Paul Erdős
Paul Erdős at a student seminar in Budapest (Fall 1992)
Born	(1913-03-26)26 March 1913 Budapest, Austria-Hungary
Died	20 September 1996(1996-09-20) (aged 83) Warsaw, Poland
Residence	Hungary United Kingdom United States Israel Then itinerant
Nationality	Hungarian
Fields	Mathematician
Institutions	Manchester Princeton Purdue Notre Dame University of Memphis[2] Then itinerant
Alma mater	Eötvös Loránd University
Doctoral advisor	Leopold Fejér
Doctoral students	Bonifac Donat Joseph Kruskal Alexander Soifer
Known for	See list
Notable awards	Wolf Prize (1983/84) AMS Cole Prize (1951)

Paul Erdős (occasionally spelled Erdos or Erdös; Hungarian: Erdős Pál, pronounced [ˈɛrdøːʃ ˈpaːl]; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. Erdős published more papers than any other mathematician in history,^[1] working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. He is also known for his "legendarily eccentric" personality.^[2][3]

Biography[link]

Paul Erdős was born in Budapest, Hungary on March 26, 1913.^[4] He was the only surviving child of Anna and Lajos Erdős (formerly Engländer);^[5] his siblings died before he was born, aged 3 and 5. His parents were both Jewish mathematicians from a vibrant intellectual community.^[6] His fascination with mathematics developed early—at the age of three, he could calculate how many seconds a person had lived.^[7]

Both of Erdős's parents were high school mathematics teachers, and Erdős received much of his early education from them. Erdős always remembered his parents with great affection. At 16, his father introduced him to two of his lifetime favorite subjects—infinite series and set theory. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Erdős later published several articles in it about problems in elementary plane geometry.^[8]

In 1934, at the age of 21, he was awarded a doctorate in mathematics.^[9] 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 among 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 generally donated to people in need and various worthy causes. He spent most of his life as a vagabond, traveling 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 colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities. (This quotation is often attributed incorrectly to Erdős himself. The German original, "Ein Mathematiker ist eine Maschine, die Kaffee in Sätze verwandelt"^[11] of the sentence is a wordplay on the double meaning of "Satz": "theorem" or "residue of coffee", lost in the English translation)^[12] 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.^[13] Erdős won the bet, but complained that during his abstinence 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 use.

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).^[14] 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!". This later inspired a book entitled Proofs from THE BOOK.

Other idiosyncratic elements of Erdős' vocabulary include:^[15]

• Children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted by that Greek letter (ε));
• Women were "bosses";
• Men were "slaves";
• People who stopped doing math had "died";
• People who physically died had "left";
• Alcoholic drinks were "poison";
• Music was "noise";
• People who had married were "captured";
• People who had divorced were "liberated";
• To give a mathematical lecture was "to preach" and
• To give an oral exam to a student was "to torture" him/her.

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").^[16]

In 1952, during the McCarthy anti-communist investigations, 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.^[17] 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 freedom 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.^[18]

During the last decades of his life, 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 UK 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 colleague Adrian Bondy.^[19][20] 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.

His life was documented in the film N Is a Number: A Portrait of Paul Erdős, made while he was still alive.

Mathematical work[link]

Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler; Erdős published more papers, while Euler published more pages.^[1] He wrote around 1,525 mathematical articles in his lifetime,^[21] mostly with co-authors. He strongly believed in and practiced mathematics as a social activity,^[22] having 511 different collaborators in his lifetime.^[23]

In terms of mathematical style, Erdős was much more of a "problem solver" than a "theory developer". (See "The Two Cultures of Mathematics"^[24] 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."^[25] Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did,^[26] a pattern that extends to other prizes.^[27] 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".^[28] In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".

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. Erdős also contributed to fields in which he had little real interest, such as topology, where he is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional.^[29]

Erdős problems[link]

Throughout his career, Erdős would offer prizes for solutions to unresolved problems.^[30] These ranged from $25 for problems that he felt were just out of the reach of current mathematical thinking, to several thousand dollars for problems that were both difficult to attack and mathematically significant. There are thought to be at least a thousand such outstanding prizes, though there is no official or comprehensive list. The prizes remain active despite Erdős's death; Ronald Graham is the (informal) administrator of solutions. Winners can get either a check signed by Erdős (for framing only) or a cashable check from Graham.^[31]

Perhaps the most mathematically notable of these problems is the Erdős conjecture on arithmetic progressions:

If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length.

If true, it would solve several other open problems in number theory (although one main implication of the conjecture, that the prime numbers contain arbitrarily long arithmetic progressions, has since been proved independently as the Green–Tao theorem.) The problem is currently worth US$5000.^[32]

The most familiar problem with an Erdős prize is likely the Collatz conjecture, also called the 3N+1 problem. Erdős offered $500 for a solution.

Collaborators[link]

His most frequent collaborators include Hungarian mathematicians András Sárközy (62 papers) and András Hajnal (56 papers), and American mathematician Ralph Faudree (50 papers). Other frequent collaborators were^[33]

For other co-authors of Erdős, see the list of people with Erdős number 1 in List of people by Erdős number.

Erdős number[link]

Because of his prolific output, friends created the Erdős number as a humorous tribute. An Erdős number describes a person's degree of separation from Erdős himself, based on their collaboration with him, or with another who has their own Erdős number. 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. Approximately 200,000 mathematicians have an assigned Erdős number,^[34] and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in light of the small world phenomenon). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics have Erdős numbers as well.^[35]

It is 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.^[36] Erdős numbers have also been assigned to an infant, a horse, and several actors.^[37]

The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1.^[38] Goffman published his observations about Erdős' prolific collaboration in a 1969 article titled "And what is your Erdős number?"^[39]

See also[link]

• List of topics named after Paul Erdős – including conjectures, numbers, prizes, and theorems

Notes[link]

References[link]

• Aigner, Martin; Günther Ziegler (2003). Proofs from THE BOOK. Berlin; New York: Springer. ISBN 3-540-40460-0. 
• Csicsery, George Paul (2005). N Is a Number: A Portrait of Paul Erdős. Berlin; Heidelberg: Springer Verlag. ISBN 3-540-22469-6.  - DVD
  • 1993 documentary film "N Is a Number: A Portrait of Paul Erdős" by George Paul Csicsery.
• Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. London: Fourth Estate Ltd. ISBN 1-85702-811-2. 
• Kolata, Gina (1996-09-24). "Paul Erdos, 83, a Wayfarer In Math's Vanguard, Is Dead". New York Times: pp. A1 and B8. http://query.nytimes.com/gst/fullpage.html?res=9C02EED7113DF937A1575AC0A960958260. Retrieved 2008-09-29. 
• Bruce Schechter (1998). My Brain is Open: The Mathematical Journeys of Paul Erdős. Simon & Schuster. ISBN 0-684-84635-7. 

External links[link]

This article's use of external links may not follow Wikipedia's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. (September 2011)

Wikiquote has a collection of quotations related to: Paul Erdős

Wikimedia Commons has media related to: Paul Erdős

• Erdos' Scholar Google profile
• Searchable collection of (almost) all papers of Erdős
• O'Connor, John J.; Robertson, Edmund F., "Paul Erdős", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Erdos.html .
• Paul Erdős at the Mathematics Genealogy Project
• Jerry Grossman at Oakland University. The Erdös Number Project
• The Man Who Loved Only Numbers - Royal Society Public Lecture by Paul Hoffman (video)
• Radiolab: Numbers, with a story on Paul Erdős

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"The Man" is a slang phrase that may refer to the government or to some other authority in a position of power. In addition to this derogatory connotation, it may also serve as a term of respect and praise.

The phrase "the Man is keeping me down" is commonly used to describe oppression. The phrase "stick it to the Man" encourages resistance to authority, and essentially means "fight back" or "resist", either openly or via sabotage.^[1]

History[link]

The earliest recorded use^{[citation needed]} of the term "the Man" in the American sense dates back to a letter written by a young Alexander Hamilton in September 1772, when he was 15. In a letter to his father James Hamilton, published in the Royal Dutch-American Gazette, he described the response of the Dutch governor of St. Croix to a hurricane that raked that island on August 31, 1772. "Our General has issued several very salutary and humane regulations and both in his publick and private measures, has shewn himself the Man." ^{[2][dubious – discuss]} In the Southern U.S. states, the phrase came to be applied to any man or any group in a position of authority, or to authority in the abstract. From about the 1950s the phrase was also an underworld code word for police, the warden of a prison or other law enforcement or penal authorities.

It was also used as a term for a drug dealer in the 1950s and 1960s and can be seen in such media as Curtis Mayfield's "No Thing On Me", William Burroughs's novel Naked Lunch, and in the Velvet Underground song "I'm Waiting for the Man", in which Lou Reed sings about going to Uptown Manhattan, specifically Lexington Avenue and 125th Street, to buy heroin.

The use of this term was expanded to counterculture groups and their battles against authority, such as the Yippies, which, according to a May 19, 1969 article in U.S. News and World Report, had the "avowed aim ... to destroy 'The Man', their term for the present system of government". The term eventually found its way into humorous usage, such as in a December 1979 motorcycle ad from the magazine Easyriders which featured the tagline, "California residents: Add 6% sales tax for The Man."

In present day, the phrase has been popularized in commercials and cinema, featuring particularly prominently as a recurring motif in the 2003 film School of Rock.^[3][4][5][6]

Use as praise[link]

The term has also been used as an approbation or form of praise. This may refer to the recipient's status as the leader or authority within a particular context, or it might be assumed to be a shortened form of a phrase like "He is the man (who is in charge)." One example of this usage dates to 1879 when Otto von Bismarck commented, referring to Benjamin Disraeli's pre-eminent position at the Congress of Berlin, "The old Jew, he is the man."^{[7][dubious – discuss]}

In more modern usage, it can be a superlative compliment ("you da man!") indicating that the subject is currently standing out amongst his peers even though they have no special designation or rank, such as a basketball player who is performing better than the other players on the court. It can also be used as a genuine compliment with an implied, slightly exaggerated or sarcastic tone, usually indicating that the person has indeed impressed the speaker but by doing something relatively trivial.

See also[link]

• Absentee landlord
• Big Brother (Nineteen Eighty-Four)
• New World Order (conspiracy theory)
• Ruling class
• The Establishment
• Power elite
• Corrections officer
• YTMND

Notes[link]

References[link]

• Lighter, J.E. (Ed.). (1997). Random House Dictionary of American Slang. New York: Random House.