Mathematics Hothouse

(From A Beautiful Mind by Sylvia Nasar)

The first bite of mathematical apple probably occurred when Nash at around age thirteen or fourteen read E. T. Bell’s extra ordinary book Men of Mathematics — an experience he alludes to in his autobiographical essay (of Nobel Prize, Economics) Bell’s book, which was published in 1937, would have given Nash the first glimpse of real mathematics, a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school or even in the entertaining but ultimately trivial calculations that Nash carried out in the course of chemistry and electrical experiments.

Men of Mathematics consists of lively — and, as it turns out, not entirely accurate — biographical sketches. Its flamboyant author, a professor of mathematics at California Institute of Technology, declared himself disgusted with “the ludicrous untruth of the traditional portrait of the mathematician” as a “slovenly dreamer totally devoid of common sense.” He assured his readers that the great mathematicians of history were an exceptionally virile and even adventuresome breed. He sought to prove his point with vivid accounts of infant precocity, monstrously insensitive educational authorities, crushing poverty, jealous rivals, love affairs, royal patronage, and many varieties of early death, including some resulting from duels. He even went so far in defending mathematicians as to answer the question : “How many of the great mathematicians have been perverts?” None, was his answer. ‘Some lived celibate lives, usually on account of economic disabilities, but the majority were happily married…The only mathematician discussed here whose life might offer something of interest to a Freudian is Pascal.’ The book became a bestseller as soon as it appeared.

What makes Bell’s account not merely charming, but intellectually seductive, are his lively descriptions of mathematical problems that inspired his subjects when they were young, and his breezy assurance that there were still deep and beautiful problems that could be solved by amateurs, boys of fourteen, to be specific. It was Bell’s essay on Fermat, one of the greatest mathematicians of all time, but a perfectly conventional seventeenth century French magistrate, whose life was “quiet, laborious and uneventful,” that caught Nash’s eye. The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory — “the higher arithmetic.” Number theory, investigates the natural relationships of those common whole numbers 1, 2, 3, 4, 5…which we utter almost as soon as we learn to talk.

For Nash, proving a theorem known as Fermat’s (Little) Theorem about prime numbers, those mysterious integers that have no divisors besides themselves and one produced an epiphany of sorts. Often mathematical geniuses, Albert Einstein and Bertrand Russell among them recount similar revelatory experiences in early adolescence. Einstein recalled the “wonder” of his first encounter with Euclid at age twelve:

“Here were assertions, as for example the intersection of three altitudes of a triangle at one point which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me.”

Nash does not describe his feelings when he succeeded in devising a proof for Fermat’s assertion that if n is any whole number and p any prime number, then n multiplied by itself p times minus p is divisible by p. But, he notes the fact in his autobiographical essay, and his emphasis on this concrete result of his initial encounter with Fermat suggests that the thrill of discovering and exercising his own intellectual powers — as much as any sense of wonder inspired by hitherto unsuspected patterns and meanings — was what made this moment such a memorable one. That thrill has been decisive for many a future mathematician. Bell describes how success in solving a problem posed by Fermat led Carl Friedrich Gauss, the renowned German mathematician, to choose between two careers for which he was similarly talented. ‘It was this discovery …which induced the young man to choose mathematics instead of philology as his life work.”…

For those readers who are interested:

1. Who wants to be a mathematician:

http://www.ams.org/publicoutreach/students/wwtbam/wwtbam

2. Resonance Journal (India):

https://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/

3. Ramanujan School of Mathematics; Super30 of Prof Anand Kumar:

http://www.super30.org/rsm.html

Cheers,

Nalin Pithwa