Why Does the World Exist?: An Existential Detective Story (21 page)

10

PLATONIC REFLECTIONS

See
Mystery
to
Mathematics
fly!
In vain! they gaze, turn giddy, rave, and die.

—ALEXANDER POPE,
The Dunciad

M
ysticism and mathematics go way back together. It was the mystical cult of Pythagoras that, in ancient times, invented mathematics as a deductive science. “All is number,” proclaimed Pythagoras—by which he seemed to mean that the world was quite literally constituted by mathematics. It is little wonder that the Pythagoreans worshipped numbers as a divine gift. (They also believed in the transmigration of souls and held the eating of beans to be wicked.)

Today, two and a half millennia later, mathematics still has a whiff of the mystical about it. A majority of contemporary mathematicians (a typical, though disputed, estimate is about two-thirds) believe in a kind of heaven—not a heaven of angels and saints, but one inhabited by the perfect and timeless objects they study:
n
-dimensional spheres, infinite numbers, the square root of –1, and the like. Moreover, they believe that they commune with this realm of timeless entities through a sort of extrasensory perception. Mathematicians who buy into this fantasy are called “Platonists,” since their mathematical heaven resembles the transcendent realm described by Plato in his
Republic
. Geometers, Plato observed, talk about circles that are perfectly round and infinite lines that are perfectly straight. Yet such perfect entities are nowhere to be found in the world we perceive with our senses. The same is true, Plato held, of numbers. The number 2, for instance, must be composed of a pair of perfectly equal units; but no two things in the sensible world are perfectly equal.

Plato concluded that the objects contemplated by mathematicians must exist in another world, one that is eternal and transcendent. And today’s mathematical Platonists agree. Among the most distinguished of them is Alain Connes, holder of the Chair of Analysis and Geometry at the Collège de France, who has averred that “
there exists, independently
of the human mind, a raw and immutable mathematical reality.” Another contemporary Platonist is René Thom, who became famous in the 1970s as the father of catastrophe theory. “
Mathematicians should have
the courage of their most profound convictions,” Thom has declared, “and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them.”

Platonism is understandably seductive to mathematicians. It means that the entities they study are no mere artifacts of the human mind: these entities are discovered, not invented. Mathematicians are like seers, peering out at a Platonic cosmos of abstract forms that is invisible to lesser mortals. As the great logician Kurt Gödel, among the staunchest of Platonists, put it, “
We do have something
like a perception” of mathematical objects, “despite their remoteness from sense experience.” And Gödel was quite sure that the Platonic realm which mathematicians imagined themselves to be perceiving was no collective hallucination. “I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception,” he declared. (Gödel also believed in the existence of ghosts, but that is another matter.)

Many physicists also feel the allure of Plato’s vision. Not only do mathematical entities seem to be “out there”—eternal, objective, immutable—they also appear to be sovereign over the physical universe. How else can we account for what the physicist Eugene Wigner famously called the “
unreasonable effectiveness
of mathematics in the natural sciences”? Mathematical beauty has time and again proved to be a reliable guide to physical truth, even in the absence of empirical evidence. “
You can recognize
truth by its beauty and simplicity,” said Richard Feynman. “When you get it right, it is obvious that it’s right.” If, in Galileo’s phrase, the “
book of nature
is written in the language of mathematics,” this could only be because the natural world itself is inherently mathematical. As the astronomer James Jeans picturesquely put it, “
God is a mathematician
.”

To a devout Platonist, though, this invocation of God is merely a bit of superfluous poetry. Who needs a creator when mathematics by itself might be capable of engendering and sustaining a universe? Mathematics feels real, and the world feels mathematical. Could it be that mathematics furnishes the key to the mystery of being? If mathematical entities do exist, as the Platonists believe, they must exist necessarily, from all eternity. Perhaps this eternal mathematical richness somehow spilled over into a physical cosmos—a cosmos of such complexity that it gave rise to conscious beings who are able to make contact with the Platonic world whence they ultimately sprang.

This is a pretty picture. But could anyone who is not in the habit of eating lotus leaves take it seriously?

I had the impression that at least one quite rigorous thinker did: Sir Roger Penrose, the emeritus Rouse Ball Professor of Mathematics at Oxford. Penrose is among the most formidable mathematical physicists alive. He has been hailed by fellow physicists, notably Kip Thorne, for bringing higher mathematics back into theoretical physics after a long period in which the two fields had ceased to communicate. In the 1960s, working with Stephen Hawking, Penrose used sophisticated mathematical techniques to prove that the expansion of the universe out of the Big Bang must have been a precise reversal of the collapse of a star into a black hole. In other words, the universe must have begun as a singularity. In the 1970s, Penrose developed the “cosmic censorship hypothesis,” which says that every singularity is cloaked by an “event horizon” that protects the rest of the universe from the breakdown of physical laws. Penrose has also been a pioneer of “twistor theory,” a beautiful new approach to reconciling general relativity with quantum mechanics. In 1994, Queen Elizabeth bestowed a knighthood on Penrose for such achievements.

Penrose also has a penchant for oddities. As a graduate student, he became obsessed with “impossible objects”—that is, physical objects that seem to defy the logic of three-dimensional space. His success in “constructing” one such impossible object, now known as the “Penrose tribar,” inspired M. C. Escher to create two of his most famous prints:
Ascending and Descending
, which depicts a gaggle of monks endlessly tromping up—or down?—a staircase that goes nowhere, and
Waterfall
, which shows a perpetually descending circuit of water. (I once heard the philosopher Arthur Danto say that every philosophy department should keep an impossible object around the office, to instill a sense of metaphysical humility.)

Penrose, I knew, is an unabashed Platonist. Over the years, in his writings and public lectures, he has made it clear that he takes mathematical entities to be as real and mind-independent as Mount Everest. Nor has he been shy about invoking the name of Plato himself. “
I imagine that
whenever the mind perceives a mathematical idea it makes contact with Plato’s world of mathematical concepts,” he wrote in his 1989 book,
The Emperor’s New Mind
. “The mental images that each [mathematician] has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the
same
eternally existing Platonic world!”

What really piqued my interest, though, was Penrose’s occasional hint that our own world was an outcropping of this Platonic world. I first noticed such hints in his second book for a popular audience,
Shadows of the Mind
, which came out in 1994 and, like its intellectually daunting predecessor, was an improbable bestseller. Penrose began by arguing, based on an appeal to Gödel’s incompleteness theorem, that the human mind had powers of mathematical discovery that exceeded those of any possible computer. Such powers, Penrose contended, must be essentially quantum in nature. And they would be understood only when physicists had discovered a theory of quantum gravity—the holy grail of contemporary physics. Such a theory would finally make sense of the bizarre interface between the quantum world and classical reality—and, in the bargain, it would reveal how the human brain leapfrogs the bounds of mechanical computation into full Technicolor consciousness.

Penrose’s ideas on consciousness did not impress many brain scientists. As the late Francis Crick irritably jibed, “
His argument is
that quantum gravity is mysterious and consciousness is mysterious and wouldn’t it be wonderful if one explained the other.” Yet there was more to Penrose’s agenda than that. The very title of his book,
Shadows of the Mind
, was a double entendre. On the one hand, it was meant to suggest that the electrical activities of our brain cells, usually thought to be the cause of our mental life, are mere “shadows” cast by deeper quantum processes going on in the brain, which are the true springs of consciousness.

On the other hand, “shadows” harkened back to Plato—specifically, to Plato’s Allegory of the Cave, from Book VII of the
Republic
. In this allegory, Plato likens us to prisoners chained in a cave and condemned to look only at the rock wall in front of them. On that wall they see a play of shadows, which they take for reality. Little do they realize that there is a world of real things behind them which is the source of these shadowy images. If one of the prisoners were to be liberated from the cave, he would initially be blinded by the sunlight outside. But as his eyes adjusted, he would come to understand his new surroundings. And what would happen if he returned to the cave to tell his fellow prisoners about the real world? Unused to the darkness after his time in the sunlight, he would be unable to make out the shadows they took for reality. His tale of a real world outside the cave would “provoke laughter.” The other prisoners would say that “he had returned from his journey aloft with his eyes ruined,” and that “it was not worthwhile even to attempt the ascent.”

This outside world in the Allegory of the Cave stands for the timeless realm of Forms, wherein genuine reality resides. For Plato, the inhabitants of this realm included abstractions like Goodness and Beauty, as well as the perfect objects of mathematics. Was Penrose, in suggesting that what we took to be reality consisted of “shadows” of such a realm, merely trafficking in neo-Platonist mysticism? Or did his almost unrivalled grasp of quantum theory and relativity, of singularities and black holes, of higher mathematics and the nature of consciousness, afford him genuine insight into the mystery of existence?

I did not have to journey far to obtain enlightenment on this matter. Waiting for the elevator one day in the lobby of the mathematics building at New York University, I saw an announcement that Penrose would soon be coming to Manhattan. He had been invited to deliver an endowed series of lectures on his contributions to theoretical physics. I went home and called his publicist at Oxford University Press to see if an interview could be arranged. A couple of days later, she rang back to say that “Sir Roger” had agreed to make a little time for me to talk about philosophy.

As it happened, Penrose was being put up in a stately apartment building fronting the west side of Washington Square, just steps away from my own Greenwich Village residence. On the appointed day, I set out across the square, which, given the glorious spring weather, was a typically cacophonous buzz of activity. Here, a pick-up jazz combo played for people lounging in the grass; there, a would-be Bob Dylan was wailing away over his guitar. By the big fountain in the middle of the square, banjee boys did improvised gymnastics for an earnest-looking audience of European tourists, while the dogs in the nearby dog run capered and barked.

I exited the square at the northwest corner, where the chess hustlers congregate over the outdoor chess tables, waiting for naive passersby to join them in a game and lose a little money. Glancing up at the old Earle Hotel near the corner, I recalled reading somewhere that it was during a stay in that very hotel many decades ago that the Mamas and the Papas had written their hit song “California Dreamin’.” Inevitably, that was the tune running through my head as I entered the lobby of the building where Penrose was staying, which was vaguely Moorish in decor. The liveried doorman told me to take the elevator to the penthouse.

Sir Roger himself answered the door. He was an elfin man who, with his thick head of auburn hair, looked much younger than his years. (He was born in 1931.) The apartment where he was staying was quite grand, in a prewar New York sort of way. Its high ceilings were embellished with elaborate moldings, and great casement windows with leaden mullions overlooked the treetops of Washington Square. By way of small talk, I pointed out an enormous elm, reputed to be the oldest tree in Manhattan, and told Sir Roger that it was known as the “hanging tree,” since it had been used for executions in the late eighteenth century. He nodded genially at this bit of unsolicited information, then padded off to the kitchen to fetch me a cup of coffee.

Why, I briefly wondered as I took a seat on the sofa, did everyone but me seem to find caffeinated beverages more conducive than alcohol to pondering the mystery of existence?

When Sir Roger returned, I asked him whether he really believed in a Platonic world, one that exists over and above the physical world. Wouldn’t such a two-world view be a little profligate, ontologically speaking?

“Actually, there are
three
worlds,” he replied, warming to my challenge. “Three worlds! And they’re all separate from one another. There’s the Platonic world, there’s the physical world, and there’s also the mental world, the world of our conscious perceptions. And the interconnections among these three worlds are
mysterious
. The main mystery I’ve been addressing, I suppose, is how the mental world is related to the physical world: how certain types of highly organized physical objects—our brains—seem to produce conscious awareness. But another mystery—which, to a mathematical physicist, is just as deep—is the relationship between the Platonic world and the physical world. As we search for the deepest possible understanding of how the world behaves, we’re driven to mathematics. It’s almost as though the physical world is
built
out of mathematics!”