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