Why Does the World Exist?: An Existential Detective Story (23 page)

How can the romantic Platonism of Penrose, Tegmark, and others survive Russell’s cold cynicism? Well, if neither logic nor feeling can underwrite the existence of timeless mathematical Forms, then perhaps science can. Our best scientific theories of the world, after all, incorporate quite a lot of mathematics. Take Einstein’s general theory of relativity. In describing how the shape of spacetime is determined by the way matter and energy are distributed throughout the universe, Einstein’s theory invokes a host of mathematical entities, like “functions,” “manifolds,” and “tensors.” If we believe that the theory of relativity is true, then aren’t we committed to the existence of these entities? Isn’t it intellectually dishonest to pretend they aren’t real if they are indispensable to our scientific understanding of the world?

That, in a nutshell, is the so-called Indispensability Argument for mathematical existence. It was originally proposed by Willard Van Orman Quine, the dean of twentieth-century American philosophy and the man who famously declared, “
To be is
to be the value of a variable.” Quine was the ultimate “naturalist” philosopher. For him, science was the final arbiter of existence. And if science inescapably refers to mathematical abstractions, then those abstractions
exist
. Although we don’t observe them directly, we need them to explain what we do observe. As one philosopher put it, “
We have the same
kind of reason for believing in numbers and some other mathematical objects as we have for believing in dinosaurs and dark matter.”

The Indispensability Argument has been called the only argument for mathematical existence that is worth taking seriously. But even if it is valid, it provides scant comfort for Platonists like Penrose and Tegmark. It robs mathematical Forms of their transcendence. They become mere theoretical posits that help explain our observations. They are on par with physical entities like subatomic particles, since they occur in the same explanations. How can they be responsible for the
existence
of the physical world if they themselves are part of the very
fabric
of that world?

And it gets worse for the Platonists. Mathematics, it turns out, may not be indispensable to science after all. It may be that we can explain how the physical world works without invoking abstract mathematical entities, just as we have learned to do so without invoking God.

One of the first to raise this possibility was the American philosopher Hartry Field. In his 1980 book,
Science without Numbers
, Field showed how Newton’s theory of gravitation—which, on the face of it, is mathematical through and through—could be reformulated so that it made no reference whatsoever to mathematical entities. Yet the numbers-free version of Newton’s theory would yield exactly the same predictions, though in a rather more roundabout way.

If the program of “nominalizing” science—that is, of stripping away its mathematical trappings—could be extended to theories like quantum mechanics and relativity, it would mean that Quine was wrong. Mathematics is not “indispensable.” Its abstractions need play no role in our understanding of the physical world. They are just a glorified accounting device—nice in practice (since they lead to shorter derivations), but dispensable in theory. To creatures of greater intelligence elsewhere in the cosmos, they might not be necessary at all. Far from being timeless and transcendent, numbers and other mathematical abstractions would be exposed as mere terrestrial artifacts. We could banish them from our ontology the way the protagonist of Bertrand Russell’s story “The Mathematician’s Nightmare” did—with a cry of “
Avaunt! You
are only Symbolic Conveniences!”

But would that spell the doom of Platonism as a resolution to the mystery of existence? Maybe not. Recall that there was something missing from Roger Penrose’s Platonic scheme. The worlds of matter and consciousness were “shadows,” he held, of the Platonic world of mathematics. But what, in this metaphor, was the source of the illumination that allowed the Forms to cast their shadows? Sir Roger conceded that it was a “mystery” how mathematical abstractions could be creatively effective. Such abstractions are supposed to be causally inert: they neither sow nor reap. How could mere passive patterns, however perfect and timeless, reach out and make a world?

Plato himself had no such lacuna in his scheme. For him there
was
a source of light, a metaphorical Sun. And that was the Form of the Good. Goodness, in Plato’s metaphysics, stands above the lesser Forms, including the mathematical ones. Indeed, it stands above the Form of Being: “the Good is itself not existence, but far beyond existence in dignity,” as Socrates tells us in Book VI of Plato’s
Republic
. It is the Form of the Good that “bestows existence upon things”—not by free choice, the way the Christian God is supposed to have done, but by logical necessity. Goodness is the ontological Sun. It shines beams of Being on the lesser Forms, and they in turn cast a shadowy play of Becoming—which is the world we live in.

So that is Plato’s vision of the Good as a sunlike source of reality. Should we dismiss it as a woolly poetical conceit? It seems even less helpful than Penrose’s own mathematical Platonism at resolving the mystery of existence. Who could imagine that abstract Goodness might bear creative responsibility for a cosmos like ours, which is un-good in so many ways? Yet I was surprised to find that there was at least one thinker who did imagine precisely such a thing. And I was still more surprised to discover that he had managed to convince some of the world’s leading philosophers that he might not be entirely daft in doing so. Yet somehow I wasn’t surprised to learn that he lived in Canada.

Interlude

It from Bit?

M
athematical Platonism turned out to be a nonstarter as an ultimate explanation of being. But its shortcomings invite deeper reflection on the nature of reality.

Of what does reality, at the most fundamental level, consist? It was Aristotle who supplied the classic answer to this question:

Reality
=
Stuff + Structure

This Aristotelian doctrine is known as “hylomorphism,” from the Greek
hyle
(stuff) and
morphe
(form, structure). It says that nothing really exists unless it is a composite of structure and stuff. Stuff without structure is chaos—tantamount, in the ancient Greek imagination, to nothingness. And structure without stuff is the mere ghost of being, as ontologically wispy as the smile of the Cheshire Cat.

Or is it?

Over the last few centuries, science has relentlessly undermined this Aristotelian understanding of reality. The better our scientific explanations get, the more that “stuff” tends to drop out of the picture. The dematerialization of nature began with Isaac Newton, whose theory of gravity invoked the seemingly occult notion of “action at a distance.” In Newton’s system, the Sun reached out and exerted its gravitational pull on Earth, even though there was nothing but empty space between them. Whatever the mechanism of influence between the two bodies might be, it seemed to involve no intervening “stuff.” (Newton himself was coy on how this could be, declaring
Hypotheses non fingo
—“I frame no hypotheses.”)

If Newton dematerialized nature on the largest of scales, from the solar system on up, modern physics has done the same on the smallest of scales, from the atom on down. In 1844, Michael Faraday, observing that matter could be recognized only by the forces acting on it, asked, “
What reason
is there to suppose that it exists at all?” Physical reality, Faraday proposed, actually consists not of matter but of
fields
—that is, of purely mathematical structures defined by points and numbers. In the early twentieth century, atoms, long held up as paragons of solidity, were discovered to be mostly empty space. And quantum theory revealed that their subatomic constituents—electrons, protons, and neutrons—behaved more like bundles of abstract properties than like little billiard balls. At each deeper level of explanation, what was thought to be stuff has given way to pure structure. The latest development in this centuries-long trend toward the dematerialization of nature is string theory, which builds matter out of pure geometry.

The very notion of
impenetrability
, so basic to our everyday understanding of the material world, turns out to be something of a mathematical illusion. Why don’t we fall through the floor? Why did the rock rebound when Dr. Johnson kicked it? Because two solids can’t interpenetrate each other, that’s why. But the reason they can’t has nothing to do with any sort of intrinsic stufflike solidity. Rather, it’s a matter of numbers. To squash two atoms together, you’d have to put the electrons in those atoms into numerically identical quantum states. And that is forbidden by something in quantum theory called the “Pauli exclusion principle,” which allows two electrons to sit directly on top of each other only if they have opposite spins.

As for the sturdiness of individual atoms, that too is essentially mathematical. What keeps the electrons in an atom from collapsing into the nucleus? Well, if the electrons were sitting right on top of the nucleus, we’d know exactly where each electron was (right in the center of the atom) and how fast it was moving (not at all). And that would violate Heisenberg’s uncertainty principle, which does not permit the simultaneous determination of a particle’s position and momentum.

So the solidity of the ordinary material objects that surround us—tables and chairs and rocks and so forth—is a joint consequence of the Pauli exclusion principle and Heisenberg’s uncertainty principle. In other words, it comes down to a pair of abstract mathematical relations. As the poet Richard Wilbur wrote, “
Kick at the rock
, Sam Johnson, break your bones: / But cloudy, cloudy is the stuff of stones.”

At its most fundamental, science describes the elements of reality in terms of their relations to one another, ignoring any stufflike quiddity those elements might possess. It tells us, for example, that an electron has a certain mass and charge, but these are mere propensities for the electron to be acted upon in certain ways by other particles and forces. It tells us that mass is equivalent to energy, but it gives us no idea of what energy really
is
—beyond being a numerical quantity that, when calculated correctly, is conserved in all physical processes. As Bertrand Russell noted in his 1927 book,
The Analysis of Matter
, when it comes to the intrinsic nature of the entities making up the world, science is silent. What it presents us with is one great relational web: all structure, no stuff. The entities making up the physical world are like the pieces in a game of chess: what counts is the role defined for each piece by a system of rules that say how it can move, not the stuff that the piece is made of.

The physicist’s view of reality, by the way, is remarkably akin to the view of language proposed over a century ago by Ferdinand de Saussure, the father of modern linguistics. Language, Saussure maintained, is a purely relational system. Words have no inner essence. The intrinsic character of the noises we make when talking is irrelevant to communication; the important thing is the system of
contrasts
among those noises. This is what Saussure meant when he wrote that “
in language
, there are only differences
without positive terms
.” Saussure’s elevation of structure over stuff was the inspiration for the structuralist movement that swept aside existentialism in France in the late 1950s. It was taken up in anthropology by Claude Lévi-Strauss and in literary theory by Roland Barthes. Its extension to the universe as a whole might well be called “cosmic structuralism.”

If reality were indeed pure structure, that would open up radically new ways of thinking about it. One of these is the way of Penrose and Tegmark. On their view, reality is in essence
mathematical
. Mathematics, after all, is the science of structure; it neither knows of, nor cares about, stuff. Worlds that are structurally the same but made out of different stuff are identical in the eyes of the mathematician. Such worlds are called “isomorphic,” from the Greek words
isos
(same) and
morphe
(form). If the universe is structure all the way down, then it can be exhaustively characterized by mathematics. And if mathematical structures have an objective existence, then the universe must
be
one of these structures. That, at least, seems to be Tegmark’s meaning when he says that “
all mathematical structures
exist physically.” If there is no ultimate stuff to reality, then mathematical structure is tantamount to physical existence. Who needs flesh when bones are enough?

A somewhat different take on a stuff-less reality is to see it as consisting not of mathematics but of
information
. This view is summed up in a slogan coined by the late physicist John Archibald Wheeler: “it from bit.” (Wheeler—who collaborated with Albert Einstein and taught Richard Feynman—had a gift for such coinages; he also gets credit for “black hole,” “wormhole,” and “quantum foam.”)

The it-from-bit story goes as follows. At bottom, science tells us only about
differences
: how differences in the distribution of mass/energy are associated with differences in the shape of spacetime, for example, or how differences in the charge of a particle are associated with differences in the forces it feels and exerts. States of the universe can thus be seen as pure information states. As the British astrophysicist Sir Arthur Eddington once put it, “
Our knowledge
of the nature of the objects treated in physics consists solely of readings of pointers on instrument dials.” The “medium” in which these information states are realized, whatever it might be, plays no role at all in the explanation of physical phenomena. Therefore, it can be dispensed with altogether—shaved away by Occam’s razor. The world is nothing but a flux of pure differences, without any underlying substance. Information (“bit”) suffices for existence (“it”).

Some it-from-bit proponents stretch this logic still further. They look on the universe as a giant computer simulation. Among those who have taken this view are Ed Fredkin and Stephen Wolfram, both of whom hypothesize that the universe is a “cellular automaton” that uses a simple program to generate complex physical outcomes. Perhaps the most radical cosmos-as-computer advocate is the American physicist Frank Tipler. The striking thing about Tipler’s vision is that it involves no actual computer: his cosmos is all software, no hardware. A computer simulation, after all, is just the running of a program; and a program, in essence, is a rule that transforms an input string of numbers into an output string of numbers. So any computer simulation—say, the simulation of the physical universe—corresponds to sequences of strings of numbers: a pure mathematical entity. And if mathematical entities have an eternal Platonic existence, then, on Tipler’s view, the existence of the world has been fully explained: “
at the most basic ontologica
l level,” he declares, “the physical universe is a concept.”

And what of the simulated beings who are somehow a part of that “concept”—beings like us? Would they realize that time was an illusion, that they were mere frozen bits of an eternal Platonic videotape? Not at all, according to Tipler. They would have no way of knowing that their reality consisted in being “a sequence of numbers.” Yet, oddly enough, it is their simulated mental states that endow the overall mathematical concept of which they are a part with physical existence. For, as Tipler writes, “this is exactly what we mean by existence, namely, that thinking and feeling beings think and feel themselves to exist.”

The picture of the universe as an abstract program—it from bit—strikes some thinkers as strangely beautiful. And it seems consistent with the way science represents nature, as a network of mathematical relations. But is that truly all there is? Is the world devoid of ultimate stuff? Is it indeed structure all the way down?

There is one aspect of reality that doesn’t seem to have a place in this metaphysical picture: our own consciousness. Think of the way a pinch feels, a tangerine tastes, a cello sounds, or the rosy dawn sky looks. Such qualitative experiences—philosophers call them “qualia” (the plural of the Latin singular
quale
)—have an inner nature that goes beyond their role in the causal web. So, at least, philosophers like Thomas Nagel have argued. “
The subjective features
of conscious mental processes—as opposed to their purely physical causes and effects—cannot be captured by the purified form of thought suitable for dealing with the physical world that underlies appearances,” Nagel writes.

One way of making this point vivid is due to the Australian philosopher Frank Jackson. Imagine, says Jackson, a scientist named Mary who knows everything there is to know about color: the neurobiological processes by which we perceive it, the physics of light, the composition of the spectrum, and so forth. But imagine further that Mary has lived her entire life in a black-and-white environment, that she has never actually
seen
a color herself. Despite her complete scientific understanding of color, there is something of which Mary is ignorant: what colors look like. She does not know what it is like to experience the color red. It follows that there is something to this experience—something subjective and qualitative—that is not captured by the objective, quantitative facts of science.

Nor, it would seem, can this subjective aspect of reality be captured by a computer simulation. Consider the theory called “functionalism,” which holds that states of the mind are essentially computational states. What defines a mental state, according to functionalism, is not its intrinsic nature, but rather its place in a computer flowchart: the way it is causally related to perceptual inputs, to other mental states, and to behavioral outputs. Pain, for example, is defined as a state that is caused by tissue damage and that, in turn, causes withdrawal behavior and certain vocalizations, like “ouch.” Such a flowchart of causal connections can be implemented in a software program, which, if run on a computer, would simulate
being in pain
.

But would this simulation duplicate what seems most real to us about pain: the horrible way it
feels
? To the philosopher John Searle, the very idea seems “
frankly, quite crazy
.” “Why on earth,” he asks, “would anyone in his right mind suppose a computer simulation of mental processes actually had mental processes?” Suppose, Searle says, the program simulating the experience of pain were to be run on a computer consisting of old beer cans tied together with string and powered by windmills. Can we really believe, Searle asks, that such a system would feel pain?

The philosopher Ned Block has come up with another thought experiment along the same lines. He invites us to imagine what would happen if the population of China were to simulate the brain’s program. Suppose we got each Chinese person to mimic the activity of a particular brain cell. (There are only around a hundredth as many Chinese as there are cells in the human brain, but no matter.) Synaptic connections among the different cells could be simulated by cell-phone links among the Chinese. Would the nation of China, if it were to mimic the brain’s software in this way, then have conscious states over and above those of its individuals? Could it experience, say, the taste of peppermint?

The conclusion the philosophers who come up with these thought experiments want us to draw is that there is more to consciousness than the mere processing of information. If this is true, then science, insofar as it describes the world as a play of information states, would seem to leave out a part of reality: the subjective, irreducibly qualitative part.