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 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