SIAM News Blog

From the Foundations of Mathematics to Eugenics and Beyond

By James Case

When Einstein Walked with Gödel: Excursions to the Edge of Thought. By Jim Holt. Courtesy of Farrar, Straus and Giroux.
When Einstein Walked with Gödel: Excursions to the Edge of Thought. By Jim Holt. Farrar, Straus and Giroux, New York, NY, May 2018. 384 pages, $28.00.

The dust jacket of this collection of lightly-edited essays features an image of Albert Einstein and Kurt Gödel, dressed in overcoats and walking together. They regularly walked between their Princeton, NJ homes and offices at the Institute for Advanced Study in both good and bad weather. This custom began soon after Gödel’s arrival at the institute in 1940 and continued until Einstein’s 1955 death became imminent. While one cannot truly ascertain their topics of conversation, it seems safe to assume that general relativity was a recurrent theme. A student of both math and physics at the University of Vienna, Gödel was naturally curious about the subject and—during his acquaintance with the master—discovered a new set of solutions to Einstein’s field equations, which described a rotating universe that could accomodate time travel. This unexpected development inspired Jim Holt to reprint (from Lapham’s Quarterly) his own related essay, “Time — the Grand Illusion?” in his most recent book.

When Einstein Walked with Gödel is a collection of New Yorker-style essays,1 some 12 to 15 pages long, grouped in nine parts. The sole exception is Part VIII, which offers 15 op-ed-length “quick studies.” One such essay, “The Cruel Law of Eponymy,” bemoans the fact that mathematical and scientific discoveries—such as Pythagoras’ theorem and the Gaussian distribution—are seldom named for their actual discoverers, but rather for a subsequent developer. A more mundane example is the flush toilet, which was not invented in the 19th century by plumber Thomas Crapper, but by Sir John Harrington at the court of Queen Elizabeth I. Crapper merely made them available to middle-class buyers. Holt’s other short essays concern Heisenberg’s uncertainty principle, the law of least action, and Emmy Noether’s beautiful theorem. The latter, he explains, reveals an unexpected duality between the symmetries of a physical system and the conservation laws that apply to it. Richard Feynman calls her discovery “a most profound and beautiful thing,” one that “most physicists still find somewhat staggering.”

Yet another of Holt’s quick studies examines overconfidence in light of the Monty Hall problem. Named for the original host of the game show Let’s Make a Deal, the problem was first publicized by Parade columnist Marilyn vos Savant. Contestants on the show were offered three doors, one of which harbored a valuable prize (the other two concealed relatively worthless consolation prizes). After a contestant chose what he/she thought was most likely the winning door, the host opened a different door, revealing a worthless consolation prize. The contestant then had to decide whether to revise his/her guess before the host opened a second door. Should the contestant (A) assume his/her original guess was mistaken and switch, or (B) assume it was correct and stand pat? Since the initial guess was a one-in-three proposition, it was twice as likely to be wrong as right; hence option A. More than a few professional mathematicians, including Paul Erdős himself, have refused to accept this seemingly-counterintuitive conclusion until shown a formal proof.

The essay quotes a variety of psychological studies finding that people often claim to be “absolutely certain” of issues they know little about, while better-informed individuals are more likely to concede doubt. In one such study, experimenters administered tests in logic, English grammar, and humor (where respondents’ ratings of jokes were judged against the opinions of a panel of professional comedians) to a group of subjects. In their exit interviews, the lowest scorers tended to express confidence in their performance, while the highest scorers were more skeptical of themselves. For this and other reasons, Holt concludes that high levels of confidence correlate with high levels of overconfidence.

Part II of the book—entitled “Numbers in the Brain, in Platonic Heaven, and in Society”—is more representative of its overall content and consists of three essays. The first addresses the neuroscience of mathematics and features the work of French researcher Stanislas Dehaene. Dehaene sought to discover the way in which numbers are coded in our minds, and used brain scanning techniques to determine where our numerical skills reside among the folds and crevices of our cerebral cortices. He also examined the extent to which certain natural languages, such as Chinese, facilitate the learning and use of arithmetic.

The second essay pertains to the Riemann zeta function and what Holt calls “the laughter of the primes.” After an introductory discourse on laughter—as practiced by children, chimpanzees, and the rest of us—he explains prime numbers and reveals that infinitely many of them exist. He then introduces Leonhard Euler’s discovery

\[\zeta(s) =1+2^{-s}+3^{-s}+4^{-s}+...= \frac{1}{1-2^{-s}} \times \frac {1}{1-3^{-s}} \times \frac{1}{1-5^{-s}} \times \frac{1}{1-7^{-s}} \times \frac {1}{1-11^{-s}} \times \dotsc .\]

It had little immediate impact, because Euler never noticed his result’s application to both complex and real values of the exponent \(s\). Full implications did not begin to emerge until Bernhard Riemann entered the field.

The laughter in Holt’s rendition relates to the fact that each zero of the zeta function is associated with a musical note, the amplitude and frequency of which depend on the zero’s location in the (complex) -plane. Only if all of the zeros lie in the vertical strip \(0<IM(s)<1\) will the “orchestra of the primes” be in balance, with no one note drowning out another. Riemann’s hypothesis then asserts that all of the zeta function’s complex zeros lie on the center line of that strip.

The final essay in Part II is entitled “Sir Francis Galton, the Father of Statistics…and Eugenics.” One of the 19th century’s great innovators, Galton is deemed the father of fingerprinting. He also coined the phrase “nature versus nurture.” His motto became, “Whatever you can, count.” As a young man in search of a career, Galton heeded the advice of his cousin Charles Darwin to “read mathematics like a house on fire.” He exerted himself at the University of Cambridge until suffering a breakdown from overwork.

Keenly aware of Darwin’s theory of natural selection and the results achieved by selectively breeding farm animals, Galton never doubted that nature would dominate nurture in the development of human talent and virtue. He believed that one could improve and enhance humanity itself by selective breeding, early marriage, and high fertility among the “genetic elite.” His work in statistics and eugenics were not separate endeavors, but rather parts of a single program “for the improvement of the breed.”

In Galton’s day, statistical inference was a dreary enterprise riddled with population numbers, trade figures, and the like. The field was entirely devoid of mathematical interest, save for the ubiquity of the Gaussian (normal) distribution. Galton invented his eponymous “Galton Board” (available from Amazon for $49.99) to demonstrate the connection between sums of random variables and normal distribution. It was but one of his many contributions to science.

Though primarily interested in the inheritance of intelligence, Galton wisely concentrated on more readily measureable characteristics. After obtaining the heights of 205 pairs of parents and 928 of their offspring, he plotted the average height of each couple against that of their children and drew a straight line through the resulting cloud of data points in an attempt to quantify the evident trend. Finding the slope of the line to be two-thirds, he concluded that exceptionally tall/short parents tend to have slightly less exceptionally tall/short children. He dubbed this tendency “regression toward mediocrity” and coined the term “correlation” to describe the observed relationship. Regression and correlation, says Holt, spawned a genuine revolution in science by drawing attention to “statistical” laws. Researchers previously assumed scientific laws to be deterministic.

Other parts of When Einstein Walked with Gödel concern pure and applied mathematics, maps, infinity, computers, the cosmos, and the nature of truth itself. Holt discusses all in a manner that is both erudite and accessible, interspersing passages of mathematical exposition with items of personal interest—including a healthy dose of plain old gossip—about prominent mathematicians.

1 Several of these did originally appear in the New Yorker, and several others appeared in the New York Review of Books. In his acknowledgements section, Holt names the publications in which each essay originally appeared.

James Case writes from Baltimore, Maryland.

blog comments powered by Disqus