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

by Jim Holt

The idea of the infinitely small—the infinitesimal—raises the question of whether reality is more like a barrel of molasses (continuous) or a heap of sand (discrete).

Both Gödel and Einstein insisted that the world is independent of our minds yet rationally organized and open to human understanding.

But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying when it asserts, “I am not provable”? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete, because there is at least one true proposition about numbers (the one that says “I am not provable”) that cannot be proved within it. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

Some thinkers (like the physicist Roger Penrose) have taken this theme further, maintaining that Gödel’s incompleteness theorems have profound implications for the nature of the human mind. Our mental powers, it is argued, must outstrip those of any computer, because a computer is just a logical system running on hardware and our minds can arrive at truths that are beyond the reach of a logical system.

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.

As the mathematician Hermann Weyl memorably put it, “The objective world simply is; it does not happen.

Hawking says that asking what came before the big bang is as silly as asking what’s north of the North Pole. The answer, of course, is nothing.

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. It is also pretty clear that the nature of memory has something to do with the feeling that we are moving in time. The past and the future might be equally real, but—for reasons traceable, oddly enough, to the second law of thermodynamics—we cannot “remember” events in the future, only ones in the past. Memories accumulate in one temporal direction and not in the other.

Time is nature’s way to keep everything from happening all at once.” In a footnote, Wheeler writes that he discovered this quotation among graffiti in the men’s room at the Old Pecan Street Café in Austin, Texas.

If there is one proposition about time that all scientifically inclined thinkers can agree on, it might be one attributed to the nonscientist Hector Berlioz, who is said to have quipped, “Time is a great teacher, but unfortunately it kills all its pupils.”

Even Arabic numerals follow this logic: 1 is a single vertical bar; 2 and 3 began as two and three horizontal bars tied together for ease of writing.

In general, things that have been around for a long time will likely be around for a lot longer. Conversely, things of recent origin likely won’t be. Both of these conclusions flow from the “Copernican principle,” which says, in essence, you’re not special. If there’s nothing special about our perspective, we’re unlikely to be observing any given thing at the very beginning or the very end of its existence. Let’s say you go to see a Broadway play. No one can be sure exactly how long the play will run. It could be anything from a few nights to many years. But you do know that of all the people who will see it, 95 percent of them will be among neither the first 2.5 percent nor the last 2.5 percent to do so. Therefore, if you’re not “special”—that is, if you’re just a random member of the play’s total audience—you can be 95 percent sure that you don’t fall into either of these two tails. What this means is that if the play has already had n performances at the point in its run when you happen to see it, you can be 95 percent sure that it has no more than 39 × n performances to go and no fewer than n ÷ 39. (This is a matter of elementary arithmetic: the upper

limit keeps you out of the first 2.5 percent of the total audience, and the lower limit keeps you out of the last 2.5 percent.) With nothing more than the Copernican principle and a grade school calculation, you can come up with a 95 percent confidence interval for the longevity of something like a Broadway play. That is fairly amazing.

In recent years, though, practitioners of the fanciful art of evolutionary psychology have been more inventive in coming up with Darwinian rationales for laughter. One of the most plausible is from the neuroscientist V. S. Ramachandran. In his 1998 book, Phantoms in the Brain (written with Sandra Blakeslee), Ramachandran advanced what might be called the “false alarm” theory of laughter. A seemingly threatening situation presents itself; you go into the fight-or-flight mode; the threat proves spurious; you alert your (genetically close-knit) social group to the absence of actual danger by emitting a stereotyped vocalization—one that is amplified as it passes, contagiously, from member to member.

And in the evolution of humor over the millennia, the perception of incongruity has played a more and more dominant role. Laughter, at its highest, is now regarded as the expression of an intellectual emotion. Indeed, the upper bound of the evolution of jocularity may be the Jewish joke, where a Talmudic playfulness toward language and logic reigns. (Think of your favorite Groucho Marx or Woody Allen line.) On this intellectualist view, the greatest stimulus to laughter is pure, abstract incongruity. Every good joke, as Schopenhauer held, is a disrupted syllogism. (For example, “The important thing is sincerity. If you can fake that, you’ve got it made.”) And incongruity is the opposite of boring old tautology. And just as universal.

The Riemann zeta hypothesis, when it is finally dispatched in the aeons to come, will provide just such a resolution. Amid peals of laughter, the Platonic otherness of the primes will dissolve into trivial tautology. It is sobering to think that what is today regarded as the hardest problem ever conceived by the human mind may well be, in the Year Million, a somewhat broad joke, fit for schoolchildren.

Methodical in his observations, he became adept in the use of the sextant, at one point employing this navigational device to measure from afar the curves of an especially buxom native woman—“a Venus among Hottentots.”

When the king dispatched his niece, smeared in butter and red ocher, to his guest’s tent to serve as a wife for the night, Galton, wearing his one clean suit of white linen, found the naked princess “as capable of leaving a mark on anything she touched as a well-inked printer’s roll … so I had her ejected with scant ceremony.”

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.”)

“Regression to the mean motivates almost every variety of risk-taking and forecasting. It is at the root of homilies like ‘What goes up must come down,’ ‘Pride goeth before a fall,’ and ‘From shirtsleeves to shirtsleeves in three generations.’”

(It is a sobering thought that we might generally tend to overrate censure and underrate praise because of the regression fallacy.)

These have been likened to the yin and yang of mathematics: geometry is space, algebra is time; geometry is like painting, algebra is like music; and so on. Less fancifully, geometry is about form, algebra is about structure—in particular, the structure that lurks within equations. And as Descartes showed with his invention of “Cartesian coordinates,” equations can describe forms: x2 + y2 = 1, for example, describes a circle of radius 1. So algebra and geometry turn out to be intimately related, exchanging what André Weil called “subtle caresses.”

traveling across Nazi Germany in a padlocked train. “Of the people we knew, we alone moved to France and survived,” Mandelbrot writes, adding that many of their neighbors in the Warsaw ghetto “had been detained by their precious china, or inability to sell their Bösendorfer concert grand piano.”

The most frequently occurring word in written English is “the,” followed by “of” and then “and.” Zipf ranked all the words in a large variety of written texts in this way and then plotted their frequency of usage. The resulting curve had an odd shape. Instead of falling gradually from the most common word to the least common, as one might expect, it plunged sharply at first and then leveled off into a long and gently sloping tail—rather like the path of a ski jumper. This shape indicates extreme inequality: a few hundred top-ranked words do almost all the work, while the large majority languish in desuetude. (If anything, Zipf underestimated this linguistic inequality: he was using James Joyce’s Ulysses, rich in esoteric words, as one of his main data sources.) The “law” Zipf came up with was a simple yet precise numerical relation between a word’s rank and its frequency of usage.

perhaps information becomes less valuable when it’s more widely available.

Rather, it is a consequence of the “anthropic principle”: if our universe weren’t the way it is, we wouldn’t be here to observe it.

Truth, after all, is a relationship between a theory and the world, whereas beauty is a relationship between a theory and the mind.

Hence what has been called the best explanation of quantum mechanics in five words or fewer: “Don’t look: waves. Look: particles.”

inscription that someone once said should be on all churches: “Important if true.”

is quite similar to that of other mammal species, which tend to go extinct around 2 million years after appearing).

“The more the universe seems comprehensible, the more it also seems pointless.”

In particular (as the philosopher Thomas Nagel has observed), it does not matter now that in a trillion trillion years nothing we do now will matter.

Lawrence Krauss said, “that, sitting in a place on the edge of nowhere in a not especially interesting time in the history of the universe, we can, on the basis of simple laws of physics, draw conclusions about the future of life and the cosmos. That’s something we should relish, regardless of whether we’re here for a long time or not.”

Now, suppose we construct two cosmic scales, one for size and one for longevity. The size scale will extend from the smallest possible size, the Planck length, to the largest possible size, the radius of the observable universe. The longevity scale will extend from the briefest possible life span, the Planck time, to the longest possible life span, the current age of the universe. Where do we rank on these two scales? On the cosmic size scale, humans, at a meter or two in length, are more or less in the middle. Roughly speaking, the observable universe dwarfs us the way we dwarf the Planck length. On the longevity scale, by contrast, we are very close to the top. The number of Planck times that make up a human lifetime is very, very much more than the number of human lifetimes that make up the age of the universe. “People talk about the ephemeral nature of existence,” the physicist Roger Penrose has commented, “but [on such a scale] it can be seen that we are not ephemeral at all—we live more or less as long as the Universe itself!”

The sun still has half its ten-billion-year life span to go. The earth’s population might stabilize at fifteen billion or so, and our successors could even colonize other parts of the galaxy, allowing a far greater increase in their numbers. But think what that means: nearly every human who will ever exist will live in the distant future. This would make us unusual in the extreme. Assume, quite conservatively, that a billion new people will be born every decade until the sun burns out. That makes a total of 500 quadrillion people. At most, 50 billion people have either lived in the past or are living now. Thus we would be among the first 0.00001 percent of all members of the human species to exist. Are we really so special?

more than seven billion of the fifty billion humans who have ever lived are alive today,

Broadly speaking, philosophy has three concerns: how the world hangs together (metaphysics), how our beliefs can be justified (epistemology), and how to live (ethics).

Heinrich Heine: “God will pardon me. It’s his métier.”

As Epicurus put it, where death is, I am not; where I am, death is not.

am very tired because my liver is acting up which is a pity, for my desire to lay the broads here (and there are some fine specimens walking around on campus) is considerably reduced,” Feyerabend wrote from Berkeley.

One view of scientific progress is that it consists in replacing teleological (final cause) explanations with mechanistic (efficient cause) explanations.

The idea that a contradiction is bad because absolutely anything follows from it might seem strange to a non-logician. Bertrand Russell was once trying to get this very point across at a public lecture when a heckler interrupted him. “So prove to me that if two plus two is five, I’m the Pope,” the heckler said. “Very well,” Russell replied. “From ‘two plus two equals five’ it follows, subtracting three from each side, that two equals one. You and the Pope are two, therefore you are one.”

Heisenberg, one of the inventors of quantum mechanics, was the leader of Hitler’s atomic bomb project during World War II. After the war, he claimed that he had deliberately sabotaged the Nazi bomb effort. Many believed him. But it seems more likely that his failure was due not to covert heroism but to incompetence.

David A. Dunning and Justin Kruger drew a poignant conclusion from their research: The most incompetent people have the most inflated notion of their abilities. “Not only do they reach erroneous conclusions and make unfortunate choices,” the two psychologists observed, “but their incompetence robs them of the ability to realize it.”

that “no scientific discovery is named after its original discoverer,” was so dubbed by the historian/statistician Stephen Stigler. An

says the panpsychist, there is consciousness. In David Chalmers’s slogan, “Experience is information from the inside; physics is information from the outside.”

The evolutionary biologist E. O. Wilson, in a 1985 paper written with the philosopher Michael Ruse, put the point starkly: ethics “is an illusion fobbed off on us by our genes to get us to cooperate,” and “the way our biology enforces its ends is by making us think that there is an objective higher code to which we are all subject.”

If the after-death options are either a beatific vision (God) or oblivion (no God), then it is poignant to think that believers will never discover that they are wrong, whereas Dawkins and fellow atheists will never discover that they are right.

It is the final proof of God’s omnipotence that he need not exist in order to save us.” 22

Singer is an avowed utilitarian. In its purest form, utilitarianism says that we should seek to act in a way that brings about the greatest happiness in the world. Now, one test of an ethical principle is that it be “universalizable.” We ask what the world would be like if everybody acted on this principle. What if everyone devoted himself to the happiness of others? Then, on average, everyone would be less happy, because everyone would be subordinating his own happiness to the needs of others. And if everyone gave all his money to Oxfam, the resulting contraction in consumer demand would cause the world economy to collapse, leading to enormous suffering. So these principles of doing good, you might think, are collectively self-defeating.