Paul Erdos
Encyclopedia
Paul Erdős was a Hungarian
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, classical analysis, approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...

, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. He is also known for his "legendarily eccentric" personality.

Biography

Paul Erdős was born in Budapest
Budapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...

, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

 on March 26, 1913. He was the only surviving child of Anna and Lajos Erdős (formerly Engländer); his siblings died before he was born, aged 3 and 5. His parents were both Jewish mathematicians from a vibrant intellectual community. His fascination with mathematics developed early
Child prodigy
A child prodigy is someone who, at an early age, masters one or more skills far beyond his or her level of maturity. One criterion for classifying prodigies is: a prodigy is a child, typically younger than 18 years old, who is performing at the level of a highly trained adult in a very demanding...

—at the age of three, he could calculate how many seconds a person had lived.

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
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL
Középiskolai Matematikai és Fizikai Lapok
Középiskolai Matematikai és Fizikai Lapok is a Hungarian mathematics and physics journal for high school students...

, the Mathematical and Physical Monthly for Secondary Schools. Erdős later published several articles in it about problems in elementary plane geometry.

In 1934, at the age of 21, he was awarded a doctorate in mathematics. Because anti-Semitism
Anti-Semitism
Antisemitism is suspicion of, hatred toward, or discrimination against Jews for reasons connected to their Jewish heritage. According to a 2005 U.S...

 was increasing, he moved that same year to Manchester
Manchester
Manchester is a city and metropolitan borough in Greater Manchester, England. According to the Office for National Statistics, the 2010 mid-year population estimate for Manchester was 498,800. Manchester lies within one of the UK's largest metropolitan areas, the metropolitan county of Greater...

, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

. 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
Philanthropist
A philanthropist is someone who engages in philanthropy; that is, someone who donates his or her time, money, and/or reputation to charitable causes...

 to people in need and various worthy causes. He spent most of his life as a vagabond
Vagabond (person)
A vagabond is a drifter and an itinerant wanderer who roams wherever they please, following the whim of the moment. Vagabonds may lack residence, a job, and even citizenship....

, 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
Alfréd Rényi
Alfréd Rényi was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory.-Life:...

 said, "a mathematician is a machine for turning coffee into theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s", and Erdős drank copious quantities. (This quotation is often attributed incorrectly to Erdős himself. The German original of the sentence is a wordplay on the double meaning of "Satz": "theorem" or "residue of coffee", lost in the English translation) After 1971 he also took amphetamine
Amphetamine
Amphetamine or amfetamine is a psychostimulant drug of the phenethylamine class which produces increased wakefulness and focus in association with decreased fatigue and appetite.Brand names of medications that contain, or metabolize into, amphetamine include Adderall, Dexedrine, Dextrostat,...

s, despite the concern of his friends, one of whom (Ron Graham
Ronald Graham
Ronald Lewis Graham is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years"...

) bet him $500 that he could not stop taking the drug for a month. 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
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

 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
Fascism
Fascism is a radical authoritarian nationalist political ideology. Fascists seek to rejuvenate their nation based on commitment to the national community as an organic entity, in which individuals are bound together in national identity by suprapersonal connections of ancestry, culture, and blood...

" (SF). He accused the SF of hiding his socks and Hungarian passport
Passport
A passport is a document, issued by a national government, which certifies, for the purpose of international travel, the identity and nationality of its holder. The elements of identity are name, date of birth, sex, and place of birth....

s, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof
Mathematical beauty
Many mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...

 he would exclaim, "This one's from The Book!". This later inspired a book entitled Proofs from THE BOOK
Proofs from THE BOOK
Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem...

.

Other idiosyncratic elements of Erdős' vocabulary include:
  • Children were referred to as "epsilon
    Epsilon
    Epsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a close-mid front unrounded vowel . In the system of Greek numerals it has a value of 5. It was derived from the Phoenician letter He...

    s" (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
Imperialism
Imperialism, as defined by Dictionary of Human Geography, is "the creation and/or maintenance of an unequal economic, cultural, and territorial relationships, usually between states and often in the form of an empire, based on domination and subordination." The imperialism of the last 500 years,...

 and given a name that began with a lowercase letter. For example, the U.S. was "samland" (after Uncle Sam
Uncle Sam
Uncle Sam is a common national personification of the American government originally used during the War of 1812. He is depicted as a stern elderly man with white hair and a goatee beard...

), the Soviet Union was "joedom" (after Joseph Stalin
Joseph Stalin
Joseph Vissarionovich Stalin was the Premier of the Soviet Union from 6 May 1941 to 5 March 1953. He was among the Bolshevik revolutionaries who brought about the October Revolution and had held the position of first General Secretary of the Communist Party of the Soviet Union's Central Committee...

), and Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...

 was "israel". For his epitaph
Epitaph
An epitaph is a short text honoring a deceased person, strictly speaking that is inscribed on their tombstone or plaque, but also used figuratively. Some are specified by the dead person beforehand, others chosen by those responsible for the burial...

 he suggested, "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább").

In 1952, during the McCarthy anti-communist investigations
McCarthyism
McCarthyism is the practice of making accusations of disloyalty, subversion, or treason without proper regard for evidence. The term has its origins in the period in the United States known as the Second Red Scare, lasting roughly from the late 1940s to the late 1950s and characterized by...

, 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. 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
United States Citizenship and Immigration Services
United States Citizenship and Immigration Services is a component of the United States Department of Homeland Security . It performs many administrative functions formerly carried out by the legacy United States Immigration and Naturalization Service , which was part of the Department of Justice...

 at periodic intervals. The government changed its mind in 1963 and Erdős resumed including American universities in his teaching and travels.

Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

, then a Communist
Communism
Communism is a social, political and economic ideology that aims at the establishment of a classless, moneyless, revolutionary and stateless socialist society structured upon common ownership of the means of production...

 nation, was under the hegemony
Warsaw Pact
The Warsaw Treaty Organization of Friendship, Cooperation, and Mutual Assistance , or more commonly referred to as the Warsaw Pact, was a mutual defense treaty subscribed to by eight communist states in Eastern Europe...

 of the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

. 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 Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...

is.

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
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...

 and the UK Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

. Shortly before his death, he renounced his honorary degree from the University of Waterloo
University of Waterloo
The University of Waterloo is a comprehensive public university in the city of Waterloo, Ontario, Canada. The school was founded in 1957 by Drs. Gerry Hagey and Ira G. Needles, and has since grown to an institution of more than 30,000 students, faculty, and staff...

 over what he considered to be unfair treatment of colleague Adrian Bondy. He died "in action" of a heart attack
Myocardial infarction
Myocardial infarction or acute myocardial infarction , commonly known as a heart attack, results from the interruption of blood supply to a part of the heart, causing heart cells to die...

 on September 20, 1996, at the age of 83, while attending a conference in Warsaw
Warsaw
Warsaw is the capital and largest city of Poland. It is located on the Vistula River, roughly from the Baltic Sea and from the Carpathian Mountains. Its population in 2010 was estimated at 1,716,855 residents with a greater metropolitan area of 2,631,902 residents, making Warsaw the 10th most...

, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

. 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

Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

; Erdős published more papers, while Euler published more pages. He wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.

In terms of mathematical style, Erdős was much more of a "problem solver" than a "theory developer". (See "The Two Cultures of Mathematics" by Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated.) Joel Spencer
Joel Spencer
Joel Spencer is an American mathematician. He is a combinatorialist who has worked on probabilistic methods in combinatorics and on Ramsey theory. He received his doctorate from Harvard University in 1970, under the supervision of Andrew Gleason...

 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." Erdős never won the highest mathematical prize, the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...

, nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the Wolf Prize, where his contribution is described as "for his numerous contributions to number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 and mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, and for personally stimulating mathematicians the world over". 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
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...

 and the application of the probabilistic method
Probabilistic method
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the...

 especially stand out. Extremal combinatorics
Extremal combinatorics
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects can be, if it has to satisfy certain restrictions.For example, how many people can we invite to a party where among each...

 owes to him a whole approach, derived in part from the tradition of analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. 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
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

, along with Atle Selberg
Atle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...

, which showed how combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 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
Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets...

 that is not zero-dimensional
Zero-dimensional space
In mathematics, a zero-dimensional topological space is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space...

.

Erdős problems

Throughout his career, Erdős would offer prizes for solutions to unresolved problems. 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
Ronald Graham
Ronald Lewis Graham is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years"...

 is the (informal) administrator of solutions. Winners can get either a check signed by Erdős (for framing only) or a cashable check from Graham.

Perhaps the most 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 progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...

s 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 number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s contain arbitrarily long arithmetic progressions, has since been proved independently as the Green–Tao theorem.) The problem is currently worth US$5000.

Collaborators

His most frequent collaborators include Hungarian mathematicians András Sárközy
András Sárközy
András Sárközy is a Hungarian mathematician, working in analytic and combinatorial number theory, although his first works were in the fields of geometry and classical analysis. He has the largest number of papers co-authored with Paul Erdős...

 (62 papers) and András Hajnal
András Hajnal
András Hajnal is an emeritus professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.-Biography:Hajnal was born on 13 May 1931, in Hungary....

 (56 papers), and American mathematician Ralph Faudree
Ralph Faudree
Ralph Jasper Faudree is a mathematician, a professor of mathematics and the provost of the University of Memphis.Faudree was born in Durant, Oklahoma. He did his undergraduate studies at Oklahoma Baptist University, graduating in 1961, and received his Ph.D. in 1964 from Purdue University under...

 (50 papers). Other frequent collaborators were

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

Erdős number

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,
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.

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
Emory University
Emory University is a private research university in metropolitan Atlanta, located in the Druid Hills section of unincorporated DeKalb County, Georgia, United States. The university was founded as Emory College in 1836 in Oxford, Georgia by a small group of Methodists and was named in honor of...

 awarded them honorary degrees on the same day. Erdős numbers have also been assigned to an infant, a horse, and several actors.

The Erdős number was most likely first defined by Casper Goffman, an analyst
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 whose own Erdős number is 1. Goffman published his observations about Erdős' prolific collaboration in a 1969 article titled "And what is your Erdős number?"

See also

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

External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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