Reminiscences of Paul ErdÃ¶s (1913-1996)
Melvin Henriksen
I met Paul ErdÃ¶s shortly after his 40th birthday in April 1953 at
Purdue University in West Lafayette, Indiana. He was already a living
legend because of his substantial contributions to the theory of numbers,
the theory of sets, what is now called discrete mathematics, as well as to
many other areas of mathematics. (For example, although he had little
interest in topology, his name appears in most topology texts as the first
person to give an example of totally disconnected topological space that is
not zero-dimensional.) I was a 26-year old instructor in my first year at
Purdue. Many of my colleagues knew him well. He had been a visiting
research associate at Purdue for a couple of years during World War II, and
had visited so many universities and attended so many conferences that he
was well known to most of the others. Those that were active in research
admired his mathematical accomplishments, while others on the faculty were
amused by his eccentricities. What I remember most clearly is his
announcement to everyone that "death begins at 40".
I am not qualified to write a biography of ErdÃ¶s, but some background
seems necessary. There is an excellently written and accurate obituary of
him by Gina Kolata in the Sept. 21, 1996 issue of the New York Times,
beginning on page 1. An interview conducted in 1979 which reveals much of
his personality appeared in the volume Mathematical People edited by
D.J. Albers and G.L. Alexanderson (Birkhauser 1985). The Mathematical
Association of America (MAA) sells two videos of ErdÃ¶s, and Ronald
Graham, a long time collaborator, has edited together with Jarik Nesetril
two volumes on his mathematical work and life. (Both volumes have been
published by Springer-Verlag and were available in January 1997. They
include a detailed biographical article by Bella Bollobas.)
ErdÃ¶s was born in Budapest in 1913 of parents who were Jewish
intellectuals. His brilliance was evident by the time he was three years
old. For this reason, and perhaps because two older sisters died of scarlet
fever shortly before he was born, his parents shielded him almost
completely from the everyday problems of life. For example, he never had to
tie his own shoelaces until he was 14 years old, and never buttered his own
toast until he was 21 years old in Cambridge, England. In return for the
freedom to concentrate almost exclusively on intellectual pursuits, he paid
the price of not learning the social skills that are expected of all of us
and usually acquired in childhood.
He became internationally famous at the age of 20 when he got a
simple proof of a theorem that was originally conjectured by
Bertrand and later proved by Tchebychev: For every positive integer
n, there is a prime between n and 2n. Tchebychev's
proof was quite hard! ErdÃ¶s completed the requirements for the Ph.D.
at the University of Budapest about a year later, but had no chance of
getting a position in Hungary because he was a Jew living under a right
wing dictatorship allied with Nazi Germany. He spent some time at
Cambridge University in 1935. There, his life as a wandering mathematician
began. In fact, he had visited Cambridge three times the year before. He
liked traveling and had no trouble working while doing so. He liked people,
and except for those who could not tolerate his ignorance of the social
graces, they liked him. He tried his best to be pleasant to everyone and
was generous in giving credit and respect to his collaborators.
I do not know when he first came to the United States, but he spent the
years of World War II here, two of them at Purdue. Nor can I give a list of
the many universities he visited for any substantial length of time. By the
time I met him, he had written joint papers with many mathematicians most
of whom had established research reputations before working with
ErdÃ¶s. The only ErdÃ¶s collaborator who worked with him
unwillingly was Atle Selberg. In the late 1940s, both of them, working
independently, had obtained "elementary" proofs (meaning: proofs that did
not use complex analysis) of the prime number theorem. The theorem states
that the number of primes less than or equal to (a positive real number)
x is asymptotically equal to x/log(x). This had been
conjectured by Gauss and Legendre based on empirical data, but it had only
been proved many years later, by two French mathematicians, Jacques
Hadamard and Charles de la VallÃ©e Poussin (also working
independently). Both proofs depended heavily on complex analysis. What
Selberg and ErdÃ¶s did in their "elementary" proofs was to avoid using
complex analysis (the proofs were in no sense "easy"). In those pre-email
days, the fastest courier of mathematical news was Paul ErdÃ¶s. He told
anyone who would listen that Selberg and he had devised an elementary proof
of the prime number theorem.
Almost every number theorist knew of ErdÃ¶s, while few had heard of the
young Norwegian Selberg. So when the news traveled back to Selberg, it
appeared that ErdÃ¶s had claimed all the credit for himself. The
ensuing bitterness was not healed by the two of them writing a joint
paper. Selberg later published another elementary proof on his own, and
went on to a brilliant mathematical career, eventually becoming a permanent
member of the Institute for Advanced Study in Princeton, the Valhalla for
mathematicians. ErdÃ¶s had been a visitor there earlier, but was not
offered a membership. Exactly what happened is controversial to this day,
and reading the article by Bollobas will shed more light on this matter
than this short summary can.
ErdÃ¶s spent the academic year 1953-54 at the University of Notre Dame
in South Bend, Indiana. Arnold Ross, the chairman of the Mathematics
Department, had arranged for him to teach only one (advanced) course, and
supplied an assistant who could take over his class if he had the urge to
travel to talk with a collaborator. ErdÃ¶s had rejected organized
religion as a young man, and had been persecuted in Roman Catholic
Hungary. So we teased him about working at a Catholic institution. He said
in all seriousness that he liked being there very much, and especially
enjoyed discussions with the Dominicans. "The only thing that bothers me",
he said, "There are too many plus signs." He came by bus to West Lafayette
fairly often for short periods because he had so many friends there and
because he liked the mathematical atmosphere.
At that time, Leonard Gillman and I were trying to study the structure of
the residue class fields of rings of real-valued continuous functions on a
topological space modulo maximal ideals. We had learned quite a bit about
them, but had run into serious set-theoretic difficulties. ErdÃ¶s had
little interest in abstract algebra or topology, but was a master of
set-theoretic constructions. Without bothering him with our motivation for
asking them, we asked him a series of questions about set theory, which he
managed to answer while we could not.
He was not terribly interested when we supplied him with the motivation,
and I have often said that ErdÃ¶s never understood our paper; all he
did was the hard part. This paper by ErdÃ¶s, Gillman and Henriksen was
published in the Annals of Mathematics in 1955. Without any of us realizing
it in advance, it became one of the pioneering papers in nonstandard
analysis, and was often credited to ErdÃ¶s, et al.
ErdÃ¶s got an offer allowing him to stay indefinitely at Notre Dame on
the same generous basis. His friends urged him to accept. "Paul", we said
"how much longer can you keep up a life of being a traveling
mathematician?" (Little did we suspect that the answer was going to turn
out to be "more than 40 years.") ErdÃ¶s thanked Ross, but turned him
down. As it turned out, he would not have been at Notre Dame the next year
whatever his answer had been.
The cold war was in full swing, the United States were in the grip of
paranoia about communism, and many regarded unconventional behavior as
evidence of disloyalty. ErdÃ¶s had never applied for citizenship
anywhere he lived, and had acquired Hungarian citizenship only by accident
of birth. He belonged to no political party, but had a fierce belief in the
freedom of individuals as long as they did no harm to anyone else. All
countries who failed to follow this were classified as imperialist and
given a name that began with a small letter. For example, the U.S. was
samland and the Soviet Union was joedom (after Joseph
Stalin). He talked of an organization called the f.b.u--a combination of
the F.B.I and O.G.P.U (which later became the K.G.B) and conjectured that
their agents were often interchanged.
In 1954, ErdÃ¶s wanted to go the
International Congress of Mathematicians (held every four years), which was
to be in Amsterdam that August. As a non-citizen leaving the U.S. with
plans to return, he had to apply for a
re-entry permit. After being interviewed by an INS agent in South Bend in
early 1954, he received a letter saying that re-entry would be denied if he
left the U.S. He hired a lawyer and appealed only to be turned down
again. No reason was ever given, but his lawyer was permitted to examine a
portion of ErdÃ¶s' file and found recorded the following facts:
- He corresponded with a Chinese number theorist named Hua who had
left his position at the University of Illinois to return to (red)
China in 1949. (A typical ErdÃ¶s letter would have begun: Dear
Hua, Let p be an odd prime ...)
- He had blundered onto a radar installation on Long island in 1942
while discussing mathematics with two other non-citizens.
- His mother worked for the Hungarian Academy of Sciences, and had had
to join the communist party to hold her position.
To ErdÃ¶s, being denied the right to travel was like being denied the
right to breathe, so he went to Amsterdam anyway. He was confident that he
could easily obtain a Dutch and an English visa. The Dutch gave him a visa
good for only a few months, and England would not let him come, likely
because if they chose to deport him, the only country obligated to accept
him was communist Hungary. By then, ErdÃ¶s was a member of the
Hungarian Academy of Sciences, but he would go to Hungary only if his
friends could assure him that he would be permitted to leave. At this
point, he swallowed his pride and obtained a passport from israel (note the
punctuation) which served to give him freedom to travel anywhere in western
Europe. He was permitted to return to the United States in the summer of
1959 on a temporary visa to attend a month long conference on number theory
in Boulder, Colorado. He stopped at Purdue on his way back to Europe to
give a colloquium talk. When I picked him up at the airport, what struck me
first was that he had a suitcase! For many years, he traveled only with a
small leather briefcase containing a change of socks and underwear in
addition to a wash-and-wear shirt, together with some paper and a few
reprints. About a year later, the United States government lost its fear of
ErdÃ¶s and gave him resident alien status once more. He never had
trouble going in or out of the U.S. again.
ErdÃ¶s had lived from hand to mouth most of the time until the late
1950s. When the Russians sent Sputnik into orbit and the space race began,
there was a vast increase in government support of research. This made it
possible for his many friends and co-authors to give him research
stipends. This had little effect on his lifestyle. His suitcase was rarely
more than half full, and he gave away most of his money to help talented
young mathematicians or to offer cash prizes for solving research problems
of varying degrees of difficulty. (The cash prizes were not as costly as he
had expected. The winners would often frame his checks without cashing
them. Solving a $1000 problem would make you internationally famous, and
being able to say that you solved any of his prize problems enhanced your
reputation.) Around 1965, Casper Goffman concocted the idea of an
ErdÃ¶s number. If you had written a joint paper with him, your
ErdÃ¶s number was 1. If you had written a joint paper with someone
with ErdÃ¶s number 1, your ErdÃ¶s number is 2, and so on
inductively. There is now an ErdÃ¶s Number
Project home page on the web where you can see a list of all who have
an Edos number of 1 (there are 462 of us) and 2 (all 4566 of them,
including Albert Einstein). All in all, ErdÃ¶s wrote about 1500
research papers, and 50 or so more will appear after his death.
While we did no more joint research, we often met at conferences or when we
were both visiting the same university. Sometimes I could hardly talk to
him because he was surrounded by mathematicians eager to ask him questions,
but when I could, he inquired about mutual friends and asked about
follow-up work on our paper and progress about solving the open problems we
had posed. While he devoted his life to mathematics, he was widely read in
many areas and I almost always learned a great deal talking to him about
many non-mathematical ideas. I saw him last in Budapest last Sept. 4. He
attended the first half of a talk I gave about separate vs. joint
continuity. He apologized in advance about having to leave early because he
had made an appointment he could not break before he knew I would be
speaking. Even then, he made two helpful comments while present. Before I
left the Academy of Sciences, I stopped to say good-bye and saw him going
over a paper with a young Hungarian mathematician. He died in Warsaw of a
heart attack on Sept. 20. He worked on what he loved to do to the last!
ErdÃ¶s had a special vocabulary that he concocted and used
consistently in his speech. Some samples are:
- Children are Epsilons
- Women are Bosses
- Men are Slaves
- Married Men have been Captured
- Alcoholic Drinks are Poison
- God is The Supreme Fascist or SF
- Music is Noise.
Examples:
I asked Louise Piranian (President of the League of Women Voters
in Ann Arbor, Michigan in the early 1950s) "When will you bosses take the
vote away from the slaves?" Answer :"There is no need; we tell them how to
vote anyway."
"Wine, women, and song" becomes "Poison, bosses, and noise".
ErdÃ¶s said that the SF had a Book containing elegant proofs of all the
important theorems, and when a mathematician worked very hard, the SF could
be distracted long enough to allow her or him to take a brief
peek. Particularly elegant proofs were described as fit to be placed in the
Book.
There are many ErdÃ¶s stories that were embellished over the years and
made more delightful than the truth. For example, consider the story about
blundering into a radar installation in 1942:
- Embellished version: ErdÃ¶s, Hochschild (a German) and
Kakutani (a Japanese) drove a car out onto Long Island and held an animated
mathematical conversation in German. They walked onto a radar installation
and were apprehended by a guard who was convinced that he had caught a
group of foreign spies. They were questioned closely by military
intelligence and released with a warning when they promised never to do
such a thing again.
- Actual version: The car was driven by Arthur Stone (an
Englishman). Hochschild was supposed to come, but did not
because he had a date. They were speaking English because it was
their only western language understood by Kakutani. The guard
was satisfied as soon as they presented proper identification,
and they were visited individually and briefly a few days later
by military intelligence agents.
ErdÃ¶s liked to tell many stories about himself. In particular, when he
grew older, he claimed to be two billion years old because when he was in
high school, he was taught that the earth was two and a half billion years
old--but now we know it is four and a half billion years old.
Because he seemed to be in a state of Brownian motion, it was often hard to
locate him at any given time. ErdÃ¶s visited Claremont twice in the
1970s and could often be found at UCLA. For many years the way to contact
him was to call Ron Graham of Bell Labs on the east coast, Paul Bateman of
the University of Illinois, or Ernst Strauss at UCLA to find out where he
was. Strauss died in 1983 and was replaced by Bruce Rothschild. Paul
Bateman retired. Although Ron Graham himself traveled a great deal, until
the end he was the person most likely to know of ErdÃ¶s'
whereabouts.
With ErdÃ¶s' death we have lost one of the great mathematicians and
free spirits of this century and it is hard to imagine that we will see
anyone like him again. I feel fortunate to have had the privilege of
knowing and working with him.
Melvin Henriksen
Harvey Mudd College
Claremont CA 91711
[email protected]
Melvin Henriksen just retired as a Professor at Harvey Mudd college.
Prior to coming there in 1969, he held teaching positions at the University of
Alabama, Purdue University (where he met Paul Erdos), Wayne State University,
and Case Western Reserve University, as well as visiting the Institute for
Advanced Study, the University of California at Berkeley, Wesleyan University,
and the University of Manitoba. He has done research in algebra and general
topology. He feels privleged to have known Paul ErdÃ¶s.