Philosophers of mathematics divide roughly into "platonists" or "realists" and "nominalists" or "anti-realists." (If you have studied some philosophy, go here for background.) The realists are those who take the indispensability of mathematics for science as evidence that mathematical entities (which I shall call "M's" for short) belong in our ontology, our inventory of what-there-is, so that M's can be said to "exist." But we need to be clear about what that means. There's a sense in which M's exist whether we've discovered them or merely thought them up. If the former, they exist "objectively" like natural objects such as water and the Sun; if the latter, they are only mental entities, objects of thought, i.e., what the scholastics called

*entia rationis*

*.*So, M's clearly exist; the question is what sort of existence they have.

Given that question, mathematical realists face a further one: if M's exist objectively, so that we discover rather than invent them, how do they exist

*other*than as mere objects of thought? They aren't material objects; they aren't spirits, like God or angels; indeed, they neither act nor change. If one grants that there are different kinds of existence, there doesn't seem to be any good answer to that question. Yet some realists, such as the late W.V.O. Quine, field the difficulty by saying that existence is

*univocal*, so that it's not something you get in different sorts at all. Thus, to say of such-and-such that it "exists" is simply to say that it is the value of the variable in a true statement of the form: "For some x, x is F" or "There is an x such that x is F', where 'F' stands for a predicate. Such-and-such exists, therefore, just in case we can say something true about it; in the strict sense, that's what it

*means*to say that such-and-such exists. If that's so, then it doesn't matter whether M's are invented or discovered. Even if M's are invented by human thought, they exist objectively just like artifacts such as chairs and computers.

But that solution comes at a price: existence itself is taken as a logical rather than a metaphysical notion. Thus, to exist is to belong to a universe of discourse in an

*n-*order predicate calculus. While some 20th-century and contemporary philosophers in the tradition broadly called "analytic" don't have a problem with that, others do, and it certainly does seem counterintuitive. Rather than commit to defending mathematical realism by means of ontological univocity, then, some philosophers would rather be mathematical anti-realists and leave the broader ontological issue open.

The problem faced by anti-realists is to explain how it is that a mere human invention works so well in the discovery and expression of the way the world works, sometimes called "the laws of nature." Some try to do it by appealing to those laws themselves. Given the mathematical structure of the laws of nature, it would be surprising if evolved beings intelligent enough to understand those laws did

*not*think up M's. But it's hard to see how that appeal is any different, in the final analysis, from realism. If the laws of nature are mathematical, then M's are there to be discovered, and what we think we think up are really just the formal features of what's already there. Our intelligence does not invent M's but merely abstracts them from natural objects. That's pretty close to the view of Aristotle and St. Thomas Aquinas.

Now if, in face of that, we persist in being mathematical anti-realists, then the only explanation on offer for the fit of mathematics with nature is the kind the Pope suggests. But of course one would then have to have very good reason to be an anti-realist about M's to start with. I can't think of any.