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

by Jim Holt

What science can tell us something about is the psychology of time’s passage. 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. This seems to explain the psychological arrow of time. It does not, unfortunately, explain why that arrow seems to fly.

If all of this leaves you utterly bewildered about time, you are in eminent company. John Archibald Wheeler, one of the great physicists of the twentieth century, took to quoting this in a scientific paper: “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 you are asked to choose which of a pair of Arabic numerals—4 and 7, say—stands for the bigger number, you respond “7” in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene’s experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the École Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number. Dehaene conjectured that when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this.

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.

What would we see if, like Mr. Square, we were actually yanked out of our world into “a more spacious Space” with an extra dimension? No doubt we would be as dumbstruck by the vision of the hypersphere as Mr. Square was by that of the sphere. We would also be astonished to find, looking back upon our 3-D world, that all objects were transparent to our sight and visible from every perspective simultaneously. We could reach inside someone’s body and remove his appendix without breaking his skin (a great boon for surgeons). We could pick up a left shoe, rotate it in the fourth dimension, and restore it to the 3-D world as a right shoe.

String theorists have come up with some extremely elegant and subtle conjectures as to why, out of the nine spatial dimensions they posit, exactly three of them expanded to enormous size after the big bang while the remaining six got choked off and remained tiny. But there is another sort of explanation, one that is perhaps easier to grasp: in a world where the number of spatial dimensions was anything other than three, beings like us simply could not exist. In a space of more than three dimensions, there would be no stable planetary orbits. (This was proved a century ago by Paul Ehrenfest.) Nor would there be stable orbits for electrons within atoms. Therefore, there could be no chemistry, and hence no chemically based life-forms, in a world of more than three spatial dimensions.

Well, then, what about a world of fewer than three spatial dimensions? As noted earlier, sound waves would not propagate cleanly in 2-D Flatland, or indeed in any other space with an even number of dimensions. But the difficulties are not limited to sound; it is impossible to transmit well-defined signals of any type in a space that has an even number of dimensions. This rules out the kind of information processing that is essential to intelligent life.

During this golden age of discovery, scientists treated the infinitesimal as they would any other number, until it became convenient in their calculations to set it to zero (as Newton somewhat fishily did in cases like that of the falling rock, above). This cavalier attitude toward the infinitely small is captured by the advice of the French mathematician Jean Le Rond d’Alembert: “Allez en avant, et la foi vous viendra” (Go forward, and the faith will come to you). Still, there remained those who felt it a scandal that the edifice of modern science was being erected on such metaphysically shaky foundations. Throughout the eighteenth century, there were many efforts to answer the charges against the infinitesimal put by critics like Berkeley and to find a logical set of rules for its use. None were successful; some were simply fatuous. (Well into the next century, Karl Marx would try his hand at this problem, leaving nearly a thousand unpublished pages devoted to it.) One of the more philosophically appealing attempts was that of Bernard de Fontenelle, who tried to rationalize the infinitesimal by characterizing it as the reciprocal of the infinitely large. Though Fontenelle was ultimately defeated by formal difficulties, he was prescient in arguing that the reality of objects like the infinitesimal rested ultimately on their logical coherence, not on their existence in the natural world.

If there is no such thing as the infinitesimal, then, as Russell observed, notions like “the next moment” and “state of change” become meaningless. Nature is rendered static and discontinuous, because there is no smooth transitional element to blend one event into the next. In a rather abstract sense, things no longer “hang together.” The resulting sense of ontological discontinuity can be detected in the cultural lurch toward modernism—as witness Seurat’s pointillism, Muybridge’s stop-motion photography, the poetry of Rimbaud and Laforgue, the tone rows of Schoenberg, and the novels of Joyce.

“Science is a differential Equation. Religion is a Boundary Condition.”

There is a reason the new media have caught on, Pinker observes: “Knowledge is increasing exponentially; human brainpower and waking hours are not.” Without the Internet, how can we possibly keep up with humanity’s ballooning intellectual output?

According to the Copenhagen interpretation, the very act of observation causes the spread-out probability wave to collapse into a sharply located particle. Hence what has been called the best explanation of quantum mechanics in five words or fewer: “Don’t look: waves. Look: particles.”

Hearing this news made me think of the inscription that someone once said should be on all churches: “Important if true.”

He bases these predictions on what he calls the Copernican principle, which says, in essence, you’re not special. “If life in the universe is going to last a long time, why do we find ourselves living when we do, only 14 billion years after the beginning?” Gott said to me, speaking in an improbable Tennessee accent whose register occasionally leaped up an octave, like Don Knotts’s. “And it is a disturbing fact that we as a species have only been around for 200,000 years. If there are going to be many intelligent species descended from us flourishing in epochs far in the future, then why are we so lucky to be the first?” Doing a quick back-of-the-envelope calculation, Gott determined that it was 95 percent likely that humanity would last more than 5,100 years but would die out before 7.8 million years (a longevity that, coincidentally, is quite similar to that of other mammal species, which tend to go extinct around 2 million years after appearing).

Why should we want the universe to last forever, anyway? Look—either the universe has a purpose or it doesn’t. If it doesn’t, then it is absurd. If it does have a purpose, then there are two possibilities: either this purpose is eventually achieved, or it is never achieved. If it is never achieved, then the universe is futile. But if it is eventually achieved, then any further existence of the universe is pointless. So, no matter how you slice it, an eternal universe is either (a) absurd, (b) futile, or (c) eventually pointless.

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!”

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?

But suppose, contrariwise, that humanity will be wiped out imminently, that some sort of apocalypse is around the corner. Then it is quite reasonable, statistically speaking, that our moment is the present. After all, more than seven billion of the fifty billion humans who have ever lived are alive today, and with no future epochs to live in, this is far and away the most likely time to exist. Conclusion: doom soon.

Cicero said that to philosophize is to learn how to die—a pithy statement, but a misleading one. There is more to philosophizing than that. Broadly speaking, philosophy has three concerns: how the world hangs together (metaphysics), how our beliefs can be justified (epistemology), and how to live (ethics).

The third points out that your nonexistence after your death is merely the mirror image of your nonexistence before your birth. Why should you be any more disturbed by the one than by the other?

For each symmetry that a theory possesses, there is a corresponding law of conservation that must hold in the world it describes. A conservation law is one that states that some quantity can be neither created nor destroyed. If a particular theory is symmetric under space translation—that is, if its equations remain unchanged when the spatial perspective is shifted—that implies the law of conservation of momentum. Similarly, if the theory is symmetric under time translation, that implies the law of conservation of energy. Symmetry under shifts in orientation implies the law of conservation of angular momentum.

“This would vitiate all science” is what W. V. Quine, one of the preeminent logicians of the twentieth century, would say: “Any conjunction of the form p and not-p logically implies every sentence whatever; therefore acceptance of one sentence and its negation as true would commit us to accepting every sentence as true, and thus forfeiting all distinction between true and false.” To see what Quine means, suppose you believe both p and not-p. Because you believe p, you must also believe p or q, where q is any arbitrary proposition. But from p or q and not-p, it obviously follows that q. Hence any arbitrary proposition is true. (The Latin phrase for this is ex contradictione quodlibet.) 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.”

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

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.

Unlike the bullshitter, the liar at least cares about the truth: he is anxious that his assertions be correlated with the truth, albeit negatively; so he is concerned to get the picture right, if only to invert that picture.