By Bruce MacLennan | April 01, 2014 at 06:11 PM EDT |
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This blog entry is about the philosophy of mathematics, so if you are not very interested in mathematics, you might want to skip it, since it is not especially relevant to the practice of Hypatia’s philosophy. Platonism is perhaps the first philosophy of mathematics, but it has been unpopular since the late nineteenth century. However, Balaguer’s Platonism and Anti-Platonism in Mathematics (Oxford, 1998) has done much to bring the debate up to date. Roughly speaking, the Platonic philosophy of mathematics asserts that mathematical objects (numbers, geometrical forms, sets, etc.) exist independently of us in the realm of Platonic Forms or Ideas, and that mathematics is the study of such objects. (Specialists will know that Platonists disagreed about whether the “mathematicals” are ontologically posterior, equal, or prior to the Ideas, but we can ignore that issue here.)
Do mathematical objects exist? Are they real? When we ask such questions we are usually interested in distinguishing objectively real existents, potentially relevant to all people, from personal, subjective, ephemeral experiences and thoughts, which may be very important to us as individuals, but less so to others. We expect the objects of science to be public in that independent, suitably trained observers will reach the same conclusions about them, and we expect them to be stable (if not eternal) so that the scientific knowledge we acquired yesterday will still be true tomorrow. Now, mathematical objects are both public and stable (eternal, in fact), which is what makes a science of mathematics possible. For example, the Pythagorean Theorem has been discovered independently by several cultures, and the theorems proved by Euclid are still true today. Right triangles exist. They are objectively real. Not physically, of course, but Platonically.
What about non-Euclidean geometries? The answer, I think, is what Balaguer calls “full-blooded Platonism.” In this view, any self-consistent domain of mathematical objects exists (i.e., is public and eternal). In particular, any domain that can be described by a consistent set of axioms exists. Such axiom systems should be considered theoretical explications of an independently existing domain of objects, akin to other scientific theories, and so we can have several axiom systems for the same mathematical domain. If two axiom systems lead to different conclusions about the existence, identity, or properties of their objects, then they are describing different mathematical domains. (Often, however, one can be subsumed under the other, or both of them can be subsumed under a third, more general set of axioms.)
Ah, consistency! There’s the rub! We cannot prove the consistency of any but the simplest axiom systems. While this is an important theoretical result, in practice I do not think it is so important. In an inconsistent system we can prove every proposition. Therefore, if for any theorem P that we have proved, we are persistently unable to prove not-P, then we can be confident the system is consistent.
Now, Gödel’s incompleteness theorem tells us that any consistent axiom system that is powerful enough to express arithmetic must be incomplete, that is, that there are propositions P expressible in the language for which neither P nor not-P is provable. While this is a very important theoretical result, I think its practical import is that mathematical theories, like other scientific theories, are always subject to revision and extension. Different extensions may describe different mathematical domains. This also implies that the unextended axioms described multiple domains, but this is in fact the norm. In fact, any theory that describes an infinite domain will equally well describe an infinite number of other domains. In this sense our mathematical theories cannot in general pick out a unique domain from all those that exist. But because they are all described by the same axioms, it’s not a serious problem.
Finally, I do not think it matters much which kind of logic we use (first-order, higher-order, with or without equality, intuitionistic, etc.). These are just different vehicles for expressing a theory of the domain, with differing advantages and disadvantages, analogous to the wave and matrix formulations of quantum mechanics. If the theories lead to inconsistent theorems, however, then they are describing different domains.
