How did John von Neumann become so good at mathematics? by Qiaochu Yuan

Answer by Qiaochu Yuan:

von Neumann’s brain worked much better than yours, mine, or anyone else’s. He probably could’ve been good at anything, and chose to be good at mathematics. Here are some selections from his completely absurd Wikipedia article (John von Neumann – Wikipedia; seriously, read the whole thing), all emphasis mine:

Von Neumann was a child prodigy. As a 6 year old, he could multiply and divide two

8-digit numbersin his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her: "What are you calculating?"Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement, that included Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Dennis Gabor (b. 1900), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913). Collectively, they were sometimes known as Martians. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that

von Neumann was the only genius.At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was

brought to tears.His reputed powers of

memorization and recallallowed him to quickly memorize the pages oftelephone directories, and recite the names, addresses and numbers therein.Von Neumann held a lifelong passion for ancient history, being renowned for his prodigious historical knowledge. A professor of Byzantine history at Princeton once said that

von Neumann had greater expertise inByzantine historythan he did.The Nobel Laureate Hans Bethe speculated: "I have sometimes wondered whether a brain like von Neumann's does not indicate a

species superior to that of man".Eugene Wigner wrote that, seeing von Neumann's mind at work, "one had the impression of a

perfect instrumentwhose gears were machined to mesh accurately to a thousandth of an inch."Paul Halmos states that "von Neumann's

speed was awe-inspiring."Israel Halperin said: "Keeping up with him was …

impossible. The feeling was you were on a tricycle chasing a racing car."Edward Teller admitted that he "never could keep up with him".

Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and

I sometimes wondered if he used the same principle when he talked to the rest of us."When George Dantzig brought von Neumann an unsolved problem in linear programming "as I would to an ordinary mortal", on which there had been

no published literature, he was astonished when von Neumann said "Oh, that!", before offhandedly giving alecture of over an hour, explaining how to solve the problem using thehitherto unconceivedtheory of duality.George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said "Johnny was the only student I was ever

afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper.Herman Goldstine wrote: “One of [von Neumann’s] remarkable abilities was his power of

absolute recall. As far as I could tell, von Neumann was ableon once reading a book or article to quote it back verbatim; moreover, he could do ityears later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me howA Tale of Two Citiesstarted. Whereupon, without any pause, he immediatelybegan to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.”

How did John von Neumann become so good at mathematics?