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“MY BRAIN IS OPEN,” Paul Erdős announced from a colleague’s doorstep. Five foot six, frail with white hair, a gray jacket, and thick glasses, Erdős had arrived, perhaps without warning, holding a half- empty suitcase and a plastic bag. Together these containers held almost all of his possessions. Here was the world’s most prolific mathematician with, by the end of his life, roughly 1,500 mathematical papers to his name. It was time to do math. In Paul Hoffman’s book, The Man Who Loved Only Numbers: The Story of Paul Erdős and The Search for Mathematical Truth, Louise Straus, wife of late Ernst Straus, describes him like this:
[W]e never knew how many days he was going to stay. I remember during the night hearing crashing sounds. The windows had no sash cords. If you opened the lock, they’d come crashing down. He was such an intelligent man but he could never figure out how to gently lower the windows. He was the real absent-minded professor. He couldn’t figure out how to manage the shower. He could never shut the faucets off. Water ran out on the floor. The linoleum buckled, and the door wouldn’t shut again. He’d go outside to the pay phone and drop coins in it all night, calling mathematicians around the world and asking nearby friends to come over to our place. ‘I’m at the Straus house,’ he’d tell them. He never asked us first if we wanted more guests. He’d just invite all the mathematicians over. But, I must say, my husband loved it. They knocked all sorts of ideas around. A lot of good mathematics came out of it.
Erdős was always doing math. At conferences, he’d skip the lectures, gathering mathematicians into a hotel room to orchestrate multifaceted problem solving sessions. Hoffman, who had the pleasure of attending one of these sessions, describes a scene of mathematicians “sprawled across two double beds and the floor,” while Erdős, seated in a chair, spurted “flashes of insight,” trying to squeeze every last drop of mathematical truth from their brains. Erdős had a knack for pairing mathematicians with the perfect problem, a problem that lay at the edge of his or her ability. To add incentive, he put bounties on problems: five bucks to answer this easy question, but a thousand for this one here because it involves a complex arithmetic progression. “We had this problem that had gotten under our skin,” his good friend and fellow mathematician Ronald Graham recounts, “and at first we couldn’t do anything to it. Well, at least do something, Paul would urge. Crack it. Break it open a little bit. Then someone else will figure out how to break it open a little bit more.”
Eleven years before The Man Who Loved Only Numbers was published, Hoffman wrote a profile of Erdős for The Atlantic of almost the same title: “The Man Who Loves Only Numbers.” Hoffman embedded himself in Erdős’s world, interviewing him and his colleagues and performing exhaustive research. The essay is a finely crafted portrait of Erdős late in his career. It won the first National Magazine Award for Feature Writing. For the book, Hoffman dug in for a degree of personal involvement that is hard to imagine. He tracked Erdős for a decade. He studied mathematical theorems and the mathematicians behind them. He even slept in Erdős’s bedroom at Ronald Graham’s house — an addition Graham constructed especially for Erdős. “I’m not sure what I expected,” Hoffman writes, “but the experience did nothing to improve my mathematical ability.” Hoffman delves into the beauty and spirituality Erdős saw in mathematics and the conjectures, theorems, and proofs he loved.
Most of The Atlantic essay is repurposed word for word in The Man Who Loved Only Numbers; only the verb tense has been changed from present to past, a decision that carries particular weight when we consider that Erdős died while Hoffman was writing and publishing the book. Thus, Hoffman is, in a way, delivering the final dispatches from the great mathematician. He gathered nuggets of Erdős’s musings and remembrances of his colleagues, reflections that give us insight into who he was. In the acknowledgements, Hoffman states that the book “is in large part a work in oral history based on the recollections of Erdős, his collaborators, and their spouses,” which may explain why, at times, the narrative reads as if Hoffman is filling the gaps between interview snippets. The result is that we only glimpse who Erdős was as a person. For instance, we never come to understand why Erdős had a love of children. “The younger the child was,” Hoffman writes, “the deeper his connection.” But why? And why did he “forge a special bond with people he perceived as vulnerable”? I strongly suspect it has something to do with Erdős’s tragic past — surviving various wars and familial hardships, such as his father’s six-year stint in a Russian prison camp, or being a Jew in Europe during the rise of Nazism — but unfortunately the book avoids an exploration of those details, an exploration a conventional biography would be apt to include.
Meanwhile, the book’s swerving narrative connects Erdős’s quips and anecdotes to the lives of other mathematicians, such as Kurt Gödel and Georg Cantor, and events in mathematical history, such as the late 19th century shake-up over Euclid’s parallel postulate. The risk here is that we will lose sight of Erdős; either we’re anxiously awaiting his return or forgetting him entirely. Regarding the former, an entire chapter is spent sharing biographical details of Ronald Graham — who through interviews, as it turns out, actually informs much of our understanding of Erdős. And regarding the latter,a large portion of another chapter covers the fascinating, enigmatic story of Fermat’s Last Theorem, its posthumous discovery followed by the struggle to prove it, which includes Andrew Wiles’s eight years of “clandestine” work on the proof.
But even with these shortcomings, the book joyfully engages with Erdős’s eccentricities and rich history and enlightens us on the mathematics in which he immersed himself. It provides a broad and fairly deep survey of Erdős’s mathematical landscape that is difficult — if not impossible — to find in other books. More importantly, Hoffman grapples with Erdős’s motivations, revealing his lifelong pursuit to reduce mathematical complexity to what Hoffman refers to as “short, clever solutions,” solutions Erdős would consider to be beautiful. There is the famous Four Color Map Theorem, for example, which declares that no more than four colors are needed to color a map such that adjacent countries never share a color. Erdős, like many mathematicians, wasn’t satisfied with Appel and Haken’s 1976 proof. To prove the theorem, they had used a computer to analyze thousands of maps. To this, Erdős said, “I’m not an expert on the four-color problem, but I assume the proof is true. However, it’s not beautiful. I’d prefer to see a proof that gives insight into why four colors are sufficient.” And then later, explaining what determines if a mathematical concept is beautiful, Erdős says, “It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”
By his teenage years Erdős was already turning mathematics into a social event. He would meet friends in Budapest’s City Park by the bronze statue of Anonymous. Under the fascist regime, speaking freely was dangerous, so he spoke in code. Children and other small things became “epsilons,” the Greek letter denoting small mathematical quantities; women became “bosses,” men “slaves.” The habit continued later in life when USA became “Sam” and Russia “Joe,” for Joseph Stalin. International news, Hoffman writes, became “the Sam and Joe Show,” a joke that resonates when you consider that Erdős spent a decade being denied passage to the United States due to nonexistent ties to communism.
While Erdős was on fellowship in the United States, Horthy, Hungary’s dictator, joined the Nazis. The mail routes were seized, and Erdős went four years without hearing from his parents. Then one day he received a telegram: his mother and cousin were alive, but the rest of his family had been killed by the Nazis, with the exception of his father who’d died of a heart attack years earlier. Erdős’s hatred of fascism surfaced in his sense of humor. Hoffman notes that Erdős “applied [the word ‘fascism’] liberally to everything he didn’t like.” He referred to God as the Supreme Fascist, or “SF.” Melvin Henriksen, a colleague, recalls an incident when Erdős was scratched by a kitten. After he returned the kitten to its box, he exclaimed, “Fascist cat!”
But perhaps the most captivating Erdősism — and a motivating theme behind Hoffman’s book — was Erdős’s concept of “the Book.” “I’m not qualified to say whether or not God exists,” Erdős says, “I kind of doubt He does. Nevertheless, I’m always saying that the SF has this transfinite Book — transfinite being a concept in mathematics that is larger than infinite — that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect.”
Erdős had come up with the romantic notion that by deriving an elegant proof you are in essence revealing the equations of God himself, excerpts from the instruction manual of the universe. The Book contained all of mathematics’s beautiful proofs and theorems, which lay mostly undiscovered. Conveying this beauty is one of The Man Who Loved Only Numbers’s great successes. Hoffman includes number grids, equations, geometric shapes, letters of correspondence, photographs, various numerical series, fractions, and irrational numbers extended to more than 20 decimal places. Maybe a quarter of the book is devoted to mathematical explanations. We get the history and struggle behind numerous mathematical problems and watch as mathematicians break them open. The goal is total immersion in Erdős’s world so that we can understand, if only a little, the spiritual connection Erdős had with mathematics.
“A mathematician,” Erdős would say, “is a machine for turning coffee into theorems.” Apparently Erdős turned amphetamines into theorems as well, a phenomenon triggered by the death of his mother. In the final seven years of her life, the two had been so close that she traveled with him almost everywhere. “At night he held her hand until she fell asleep,” Hoffman writes. When his mother died of a bleeding ulcer in 1971, mathematics was all he had left. During the final quarter century of his life Erdős dedicated 19 hours of each day to solving mathematical problems, a pursuit fueled by 10- and 20-milligram pills of Benzedrine. He became a nomad, drawing his income mostly from his 25-country lecture circuit. At one point, Hoffman shares Erdős’s busy lecture schedule: “He’ll be going in quick succession to Memphis, Boca Raton, San Juan, Gainesville, Haifa, Tel Aviv, Montreal, Boston, Madison, DeKalb, Chicago, Champaign, Philadelphia, and Graham’s house.” Graham bet him $500 that he couldn’t go a month without the pills. After conquering the challenge, Erdős told Graham, “You’ve shown 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.”
Hoffman tells us that at 30 years old he had a hard time keeping up with Erdős, who, at the time, was 44 years older. Hoffman doesn’t judge Erdős on his liberal use of the pills, but he makes it clear that he depended on them. One of Erdős’s fears was that he’d lose his mathematical edge as he grew older, and pills, it seems, were his solution — indeed, he was productive until the end of his life.
As an homage to Erdős’s sprawling influence, his friends established a measure of any given mathematician’s “collaborative distance” from him: If you’ve co-authored a paper with him, you’ve earned an Erdős number of one. If you’ve published with a mathematician who has co-authored a paper with him, you get an Erdős number of two, and so on. In mathematics, a high Erdős number is something to brag about. On September 20, 1996, Erdős died at the age of 83. Short of any posthumous anomalies, the number of mathematicians with an Erdős number of one will forever remain at 511. Though Hank Aaron and Erdős autographed the same baseball while collecting their honorary degrees at Emory University, so, as mathematician Carl Pomerance has joked, “Hank Aaron has Erdős number one.”Thus the number should be bumped to 512.
Hoffman’s book offers so much: an assemblage of Erdős’s personal anecdotes; mathematical insights and motivations; reflections from Erdős’s colleagues; a history of Hungary reaching back to the pre-ninth century Magyars. But what it lacks is the continuity of Erdős as the subject and a deeper examination of who he was. More than twice while reading I paused to ask myself if the book stretches farther than what should be considered a biography of Paul Erdős. But this, I suspect, is the point. A biography of Erdős should include his mathematical reach, and Hoffman, in the spirit of Erdős, refuses to gloss over the mathematical beauty that Erdős saw.
The scene back in that conference hotel room where Erdős had arranged those six mathematicians on a binge problem-solving session offers a microcosm of the book’s approach; it’s one of many anecdotes that reveals a lot about the social interplay between Erdős and his colleagues. In response to a sarcastic comment Erdős makes regarding death, one of the mathematicians looks up and says, “In ten years, I want you to talk to the SF on my behalf.”
“What do you want from the SF?” Erdős says.
“I want to see the Book.”
“No one ever sees the Book. At most, you get glimpses.”
And this is what Hoffman has given us — glimpses, hundreds of them, so we are able to construct a prismatic vision of Erdős in numbers.