When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvelously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)

One might make an analogy to the Seven Wonders of the Ancient World. When this list was drawn up (the earliest extant version dates to about 140 B.C.E.), the oldest wonder on it by far was the pyramid of Giza, which went back to about 2500 B.C.E. The other six wonders—the Hanging Gardens of Babylon, the temple of Artemis at Ephesus, the statue of Zeus at Olympia, the mausoleum at Halicarnassus, the Colossus of Rhodes, and the lighthouse at Alexandria—were almost two millennia newer. And which of the Seven Wonders still survives today? The pyramid of Giza. All the rest have disappeared, done in by fires or earthquakes.

The one earmark of civilization that is likely to be recognized across the universe is number. In Carl Sagan’s science-fiction novel Contact, aliens in the vicinity of the star Vega beam a series of prime numbers toward earth. The book’s heroine (played by Jodie Foster in the movie version of Contact) works for SETI (Search for Extraterrestrial Intelligence). She realizes, with a frisson, that the prime-number pulses her radio telescope is picking up must have been generated by some form of intelligent life. But what if aliens beamed jokes at us instead of numbers? We probably wouldn’t be able to distinguish the jokes from the background noise. Indeed, we can barely distinguish the jokes in Shakespeare’s plays from the background noise. (Seriously, have you ever laughed during a Shakespeare play?) Just as nothing is more timeless than number, the core of mathematics, nothing is more parochial and ephemeral than humor, the core of laughter.

Bertrand Russell penned a gushing tribute to the glories of mathematics. “Rightly viewed,” Russell wrote, mathematics “possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” These lines, which play up the transcendent image of mathematics, are often quoted in mathematical popularizations. What one seldom encounters in such books, however, is the rather different view that Russell expressed in his late eighties, when he dismissed his (relatively) youthful rhapsodizing as “largely nonsense.” Mathematics, the aged Russell wrote, “has ceased to seem to me non-human in its subject-matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal.” So, in the course of his life, Russell underwent an evolution in his thinking about mathematics. I think our civilization will have undergone a similar evolution by the Year Million.

Prime numbers seem to crop up almost at random, sprouting like weeds among the rest of the numbers. “There is no apparent reason why one number is a prime and another not,” declared the mathematician Don Zagier in his inaugural lecture at Bonn University in 1975. “To the contrary, upon looking at these numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation.”

Those who blithely assume the truth of the Riemann conjecture should keep in mind an interesting pattern in the history of mathematics: whereas long-standing conjectures in algebra (like Fermat’s theorem) typically turn out to be true, long-standing conjectures in analysis (like the Riemann conjecture) often turn out to be false.

Eugenics, as would appear obvious from its subsequent career, is an evil thing to have fathered. Enthusiasm for Galton’s ideas led to the forcible sterilization of hundreds of thousands of people in the United States and Europe deemed genetically unfit and contributed to Nazi racial policies that culminated in the Holocaust. Today, most of us can see that the notion of uplifting humanity by getting “desirables” to breed more and “undesirables” to breed less was misbegotten from the start, both on scientific and on ethical grounds. We look back at its progenitor with a slight shudder of revulsion and chalk it up to moral progress that we have put the Galtonian gospel behind us. But perhaps we should not be so smug. In the new age of genetic engineering, it is becoming apparent that the eugenic temptation has not gone away at all; it is simply taking a new form, one that may prove harder to resist. If Galton, as our image of him would have it, was a talented and fundamentally decent man seduced by an evil concept, it might be of more than historical interest to reconsider how he went wrong.

Galton might have puttered along for the rest of his life as a minor gentleman-scientist had it not been for a dramatic event: the publication of Darwin’s Origin of Species in 1859. Reading his cousin’s book filled Galton with a sense of clarity and purpose. One thing in it struck him with special force: to illustrate how natural selection shaped species, Darwin cited the breeding of domesticated plants and animals by farmers to produce better strains. Perhaps, Galton dreamed, human evolution could be deliberately guided in the same way. “If a twentieth part of the cost and pains were spent in measures for the improvement of the human race that is spent on the improvements of the breed of horses and cattle, what a galaxy of genius might we not create!” he wrote in an 1864 magazine article, his opening eugenic salvo. (It was two decades later that he coined the actual word “eugenics,” from the Greek for “wellborn.”)

It was Galton who coined the phrase “nature versus nurture,” which still reverberates in debates today.

What made him so sure that nature dominated nurture in determining a person’s talent and temperament? The idea first arose in his mind at Cambridge, where he noticed that the top students had relatives who had also excelled there; surely, he reasoned, such runs of family success were not a matter of mere chance. His hunch was strengthened during his travels, which gave him a vivid sense of what he called “the mental peculiarities of different races.”

In his 1869 book, Hereditary Genius, he assembled long lists of “eminent” men—judges, poets, scientists, even oarsmen and wrestlers—to show that excellence ran in families. To counter the objection that social advantages, rather than biology, might be behind this, he used the adopted sons of popes as a kind of control group. His case that mental ability was largely hereditary elicited skeptical reviews, but it impressed Darwin. “You have made a convert of an opponent in one sense,” he wrote to Galton, “for I have always maintained that, excepting fools, men did not differ much in intellect, only in zeal and hard work.”

For Quetelet, the bell curve represented accidental deviations from a sort of Platonic ideal he called l’homme moyen—the average man. When Galton stumbled upon Quetelet’s work, however, he exultantly saw the bell curve in a new light: what it described were not accidents to be overlooked but differences that reveal the variability on which evolution depends. His quest to find the laws that governed how these differences were transmitted from one generation to the next led to what has been justly deemed Galton’s greatest gifts to science: regression and correlation.

Yet, as straightforward as it seems, the idea of regression has been a snare for the sophisticated and the simple alike. The most common misconception is that it implies convergence over time. If very tall parents tend to have somewhat shorter children, and very short parents tend to have somewhat taller children, doesn’t that mean that eventually everyone should be the same height? No, because regression works backward in time as well as forward: very tall children tend to have somewhat shorter parents, and very short children tend to have somewhat taller parents.

To mistake regression for a real force that causes talent or quality to dissipate over time, as so many have, is to commit what has been called Galton’s fallacy. In 1933, a Northwestern University professor named Horace Secrist produced a book-length example of the fallacy in The Triumph of Mediocrity in Business, in which he argued that since highly profitable firms tend to become less profitable and highly unprofitable ones tend to become less unprofitable, all firms will soon be mediocre. A few decades ago, the Israeli Air Force came to the conclusion that blame must be more effective than praise in motivating pilots, because poorly performing pilots who were blamed made better landings subsequently, whereas high performers who were praised made worse ones. (It is a sobering thought that we might generally tend to overrate censure and underrate praise because of the regression fallacy.) In 1990, an editorialist for The New York Times erroneously argued that the regression effect alone would ensure that racial differences in IQ would disappear over time.

Galton’s eugenic proposals were benign compared with those of famous contemporaries who rallied to his cause. H. G. Wells, for instance, was an unabashed advocate of negative eugenics, declaring that “it is in the sterilisation of failures, and not in the selection of successes for breeding, that the possibility of an improvement of the human stock lies.” George Bernard Shaw championed eugenic sex as an alternative to prescientific procreation through marriage. “What we need,” Shaw said, “is freedom for people who have never seen each other before, and never intend to see each other again, to produce children under certain definite public conditions, without loss of honor.” Although Galton was a conservative, his creed caught on with progressive figures like Harold Laski, John Maynard Keynes, and Sidney and Beatrice Webb. In the United States, New York disciples founded the Galton Society, which met regularly in the American Museum of Natural History, and popularizers helped the rest of the country become eugenic-minded. “How long are we Americans to be so careful for the pedigree of our pigs and chickens and cattle—and then leave the ancestry of our children to chance or to ‘blind’ sentiment?” asked a placard at an exposition in Philadelphia.

It was in Germany that eugenics took its most horrific form. Galton’s creed had aimed at the uplift of humanity as a whole; although he shared the racial prejudices that were common in the Victorian era, the concept of race did not play much of a role in his theorizing. German eugenics, by contrast, quickly morphed into Rassenhygiene—“race hygiene.” The Aryan race was in a struggle for domination with inferior races, German race hygienists believed, and its genetic material must not be allowed to deteriorate through the unfettered reproduction of the unfit. Under Hitler, nearly 400,000 people with putatively hereditary conditions like feeblemindedness, alcoholism, and schizophrenia were forcibly sterilized. In time, many were simply murdered. The Nazis also took “positive” eugenic measures; the Lebensborn (well of life) program coupled unmarried women who were deemed highly fit with members of the SS in hopes of producing Aryan offspring of the highest quality.

Galtonian eugenics was wrong because it was based on faulty science and carried out by coercion. But Galton’s goal, to breed the barbarism out of humanity, was not despicable. The new eugenics, by contrast, is based on a relatively sound (if still largely incomplete) science, and it is not coercive; it might be called “laissez-faire” eugenics, because decisions about the genetic endowment of children would be left up to their parents. Indeed, the only coercive policy that has been contemplated is the one in which the state outlaws these technologies, thereby enforcing the natural genetic order. (Europe, unlike the United States, has already banned germ-line manipulation.) It is the goal of the new eugenics that is morally cloudy. If its technologies are used to shape the genetic endowment of children according to the desires (and financial means) of their parents, the outcome could be a “GenRich” class of people who are smarter, healthier, and handsomer than the underclass of “Naturals.” The ideal of individual enhancement, rather than species uplift, is in stark contrast to the Galtonian vision of eugenics.

“The improvement of our stock seems to me one of the highest objects that we can reasonably attempt,” Galton declared in his 1904 address on the aims of eugenics. “We are ignorant of the ultimate destinies of humanity, but feel perfectly sure that it is as noble a work to raise its level … as it would be disgraceful to abase it.” It might be right to dismiss this (as Martin Brookes does) as a “blathering sermon.” But Galton’s words possess a certain rectitude when set beside the new eugenicists’ talk of a “posthuman” future of designer babies. Galton, at least, had the excuse of historical innocence.

To Hardy, mathematics was first and foremost a creative art. “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful,” he wrote in his classic 1940 book A Mathematician’s Apology. “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

Mathematics is abstract and difficult; its beauties would seem to be inaccessible to most of us. As the German poet Hans Magnus Enzensberger has observed, mathematics is “a blind spot in our culture—alien territory, in which only the elite, the initiated few have managed to entrench themselves.” People who are otherwise cultivated will proudly confess their philistinism when it comes to mathematics. The problem is that they have never been introduced to its masterpieces. The mathematics taught in school, and even in college (through, say, introductory calculus), is mostly hundreds or thousands of years old, and much of it involves routine problem solving by tedious calculation.

“How can it be,” Einstein asked in wonderment, “that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” Frenkel’s take on this is very different from Einstein’s. For Frenkel, mathematical structures are among the “objects of reality”; they are every bit as real as anything in the physical or mental world. Moreover, they are not the product of human thought; rather, they exist timelessly, in a Platonic realm of their own, waiting to be discovered by mathematicians. The conviction that mathematics has a reality that transcends the human mind is not uncommon among its practitioners, especially great ones like Frenkel and Langlands, Sir Roger Penrose and Kurt Gödel. It derives from the way that strange patterns and correspondences unexpectedly emerge, hinting at something hidden and mysterious. Who put those patterns there? They certainly don’t seem to be of our making.

Upon the premiere of Rites of Love and Math in Paris in 2010, Le Monde called it “a stunning short film” that “offers an unusual romantic vision of mathematicians.” The “formula of love” used in the film was one that Frenkel himself discovered (in the course of investigating the mathematical underpinnings of quantum field theory). It is beautiful yet forbidding. The only numbers in it are 0, 1, and ∞. Isn’t love like that?

When his son’s high school teacher sought help for a computer class, Mandelbrot obliged, only to find that soon students all over Westchester County were tapping into IBM’s computers by using his name. “At that point, the computing center staff had to assign passwords,” he says. “So I can boast—if that’s the right term—of having been at the origin of the police intrusion that this change represented.”

When Mandelbrot took a price chart and zoomed in from a year to a month to a single day, the wiggliness of the line did not change. In other words, price histories were self-similar—like a cauliflower. “The very heart of finance,” Mandelbrot concluded, “is fractal.” The fractal model of financial markets that Mandelbrot went on to develop has never caught on with finance professors, who still by and large cling to the efficient market hypothesis. If Mandelbrot’s analysis is right, reliance on orthodox models is dangerous. And so it has proved, on more than one occasion. In the summer of 1998, for example, Long-Term Capital Management—a hedge fund founded by two economists who had been awarded Nobel Prizes for their work in portfolio theory and staffed with twenty-five Ph.D.’s—blew up and nearly took down the world’s banking system when an unforeseen Russian financial crisis foiled its models.

For the mathematical physicist Sir Roger Penrose, this uncovenanted richness is a striking example of the timeless Platonic reality of mathematics. “The Mandelbrot set is not an invention of the human mind: it was a discovery,” Penrose has written. “Like Mount Everest, the Mandelbrot set is just there!”

What Dawkins brings to this approach is a couple of fresh arguments—no mean achievement, considering how thoroughly these issues have been debated over the centuries—and a great deal of passion. The book fairly crackles with brio. Yet reading it can feel a little like watching a Michael Moore movie. There is lots of good, hard-hitting stuff about the imbecilities of religious fanatics and frauds of all stripes, but the tone is smug and the logic occasionally sloppy.

Blackburn argues that this kind of over-demandingness threatens to undermine ethics itself. “The center of ethics must be occupied by things we can reasonably demand of each other,” he writes. Our duty to help others cannot be infinite. The moral principles we adopt must not reduce us to slaves of the impersonal good. It may be praiseworthy to give away all your money in order to save starving children abroad, or to quit your Park Avenue medical practice and join Doctors Without Borders, or to invite the homeless to crash in your apartment, but it is not obligatory.

“The liar still cares about the truth. The bullshitter is unburdened by such concerns.”

She then proceeds to apply the term “bullshit” to every kind of trickery by which powerful, moneyed interests attempt to gull the public. “Most of what passes for news,” Penny submits, “is bullshit”;