The Challenge from Aquinas That Changed Mathematics

A few years back, in my post If You Really Want to Get Into Trouble, Read the Mediaevals, I quoted Carl Boyer’s A History of Mathematics as follows:

The son Georg (Cantor) took a strong interest in the finespun arguments of medieval theologians concerning continuity and the infinite, and this militated against his pursuing a mundane career in engineering as suggested by his father. 

Until fairly recently I didn’t have the information to “flesh this out,” but David Foster Wallace, in his book Everything and More, did just that:

Elsewhere in Summa Theologiae, though, Thomas (Aquinas) advances a more original argument:

The existence of an actually infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the numbers of things in them. Now, no number is infinite, for number results from counting through a set in units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited. (Summa Theologiae, I.a., 7.4)

This passage gets quoted by G. Cantor himself in his “Mitteilungen zur Lehre vom Transfiniten,” (Contributions to the Study of the Transfinite) wherein he calls it history’s only really significant objection to the existence of an actual ∞. For our purposes, there are two significant things about Thomas’s argument: (1) It treats of ∞ in terms of “sets of things,” which is what Cantor and R. Dedekind will do 600 years hence (plus Thomas’ third sentence is pretty much exactly the way Cantor will define a set’s cardinal number.) (2) Even more important, it reduces all of Aristotle’s metaphysical distinctions and complications to the issue of whether infinite numbers exist. It’s easy to see that what Cantor really likes here is feature (2), which makes the argument a kind of tailormade challenge, since the only really plausible rebuttal to Thomas will consist in someone giving a rigorous, coherent theory of infinite numbers and their properties.

David Foster Wallace, Everything and More, pp. 92-94

There are a few things worth noting here:

• Aquinas didn’t actually argue that the infinite didn’t exist, he argued that it was restricted to God.
• Wallace really paraphrases Aquinas; for a more exact translation, read it here.
• It’s tempting to dismiss Aquinas because Cantor disproved him; however, that’s based on the concept of “science” that’s set forth these days. Today we are told that science didn’t begin until we broke off the shackles of religious and philosophical limitations, something that “happened” around the Renaissance. We’ve been doing science from the beginning of civilisation. For example, how is it possible to discuss non-Euclidean geometry, another advance in the nineteenth century, unless we had Euclidean geometry to start with? It’s the same here. Our advances start with what we have, and they don’t always proceed in the nice, straight lines that we’d like to think that they do.
• It’s highly unlikely that the musings of modern or post-modern theologians will inspire the kinds of advances that Aquinas inspired in Cantor.
• As noted in this piece, Cantor’s theory of transfinite numbers was controversial, but ultimately won the day. David Hilbert’s famous quote that “From the paradise created for us by Cantor, no one will drive us out” is still valid. Today there are people who are trying to drive us out of that paradise by challenging the whole concepts of infinites and infinitesimals, saying that these don’t exist in the physical world. In a sense they are going back to Aquinas, which is an amusing thought.