The doorbell rings, and you’re surprised to find your colleague on the stoop. He’s slight, elderly, buzzing from amphetamines, unkempt and uninvited. He shoulders past you into your living room, a single suitcase containing all of his worldly possessions in tow, and declares, “My brain is open.” You have no idea how long he intends to stay because he doesn’t have a house of his own to return to. You’re expected to do his laundry and cook his meals because he can’t be bothered to learn to take care of himself. In exchange, you’ll receive a sleepless, whirlwind encounter, communing with one of the greatest mathematical minds of the 20th century. Your involuntary hospitality will probably result in an academic publication with your name on it. This was many people’s experience of Paul Erdős, the most prolific mathematician of all time.
Erdős (pronounced “air-duhsh”—the mark on the “ő” is a Hungarian double acute accent, not an umlaut) was born in Budapest in 1913 to two high school math teachers. He was a pampered prodigy. By age four he could calculate in his head how many seconds a person had been alive, and at age 21 he buttered his own bread for the first time. That same year he earned his Ph.D. in math. His subsequent fellowship position at Princeton University was cut short because, according to The Man Who Loved Only Numbers, Paul Hoffman’s biography of Erdős, his colleagues there “found him uncouth and unconventional.” Thus began his nomadic life, in which he flitted among brief academic stints, conferences and friends’ guest rooms. As he would say, “Another roof, another proof.”
Erdős was a notoriously bad house-guest. In Hoffman’s book, mathematician Michael Jacobson of the University of Colorado Denver recounts a story in which Erdős came to his home, and they worked on math until 1:00 A.M., when Jacobson finally succumbed to exhaustion. Erdős, who tended to put in 19-hour days, stayed up and, at 4:30 A.M., banged pots in the kitchen incessantly to wake up his host. Jacobson eventually teetered downstairs in his bathrobe. He described the ensuing interaction with his colleague to Hoffman: “What were the first words out of his mouth? Not ‘Good morning’ or ‘How’d you sleep?’ but ‘Let n be an integer.’”
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Erdős’s single-minded obsession with math led to his authorship of a whopping 1,500-plus academic publications, more than any other mathematician in history. Quick aside: some contend that 18th-century mathematician Leonhard Euler was the most prolific of all time. Indeed, Euler produced more pages of math, whereas Erdős produced more papers. So who holds the crown depends on the unit of measure, but there is no controversy about these men being the two top contenders.
One of Erdős’s most notable contributions was to something called the probabilistic method. To understand its value, imagine that you’re planning a mixer for 100 people, and because mixers work best when some partygoers already know one another and some don’t, you want to guarantee that no six guests in any grouping are all friends or all strangers. Is that even possible? If you try to prevent groups of strangers by inviting many friends, then it becomes harder to avoid cliques of friends, yet too many strangers will yield the opposite problem.
Mathematicians often want to prove the existence of a thing with certain properties, such as our desired party of 100 people. A natural way to do it would be to give an explicit example of such a thing (for instance, come up with a guest list with no groups of six mutual friends or strangers). This task may be quite difficult in practice, however.
Instead Erdős pioneered an ingenious alternative. He suggested that rather than trying to design the guest list by hand, you should just pick 100 names (or whatever type of object you’re trying to find) completely at random. Then change your question to: What is the probability that my randomly chosen object has my desired properties? If you can prove that the probability is anything greater than zero, then voilà! Your object must exist; if it didn’t, the probability would be zero.
Changing the question to one about probability often makes it easier to answer. That’s in part because you can now apply a rich set of tools from probability theory. Interestingly, because the probabilistic method circumvents the need to construct your object, you often end up knowing that something exists without a clue about what it looks like. Erdős cracked many stubborn math problems with the probabilistic method, including a more general version of our mixer problem. Today the method is considered an essential technique in every researcher’s tool kit.
Much of Erdős’s success sprang from his belief in math as a social activity. He had so many collaborators that the field invented the Erdős number, a measure of authorship distance from Paul Erdős, which serves as a badge of honor for scholars. Everyone with whom Erdős co-authored a paper has an Erdős number of one, all of their co-authors have a two, and so on. You might have heard of the Bacon number, an actor’s co-starring distance from Kevin Bacon, but Erdős’s recognition as the center of his network predates Bacon’s by 25 years.
Researchers have devoted a surprising amount of effort to investigating the Erdős number, both as a lightweight amusement and as a serious tool for understanding connectivity patterns in authorship networks. Here are some curious facts about it:
Among the more than 250,000 mathematicians who share an authorship chain with Erdős, the median number of hops required to reach him is five. (I’m proud to have an Erdős number of three.)
Many prominent figures beyond math have Erdős numbers: Noam Chomsky (four), Angela Merkel (five), Stephen Hawking (four) and Elon Musk (four), for example.
If one is in a playful mood, it might be argued that Baseball Hall of Famer Hank Aaron has an Erdős number of one because the two men signed the same baseball when they received honorary degrees together from Emory University.
Actor Natalie Portman boasts the rare distinction of having an Erdős number (five) and a Bacon number (two) because of her neuroscience publication as an undergraduate (written under her birth name, Natalie Hershlag).
Someone once tried to sell an Erdős number on eBay. The winner would have gotten to collaborate with the seller, whose Erdős number was four. Several people placed substantial bids, but the auction was snagged at the last second for $1,031 by a mathematician with no intention of paying up, who called the stunt a “mockery” of the system.
Erdős’s legacy lives on not just through his publications but in the many conjectures he left behind. Sometimes the hardest thing in math is asking the right questions, and he had a keen talent for pinpointing important problems. He issued personal monetary prizes for solutions to many problems despite having little of his own money. What he collected through speaking fees, awards and part-time appointments he typically donated to unhoused people, charities and aspiring researchers. He once gave $1,000 to a talented high school student struggling to meet tuition for Harvard University. Ten years later that student felt ready to pay the money back, but Erdős instead insisted, “Do with the $1,000 what I did.”
Paul Erdős was a man devoted to exactly one thing. He never married or had children—in fact, he was celibate his entire life. He had very few hobbies, didn’t drive, and didn’t have a permanent residence or a steady job. Erdős died in 1996 at the age of 83 at a math conference in Warsaw. He died doing what he loved, largely because he never did anything else.