Chapter 0 - The Two-And-A-Half-Billion-Year-Old Man - Notes (Completed on )
Paul Erdős was an extremely eccentric mathematician who died in 1996.
Erdős wrote or coauthored 1475 academic papers, working till he died.
Erdős was itinerant, lived out of a suitcase, and stayed in the homes of other mathematicians.
"...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."
"At five foot six, 130 pounds, Erdős had the wizened, cadaverous look of a drug addict..."
"Some French socialist said that private property was theft," Erdős recalled. "I say that private property is a nuisance."
Erdős was extremely generous with money. Story from D.G. Larman - "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. Erdős removed a small amount from the pay packet to cover his own frugal needs and gave the remainder to the beggar."
Erdős lent $1,000 to a high school student named Glen Whitney to help cover his tuition at Harvard. Erdős said to only pay him back when it would not be a problem. Ten years later, Whitney had the money to repay him - Erdős replied, "Tell him to do with the thousand dollars what I did".
Erdős number: 1 = you co-authored a paper with Erdős, 2 = you wrote a paper with someone with an Erdős number of 1, etc.
In 1979, Erdős' friend and mathematician Ronald Graham bet Erdős he couldn't stop using amphetamines for a month. Erdős won the bet and promptly told Graham that he had set back mathematics for a month.
Erdős placed bounties on problems he couldn't solve, paying out three or four thousand dollars.
Erdős had no life skills - whilst a brilliant mathematician and a prodigy, he didn't know how to cook, and was allegedly 21 when he buttered bread for the first time.
Erdős had very limited interests outside of mathematics. Mathematicians and friends would take him to outings at museums, MoMA, a mime troupe etc, and he hardly paid attention.
Chapter 1 - Straight from the Book - Notes (Completed on )
Erdős believed in the "Book" - essentially a book of the most elegant proofs of all mathematical theorems.
When Erdős was 18, as a college freshman, he came up with a proof for the problem: given n > 1, a prime can always be found in (n, 2n). His proof was considered more elegant than Chebyshev's earlier proof.
"When I was a graduate student," Joel Spencer said, "I thought only third-rate mathematicians would have these fights. But it's actually first-rate mathematicians."
Erdős was extremely generous in sharing ideas; number theorist Richard Guy referred to him as "the problem poser par excellence."
The beauty of a proof is extremely important to mathematicians. Erdős on the four-color theorem: "...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."
Erdős was happy to talk mathematics with anyone; the book shares the example of David Williamson, a high school senior in 1985 who exchanged ideas with Erdős by letter.
The book gives the example of a problem on a blackboard in the mathematics lounge of Texas A&M in 1976; two analysts had created a thirty-page solution. Erdős, with no expertise in functional analysis, produced a two-line solution. (The story is recounted by geometer George Purdy).
The book gives an overview of Ramsey theory, an area of mathematics pioneered by Erdős and Graham.
Chapter 2 - Epszi's Enigma - Notes (Completed on )
Erdős was born in Budapest in 1913. Both of his parents were high school mathematics teachers.
Erdős had two older sisters who died at ages three and five from scarlet fever.
In 1914, World War I broke out, and Erdős' father was captured and sent to Siberia.
After Hungary lost World War I, Hungary was occupied by its neighbors, and subsequently taken over by a communist regime. After the Communist government fell, Hungary was taken over by a right-wing authoritarian regime under Miklós Horthy.
The "White Terror" (1919-1921) was a period of violence - explicitly targeting, torturing, and massacring enemies of the regime. Many Jews were targeted in the violence. Edward Teller, John von Neumann, Leó Szilárd, Eugene Wigner and others fled the instability of Hungary to Germany. Erdős and his family, including his recently returned father, survived.
Erdős completed his PhD in mathematics in 1934 from Pázmány Péter University in Budapest.
Erdős and a group of mathematicians met in the city park by a statue of Anonymous, a famous Hungarian historian. They constantly talked about mathematics, and came up with results on various problems like the Happy End Problem.
Erdős left Hungary in 1934 for Manchester, England, completing a four-year postdoctoral fellowship. He quickly met G. H. Hardy, the legendary English number theorist.
Hardy allegedly spent four hours every day on mathematics from 9AM - 1PM, later playing sports like cricket and tennis, and participating in discussions with other English intellectuals.
The book goes on to discuss Hardy and Ramanujan - I will skip those notes as I plan to read the biography of Ramanujan later.
In 1936, Vázsonyi was working on finding the necessary and sufficient conditions for Euler's Königsberg Theorem to apply to infinite graphs. Vázsonyi found the necessary condition but was stuck on the sufficient condition. He called Erdős - allegedly, 20 minutes after the call, Erdős was able to find the sufficient condition.
Chapter e - Problems with Sam and Joe - Notes (Completed on )
Erdős was afraid of the political turmoil in Europe in 1938 and decided to go to the US, first for a fellowship at the Institute for Advanced Study at Princeton.
Erdős developed a friendship with Stanisław Ulam, a Polish prodigy. Ulam invited Erdős to Madison, Wisconsin after Erdős' stint at the IAS.
Erdős tried to work on the Manhattan Project, but was turned down. The book attributes his rejection to Erdős not agreeing to secrecy, his eccentric nature, and an existing FBI file on him.
In 1943, Erdős took a job at Purdue University. He was cherished by the department not just because of his genius, but also his cosmopolitan nature and wide-ranging interests in scientific culture and world politics.
In 1944, the Nazis invaded Hungary, deporting many Jews to Auschwitz and killing many others. The Soviets liberated Budapest in 1945, and while Erdős' mother survived, most of her siblings were killed and Erdős' father had died earlier of a heart attack.
Erdős cared deeply about making sure his colleagues were sharp mathematically. For Ulam, Jon Folkman and others, Erdős would pester them with problems and try to help them regain their confidence.
The book then goes into detail about the formalization of mathematics, covering Bertrand Russell, Gödel, Hilbert and others.
Erdős knew Einstein but they were not close friends. Erdős became good friends with Ernst Straus, Einstein's assistant.
Erdős went back and forth between England and the US before getting a job at the University of Notre Dame in 1952.
In 1954, Erdős wanted to go to a math conference in Amsterdam but was denied a reentry visa due to the Red Scare. He resigned from Notre Dame, gave up his green card out of principle and went to Amsterdam.
Erdős encountered issues with European countries too, and ultimately found refuge in Israel, where he initially worked for three months.
In 1955, Erdős was issued a special passport that allowed him to travel in and out of Hungary whenever he wanted.
Chapter 3 - Einstein vs Dostoevsky - Notes (Completed on )
Erdős collaborated with young prodigies as young as 14, and loved children. Examples include Louis Pósa, László Lovász, and others. He cared about mentoring the next generation of mathematicians.
Erdős continued to face difficulties in coming to the US. In 1963, after more than a hundred mathematicians petitioned the US State Department, Erdős was able to return.
Erdős had a special relationship with his mother, who, despite being 84 in 1964, would travel with him nearly everywhere.
Erdős did not have any interest in romantic relationships or sleeping with women.
When Erdős' mother died in 1971, he took lots of antidepressants and amphetamines, and allegedly put in nineteen-hour days.
"But wasn't that about five years ago? He said, 'Yes, it was. But I miss her very much.'"
Chapter π - Dr. Worst Case - Notes (Completed on )
Ronald Graham was born in 1935 in California. He had a very different upbringing than the average Hungarian math prodigy. His mother was a burner and his father was a welder. They moved very frequently, and he was a good athlete.
Graham developed an interest in math, skipping a few grades and enrolling at the University of Chicago in 1951 at age fifteen.
He didn't do well in college, and hadn't even completed a course in calculus, so he switched to electrical engineering when he transferred to Berkeley.
Worried about the draft, he signed up for the Air Force, ultimately working as a type of telephone operator. He attended classes at the University of Alaska during the day and worked at night.
In 1959, Graham earned a degree in physics from the University of Alaska, and then returned to Berkeley in 1959.
Graham developed an interest in unit fractions, writing his doctoral thesis on the subject. He earned his PhD in mathematics in 1962, and then went off to work at Bell Labs.
Graham would later divorce his second wife and marry Fan Chung in 1983. Chung graduated near the top of her class and earned the highest score in the qualifying exams as a graduate student at the University of Pennsylvania.
Some mathematicians care deeply that their work is abstract and disconnected from the real world. Hardy: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
Many seemingly abstract and "useless" mathematical discoveries show up many years later in real-world phenomena. The book gives the example of Riemannian geometry and the idea of curved space, which the book alleges Einstein later said is the shape of the universe.
Upon hearing that probability theorist Mark Kac had published a paper in a physics journal, Erdős sent him a postcard: "I am praying for your soul."
Graham worked on many problems related to scheduling and efficiency, performing worst case analysis for the Nike missile defense system and NASA.
The book discusses the traveling salesman problem, alludes to the P versus NP problem, and the bin packing problem, ultimately calling Graham the father of "worst case analysis".
Story of Erdős at a math conference in 1987: "Erdős had taken over someone else's room at the Marriott and was working on six problems with six different mathematicians, who were sprawled across two double beds and the floor."
Funny Erdős lines: "Soon I will be cured of the incurable disease of life"; "Television is something the Russians invented to destroy American education."
The book lists the dizzying itinerary of Erdős: "Memphis, Boca Raton, San Juan, Gainesville, Haifa, Tel Aviv, Montreal, Boston, Madison, De Kalb, Chicago, Champaign-Urbana, Philadelphia, and back to Newark".
Chapter 4 - Marginal Revenge - Notes (Completed on )
Ruth-Aaron pairs: Pairs of consecutive numbers that share the same prime factor sum.
Carl Pomerance, a mathematician at the University of Georgia, wrote a paper on the topic. A few days later, Erdős called him to share his ideas; they ended up writing twenty-one papers together.
The book gives an overview of Fermat's Last Theorem - since I just read "Fermat's Enigma", a book on the efforts to solve the problem, I will skip these parts.
Erdős recounts that he wanted to die like Euler, doing math until his final moments.
Chapter 5 - "God Made the Integers" - Notes (Completed on )
The book raises the question - what proofs are considered important? Graham estimated there to be "upwards of a quarter million theorems" published in a year (at that time).
The book points out that seemingly recreational math, like Smith numbers, can end up having bigger connections and relationships to other parts of mathematics (prime repunits).
Carl Friedrich Gauss was born in 1777 in Germany. He was a prodigy, and the book goes over examples of his alleged abilities during his youth.
Gauss conjectured the Prime Number Theorem in the late 1790s. Erdős and Selberg found a proof using elementary methods in 1949.
The book provides a broad mathematical overview of different ideas, like e^(iπ) + 1 = 0, and the distinction of doing arithmetic using Roman numerals vs Hindu-Arabic numerals.
The book then goes into a discussion of Cantor and different infinities, and Cantor's diagonalization argument.
The book claims that one of Erdős' most important contributions to mathematics was the creation of the "probabilistic method" of proof.
"The irony is that Erdős, who shunned computers, made a major contribution to the theory of computing."
Chapter 6 - Getting the Goat - Notes (Completed on )
The book introduces Marilyn vos Savant and the Monty Hall problem. There are three doors: two of the doors have a goat behind them; one has a car behind it. After you choose a door, the host opens one of the other doors, revealing a goat. The host knows what is behind all of the doors, and always reveals a goat. Should you switch doors and why?
vos Savant argued that it is optimal to switch - your probability increases from 1/3 to 2/3. Mathematicians flooded her with mail, telling her she was wrong and insulting her.
This problem was posed to Erdős by Vázsonyi. Erdős believed that both options were the same and grew upset when told by Vázsonyi that it is optimal to switch.
Vázsonyi wrote a computer program using a Monte Carlo simulation that showed vos Savant was correct - Erdős didn't find the explanation compelling.
Erdős ended up asking Graham for an explanation, which he accepted. Graham recounts that Erdős could be difficult to work with and communicate with.
"...owing to his often-repeated comment that if he could no longer do mathematics, he would have to commit suicide."
Erdős' health declined - he lost sight in one eye and developed a heart condition. Even still, he did mathematics. "He had papers spread out all over the hospital bed and a parade of mathematicians coming in and out...He drove doctors and nurses crazy." (Ralph Faudree)
In spite of his health problems, he continued to travel, lecture and do mathematics.
In one story, Erdős began asking a question to a lecturer, but collapsed partway through. Surgeons inserted a pacemaker into Erdős and at the conference's closing banquet (the same day), Erdős came back and asked the lecturer the remainder of his question. "Mathematics was his life."
Chapter 7 - Survivors' Party - Notes (Completed on )
Erdős first visited Memphis in 1974 and would visit the city every year until 1996. He wrote forty-six papers with Faudree, and many others with other Memphis mathematicians, including the Hungarian prodigy Béla Bollobás.
The Memphis conference was at the same time as Erdős' birthday, his "birthday wake".
In 1997, after Erdős died, Bollobás opened the first Paul Erdős Memorial Lecture.
Erdős' memorial service in Hungary in 1996 was attended by over five hundred people.
Faudree recounts Erdős' work ethic: "he only needed three hours of sleep. The first time he stayed, the clock was set wrong. It said 7:00 but it was really 4:30AM. He thought we should be up working, so he turned on the TV full blast."
Ronald Gould, a mathematician at Emory: "Even though he only got three hours of sleep a night, he did take little naps throughout the day. But he was doing math even then."
The book includes many funny stories about Erdős' antics and being a bad houseguest, making a mess everywhere.
Chapter ∞ - "We mathematicians are all a little bit crazy" - Notes (Completed on )
Many mathematicians experienced severe mental health problems - Sidon, John Nash, Cantor, Gödel, Ted Kaczynski, Bertrand Russell and others.
Einstein: "I believe with Schopenhauer that one of the strongest motives that lead men to art and science is to escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one's own ever-shifting desires."
The author of the book (Paul Hoffman) shadowed Erdős and published an article in The Atlantic. After Hoffman shared the article with Erdős: "You shouldn't have mentioned the stuff about Benzedrine. It's not that you got it wrong. It's just that I don't want kids who are thinking about going into mathematics to think that they have to take drugs to succeed."
"That was Erdős, always thinking about the epsilons."