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

by Jim Holt

For little children, however, time goes quite slowly. Owing to the endless novelty of a child’s experience, a single summer can stretch out into an eternity. It has been estimated that by the age of eight, one has subjectively lived two-thirds of one’s life.

Irresistibly, irreversibly, we are being borne toward our deaths at the stark rate of one second per second.

Our conscious now—what William James dubbed the “specious present”—is actually an interval of about three seconds. That is the span over which our brains knit up arriving sense data into a unified experience.

Hector Berlioz, who is said to have quipped, “Time is a great teacher, but unfortunately it kills all its pupils.”

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.

“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.”

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.”

G. H. Hardy bragged in his famous book A Mathematician’s Apology. “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.”

It was Galton who coined the phrase “nature versus nurture,” which still reverberates in debates today. (It was probably suggested by Shakespeare’s Tempest, in which Prospero laments that his adopted son, Caliban, is “A devil, a born devil, on whose nature / Nurture can never stick.”)

Oddly enough, the bell curve—also known as the normal or Gaussian distribution (after Carl Friedrich Gauss, one of its multiple discoverers)—first arose in astronomy.

Around the middle of the nineteenth century, a sort of revolution occurred in mathematics: the emphasis shifted from science-bound calculation to the free creation of new structures, new languages.

What he found, as the computer produced increasingly detailed pictures of this figure, was utterly unexpected: a wondrous world of beetle-shaped blobs surrounded by exploding buds, tendrils, curlicues, stylized sea horses, and dragon-like creatures, all bound together by a skein of rarefied filaments. At first he suspected that the geometric riot he was seeing was due to faulty equipment. But the more the computer zoomed in, the more precise (and fantastic) the pattern became; indeed, it could be seen to contain an infinite number of copies of itself, on smaller and smaller scales, each fringed with its own rococo decorations. This was what came to be dubbed the Mandelbrot set.

One might imagine, Wittgenstein said, that the theory of infinite sets was “created by a satirist as a kind of parody of mathematics.”

Swift: “So, naturalists observe, a flea / Hath smaller fleas that on him prey; / And these have smaller still to bite ’em, / And so proceed ad infinitum.”

Perhaps, Berkeley derisively concluded, we are best off thinking of infinitesimals as “ghosts of departed quantities.”

Yet it remains a meaningful question whether the infinitesimal has physical reality—whether the infinitely small plays a role in the architecture of nature.

The computer, one might well conclude, was conceived in sin.

“The digital universe and the hydrogen bomb were brought into existence at the same time,” the historian of science George Dyson has observed.

“To read a book was to practice an unnatural process of thought, one that demanded sustained, unbroken attention to a single, static object,” he writes. As written culture superseded oral culture, chains of reasoning became longer and more complex but also clearer.

In 1903, Bertrand Russell described his “unyielding despair” at the thought that “all the labors of the ages, all the devotion, all the inspiration, all the noonday brightness of human genius, are destined to extinction in the vast death of the solar system, and that the whole temple of Man’s achievement must inevitably be buried beneath the debris of a universe in ruins.”

Cultivating indifference to death is not only philosophically unsound. It can be morally dangerous. If my own death is nothing, then why get worked up over the deaths of others?

It is sometimes argued that consciousness is an “emergent” property—that it arises from the interactions of neurons in our brains the way, say, liquidity arises from the interactions of nonliquid molecules.

Evelyn Waugh once observed, “Humility is not a virtue propitious to the artist. It is often pride, emulation, avarice, malice—all the odious qualities—which drive a man to complete, elaborate, refine, destroy, renew his work until he has made something that gratifies his pride and envy and greed. And in doing so he enriches the world more than the generous and good, though he may lose his own soul in the process. That is the paradox of artistic achievement.”

“Truths are illusions that we have forgotten are illusions.”

(Oscar Wilde, in his dialogue “The Decay of Lying,” suggests that the proper aim of art is “the telling of beautiful untrue things.”)