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The Man Who Loved Only Numbers

The Story of Paul Erdös and the Search for Mathematical Truth

By PAUL HOFFMAN

Publisher
Read the Review
THE TWO-AND-A-HALF-BILLION-

YEAR-OLD MAN

Vegre nem butulok tovabb

(Finally I am becoming stupider no more)
--the epitaph Paul Erdös wrote for himself
Paul Erdös was one of those very special geniuses,
the kind who comes along only once in
a very long while yet he chose, quite consciously
I am sure, to share mathematics with mere
mortals--like me. And for this, I will always
be grateful to him. I will miss the times he
prowled my hallways at 4:00 . and came to
my bed to ask whether my "brain is open." I
will miss the problems and conjectures and the
stimulating conversations about anything and
everything. But most of all, I will just miss
Paul, the human. I loved him dearly.
--Tom Trotter
It was dinnertime in Greenbrook, New Jersey, on a cold
spring day in 1987, and Paul Erdös, then seventy-four,
had lost four mathematical colleagues, who were sitting
fifty feet in front of him, sipping green tea. Squinting,
Erdös scanned the tables of the small Japanese restaurant,
one arm held out to the side like a scarecrow's. He was
angry with himself for letting his friends slip out of
sight. His mistake was to pause at the coat cheek while
they charged ahead. His arm was flapping wildly now,
and he was coughing. "I don't understand why the SF
has seen fit to send me a cold," he wheezed. (The SF is
the Supreme Fascist, the Number-One Guy Up There,
God, who was always tormenting Erdös by hiding his
glasses, stealing his Hungarian passport, or, worse yet,
keeping to Himself the elegant solutions to all sorts of
intriguing mathematical problems.) "The SF created us to
enjoy our suffering," Erdös said. "The sooner we die, the
sooner we defy His plans."
Erdös still didn't see his friends, but his anger dissipated--his
arm dropped to his side--as he heard the high-pitched
squeal of a small boy, who was dining with his
parents. "An epsilon!" Erdös said. (* Epsilon* was Erdös's word
for a small child; in mathematics that Greek letter is used
to represent small quantities.) Erdös moved slowly toward
the child, navigating not so much by sight as by the sound
of the boy's voice. "Hello," he said, as he reached into his
ratty gray overcoat and extracted a bottle of Benzedrine.
He dropped the bottle from shoulder height and with the
same hand caught it a split second later. The epsilon was
not at all amused, but perhaps to be polite, his parents
made a big production of applauding. Erdös repeated the
trick a few more times, and then he was rescued by one
of his confederates, Ronald Graham, a mathematician at
AT&T, who called him over to the table where he and
Erdös's other friends were waiting.
The waitress arrived, and Erdös, after inquiring about
each item on the long menu, ordered fried squid balls.
While the waitress took the rest of the orders, Erdös turned
over his placemat and drew a tiny sketch vaguely resembling
a rocket passing through a hula-hoop. His four dining
companions leaned forward to get a better view of the
world's most prolific mathematician plying his craft.
"There are still many edges that will destroy chromatic
number three," Erdös said. "This edge destroys bipartiteness."
With that pronouncement Erdös dosed his eyes and
seemed to fall asleep.
Mathematicians, unlike other scientists, require no laboratory
equipment--a practice that reportedly began with
Archimedes, who, after emerging from his bath and rubbing
himself with olive oil, discovered the principles of
geometry by using his fingernails to trace figures on his
oily skin. A Japanese restaurant, apparently, is as good a
place as any to do mathematics. Mathematicians need only
peace of mind and, occasionally, paper and pencil. "That's
the beauty of it," Graham said. "You can lie back, close
your eyes, and work. Who knows what problem Paul's
thinking about now?"
"There was a time at Trinity College, in the 1930s I
believe, when Erdös and my husband, Harold, sat thinking
in a public place for more than an hour without
uttering a single word," recalled Anne Davenport, the
widow of one of Erdös's English collaborators. "Then Harold
broke the long silence, by saying, `It is not nought. It
is one.' Then all was relief and joy. Everyone around them
thought they were mad. Of course, they were."
*
Before Erdös died, on September 20, 1996, at the age of
eighty-three, he had managed to think about more problems
than any other mathematician in history. He wrote
or co-authored 1,475 academic papers, many of them monumental,
and all of them substantial. It wasn't just the
quantity of work that was impressive but the quality:
"There is an old saying," said Erdös. "* Non numerantur, sed
ponclerantur* (They are not counted but weighed). In the
old [Hungarian] parliament of noblemen, they didn't count
the votes: they weighed them. And this is true of papers.
You know, Riemann had a very short list of papers, Godel
had a short list. Gauss was very prolific, as was Euler, of
course." Even in his seventies there were years when Erdös
published fifty papers, which is more than most good mathematicians
write in a lifetime. He proved that mathematics
isn't just a young man's game.
Erdös (pronounced "air-dish") structured his life to
maximize the amount of time he had for mathematics. He
had no wife or children, no job, no hobbies, not even a
home, to tie him down. He lived out of a shabby suitcase
and a drab orange plastic bag from Centrum Aruhaz ("Central
Warehouse"), a large department store in Budapest. In
a never-ending search for good mathematical problems
and fresh mathematical talent, Erdös crisscrossed four continents
at a frenzied pace, moving from one university or
research center to the next. His modus operandi was to
show up on the doorstep of a fellow mathematician, declare,
"My brain is open," work with his host for a day or
two, until he was bored or his host was run down, and then
move on to another home.
Erdös's motto was not "Other cities, other maidens" but
"Another roof, another proof." He did mathematics in more
than twenty-five different countries, completing important
proofs in remote places and sometimes publishing them in
equally obscure journals. Hence the limerick, composed by
one of his colleagues:

A conjecture both deep and profound

Is whether the circle is round.

In a paper of Erdös

Written in Kurdish

A counterexample is found.
When Erdös heard the limerick, he wanted to publish a
paper in Kurdish but couldn't find a Kurdish math journal.
*
Erdös first did mathematics at the age of three, but for the
last twenty-five years of his life, since the death of his
mother, he put in nineteen-hour days, keeping himself fortified
with 10 to 20 milligrams of Benzedrine or Ritalin,
strong espresso, and caffeine tablets. "A mathematician,"
Erdös was fond of saying, "is a machine for turning coffee
into theorems." When friends urged him to slow down, he
always had the same response: "There'll be plenty of time
to rest in the grave."
Erdös would let nothing stand in the way of mathematical
progress. When the name of a colleague in California
came up at breakfast in New Jersey, Erdös
remembered a mathematical result he wanted to share with
him. He headed toward the phone and started to dial. His
host interrupted him, pointing out that it was 5:00 . on
the West Coast. "Good," Erdös said, "that means he'll be
home."
When challenged further in situations like this, Erdös
was known to respond, "Louis the Fourteenth said, `I am
the state'; Trotsky could have said, `I am society'; and I say,
`I am reality.'" No one who knew him would disagree.
"Erdös had a childlike tendency to make his reality overtake
yours," a friend said. "And he wasn't an easy houseguest.
But we all wanted him around--for his mind. We
all saved problems up for him."
To communicate with Erdös you had to learn his language.
"When we met," said Martin Gardner, the mathematical
essayist, "his first question was `When did you
arrive?' I looked at my watch, but Graham whispered to
me that it was Erdös's way of asking, `When were you
born?'" Erdös often asked the same question another way:
"When did the misfortune of birth overtake you?" His
language had a special vocabulary--not just "the SF" and
"epsilon" but also "bosses" (women), "slaves" (men), "captured"
(married), "liberated" (divorced), "recaptured" (remarried),
"noise" (music), "poison" (alcohol), "preaching"
(giving a mathematics lecture), "Sam" (the United States),
and "Joe" (the Soviet Union). When he said someone had
"died," Erdös meant that the person had stopped doing
mathematics. When he said someone had "left," the person
had died.
*
At five foot six, 130 pounds, Erdös had the wizened, cadaverous
look of a drug addict, but friends insist he was frail
and gaunt long before he started taking amphetamines. His
hair was white, and corkscrew-shaped whiskers shot out at
odd angles from his face. He usually wore a gray pinstriped
jacket, dark trousers, a red or mustard shirt or pajama
top, and sandals or peculiar pockmarked Hungarian
leather shoes, made especially for his flat feet and weak
tendons. His whole wardrobe fit into his one small suitcase,
with plenty of room left for his dinosaur of a radio. He
had so few clothes that his hosts found themselves washing
his socks and underwear several times a week. "He could
buy more," one of his colleagues said, "or he could
wash them himself. I mean, it takes zero IQ to learn how
to operate a washing machine." But if it wasn't mathematics,
Erdös wouldn't be bothered. "Some French socialist
said that private property was theft," Erdös recalled. "I say
that private property is a nuisance."
The only possessions that mattered to him were his
mathematical notebooks. He filled ten of them by the time
he died. He always carried one around with him, so that
he could record his mathematical insights on a moment's
notice. "Erdös came to my twins' bar mitzvah, notebook in
hand," said Peter Winkler, a colleague of Graham's at
AT&T. "He also brought gifts for my children--he loved
kids--and behaved himself very well. But my mother-in-law
tried to throw him out. She thought he was some guy
who wandered in off the street, in a rumpled suit, carrying
a pad under his arm. It is entirely possible that he proved
a theorem or two during the ceremony."
All of his clothes, including his socks and custommade
underwear, were silk, because he had an undiagnosed
skin condition that was aggravated by other kinds
of fabric. He didn't like people to touch him. If you extended
your hand, he wouldn't shake it. Instead, he'd
limply flop his hand on top of yours. "He hated it if I
kissed him," said Magda Fredro, a first cousin who was
otherwise very close to him. "And he'd wash his hands
fifty times a day. He got water everywhere. It was hell
on the bathroom floor."
Although Erdös avoided physical intimacy, and was
always celibate, he was friendly and compassionate. "He
existed on a web of trust," said Aaron Meyerowitz, a
mathematician at Florida Atlantic University. "When I
was a graduate student and we had never met before, I
gave him a ride. I didn't know the route and asked him
if he wanted to navigate with a map. He didn't want to
[and probably didn't know how to]. He just trusted that
I, a total stranger, would get him there."
What little money Erdös received in stipends or lecture
fees he gave away to relatives, colleagues, students, and
strangers. He could not pass a homeless person without
giving him money. "In the early 1960s, when I was a student
at University College London," recalled D. G. Larman,
"Erdös came to visit us for a year. After collecting his first
month's salary he was accosted by a beggar on Euston station,
asking for the price of a cup of tea. Erdös removed a
small amount from the pay packet to cover his own frugal
needs and gave the remainder to the beggar." In 1984 he
won the prestigious Wolf Prize, the most lucrative award
in mathematics. He contributed most of the $50,000 he
received to a scholarship in Israel he established in the
name of his parents. "I kept only seven hundred and twenty
dollars," Erdös said, "and I remember someone commenting
that for me even that was a lot of money to keep."
Whenever Erdös learned of a good cause--a struggling
classical music radio station, a fledgling Native American
movement, a camp for wayward boys--he promptly made
a small donation. "He's been gone a year," said Graham,
"and I'm still getting mail from organizations he gave donations
to. Today I got a postcard from an Israeli girls'
home."
In the late 1980s Erdös heard of a promising high
school student named Glen Whitney who wanted to
study mathematics at Harvard but was a little short of
the tuition. Erdös arranged to see him and, convinced of
the young man's talent, lent him $1,000. He asked Whitney
to pay him back only when it would not cause financial
strain. A decade later Graham heard from Whitney,
who at last had the money to repay Erdös. "Did Erdös
expect me to pay interest?" Whitney wondered. "What
should I do?" he asked Graham. Graham talked to Erdös.
"Tell him," Erdös said, "to do with the thousand dollars
what I did."
*
Erdös was a mathematical prodigy. At three he could multiply
three-digit numbers in his head, and at four he discovered
negative numbers. "I told my mother," he recalled,
"that if you take 250 from 100, you get -150. My second
great discovery was death. Children don't think they're ever
going to die. I was like that too, until I was four. I was in
a shop with my mother and suddenly I realized I was
wrong. I started to cry. I knew I would die. From then on,
I've always wanted to be younger. In 1970, I preached in
Los Angeles on `my first two and a half billion years in
mathematics.' When I was a child, the Earth was said to
be two billion years old. Now scientists say it's four and a
half billion. So that makes me two and a half billion. The
students at the lecture drew a timeline that showed me
riding a dinosaur. I was asked, `How were the dinosaurs?'
Later, the right answer occurred to me: `You know, I don't
remember, because an old man only remembers the very
early years, and the dinosaurs were born yesterday, only a
hundred million years ago.'"
Erdös loved the dinosaur story and repeated it again
and again in his mathematical talks. "He was the Bob Hope
of mathematics, a kind of vaudeville performer who told
the same jokes and the same stories a thousand times," said
Melvyn Nathanson at a mathematical memorial service for
Erdös in Budapest. "When he was scheduled to give yet
another talk, no matter how tired he was, as soon as he
was introduced to an audience, the adrenaline (or maybe
amphetamine) would release into his system and he would
bound onto the stage, full of energy, and do his routine for
the thousand and first time."
In the early 1970s, Erdös started appending the initials
. to his name, which stood for Poor Great Old
Man. When he turned sixty, he became .,
the . for Living Dead. At sixty-five he graduated
to ., the . for Archeological Discovery.
At seventy he became ., the
. for Legally Dead. And at seventy-five he was
., the . for Counts Dead. In
1987, when he was seventy-four, he explained: "The Hungarian
Academy of Sciences has two hundred members.
When you turn seventy-five, you can stay in the academy
with full privileges, but you no longer count as a member.
That's why the . Of course, maybe I won't have to face
that emergency. They are planning an international conference
for my seventy-fifth birthday. It may have to be
for my memory. I'm miserably old. I'm really not well. I
don't understand what's happening to my body--maybe
the final solution."
Erdös outlived most of his friends and, to his dismay,
watched some of them lose their minds. His college thesis
adviser, Leopold Fejer, one of the strongest mathematicians
in Hungary, was burned out by the age of thirty. "He still
did very good things, but he felt that he didn't have any
significant new ideas," said Erdös. "When he was sixty, he
had a prostate operation and after that he didn't do very
much. Then he was on an even keel for fifteen or sixteen
years, and then he became senile. There was some disturbance
of the circulation. It was very sad because he knew
he was senile and he said things like, `Since I became a
complete idiot....' He was very well kept in the hospital
but died of a stroke in 1959."
When Paul Turan, his closest friend, with whom he
had written thirty papers, died in 1976, Erdös had an image
of the SF assessing the work he had done with his collaborators.
On one side of a balance the SF would place the
papers Erdös had co-authored with the dead; on the other
side the papers written with the living. "When the dead
side tips the balance," Erdös said, "I must die too." He
paused for a moment and then added, "'It's just a joke of
mine."
Perhaps. But for decades Erdös vigorously sought out
new, young collaborators and ended many working sessions
with the remark, "We'll continue tomorrow if I
live." With 485 co-authors, Erdös collaborated with more
people than any other mathematician in history. Those
lucky 485 are said to have an Erdös number of 1, a coveted
code phrase in the mathematics world for having
written a paper with the master himself. If your Erdös
number is 2, it means you have published with someone
who has published with Erdös. If your Erdös number is
3, you have published with someone who has published
with someone who has published with Erdös. Einstein
had an Erdös number of 2, and the highest known Erdös
number of a working mathematician is 7. The great unwashed
who have never written a mathematical paper
have an Erdös number of [infinity].
"I was told several years ago that my Erdös number
was 7," Caspar Goffman at Purdue wrote in 1969. But "it
has recently been lowered to 3. Last year I saw Erdös in
London .... When I told him the good news that my Erdös
number had just been lowered, he expressed regret that he
had to leave London that same day. Otherwise an ultimate
lowering might have been accomplished."
With Erdös's death, the No. 1 Club's membership will
hardly grow, except for the admission of a few stragglers
who had joint papers with him in the works that should
be published soon. "When these papers come out," said
Graham, "we'll scrutinize them carefully to make sure no
one is pretending to have worked with Erdös." And those
who could have worked with him but didn't are having
regrets. "One evening in the seventies," recalled MIT
mathematician Gian-Carlo Rota, "I mentioned to Paul a
problem in numerical computation I was working on.
Instantly, he gave me a hint that eventually led to the
complete solution. We thanked him for his help in the
introduction to our paper, but I will always regret not
having included his name as a co-author. My Erdös number
will now permanently remain equal to two."
The mathematical literature is peppered with tongue-in-cheek
papers probing the properties of Erdös numbers,
and Jerrold Grossman at Oakland University in Rochester,
Michigan, runs an Internet site, called the Erdös
Number Project, which tracks the coveted numbers. One
of Erdös's specialties was graph theory. By a * graph* ,
mathematicians don't mean the kind of chart Ross Perot
waved at the TV cameras. They mean any group of
points ("vertices" is the lingo) connected by lines
("edges"). So a triangle, for example, is a graph with
three vertices and three edges. Now take Erdös's 485 collaborators
and represent them by 485 points on a sheet of
paper. Draw an edge between any two points whenever
the corresponding mathematicians published together.
The resulting graph, which at last count had 1,381 edges,
is the Collaboration Graph.
Some of Erdös's colleagues have published papers about
the properties of the Collaboration Graph, treating it as if
it were a real mathematical object. One of these papers
made the observation that the graph would have a certain
very interesting property if two particular points had an
edge between them. To make the Collaboration Graph
have that property, the two disconnected mathematicians
immediately got together, proved something trivial, and
wrote up a joint paper.
"I wrote a paper once about the Collaboration Graph,"
said Graham, "that filled, I claimed, a much-needed gap
in the mathematical literature. Well, if the gap was
much needed, I shouldn't have written the paper!" There
is a tradition of writing these papers under pseudonyms.
"I've used the name Tom Odda," said Graham. Tom
Odda? "Look it up in * Maledicta, the Journal of Verbal
Aggression* ," said Graham. "You'll find it under Mandarin
terms of abuse. Tom Odda means * your mother's* ______,
where the blank is too unmentionable even for * Maledicta*
to fill in."
*
Though he was confident of his skill in mathematics, outside
that arcane world Erdös was very nearly helpless. After
his mother's death, the responsibility of looking after him
fell chiefly to Ronald Graham, who spent almost as much
time in the 1980s handling Erdös's affairs as he did overseeing
the seventy mathematicians, statisticians, and computer
scientists at AT&T Bell Labs. Graham was the one
who called Washington when the SF stole Erdös's visa; and
during Erdös's last few years, he said, "the SF struck with
increasing frequency." Graham also managed Erdös's
money, and was forced to become an expert on currency
exchange rates because honoraria from Erdös's lectures
dribbled in from four continents. "I signed his name on
checks and deposited them," Graham said. "I did this so
long I doubt the bank would have cashed a cheek if he had
endorsed it himself."
On the wall of Graham's old office, in Murray Hill,
New Jersey, was a sign: ANYONE WHO CANNOT COPE WITH
MATHEMATICS IS NOT FULLY HUMAN. AT BEST HE IS A TOLERABLE
SUBHUMAN WHO HAS LEARNED TO WEAR SHOES,
BATHE, AND NOT MAKE MESSES IN THE HOUSE. Near the
sign was the "Erdös Room," a closet full of filing cabinets
containing copies of more than a thousand of Erdös's articles.
"Since he had no home," Graham said, "he depended
on me to keep his papers, his mother having done it earlier.
He was always asking me to send some of them to one
person or another." Graham also handled all of Erdös's incoming
correspondence, which was no small task, because
many of Erdös's mathematical collaborations took place by
mail. He sent out 1,500 letters a year, few of which dwelt
on subjects other than mathematics. "I am in Australia," a
typical letter began. "Tomorrow I leave for Hungary. Let
* k* be the largest integer...."
Graham had less success influencing Erdös's health.
"He badly needed a cataract operation," Graham said. "I
kept trying to persuade him to schedule it. But for years
he refused, because he'd be laid up for a week, and he
didn't want to miss even seven days of working with mathematicians.
He was afraid of being old and helpless and
senile." Like all of Erdös's friends, Graham was concerned
about his drug-taking. In 1979, Graham bet Erdös $500 that
he couldn't stop taking amphetamines for a month. Erdös
accepted the challenge, and went cold turkey for thirty
days. After Graham paid up--and wrote the $500 off as a
business expense--Erdös said, "You've showed me I'm not
an addict. But I didn't get any work done. I'd get up in
the morning and stare at a blank piece of paper. I'd have
no ideas, just like an ordinary person. You've set mathematics
back a month." He promptly resumed taking pills,
and mathematics was the better for it.
In 1987 Graham built an addition onto his house in
Watchung, New Jersey, so that Erdös would have his
own bedroom, bathroom, and library for the month or so
he was there each year. Erdös liked staying with Graham
because the household contained a second strong mathematician,
Graham's wife, Fan Chung, a Taiwanese emigre
who today is a professor at the University of
Pennsylvania. When Graham wouldn't play with him,
Chung would, and the two co-authored thirteen papers,
the first in 1979.
Back in the early 1950s, Erdös started spurring on his
collaborators by putting out contracts on problems he
wasn't able to solve. By 1987, the outstanding rewards totaled
about $15,000 and ranged from $10 to $3,000, reflecting
his judgment of the problems' difficulty. "I've
had to pay out three or four thousand dollars," Erdös said
then. "Someone once asked me what would happen if all
the problems were solved at once. Could I pay? Of course
I couldn't. But what would happen to the strongest bank
if all the creditors asked for their money back? The bank
would surely go broke. And a run on the bank is much
more likely than solutions to all my problems." Now that
he's gone, Graham and Chung have decided to pay the
cash prizes themselves for Erdös's problems in graph theory,
about which they have published a book. More than
one hundred graph theory problems have a contract on
them, for a total of more than $10,000. Andrew Beal, a
Dallas banker and amateur mathematician, has offered to
help bankroll Erdös's problems in other fields.
Graham and Erdös would seem an unlikely pair. Although
Graham is one of the world's leading mathematicians,
he did not, like Erdös, forsake body for mind.
Indeed, he continues to push both to the limit. At six foot
two, with blond hair, blue eyes, and chiseled features,
Graham looks at least a decade younger than his sixty-two
years. He can juggle six balls and is a past president
of the International Jugglers Association. He is an accomplished
trampolinist, who put himself through college as
a circus acrobat. ("Trampolining is just like juggling,"
said Graham, exhibiting a mathematician's tendency to
generalize, "only there's just one object to juggle--yourself.")
He has bowled two 300 games, is vicious with a
boomerang, and more than holds his own at tennis and
Ping-Pong.
While Erdös could sit for hours, Graham is always
moving. In the middle of solving a mathematical problem
he'll spring into a handstand, grab stray objects and juggle
them, or jump up and down on the super-springy pogo stick
he keeps in his office. "You can do mathematics anywhere,"
Graham said. "I once had a flash of insight into a stubborn
problem in the middle of a back somersault with a triple
twist on my trampoline."
"If you add up Ron's mathematical theorems and his
double somersaults," one of his colleagues said, "he'd surely
have a record." Graham, in fact, does hold a world record--one
no less peculiar. He was cited in * The Guinness Book of
World Records* for coming up with the largest number ever
used in a mathematical proof. The number is incomprehensibly
large. Mathematicians often try to suggest the
magnitude of a large number by likening it to the number
of atoms in the universe or the number of grains of sand
in the Sahara. Graham's number has no such physical analogue.
It can't even be expressed in familiar mathematical
notation, as, say, the number 1 followed by .a zillion zeroes.
To cite it, a special notation had to be invoked in which
exponents are heaped on exponents to form a staggering
leaning tower of digits.
Besides staying on the cutting edge of mathematics
and acrobatics, Graham found time to learn Chinese and
take up the piano. Neither his wife nor his coworkers
understand how he does it. "It's easy," Graham said.
"There are a hundred and sixty-eight hours in * every*
week."
Erdös and Graham met in 1965 in Boulder, Colorado,
at a conference on number theory, and immediately began
collaborating, writing twenty-seven papers and one
book together. That meeting was also the first of many
spirited athletic encounters the two men had. "I remember
thinking when we met that he was kind of an old
guy," Graham said, "and I was amazed that he beat me
at Ping-Pong. That defeat got me to take up the game
seriously." Graham bought a machine that served Ping-Pong
bails at very high speeds and went on to become
Ping-Pong champion of Bell Labs. Even when Erdös was
in his eighties, they still played occasionally. "Paul loved
challenges," said Graham. "I'd give him nineteen points
and play sitting down. But his eyesight was so bad that I
could just lob the ball high into the air and he'd lose
track of it."
In later years Erdös came up with novel athletic contests
at which he'd seem to have more of a chance, though
he invariably lost. "Paul liked to imagine situations," Graham
said. "For example, he wondered whether I could
climb stairs twice as fast as he could. We decided to see. I
ran a stopwatch as we both raced up twenty flights in an
Atlanta hotel. When he got to the top, huffing, I punched
the stopwatch but accidentally erased the times. I told him
we'd have to do it again. `We're * not* doing it again,' he
growled, and stormed off.
"Another time, in Newark Airport, Erdös asked me
how hard it was to go up a down escalator. I told him it
could be done, and I demonstrated. `That was harder
than I thought,' I said. `That looks easy,' he said. `I'm
sure you couldn't do it,' I said. `That's ridiculous,' he said.
`Of course I can.' Erdös took about four steps up the escalator
and then fell over on his stomach and slid down
to the bottom. People were staring at him. He was wearing
this ratty coat and looked like he was a wino from
the Bowery. He was indignant afterward. `I got dizzy,' he
said."
Erdös and Graham were like an old married couple,
happy as clams but bickering incessantly, following scripts
they knew by heart though they were baffling to outsiders.
Many of these scripts centered on food. When Erdös was
feeling well, he got up about 5:00 . and started banging
around. He'd like Graham to make him breakfast, but Graham
thought he should make his own. Erdös loved grapefruit,
and Graham stocked the refrigerator when he knew
Erdös was coming. On a visit in the spring of 1987, Erdös,
as always, peeked into the refrigerator and saw the fruit.
In fact, each knew that the other knew that the fruit was
there.
"Do you have any grapefruit?" Erdös asked.
"I don't know," Graham replied. "Did you look?"
"I don't know where to look."
"How about the refrigerator?"
"Where in the refrigerator?"
"Well, just look."
Erdös found a grapefruit. He looked at it and looked at
it and got a butter knife. "It can't be by chance," Graham
explained, "that he so often used the dull side of the knife,
trying to force his way through. It'll be squirting like mad,
all over himself and the kitchen. I'd say, `Paul, don't you
think you should use a sharper knife?' He'd say, `It doesn't
matter,' as the juice shoots across the room. At that point
I give up and cut it for him."
Graham was not the only one who had to put up with
Erdös's kitchen antics. "Once I spent a few days with Paul,"
said Janos Path, a fellow Hungarian emigre. "When I entered
the kitchen in the evening, I was met with a horrible
sight. The floor was covered by pools of blood-like red liquid.
The trail led to the refrigerator. I opened the door,
and to my great surprise saw a carton of tomato juice on
its side with a gaping hole. Paul must have felt thirsty and,
after some reflection, decided to get the juice out of the
carton by stabbing it with a big knife."
In mathematics, Erdös's style was one of intense curiosity,
a style he brought to everything else he confronted.
Part of his mathematical success stemmed from his willingness
to ask fundamental questions, to ponder critically
things that others had taken for granted. He also asked
basic questions outside mathematics, but he never remembered
the answers, and asked the same questions again and
again. He'd point to a bowl of rice and ask what it was
and how it was cooked. Graham would pretend he didn't
know; others at the table would patiently tell Erdös about
rice. But a meal or two later Erdös would be served rice
again, act as though he'd never seen it, and ask the same
questions.
Erdös's curiosity about food, like his approach to so
many things, was merely theoretical. He never actually
tried to cook rice. In fact, he never cooked anything at all,
or even boiled water for tea. "I can make excellent cold
cereal," he said, "and I could probably boil an egg, but I've
never tried." He was twenty-one when he buttered his first
piece of bread, his mother or a domestic servant having
always done it for him. "I remember clearly," he said. "I
had just gone to England to study. It was teatime, and
bread was served. I was too embarrassed to admit that I
had never buttered it. I tried. It wasn't so hard." Only ten
years before, at the age of eleven, he had tied his shoes for
the first time.
His curiosity about driving was legendary in the mathematics
community, although you never found him behind
the wheel. He didn't have a license and depended on a
network of friends, known as "Uncle Paul sitters," to chauffeur
him around. But he was constantly asking what street
he was on and questioning whether it was the right one.
"He was not a nervous wreck," Graham said. "He just
wanted to know. Once he was driving with Paul Turan's
widow, Vera Sos. She had just learned to drive, and Paul
was doing his usual thing, `What about this road?' `What
about that road?' `Shouldn't we be over there?' Vera was
distracted and she plowed into the side of a car that must
have been going forty or fifty miles an hour. She totaled
it, and vowed that she would never drive with Erdös
again."
But outside mathematics, Erdös's inquisitiveness was
limited to necessities like eating and driving; he had no
time for frivolities like sex, art, fiction, or movies. Erdös last
read a novel in the 1940s, and it was in the 1950s that he
apparently saw his last movie, * Cold Days* , the story of an
atrocity in Novi Sad, Yugoslavia, in which Hungarians brutally
drowned several thousand Jews and Russians. Once in
a while the mathematicians he stayed with forced him to
join their families on nonmathematical outings, but he accompanied
them only in body. "I took him to the Johnson
Space Center to see rockets," one of his colleagues recalled,
"but he didn't even look up." Another mathematician took
him to see a mime troupe, but he fell asleep before the performance
started. Melvyn Nathanson, whose wife was a
curator at the Museum of Modern Art in New York,
dragged Erdös there. "We showed him Matisse," said Nathanson,
"but he would have nothing to do with it. After
a few minutes we ended up sitting in the Sculpture Garden
doing mathematics."
(C) 1998 Paul Hoffman All rights reserved. ISBN: 0-7868-6362-5

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Although it’s not recommended to go on a juice fast and work out, there are a few viable options. You can include one lite meal of fresh veggies and a protein such as a chicken breast, small steak, pork chop, or tuna. This will give you plenty of protein to maintain your muscle mass during a fast. You can also add a pure unflavored protein isolate to your juices for the extra protein. Either of these options are great if you lead a very active lifestyle or plan to work out and exercise while juice fasting.

Go easy on the juice especially at first – and stop using any ingredients that may cause problems or issue. And – use caution for excessive calories and sugar from juicing.

This is a Juicing FAQ of the top 21 questions people ask me about Juice Fasting. This is the biggest Juicing FAQ around! -

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