by Jim Holt
But gradually the spell that Sir Roger had cast on me began to wear off. How could the solemn mathematical abstractions in Plato’s heaven have given rise to the gaiety of life in Washington Square? Do such abstractions really hold the answer to the mystery of why there is Something rather than Nothing?
The scheme of being that Penrose had conjured up for me seemed almost miraculously self-creating and self-sustaining. There are three worlds: the Platonic world, the physical world, and the mental world. And each of the worlds somehow engenders one of the others. The Platonic world, through the magic of mathematics, engenders the physical world. The physical world, through the magic of brain chemistry, engenders the mental world. And the mental world, through the magic of conscious intuition, engenders the Platonic world—which, in turn, engenders the physical world, which engenders the mental world, and so on, around and around. Through this self-contained causal loop—Math creates Matter, Matter creates Mind, and Mind creates Math—the three worlds mutually support one another, hovering in midair over the abyss of Nothingness, like one of Penrose’s impossible objects.
Yet, despite what this picture might suggest, the three worlds are not ontologically coequal. It is the Platonic world, in Penrose’s vision, that is the fons et origo of reality. “To me the world of perfect forms is primary, its existence being almost a logical necessity—and both the other worlds are its shadows,” he wrote in Shadows of the Mind. The Platonic world, in other words, is compelled to exist by logic alone, and the contingent world—the world of matter and mind—follows as a shadowy by-product. That’s Penrose’s solution to the puzzle of existence.
And it left me with two misgivings. Is the existence of the Platonic world really assured by logic itself? And even if it is, what then makes it cast shadows?
As to the first, I couldn’t help noticing what looked like a failure of nerve on Penrose’s part. The existence of the Platonic world, he said, is “almost a logical necessity.” Why this “almost”? Logical necessity is not a thing that admits of degree. It is all or nothing. Penrose makes much of the alleged fact that the Platonic world of mathematics is “eternally existing,” that its reality is “profound and timeless.” But the same, one might note, would be true of God—if God existed. Yet God is not a logically necessary being; his existence can be denied without contradiction. Why should mathematical objects be superior to God in this respect?
The belief that the objects of pure mathematics exist necessarily has been called an “ancient and honorable” one, but it doesn’t hold up terribly well under scrutiny. It seems to be based on two premises: (1) mathematical truths are logically necessary; and (2) some of those truths assert the existence of abstract objects. As an example, consider proposition twenty in Euclid’s Elements, which says that there are infinitely many prime numbers. This certainly looks like an existence claim. Moreover, it appears to be true as a matter of logic. Indeed, Euclid proved that denying the existence of an infinity of primes led straight to absurdity. Suppose there were only finitely many prime numbers. Then, by multiplying them all together and adding 1, you would get a new number that was bigger than all the primes and yet divisible by none of them—contradiction!
Euclid’s reductio ad absurdum proof of the infinity of prime numbers has been called the first truly elegant bit of reasoning in the history of mathematics. But does it give any grounds for believing in the existence of numbers as eternal Platonic entities? Not really. In fact, the existence of numbers is presupposed by the proof. What Euclid really showed was that if there are infinitely many things that behave like the numbers 1, 2, 3, . . . , then there must be infinitely many things among them that behave like prime numbers. All of mathematics can be seen to consist of such if-then propositions: if such-and-such a structure satisfies certain conditions, then that structure must satisfy certain further conditions. These if-then truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition “2 + 2 = 4,” for example, tells you that if you had two unicorns and you added two more unicorns, then you would end up with four unicorns. But this if-then proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all.
Mathematicians essentially make up complex fictions. Some of these fictions have analogues in the physical world; they compose what we call “applied mathematics.” Others, like those positing higher infinities, are purely hypothetical. Mathematicians, in creating their imaginary universes, are constrained only by the need to be logically consistent—and to create something of beauty. (“ ‘Imaginary universes’ are so much more beautiful than the stupidly constructed ‘real’ one,” declared the great English number theorist G. H. Hardy.) As long as a collection of axioms does not lead to a contradiction, then it is at least possible that it describes something. That is why, in the words of Georg Cantor, who pioneered the theory of infinity, “the essence of mathematics is freedom.”
So the existence of mathematical objects is not mandated by logic, as Penrose seemed to believe. It is merely permitted by logic—a much weaker conclusion. Practically anything, after all, is permitted by logic. But for some modern-day Platonists of an even more radical stripe, that seems to be permission enough. As far as they are concerned, self-consistency alone guarantees mathematical existence. That is, as long as a set of axioms does not lead to a contradiction, then the world it describes is not only possible—it is actual.
One such radical Platonist is Max Tegmark, a young Swedish-American cosmologist who teaches at MIT. Tegmark believes, like Penrose, that the universe is inherently mathematical. Also like Penrose, he believes that mathematical entities are abstract and immutable. Where he goes beyond Sir Roger is in holding that every consistently describable mathematical structure exists in a genuine physical sense. Each of these abstract structures constitutes a parallel world, and together these parallel worlds make up a mathematical multiverse. “The elements of this multiverse do not reside in the same space but exist outside of space and time,” Tegmark has written. They can be thought of as “static sculptures that represent the mathematical structure of the physical laws that govern them.”
Tegmark’s extreme Platonism furnishes a very cheap resolution to the mystery of existence. It is basically, as he concedes, a mathematical version of Robert Nozick’s principle of fecundity, which says that reality encompasses all logical possibilities, that it is as rich and variegated as it can be. Anything that is possible must actually exist—hence the triumph of Something over Nothing. What makes such a principle compelling for Tegmark is the peculiar ontological muscle that mathematics seems to possess. Mathematical structures, he says, “have an eerily real feel to them.” They are fruitful in uncovenanted ways; they surprise us; they “bite back.” We get more out of them than we seem to have put into them. And if something feels so real, it must be real.
But why should we be swayed by this “real feel,” no matter how eerie? Tegmark and Penrose may be swayed, but another great physicist, Richard Feynman, was decidedly not. “It’s just a feeling,” Feynman once said dismissively, when asked whether the objects of mathematics had an independent existence.
Bertrand Russell came to take an even sterner view of such mathematical romanticism. In 1907, when he was in his relatively youthful thirties, Russell penned a gushing tribute to the transcendent glories of mathematics. “Rightly viewed,” he wrote, mathematics “possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.” Yet by his late eighties he had come to view his callow rhapsodizing as “largely nonsense.” 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.”
How can the romantic Platonism of Pe
nrose, 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?
Mathematical 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 rebou
nd 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.”