to the sun if you could travel by train. So I had a very good feeling for numbers when I was a small child. NARRATOR The Hungarian Academy of Sciences dates from the 19th century, when Hungarian mathematics began to flourish. Erds grew up in an ideally supportive environment, his natural talent for math was actively promoted by his parents and teachers. 7 INSERT LOWER THIRD LEOPOLD FEJR (1880-1959) NARRATOR A national network of high school math journals and monthly competitions helped organize the work of gifted young people. Mrta Svd was one of the few who solved a problem and earned a prize. MRTA SVD The main thing was to solve problems.The reward of it was that if your pictures appeared at the end of the year as... as industrious problem solvers, well, you felt the top of the world, so... So when I got to the universityyou know we knew each other; it was a family of young mathematicians. He came out with the real thing when he was about 20, 19 or 20, because he proved what was called Chebyshevs Theorem. He gave a seminar where he proved that there is a prime between n and 2n. It was not bad for a youngster. NARRATOR Prime numbers are numbers which can only be divided by themselves and the number one, like 3, 5, 7, 11 and 13. Simply put, Chebyshevs Theorem states that there is always a prime number between any number and its double. The prime number between 2 and two times 2, or 4, is 3. The prime number between 5 and two times 5, or 10, is 7. The prime numbers between 19 and two times 19, or 38, are 23, 29, 31 and 37. Young Erdss mathematical proof was more elegant than a proof advanced by Chebyshev sixty years earlier, prompting the rhyme: Chebyshev said it and Ill say it again, Theres always a prime between N and two N MRTA SVD He did his Ph. D. simultaneously with his undergraduate, finishing his course. And I dont think he sat for things like Philosophy and Education, for which we ordinary mortals had to sit, after the fifth year. I mean, Paul didnt bother about a thing like this, you know. NARRATOR Erds and his friends from the university met on Sundays in a Budapest park to discuss problems. Their favorite spot was the statue of Anonymous, a medieval historian. PAUL ERDS Gallai remembered. MRTA SVD Na. PAUL ERDS 8 I told him this problem here. MRTA SVD Yes. PAUL ERDS If you have n points in the plane not all on a line, then there is a line which goes through exactly two of the points. NARRATOR Several members of the group later became famous mathematicians, and most of them co-authored papers with Erds. MARTA SVD Well, and he has been quoted. Yes. PAUL ERDS Gallai was the first. Gallai-Sylvester theorem. MARTA SVD Yes, Gallai, yeah, yeah, yeah, yeah. MARTA SVD Tibor! INSERT LOWER THIRD TIBOR GALLAI NARRATOR It was here in 1933... PAUL ERDS regsgednek le kell lni. (Sit your old self down.) NARRATOR ...that Esther Klein, a member of the Anonymous Group, posed a problem that led to the rediscovery of an important mathematical theory. Erds and George Szekeres, another member of the group, expanded on Kleins proof in what Erds calls the Happy Ending Paper, because soon after it was published, Szekeres and Klein married. BLA BOLLOBS Erds knows about more problems than anybody else and he not only knows about various problems and conjectures, but he also knows the tastes of various mathematicians. 9 INSERT LOWER THIRD BLA BOLLOBS Trinity College University of Cambridge BLA BOLLOBS So if I get a letter from him giving me three of his conjectures and two of his problems, then its sure that these are exactly the kind of conjectures and problems I am interested in and these are exactly the kind of questions I may be able to answer. Of course, this applies not only to me, but to everybody else. So Erds has an amazing ability to match problems with people, which is why so many mathematicians benefit from his presence. Every letter is likely to inspire you to do some work, or every phone call will give you some problems you are... you are interested in. NARRATOR Erds likes to work intensely with his collaborators, dropping in on them for a few days at a time. PAUL ERDS Maybe there are sequences which you cant extend except to take the density to be one. BLA BOLLOBS Yes... NARRATOR Bla Bollobs, a professor at Cambridge, is one of dozens of colleagues Erds has cultivated from childhood. BLA BOLLOBS Okay. Yes. PAUL ERDS Then both densities will be one. NARRATOR Bollobs was fourteen when he won a math competition in Hungary and first came to Erdss attention. Erds gave him new problems to solve and began to mold the directions Bollobs would pursue in math. Today, Bollobs is a leading figure in Random Graphs, a field Erds pioneered in the fifties. GABRIELA BOLLOBS I do your chin. BLA BOLLOBS He has a tremendous feel for what is interesting. So he will say, why dont you try to do this, why dont you try to do that. So, as if somebody tried to tell you, why dont you 10 chisel there. If you do, you will find a beautiful part. And indeed people do it, and more often than not, they do get much more exciting things than one would expect it at the beginning. PAUL ERDS Somebody asked me a problem and sometimes I see immediately what you have to do. And then later one has to carry out the details. And very often it works immediately, that you know that you are on the right track. Of course, sometimes you are deceived. TOMASZ LUCZAK And this is the largest independent set. Okay? It gives you that, say, hypergraph with n vertices, e edges, has roughly the same independent set of the same size. And you will create a small clique and.... JOEL SPENCER When mathematics appears in print its formal, its pure, its theorem, proof, theorem, proof, corollary. But when were doing mathematics, its a completely different thing; its three or four people sitting around with cups of coffee, a pad of paper, throwing ideas back and forth, making a lot of wild conjectures, most of which turn out to be completely false. The Hungarian definition of a mathematician is that a machine that turns coffee into theorems. NARRATOR Among the prominent American mathematicians who take care of Erds in the United States, are Ronald Graham, Director of Mathematical Research at AT&T Bell Laboratories, and Fan Chung, a fellow at Bellcore and Harvard professor. NARRATOR The Grahams provide Erds with an American base. They organize his travels, keep his financial affairs in order, and, like other mathematicians around the world, attend to his personal needs. Having Paul Erds as a house guest also presents the opportunity to work with him. FAN CHUNG ... because a cordal... PAUL ERDS I dont... I would be willing to listen to the proof, but I dont think it will help for this conjecture. FAN CHUNG Yeah. PAUL ERDS How much do you think would be needed for bounded edge density? 11 FAN CHUNG Im going to start with the grid that all the points on the outside, Im going to put a clique on it. First, I put I put all the edges joining any two vertices outside. PAUL ERDS Mm. Hm. FAN CHUNG So what I will do is, I partition into four pieces... RONALD GRAHAM (v.o.) Paul often says, when asked, "Well what are the really important things that you want to do in life?" He says, for him: to find new results, and try to prove them. Of course, he has a side comment here, which is a kind of a running joke about what he calls the SF, which...by that he means God, or whoever's watching down. And he has a perverse view of the almighty, feeling that it's his job to try to make people unhappy, to try to annoy them. And so, to get even part of his mission is to annoy the...as he calls the...the SF, the Supreme Fascist. PAUL ERDS SF means supreme fascist. This would show that God is bad. I dont claim that this is correct or that God exists. This is just sort of half a joke. When they ask you... As a joke I said, What is the purpose of life? I said, Prove and conjecture and keep the SFs score low. Now the game with the SF is defined as follows: If you do something bad, the SF gets at least two points. If you dont do something good which you could have done, the SF gets at least one point. And if you are okay, then nobody gets any point. And the aim is to keep the SFs score low. VERA SS Mathematicians are rather lucky, because one can be much more independent from real life. To do mathematics one needs only paper and pen, and sometimes not even that. MELVYN NATHANSON Erds and I wrote our first joint paper just about 20 years ago when I was getting out of graduate school. INSERT LOWER THIRD MELVYN NATHANSON City University of New York 12 MELVYN NATHANSON And I think over the last 20 years at universities and at conferences Ive probably heard him lecture at least fifty times. Pauls interested in a lot more than just mathematics. I used to be involved in some foreign policy things and he will come to my house and ask, "where is the latest issue of Foreign Affairs?" And have I read this article? I lived in Russia and he wants to know whats happening there. I work in New York; hes interested in social problems in New York City. Hes interested in why the universities and the schools in America dont work right, about budgets for research in America. Hes interested in history, in literature and hes interested... I mean, hes interested intensely in mathematics. This is what he does, this is his life, but its not a monomaniacal interest in mathematics. He's extraordinarily broad and sort of cultured in this old-fashioned European way. (Card) SAM & JOE SAM = USA (Uncle Sam) JOE = USSR (Josef Stalin) PAUL ERDS Sam is the USA, Uncle Sam; and Joe is the USSR, after Stalin. So I made once this joke, And Sam and Joe went up the hill to fetch a pail of water. And Sam fell down and broke his crown and Joe came tumbling after. When I told that to a little child forty years ago, she corrected me very seriously and said, No Erds, Jack and Jill went up the hill. Actually, Jack and Jill, I think, were politicians in the Elizabethan era, and I corrected her laughing, that you know, in 300 years, maybe they will sing it with Sam and Joe? But now there is some chance that they wont after all, they wont fall down and break their crown. PAUL ERDS You know, when I first met Landau in 1935 in Cambridge, he told me, Wir matematiker sind all ein bischen meshuggeneh, which means, we mathematicians are all a little bit crazy. And in 1932, I met a Hungarian mathematician called Sidorn, who worked in... mostly in trigonometric series, and he was a very good mathematician, but he was a bit crazier than the average mathematician. In fact, he was a borderline schizophrenic. They tell about him that he usually talked this way (turns his back to audience and leans forward) to you. He turned towards the wall and talked, but when he talked about mathematics he talked sense. And he even made it in a Hungarian anecdote book, because once in 1937 when Turn and I visited himhe also had persecution complexso he opened the door a crack and said, please come rather at another time and to another person. Krem jjenek inkb mskor s mshoz. That sounds better in Hungarian. But later on he was again reasonable. Now anyway, let us forget this sad story. Actually, he had a curious death. He died like Cyrano de Bergerac. A ladder fell on him and 13 broke his leg, and he died in the hospital of pneumonia. But anyway, I am sorry that he didnt live to see how popular he became. Every mathematician seems to know him now. MELVYN NATHANSON I think of him as the Bob Hope of mathematicians. You know, hes almost 80, hes been lecturing for 60 years, he has the style down, he has his anecdotesWhat is a PGOM?, an SJ? He talks about the Hungarian mathematician Sidorn. The mathematics always changes; the theorems are new, the results are new, the conjectures are new, but the style is wonderful and the style always comes through. LSZL LOVSZ Paul Erds likes to raise problems and to ask new questions. INSERT LOWER THIRD LSZLO LOVSZ Professor, Computer Science Princeton University LSZL LOVSZ And in fact he quite often offers money for the solution. PAUL ERDS I would like to offer a hundred dollars for a construction of such a sequence. This has never been accomplished. RONALD GRAHAM You should put the little zeroes there, so it doesnt turn into fifty-thousand. PAUL ERDS Okay. Oh yeah. Oh, that is some money for a problem. RONALD GRAHAM So I have the blank check here. LSZL LOVSZ Most of us have won smaller amounts from him on this basis. And if he likes a problem and he finds it very difficult, then to stimulate or to create interest, he raises the amount. And in fact on one occasion he paid two or three thousand dollars for the solution of a problem. RONALD GRAHAM There's a very nice story. I know a young mathematician who was admitted to Harvard, but who's father had enough money to pay, but didn't want to pay. And Harvard wasn't able to do anything for him. And Paul had met him in my office once and loaned the guy a thousand dollars, saying, "Look, pay me back when you can; and if you can't, do the same thing to someone else." And this was just a young mathematician he'd met for half an hour. 14 NARRATOR The search for solutions is often frustrating, but in the course of looking for answers to unsolved problems, Erds has discovered tools for opening up whole new areas of mathematics. HERBERT WILF He wants to know how many things you can do? How big is it? Is it very big or very small? INSERT LOWER THIRD HERBERT WILF University of Pennsylvania HERBERT WILF Suppose you have N points sitting on a table top and you start drawing lines between pairs of them, how many lines can you draw before you're really sure that there must be a triangle somewhere? Suppose you're trying very hard to avoid completing a triangle, and you just walk up there and draw in one more line and one more line. But at a certain point, if you draw enough lines, there must be a triangle somewhere on the table top. Well, how many lines is that? And that kind of a question in which how many things can you do until you can be very sure that there must be a certain kind of configuration sitting there, is one of the common denominators that runs through a lot of his work. NARRATOR Ramsey Theory is an area of combinatorics to which Erds has made important contributions. It is often expressed as the party problem. The party problem shows how the addition of only one element can turn a relatively simple problem into one with no solution in sight. PAUL ERDS Suppose there are six people in a party, then there are always three of them so that every two know each other or no two know each other. That is a special case of a famous theorem in combinatorics. Is that clear? Did I express myself clearly? Suppose we have six people at a party, then there are always three of them, so that either every two know each other or no two know each other. NARRATOR You can also imagine the 6 people as six points and look at all the possible lines connecting the pairs of points. Connecting all the pairs of points gives you 15 lines. Now suppose you color each of these lines in an arbitrary way with two colors, say red and blue. By the time youve finished coloring all the lines between the six points, either red or blue, you end up with at least one red triangle or with one blue triangle. There are 32,768 ways to color the 15 lines between six points red or blue. That can take a long time. An easier way to show that Ramsey Theory is true uses a mathematical proof that reduces the number of lines you have to draw. 15 Start with six points and label them A through F. From any one point there are only five lines extending to the other points. If the lines can only be red or blue, no matter how hard you try to avoid it, at least three of them must be the same, either red or blue. Now that we know that at least three lines will always be the same color, we can work with just those three lines. Now try to connect up points C and D, D and E or C and E. If you use a blue line between any of these, you will create a blue triangle. To avoid using any blue lines, all three of these lines must be red, but that gives you a red triangle. No matter how hard you try, you cannot avoid ending up with either a red triangle or a blue triangle. And that is the mathematical proof that if you are trying to connect six points with red and blue lines, you will always get at least a red or a blue triangle. PAUL ERDS Now you can ask, how many people do you need at a party that there should be four such people? The answer is eighteen. It's not quite so simple anymore. Now how many people do you need to have five such people? Well, nobody knows. It's between 41 and 55. You can ask, how is that possible that we don't know it; we have these powerful computers? But they're not good enough. NARRATOR To join four points, all connected with lines of the same color, makes the problem a bit more complex. How many points do you need to be sure that you can always find four points all connected by lines of the same color? The number of lines you have to draw to find the answer is 153, and the number of ways to color these lines either red or blue is 2153 power or 1046 power. The mathematical proof gives us the answer that you need at least 18 points to always be sure of having either a red or a blue figure joined by lines to all four points. There is as yet no answer for five points. The number is between 43 and 55. There are 10 447 power ways to color all of the lines connecting 54 points. PAUL ERDS Suppose an evil spirit would tell mankind: Either you tell me the answer for five people or I will exterminate the human race. I said as a joke, it would be best in this case to try to compute it, both by mathematics and with a computer. If he would ask for sixhow many you need for six people, the best thing would be to destroy him before he destroys us, PAUL ERDS (continued) because we couldn't do it for six people. Now if we would be so clever that we would have a mathematical proof, we could just tell the evil spirit to go to hell. (Reel 2) PAUL ERDS It follows from general principles of course that something is true. 16 JOEL SPENCER Theres got to be some function... PAUL ERDS Yeah, of course, but what function it is, he cannot decide. JOEL SPENCER Is there an upper bound? JOEL SPENCER Most mathematicians build theories, most great mathematicians. Paul Erds is a great mathematician, but he has a unique style. His style is that he asks problems. He asks specific problems. INSET LOWER THIRD JOEL SPENCER Courant Institute (NYU) JOEL SPENCER And each of these problems seem like individual questions. But in doing that he constructs theories of mathematics. As someone said about him, the problems that Paul Erds asks are just examples of the great meta-theorems that he keeps inside his mind. I don't know what actually happens inside Paul Erds's mind, but functionally that's true. By asking these questions, the mathematical theories have been developed. NARRATOR Erdss work was noticed by prominent mathematicians in both England and Germany. In 1934 Erds went to England. INSERT LOWER THIRDS OVER FIRST TWO STILL PHOTOGRAPHS G. H. HARDY (1877-1947) 17 J. E. LITTLEWOOD (1885-1977) NARRATOR Harold Davenport and his wife Anne became close friends. Mrs. Davenport still remembers Erds's arrival... ANNE DAVENPORT When he got on the train leaving Budapest to come to Western Europe for the first time, he didnt know how to cope with the meals served on the train or anything. INSERT LOWER THIRD LADY JEFFREYS ANNE DAVENPORT ANNE DAVENPORT And when he arrived in Cambridge he said to my husband, Do you think I could cut my own bread and butter it as you did? Would I be able to manage it? Hed never had to do anything for himself. LADY JEFFREYS His mother had done it. ANNE DAVENPORT His mother had done everything. PAUL ERDS So I left in '34 and then I commuted between England and Hungary. But as the political situation got worse, I had finally to leave Hungary for good. NARRATOR Hungary was being pulled into the orbit of Nazi Germany. For the circle of young Jewish mathematicians in Budapest, the signs were ominous... In September 1938 Erds left for the United States. He had been invited to Princeton to work at the Institute for Advanced Study. Erds spent World War Two in the United States, going from school to school, not knowing what had become of his parents and friends. Mrta Svd, Esther Klein and George Szekeres were among the lucky onesthey made it to Australia. Others stayed in Hungary in hiding. Those who were caught were imprisoned or killed by the Nazis. A ferocious two-month battle between Germans and Russians all but destroyed Budapest. 18 Among the few who died of natural causes was Erds's father. Miraculously, Erds's mother survived both the Nazis and the devastation. PAUL ERDS My mother lost two brothers and sisters, two brothers and two sisters who were murdered, and many of the younger generation were killed. And many of my friends were killed. 19 NARRATOR It was 1948three more yearsbefore Erds was allowed to visit his mother in Hungary. Back in the United States, his trips to Communist Hungary aroused suspicions during the anti-Communist frenzy of the McCarthy era. In 1954 he was investigated by the FBI and denied re-entry to the United States. PAUL ERDS Since I don't let Sam and Joe tell me where I am traveling, I just chose freedom and left the United States, which I still feel, that I acted in the best traditions of Americathat you don't let the government push you around. INSERT LOWER THIRD OVER FIRST STILL PHOTO BUDAPEST 1956 NARRATOR In Hungary, there was now another kind of oppression: A brutal Stalinist regime enforced its will with the help of Soviet troops. BLA BOLLOBS In the fifties and sixties Hungary was a very closed in society; it was a very claustrophobic society. INSERT LOWER THIRD BLA BOLLOBS Trinity College University of Cambridge BLA BOLLOBS The iron curtain was down and the Hungarian mathematicians had very little access to the West. They couldnt travel, and therefore it was terribly important that somebody could come from the West who knew lots and lots of the prominent mathematicians. And this was exactly Paul Erds. He was the window of the Hungarian mathematicians to the West. (Airport Announcer) HERBERT WILF Paul Erds is not a person with a fixed address. People oftenafter I tell them something about Paulwill ask me, "Well, where is he?" And my answer is that hes everywhere and hes nowhere. Hes nowhere because he doesnt have a fixed institutional affiliation and hes everywhere because he travels constantly carrying the word. If some people in England, for example, prove a beautiful new theorem, Paul will carry it all over the world like a bumblebee spreading intellectual pollen. 20 PAN AM LADY Do you have a small white card from U.S. Immigration? PAUL ERDS No, Here was my visa. PAN AM LADY Okay. 21 PAUL ERDS And they... You know I didnt even see that finally they found the record that I entered. That is here. PAN AM LADY Okay, thats not going to be a problem. PAUL ERDS Thank you. ANNE DAVENPORT You see when he left America in the fifties and couldnt go back, he brought all his worldly possessions with him. And I met him at the air terminal, and in the taxi on the way home he opened a suitcase looking for somethingone of two suitcasesit was half empty. And it wasnt in there, so he looked in the other one and that was half empty. And here was a man with all his worldly possessions in two half empty suitcases. I was rather envious in a way of this very simple life, but I was shocked to the core that if you get one hole in your sock, you throw away the pair and buy a new pair. It struck me as... outrageous. VERA SS To spend all his life, 24 hours with mathematics or with what he wants to spend, he had to have this life: Not to have a job. Not to have any restrictions, not in private life, not in academic life. So if he wants to be tomorrow in Australia, he just takes the next flight and goes there. (v.o. inside plane) He never carried checks. He never had a credit card, he never carried traveler checksnothing. He is willing to leave the airport with 20 dollars in his pocket and go from Hungary to Australia. MELVYN NATHANSON So the lifestyle is kind of a reflection of the work. The way he travels around from place to place constantly is kind of the way in which he jumps from one field of mathematics to another constantly. You know, he will come to New Jersey and stay with me in Maplewood and well talk about additive number theory. And then he will fly to Kalamazoo to meet the graph theorists and hell do graph theory. He is very quick, even at the age of 80. JOEL SPENCER In 1959 Paul wrote a paper with Alfred Rnyi, another Hungarian mathematician, entitled On the Evolution of Random Graphs. This paper started an entire field, and you look back and you can see that all the developments stemmed from this one idea of what happens when you take a graph and you throw in edges at random in a certain probability model. And since then there have been hundreds of papers and a number of books have been published on the field and its an established part of mathematics. BLA BOLLOBS 22 Now its not surprising, but at the time when it arose it was very surprising that lots of combinatorial structures can be found if you take things totally at random. And then you get a structure which has some wonderful properties. INSERT LOWER THIRD THE RANDOM RUN Poznan, Poland 1989 23 ANDRZEJ RUCINSKI Let me explain the rules one more time. Our honorary referee, Professor Paul Erds, will toss dice twice. The first time, prior to the run, saythis is just trialit will be six. (crowd reacts) Just a trial. Then during the sixth lap, when you are running the sixth lap, Paul will make another toss, and we will be notified immediately when we are complete with the sixth lap, how many laps you have to continue to run, without any stop, without any break. We continue to run until... QUESTION But how do you know who runs six laps and who runs three? ANDRZEJ RUCINSKI No. No. All run the same number of laps, or walk. PAUL ERDS Okay? ANONYMOUS Okay. (Paul throws dice. Crowd reacts) MICHAL KARONSKI Ready. VOICE On three. VOICE Go. NARRATOR Paul Erdss discovery of the probabilistic method pioneered a technique using randomness. When dealing with a large number of choices, making the choices randomly will give as good an approximation of the right answers as trying to exhaust all the possibilities. PAUL ERDS Now? Right away? TOMASZ LUCZAK Yeah, as soon as they are running. They don't know it. Wow! NARRATOR Randomly choosing how many laps to run in a race is a simplified tribute to the idea. 24 TOMASZ LUCZAK Yeah, but I show it just before they come. NARRATOR The search for solutions to problems is often more like a game, driven not by a desire to build or resolve a question, but by a need to know how the question might be resolved. (applause) The pure mathematician shies away from the practical applications of his work, but in a peculiar way the most interesting results in mathematics end up being useful to other sciences. BLA BOLLOBS A Physicist would love to find out how the world is put together; they always want to find out how things work. What makes things tick? A mathematician would like to find out how mathematical ideas are put together. What is behind the next hill? What happens if we climb that mountain? What kind of panorama shall we see? So we keep going on, trying to explore more and more of an unknown landscape, and whatever we find... Often the things we find make us very happy, make us terribly excited. But once you see the next view, there is another hill beyond the horizon and you feel a tremendous urge that you must go there, you must explore it. You must climb that mountain and find out what is behind there. FAN CHUNG In fact, I think he does not like to be alone very much at all. INSERT LOWER THIRD FAN CHUNG Bellcore & Harvard University FAN CHUNG When he is at our house we not only just have one guest. In fact, all the time, we have a stream of so-called Paul sitters. So there are all these mathematicians coming in, working with Paul. PETER WINKLER The first player might choose a vertex and color it blue. PAUL ERDS Mm hm. PETER WINKLER ...and then the adversary, who is trying not to color it successfully, chooses the antipodal vertex on the icosahedron. PAUL ERDS Yeah. And if it is the other way round, if it is second? PETER WINKLER 25 If he's second, then only two colors is enough. PAUL ERDS Yeah, always enough? Every bipartite graph? FAN CHUNG He is very demanding... Theres a certain innocence to it, really, because in Pauls mind theres only one reality. That is mathematics. JOEL SPENCER Where else do you have absolute truth? You have it in mathematics and you have it in religion. In mathematics you can really argue that this is as close to absolute truth as you can get. And so when he asks a problem: Are there an infinite number of primes?a classic Greek problemWhen that was solved in ancient Greece, when Euclid showed that there were an infinite number of primes, that's it. There are an infinite number of primes and there are no ifs ands or buts. That's as close to absolute truth as I can see you getting. PAUL ERDS ...Spinoza, was another... NARRATOR Euclid, Descartes, Euler and Newton are a few of the bright links in an unbroken lineage to which Erds constantly refers. PAUL ERDS ...the only condition was that at least formally he should declare himself a Lutheran. NARRATOR Personally transmitting this legacy from one generation to the next, adds an inspiring dimension to the quest for knowledge. Through personal anecdotes, the lore of mathematics turns the giants of the past into flesh and blood human beings. PAUL ERDS ...and he lived from polishing diamonds. TOMASZ LUCZAK Really? PAUL ERDS And Jewish philosophy. Yes, because you know, he had no job. He lived in Holland, which was politically... INSERT LOWER THIRD TOMASZ LUCZAK Adam Mickiewicz University 26 Poznan, Poland TOMASZ LUCZAK Yeah. Philosophy and mathematics are not jobs at all. PAUL ERDS Yeah. Well, not those days. TOMASZ LUCZAK (v.o.) Actually, the difference between us is 50 years, 50 years without a few days. Our first joint paper we wrote on our 100th anniversary: I was 25 and Paul was 75. Its not so easy to write a paper in mathematics. Actually, you can do it when you have enough technical ability, you can overcome some technical problems. But sometimes you need a message from the other world. Sometimes it's a flash of light, but it happens very rarely, I think. And I suppose that Paul is the man who can decipher this whisper, this feeling. MICHAL KARONSKI Definitely what strikes me always is that he is very good, a very good man. INSERT LOWER THIRD MICHAL KARONSKI Adam Mickiewicz University Poznan 27 MICHAL KARONSKI If I could fix the standard for a good man, then Paul probably would be a definite standard for me. (Music, applause) MICHAL KARONSKI (at banquet) We want to express all our deep warm feelings to you, and I want to raise this toast for you, Paul. MICHAL KARONSKI (v.o.) I guess the other question that always comes to me if I see him, whether he is lonely or not? You know, there is always a lot of people around him. He likes to have people around him. But sometimes you feel that he is really very lonely among people. INSERT LOWER THIRD MUIR WOODS California 1989 FATHER Say, Hi. PAUL ERDS Look what I can do. FATHER Tell him Hi. PAUL ERDS He may be too small to follow it. FATHER Hes pretty good, idnt he? Yes. Hes pretty good. PAUL ERDS How many teeth does he have? FATHER Eight. Just enough to bite you good. PAUL ERDS Mm hm. Bye, bye. FATHER Shes a good girl. 28 BLA BOLLOBS Paul and his mother, Anoush nni, formed a wonderful pair. They were really devoted to each other. They looked after each other. Of course, Anoush nni, when I knew her, was in her eighties. She was a little frail, but still she was certainly easily capable of looking after herself, and Erds was never a very worldly figure. In spite of that, they got on extremely well with each other, and they looked after each other very well. They worried, whether one of them was catching cold, or whether she slept enough that afternoon or that night. BLA BOLLOBS (continued) He had hundreds and hundreds of reprints, and she kept them in a very good order. She started to go around with him on his trips around the world in the sixties. Their very first trip was to Israel. She enjoyed it very much. And from then on, more often than not, she accompanied him. Being close to Erds, she was very much the Queen Mother. All the mathematicians came. Lots of young people came to see him. ANNE DAVENPORT They just understood one another completely and he was absolutely devoted to her. INSERT LOWER THIRD ANNE DAVENPORT ANNE DAVENPORT It was most touching to see the way he looked after her, but it was she who'd looked after him in his childhood, who'd looked after him and taught him mathematics and everything, but not taught him how to be a practical man of the world. RONALD GRAHAM Well, in his perverse humor, a slave is a male and a boss is a woman, a female. He feels that's the natural order of things. He never was married, although he had something of a girlfriend that he doesn't talk about much. PAUL ERDS As somebody puts it: "He likes girls, but he doesn't like the thing which they are standing for." Actually, I have an abnormality; I can't stand sexual pleasure. It's a curious abnormality; it's almost unique. HERBERT WILF Well, Paul has always been the kind of person who gives you a kind of a literal response if you ask him a question that a lot of us would consider as banter and doesnt really require a response. INSERT LOWER THIRD HERBERT WILF University of Pennsylvania HERBERT WILF 29 And in this case the question that I asked him was, Good morning Paul. How are you today? And that, of course, is the kind of question that ninety-nine point eight percent of the people will just let go with, Fine, how are you? or ignore completely or whatever. But Paul is not ninety-nine point eight percent of the people. Paul Erds stopped in his tracks and he considered very carefully the question of how he was that morning, because I had just asked him, with the same attention that he would have given to a complex mathematical problem. And after a few moments... And of course, I stopped dead and we just stood there for awhile until he got to the conclusion of how he was. And the answer was, Herbert, I am feeling very depressed this morning. And I said, Im sorry Paul. Why is that so? He said, I miss my mother. Shes dead, you know. And I said, Yes, I knew that, but that was five years ago. Wasnt it? And he said, Yes, but I miss her very much. And we just stood there in the sunshine for another minute and continued on to breakfast. VERA SS She lived here in Budapest, in the next house. They had an apartment there, but since his mother died, he never wants to stay in that apartment. Sometimes he had to go there to find some documents or some books and then he went there, but never alone. INSERT LOWER THIRD RANDOM GRAPHS '89 Poznan PAUL ERDS (at banquet) Well, let me just thank for the nice toast. And maybe I shall add that the only good wish for an old man you can say is an easy cure of the incurable disease of life. The meeting, like life will, is nearing its end, but just like life it was very pleasant. (laughter, applause) Maybe I can add one more thing. (Laughter) When I last talked to Plya he was over 97, and I told him, we will celebrate your hundredth birthday with great splendor. And Plya said laughing, Well, maybe I want to be a hundred, but not hundred and one, because old age and stupidity are too unpleasant. Unfortunately, he didnt make it. He died before he was 98. Well well see what will be my fate. Oh, one more thing. (Laughter.) Euler, when he died, he simply collapsed and said, I am finished. (Laughter.) And when I told this story, somebody callously just remarked, Well, another conjecture of Euler was proven. (Laughter.) INSERT LOWER THIRD BUDAPEST 1989 PAUL ERDS Servus. N, milyen ids ez a pici? (Hello. How old is that little one?) CHILD Kt ves. (Two years old.) PAUL ERDS 30 Nagyon des. (Very sweet.) CHILD Ez az n testvrem. (She's my sister.) PAUL ERDS Igen, nzd mit tudok csinlni. El elytem s megfogom. Egy, kett, hrom. Servustok. (Look at what I can do, I drop it, then catch it. One, two three. Goodbye.) ANNE DAVENPORT He looks a little bit like a lost child and you feel that you should help him. LADY JEFFREYS He does, yes. Yes. ANNE DAVENPORT And you know that hes not dangerous and wont do anything horrible to you, so you just feel sorry for this poor stray lamb. LADY JEFFREYS And I think also you do feel hes got a very unusual mind. To be so concentrated on purely abstract mathematics... ANNE DAVENPORT Yes. LADY JEFFREYS ...I think is extraordinary. ANNE DAVENPORT Yes. LADY JEFFREYS He does live with mathematics completely. ANNE DAVENPORT Absolutely. Absolutely. LADY JEFFREYS I think that if there were no mathematics he wouldnt exist. ANNE DAVENPORT Yes. 31 LADY JEFFREYS Hed have to invent it. JOEL SPENCER Paul talks about The Book. The Book has all the theorems of mathematics. Theorems can be proven in a lot of different ways, but in The Book there is only one proof and it is the one that is the clearest proof, the one that gives the most insight, the most aesthetic proof. It's what he calls The Book proof. And sometimes when there's a problem and somebody solves it and the proof is not so beautiful, then he'll say, "Well okay, but let's look for The Book proof; let's try to find The Book proof. And this is the sense of mathematics, that...that The Book is there, the theorems have an existence of their own. And what we're doing is we're just trying to uncover. We're trying to read the pages of The Book. We don't create mathematics. What we do is we read the pages of The Book. We discover the pages of The Book. So when he goes from university to university, and he talks about problems, and he asks everybody to try to solve these problems, it doesn't matter who solves the problem. It really doesn't matter to him, because all of us are in the same venture. We're all just trying to uncover the pages. And sometimes we succeed. Sometimes we find these beautiful theorems. INSERT TITLES OVER FINAL SHOT Paul Erds died on September 20, 1996 while attending a mathematical meeting in Warsaw, Poland 32 Mathematicians in order of appearance Paul Erds Ronald Graham Vera T. Ss Joel Spencer J.W.S. Cassels, F.R.S. Mrta Svd Tibor Gallai Bla Bollobs Fan Chung Melvyn Nathanson Lszl Lovsz Herbert Wilf Tomsz Luczak Andrzej Rucinski Peter Winkler Michal Karonski Also appearing Alice Fialowski Gabriella Bollobs (sculpting) Anne Davenport Lady Bertha Jeffreys Deborah Knous (Pan Am counter) Gene and Elise Patterson (in Muir Woods) Gyula Katona (on recorder) and various Epsilons Produced, directed and edited by George Paul Csicsery 33 Based on The Man Who Loves Only Numbers an article by Paul Hoffman Cinematography by John Knoop Music by Mark Adler Computer animation by Judith Banning and James Locker Red Dot Interactive Narrator James Locker Mathematics Consultants Donald J. Albers Gerald L. Alexanderson Ronald Graham Reuben Hersh Charles L. Silver Joel Spencer Consulting Editors Nathaniel Dorsky Leslie Asako Gladsj 34 Production Crew Budapest Sound Recordists Gyula Traub Akos Solymosi Production Manager dn Pl Gaffer Endre Vizhny Cambridge Sound Recordist Mike McDuffie Assistant Mark Bollobs New Jersey Sound Recordist John Giannini Philadelphia Sound Recordist Kelvin Walker Poznan Sound Recordist Henryk Juchacz Assistants Filip Karonski Jacek Karonski San Francisco Sound Recordist Toni Hafter Production Manager Berry Minott Assistant Camera David Collier Production Assistant Este Trk Narration recording, effects editing and sound mix Jim Kallett Opticals Michael Hinton Credit cinematography Rock Ross Sound Editor Betsy Bannerman 35 Assistant Editors Mohammed Husseini Scott Mahoy Joseph Schuster Hungarian archival and newsreel footage courtesy of Magyar Film Intzet Edit Kszegi and Andrs Surnyi Photographs by George Plya courtesy of Gerald L. Alexanderson and Birkhuser Publishing Erds family photos courtesy of Bla Bollobs Anne Davenport Paul Erds Vera T. Ss Photos from the Hungarian Revolution courtesy of Erich Lessing/Magnum and The Hulton Deutsch Collection Erds sculpture by Gabriella Bollobs Russian voice Peter DePavloff Musicians Cello Judiyaba Clarinet Larry London Piano Mark Shapiro Violin Dan Smiley Music Recording/Mix Engineer 36 Cookie Marenco/OTR Studios Film Lab and Video Transfers Monaco Film & Video Video to Film Transfer Image ATS On-line Video Editor Ed Rudolph Video Arts Video Editor Lori Muttersbach Narration Recording and Mix Music Annex Negative Cutting K. Bik Films Edge-coding Insync Subtitle Coordinator Pierre Cottrell Subtitles Laser Vido Titres Funded by American Mathematical Society Film Arts Foundation Heineman Foundation Mathematical Association of America National Science Foundation Informal Science Education Program 37 38 Special thanks to Paul Erds and Bla Bollobs Emma Bollobs Joan Brandt Linda Chan Phyllis Chinn Gabriella Csicsery Plma Csicsery Sigmund Csicsery Joseph Deken Peter Dolan Hyman Field Ronald Graham Andrs Hajnal Bob Hawk William Jaco Michal Karonski John Knoop Fran Lerner Leonard Michaels Anne O'Toole Akos Ravasz David Rose Michael Rossman Thomas Sanchez Kim Salyer Raquel Scherr Pl Schiffer Gail Silva Vera T. Ss Joel Spencer Chris Sykes Suzanne & George Szab Kati & Mikls Szemere Molly Walker Frank Warner 39 Adam Mickiewicz University AT&T Bell Laboratories Bellcore DIMACS Ferihegy Airport, Budapest Film Arts Foundation Forman Entertainment Group Institute for Advanced Study at Princeton MALEV The Mathematical Institute of the Hungarian Academy of Sciences Newark International Airport Pan Am San Francisco International Airport Trinity College, University of Cambridge United Airlines University of Pennsylvania and many more N is a Number A Portrait of Paul Erds 1993 George Paul Csicsery All Rights Reserved