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Erdish reminisces about Ulam: he was a child prodigy and a god

Erdish reminisces about Ulam: he was a child prodigy and a god

Stanis·ław Ulam (1909-1984) was a Poland United States mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, and the Teller-Ulam configuration of the hydrogen bomb was named after him and Edward Teller· After Ulam's death, his lifelong friend Paul Erdős (1913-1996), a renowned mathematician, published this memorial in 1985, with a special focus on some of the work they had done together.

撰文 | Paul Erdős

Translation | Zhang and Zhi

First, a brief introduction. For 50 years, Ulam has been my friend and collaborator. I have had countless discussions with him about mathematics and politics, and I have co-authored many papers. In this article, I will focus on the research we collaborated on, ignoring his work in physics, biology, computing, and computer science.

Ullam once wrote a very good autobiography[7], and I remember that these things I want to talk about are not mentioned in his autobiography. I hope my account is as accurate as possible.

I first met Ulam in Cambridge, United Kingdom, in 1935, and the second time in Cambridge, Massachusetts, United States, in 1938-1939, when he was a member of the Harvard University Society. But it was in 1941-1943 that we really started communicating mathematics, when I visited him twice at the University of Wisconsin, during which time we got our first joint research. Later, in 1946, I visited him in Santa Fe and Los Angeles. He was seriously ill, possibly encephalitis (which was almost the only time he got sick, and he was in good health until he died of a heart attack). He was discharged from the hospital to recuperate on an island south of Los Angeles, and I went to visit him (the whole thing is mentioned in his autobiography). I went to Los Alamos to meet him a few more times, last in 1952.

In 1963 we met again at a number theory conference in Boulder, Colorado, United States. Then we visited Aspen together. One time I was at his house when he got a call from the White House asking about his proposal for a nuclear-test-ban treaty — which Ulam strongly supported. Then in 1968 and 1970, I was a visiting professor at the University of Colorado and wrote our first collaborative papers with him on additive number theory and set theory. On that occasion in 1970, when my 90-year-old mother was with me, Françoise, the wife of Ullam, wrote a short essay for my mother. By the end of the '70s, we were often at the University of Florida together. I was going to continue our research, but I was surprised to learn that he died of a coronary heart attack in May 1984.

Ulam was extremely intelligent, he was both a prodigy and a "prodigy". The word deity was coined by Ullam and cannot be found in any dictionary. When I gave a talk on the topic of child prodigies, Ullam commented that both of us were "gods," meaning that we were still able to "prove theorems and make conjectures" in the dotage of old age. Perhaps this is a sad footnote to the good wishes of a person's destiny, and our most fervent hope for a baby is that you "be born a prodigy and grow old as a god."

Ulam was undoubtedly a child prodigy, and before the age of 20 he proved that there is a 2-valued measure (i.e., the measure of any measurable set is 0 or 1) on any infinite set, such that the measure of the whole set is 1, and the measure of any single point is 0, and the measure is finite. Alfred Tarski (1901-1983) independently discovered this theorem a few months later. Recently I discovered that Frigyes Riesz had predicted this fact 20 years earlier, and he had demonstrated it at the International Congress of Mathematicians in Rome in 1908.

Erdish reminisces about Ulam: he was a child prodigy and a god

In my opinion, this is one of the most important developments in modern mathematics, and the second starting point for this development is my collaborative paper with Tarski [4, 5], which inherits and develops Tarski's earlier work, for which I am very honored. Readers, please allow me to insert a few more recollections. I used to mistakenly think that the first unreachable cardinal might be measurable. In 1957, András Hajnal (1931-2016) and I proved a theorem from which it was easy to deduce that there were no countable additive measures on the first and many other unattainable cardinal numbers. Hajnal didn't realize this until the Hanf-Tarski and Kiesler-Tarski were published. But I'm afraid the blame lies with me, as Hanjnal says, "I'm just a young man." How could I doubt and refute 'pgom' (poor greatest old man; Translator's note: Poor great old man, referring to Erdős. Erdős likes to put this abbreviation after his signature). "Even a long time ago, I was already in old age. In fact, Hajnal also argues that the result of that oversight was that the insights in the Hanf-Kiesler-Tarski proofs were far deeper than ours, and their work soon led to the exploratory development of the large cardinality theory. If we had been the first to publish the proof, perhaps we would not have developed as rapidly as we did later.

Erdish reminisces about Ulam: he was a child prodigy and a god

Ullam's collaborative work with John C. Oxtoby (1910-1991), Barry C. Mazur (1937-), and Karol Borsuk (1905-1982) was crucial to mathematics, but I am not the best person to evaluate this work. He was the same as D. H. Hyers' (1913-1997) work on the functional equation f(x+y) = f(x)+f(y) is equally interesting, as is his work with Cornelius J. Everett (1914-1987). But since I'm writing for this journal, I should talk about his famous refactoring conjecture [the term in Harary's [6] is called "reconstruction disease"]. The first conclusion in this regard came from Paul Kelly, a student of Ulam, and the general situation is far from being resolved, and today the field is very active. Ulam has a very broad meta-problem: if in a certain structure A^2 = B^2, then is A = B true? The answer to this question is often no, but there are some exceptions. These questions have given rise to a number of interesting papers.

During his years at Los Alamos, Ulam studied how computers could be used to solve purely and applied mathematical problems, and achieved some important pioneering results. I'm not going to spend a lot of time here, not because I don't think the research is important or interesting, but because I think this part should be left to the experts in the field. I'll just mention that together with his collaborators, he came up with some interesting, rich, and unexpected conjectures in iterative functions. There is even less I can say about his contribution to interstellar voyages in Project Orion. I have the impression that Freeman Dyson (1923-2020) was very active in this project, and I hope he and others will write in more depth. There is one more interesting fact about this. Ulam has always been very proud to be one of the initiators of the project, and he regrets that the project fell through in the end (as far as I know, the project was abandoned long before the treaty banning nuclear explosions in space (the Partial Nuclear-Test-Ban Treaty) was signed. Ullam certainly does not want to violate the treaty, but wants to renegotiate). On one occasion he told me that he found in Goethe's Faust a brilliant slogan for the Orion Project: "Und was vor uns ein alter Mann gedacht und was wir dann so herrlich weitgebracht ja bis an die Sterne weit" ["The ideas of an old man have been carried forward by us, yes, as far as the stars"; Translator's note: Faust is not alter Mann, but weiser Mann]. Ulam said that the "old man" here refers to Einstein. I immediately corrected him, "No, the old man should be you, and the stars (stars) should be replaced by planets." "Ulam is always afraid of getting old, and he is proud that he can still play tennis at the age of 70, and even play well. He was fortunate enough to escape two demons - his old age and his declining intelligence, and he died of heart failure without fear or pain, and he was able to prove theorems and make conjectures before he died.

On my last visit to the University of Florida, Alexander R. Bednarek (1933-2007) told me a great story about Ulam. Maybe the story has been polished to some extent, but Marcel Riesz (1886-1969) once told me, "If you have a good story, you don't have to worry about whether it's true or not," and at least I'm sure it's true. A few years ago, Professor Gladysz of the University of Wroclaw visited Gainesville, home to the University of Florida. It just so happened that he had never met Ulam, and after Bednarek introduced them to each other, the two talked for a long time in Poland. When Ulam left, Gladysz asked Bednarek: Is Ulam the son of the famous Ullam? Bednarek was embarrassed and didn't tell the truth, but he thought that Ulam would be happy to hear it, so he told Ullam the story. Bednarek told me that the next day almost all mathematicians knew about it.

As a mathematician, Ullam was not only adept at proving interesting and profound theorems, but perhaps he was also good at formulating novel and enlightening questions and conjectures. He also made a lot of wonderful conjectures in areas that he didn't dabble in too much. I'm going to give you two or three examples from my own familiar field. Norman H. Anning (1883-1963) demonstrated with me that if x1, x2,... is an infinite set of points in a plane (or En, i.e., a high-dimensional Euclidean space), and the distances between the two are all integers, then the points must be on the same line. Ullam immediately asked, "Is it possible to have an infinite number of such points, which are not all located in a straight line, and all distances between them are rational numbers?" I replied, "Yes, Anning and I found such examples, but Euler had foreseen this a long time ago." Ullam retorted, "I don't believe that the set of points in a plane can be dense everywhere and the distance is rational. "I think his guess should be correct, but the question is probably going to be very deep." The condition that the distance between two and two is a rational number may be very harsh for an infinite set of points, but we don't know anything about it.

Erdish reminisces about Ulam: he was a child prodigy and a god

Even though Ullam was not a number theorist, he published many interesting number theory problems, many of which he presented at Boulder's 1963 Conference on Number Theory. He also independently discovered the "lucky number" with Eri Jabotinsky of Haifa (an Israel city). The resulting number has many prime-like properties).

In the '70s and '80's, Ulam and I were often at the University of Florida together, and we published many articles on combinatorics and set theory. I just want to mention here that one of the questions raised by Ullam leads to many problems and conclusions in graph theory.

The following question was first posed by our five authors in a paper [2]: let G(n) and G'(n) be two graphs with n vertices, and e(G) be the number of edges of G. Let's assume that e(G) = e(G'). U-decomposition refers to the division of a set of edges into shapes

Erdish reminisces about Ulam: he was a child prodigy and a god

and make all graphs isomorphic to . If G and G' have the same number of edges, then the above decomposition must exist. DEFINE U(G, G') AS THE SMALLEST N THAT MAKES U DECOMPOSE. cause

Erdish reminisces about Ulam: he was a child prodigy and a god

We prove it

Erdish reminisces about Ulam: he was a child prodigy and a god

We have also published a number of papers on this and related topics. This problem can be generalized to hypergraphs, and its research is still active today.

We hope that there will be many more interesting new developments. Fan-Rong (1949-) and I have only recently completed a dissertation in this direction.

Erdish reminisces about Ulam: he was a child prodigy and a god

Ulam has been a friend and collaborator of mine for 50 years, and it is clear that from now on science and society, and especially the world of mathematics, will never be the same again.

In the story of One Thousand and One Nights, the king is paid tribute to "O king, may you be immortal". A tribute to mathematicians and scientists might be more realistic: "May your theorems, O mathematicians, be immortal." "I wish and hope that Stan's theorem will have such a fate.

Erdish reminisces about Ulam: he was a child prodigy and a god

The Adventures of a Mathematician: The Autobiography of Ulam (Yilin Publishing House, November 2023)

bibliography

[1] F. R. K. Chung and P. Erdős, On unavoidable hypergraphs (to appear in J. Graph Theory) .

[2] F. R. K. Chung, P. Erdős, R. L. Graham, S. M. Ulam and F. F. Yao, Minimal decompositions of two graphs into pairwise isomorphic subgraphs. Proc. Tenth Southeastern Conf. on Combinatorics, Graph Theory and Computing (1979) 3-18.

[3] P. Erdős, Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950) 121-141.

[4] P. Erdős and A. Tarski, On families of mutually exclusive sets. Annals of Math. 44 (1943), 315-329.

[5] P. Erdős and A. Tarski, On some problems involving inaccessible cardinals. Essays on the Foundations of Mathematics, Hebrew University, Jerusalem (1961) 50-82.

[6] F. Harary, The Four Color Conjecture and other Graphical Diseases. Proof Techniques in Graph Theory, Academic Press, New York (1969).

[7] S. M. Ulam, Adventures of a Mathematician, Scribner, New York (1976).

本文经授权译自Erdös, Paul. Ulam, the man and the mathematician. J. Graph Theory 9(4), 1985: 445-449. https://doi.org/10.1002/jgt.3190090402

Erdish reminisces about Ulam: he was a child prodigy and a god

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