Paul Erdős in 1992

The Erdős number (Hungarian pronunciation: [ˈɛrdøːʃ]) describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.

The same principle has been proposed for other eminent people in other fields.

Overview[link]

The idea of the Erdős number was created by friends as a humorous tribute to the enormous output of Erdős, one of the most prolific modern writers of mathematical papers, and has become well known in scientific circles as a tongue-in-cheek measurement of mathematical prominence.

Paul Erdős was an influential and itinerant mathematician, who spent a large portion of his later life living out of a suitcase and writing papers with those of his colleagues willing to give him room and board.^[1] He published more papers during his life (at least 1,525^[2]) than any other mathematician in history.^[1]

Definition[link]

If Alice collaborates with Paul Erdős on one paper, and with Bob on another, but Bob never collaborates with Erdős himself, then Bob is given an Erdős number of 2, as he is two steps from Erdős.

To be assigned an Erdős number, an author must co-write a research paper with an author with a finite Erdős number. Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is k + 1 where k is the lowest Erdős number of any coauthor.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;^[3] these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (9267 people as of 2010^[4]), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one).

There is room for ambiguity over what constitutes a link between two authors; the Erdős Number Project web site says:

... Our criterion for inclusion of an edge between vertices u and v is some research collaboration between them resulting in a published work. Any number of additional co-authors is permitted,...

but they do not include non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The “Erdős number of the second kind” restricts assignment of Erdős numbers to papers with only two collaborators.^[5]

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 1.^[6] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"^[7]

Most frequent Erdős collaborators[link]

While Erdős collaborated with hundreds of co-authors, there were some individuals with whom he co-authored dozens of papers. This is a list of the ten persons that most frequently co-authored with Erdős and their number of papers co-authored with Erdõs (i.e. their number of collaborations).^[8]

Impact[link]

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Amongst all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean Erdős number is 4.65;^[3] almost everyone with a finite Erdős number has a number less than 8. Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers.^[9] For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers,^[10][11] and since Kleitman has an Erdős number of 1,^[12] a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for Erdős numbers within the cryptographic research community.

According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9. Similarly, many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4),^[13] William Labov (3),^[14] Mark Liberman (3),^[15] Geoffrey Pullum (3),^[16] or Ivan Sag (4).^[17]

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly written papers. The earliest person known to have a finite Erdős number is either Richard Dedekind (born 1831, Erdős number 7) or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.^[18] It seems that older historic figures such as Leonhard Euler (born 1707) do not have finite Erdős numbers.

Tompa^[19] proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

K. Dixit and colleagues argue that "for an individual researcher, a measure such as Erdős number captures the structural properties of network whereas the h-index captures the citation impact of the publications. One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking." Several author ranking systems based on eigenvector centrality have been proposed already, for instance the Phys Author Rank Algorithm.^[20]

Variations[link]

A number of variations on the concept have been proposed to apply to other fields.

Bacon number[link]

The Bacon number (as in the game Six Degrees of Kevin Bacon) is an application of the same idea to the movie industry, connecting actors that appeared in a film together to the actor Kevin Bacon. Although this is the most well-known numbering system of this type, it was conceived of in 1994, 25 years after Goffman's article on the Erdős number.

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series, The Wonder Years. Her Erdős number is 4^[23] and her Bacon number is 2.^[24] The lowest known Erdős–Bacon number is three for Daniel Kleitman, a mathematics professor at MIT; his Erdős number is 1 and his Bacon number is 2.^[25]

Shusaku number[link]

The Shusaku number represents the "distance" between a go player and Honinbo Shusaku, measured in Go opponents.^[26] Shusaku himself has the Shusaku number 0. If a player has played against Shusaku himself, that player would have a Shusaku number of 1. And so on.^[27]

See also[link]

• James Parry, target of a similar number.
• Morphy number
• Small world experiment
• Small-world network
• List of people by Erdős number
• List of topics named after Paul Erdős

References[link]

Further reading[link]

• Goffman, Casper (1969). "And What Is Your Erdös Number?". American Mathematical Monthly 76 (7): 791. DOI:10.2307/2317868. JSTOR 2317868. 
• De Castro, Rodrigo; Grossman, Jerrold W. (1999). "Famous Trails to Paul Erdős". The Mathematical Intelligencer 21 (3): 51–63. DOI:10.1007/BF03025416. MR 1709679. https://files.oakland.edu/users/grossman/enp/trails.ps.  Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, MR1744115.

External links[link]

• Jerry Grossman, The Erdös Number Project. Contains statistics and a complete list of all mathematicians with an Erdős number less than or equal to 2.
• "On a Portion of the Well-Known Collaboration Graph", Jerrold W. Grossman and Patrick D. F. Ion.
• "Some Analyses of Erdős Collaboration Graph", Vladimir Batagelj and Andrej Mrvar.
• American Mathematical Society, MR Collaboration Distance. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.
• "Theorems for Sale" (From Science News, Vol. 165, No. 24, June 12, 2004)

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2012)

"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.

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