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Paul Erdős

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Paul Erdős
Born(1913-03-26)March 26, 1913
Budapest, Hungary
DiedSeptember 20, 1996(1996-09-20) (aged 83)
Warsaw, Poland
Nationality Hungarian
Alma materUniversity of Pázmány Péter
Known forCombinatorics
Graph theory
Number theory
AwardsWolf Prize (1983/84)
AMS Cole Prize (1951)
Scientific career
FieldsMathematics
InstitutionsPrinceton
Purdue
Notre Dame
Then itinerant
Doctoral advisorLeopold Fejér
Doctoral studentsBonifac Donat
Joseph Kruskal
Alexander Soifer
Notes
Note that he has an Erdős number of zero.

Paul Erdős (Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. With hundreds of collaborators, he worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory.

Biography

Paul Erdős was born in Budapest, Hungary, as Erdős Pál. (Erdős is pronounced as IPA /ɛrdøːʃ/, similar to Air-doesh.) After his siblings died before his birth at the ages of 3 and 5, he was the only child of Anna and Lajos Erdős. His parents were both Jewish mathematicians, from a vibrant intellectual community.[1] At the age of three, he could calculate how many seconds his family's friends had lived (Hoffman 1998). Erdős showed early promise as a prodigy, and soon became regarded as a mathematical genius by his peers.

In 1934, he was awarded a doctorate in mathematics.[2] Because anti-semitism was increasing, he moved that same year to Manchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at Princeton University. At this time, he began to develop the habit of traveling from campus to campus. He would not stay long in one place and traveled back and forth between mathematical institutions until his death.

Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were in general donated to people in need and various worthy causes. He spent most of his life as a vagabond, travelling between scientific conferences and the homes of colleagues all over the world. He would typically show up at a colleague's doorstep and announce "my brain is open", staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom he (Erdős) should visit next. His working style has been humorously compared to traversing a linked list.

His colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities. (This quotation is often attributed to Erdős, but does seem to originate with Rényi.)[3] After 1971 he also took amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking the drug for a month. Erdős won the bet, but complained that mathematics had been set back by a month: "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper." After he won the bet, he promptly resumed his amphetamine habit.

He had his own idiosyncratic vocabulary: he spoke of "The Book", an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF) [4]. (When Hungarian troops set out to join the Germans in the attack of Yugoslavia in 1941, a prince of the Hungarian Catholic church was on hand to give his blessings. This made Erdős think that God was the Supreme Fascist.)[citation needed] He accused the SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from The Book!".

Other idiosyncratic elements of Erdős' vocabulary include: children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted ε); women were "bosses"; men were "slaves"; people who stopped doing math had "died"; people who died had "left"; alcoholic drinks were "poison"; music was "noise"; people who had married were "captured"; people who had divorced were "liberated" and to give a mathematical lecture was "to preach". Also, all countries which he thought failed to provide freedom to individuals as long as they did no harm to anyone else were classified as imperialist and given a name that began with a lowercase letter. For example, the U.S. was "samland" (after Uncle Sam), the Soviet Union was "joedom" (after Joseph Stalin), and Israel was "israel". For his epitaph he suggested, "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább"). (Hoffman 1998)

In 1952, during the McCarthy communist "witch-hunts," the U.S. government denied Erdős, a Hungarian citizen, a re-entry visa into the United States, for reasons that have never been fully explained.[2] Teaching at Notre Dame at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the Immigration Service at periodic intervals. The government changed its mind in 1963 and Erdős resumed including American universities in his teaching and travels.

Hungary, then a Communist nation, was under the hegemony of the Soviet Union. Although it curtailed the freedoms of its citizens, in 1956 it gave Erdős the singular privilege of being allowed to enter and exit Hungary as he pleased. Erdős exiled himself voluntarily from Hungary in 1973 as a principled protest against his country's policy of denying entry to Israelis.[3]

During the last decades of his life, Paul Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences and the U.K. Royal Society. Shortly before his death, he renounced his honorary degree from the University of Waterloo over what he considered to be unfair treatment of a colleague. He died 'in action' of a heart attack on September 20, 1996, at the age of 83, while attending a conference in Warsaw, Poland. Erdős never married and had no children.

Mathematical work

Erdős was one of the most prolific publishers of papers in mathematical history, second only to Leonhard Euler; Erdős published more papers, while Euler published more pages (Hoffman 1998). He wrote around 1,500 mathematical articles in his lifetime, mostly with co-authors. He had 511 different collaborators (The Erdős Number Project Data Files), and strongly believed in (and obviously practiced) mathematics as a social activity.

In terms of mathematical style, Erdős was much more of a "problem solver" than a "theory developer". (See The Two Cultures of Mathematics[5] by Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated.)   Joel Spencer states that his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career.[6] Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did[7], a pattern that extends to other prizes[8]. He did win the Wolf Prize, where his contribution is described as for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over.[9] In contrast, the works of the three winners before were recognized as outstanding, classic, and profound, and the three after as fundamental or seminal. Nonetheless, as one of Erdős's least favorite people reportedly said, Quantity has a quality all its own.[10]

Of his contributions, the development of Ramsey theory and the application of the probabilistic method especially stand out. Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory. Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered an elementary proof for the Prime number theorem, along with Atle Selberg, which showed how combinatorics was an efficient method of counting collections.

Collaborators

Among his frequent collaborators were

Erdős number

Main article: Erdős number

Because of his prolific output, friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Some have estimated that 90% of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in the light of the small world phenomenon). It is jokingly said that Baseball Hall of Famer Hank Aaron has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been humorously assigned to an infant, a horse and several actors. For details see the "Extended Erdős Number Project" [4]

The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1.[11] Goffman published his observations about Erdős's prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[12]

Works about Erdős

N Is a Number: A Portrait of Paul Erdős is a 1993 documentary about the life of Erdős.

Since Erdős's "leaving," a book entitled Proofs from THE BOOK has been published, intended as a collection of the most beautiful mathematical proofs in the spirit of Erdős.

Hoffman, Paul; The Man Who Loved Only Numbers. Hyperion, 1998. ISBN 0786863625 is a biography of Erdős.

Notes

  1. ^ The Budapest Jewish community of that day produced at least six remarkable thinkers besides Erdős: physicist and mathematician Eugene Wigner (Wigner Jenő in Hungarian), physicist Edward Teller (Teller Ede), physicist Leó Szilárd (Leó Szilárd), mathematician and polymath John von Neumann (Neumann János), physicist Dennis Gabor (Gábor Dénes), and philosopher Georg Lukács (Lukács György).
  2. ^ Erdős's thesis advisor at the University of Budapest was Leopold Fejér (or Fejér Lipót), who was also the thesis advisor for John von Neumann, George Pólya and Paul (Pál) Turán.
  3. ^ Biography of Alfréd Rényi by J.J. O'Connor and E.F. Robertson
  4. ^ Schechter, Bruce. My Brain Is Open: The mathematical journeys of Paul Erdos, New York : Simon & Schuster, 1998, p. 70-71
  5. ^ This essay is in Mathematics: Frontiers and Perspectives, Edited by V. I. Arnold, Michael Atiyah, Peter D. Lax and Barry Mazur, American Mathematical Society, 2000.
  6. ^ Joel Spencer, Prove and Conjecture!, a review of Mathematics: Frontiers and Perspectives. American Scientist, Volume 88, No. 6 November-December 2000
  7. ^ Paths to Erdos
  8. ^ From Trails to Erdos, by DeCastro and Grossman, in The Mathematical Intelligencer: vol. 21, no. 3 (Summer 1999), 51–63: A careful reading of Table 3 shows that although Erdos never wrote jointly with any of the 42 [Fields] medalists (a fact perhaps worthy of further contemplation)... there are many other important international awards for mathematicians. Perhaps the three most renowned...are the Rolf Nevanlinna Prize, the Wolf Prize in Mathematics, and the Leroy P. Steele Prizes. ... Again, one may wonder why KAPLANSKY is the only recipient of any of these prizes who collaborated with PAUL ERDOS. (After this paper was written, collaborator Lovasz received the Wolf prize, making 2 in all).
  9. ^ Wolf Foundation Mathematics Prize Page
  10. ^ wikiquote:Joseph Stalin#Unsourced
  11. ^ [1] Michael Golomb's obituary of Paul Erdős
  12. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly. 76.

References

See also

Wikiquote has quotations related to Paul Erdős.


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