There’s a new short book in the Cambridge Elements series — Penelope Maddy and Jouko Väänänen have written a very interesting contribution on Philosophical Uses of Categoricity Arguments. Here’s their Introduction:
Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this doesn’t exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here, we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with leading contemporary writers, Charles Parsons and the coauthors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case, we ask: What does the author set out to accomplish, philosophically? What do they actually do (or what can be done), mathematically? And does what’s done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.
Their scorecard? “Dedekind has successfully achieved his goal” (p. 6), and “In the end … Zermelo accomplished more than he set out to do -– and ultimately more than he could have realized at the time – so this application of categoricity arguments must be counted as a resounding success” (p. 15). As for Kreisel, properly read “determinateness of CH wasn’t his target in the first place. At his actual goal – elucidating the independence phenomenon – he succeeds” (p. 21). Next, “In the end, there seems room for doubt that our shared concept [of number], Parsons’s own Hilbertian intuition of the endless sequence of strokes, is as clear and determinate as we think it is. And if there is this room for doubt, formal categoricity theorems don’t seem to be the kind of thing that might conceivably help. Given these open questions, both mathematical and philosophical, Parsons’s appeal to categoricity arguments to establish “the uniqueness of the natural numbers” can’t yet be judged a success.” (p. 38, after a particularly useful discussion.) Finally, “We conclude that Button and Walsh have not succeeded in establishing that internalist … concerns over the status of CH are “difficult to sustain” (p. 49).
Along the way, we get pointers to some significant first-order results due to Väänänen, and the book concludes
Perhaps unsurprisingly, we think the first-order theorems do make an important philosophical point: an outcome that was thought to require secondorder resources – namely, categoricity theorems – can actually be achieved by suitable first-order means. … This is a useful discovery, which supports our general moral: a bit of mathematics that fails at one task might succeed (and even be aimed) at another.
I hope that’s enough to pique your interest in what does seem to be one of the best so far of the logic/philosophy of mathematics Elements; I enjoyed a quick first reading — it is only 50 small pages — and will want to return to think more carefully about some of the interpretations and arguments.
(A minor but welcome point: unlike some earlier Elements, this looks to have been properly LaTeXed so the symbols aren’t garbled.)
This little book should be readily available if your library has a suitable Cambridge Core subscription. And until the end of today the CUP version is freely available for download here. But there is also (as pointed out in a comment below) a version which looks to be more or less identical on the arXiv here.