Why Does the World Exist: An Existentia - By Jim Holt Page 0,83
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