When I presented my paper on the role of axioms in mathematics in Berlin, one comment I got from several people (which I’ve also gotten from some mathematician friends) was that the project falls apart in my second footnote. Basically, they want to treat all sets of axioms as structural, rather than allowing for a class of foundational axioms, encompassing something like Peano Arithmetic or possibly ZFC or some replacement. The idea is that any consistent system of axioms describes some mathematical structure (of course, only some of those structures are actually of interest to us), but there isn’t an intended model (like the actual natural numbers, or actual sets) to fall back on to extend these sets in a natural way. The Peano axioms show us some results that are useful when counting, but there isn’t some unique structure beyond this practice that fills in truth values for further claims. So the project of wondering whether we should look for new axioms falls apart. I wasn’t able to really say anything terribly convincing at the time, but I think I’ve got some general ideas now.
About a week later, I was discussing David Chalmers’ talk on terminological disputes with some friends in the department here, and I realized that this is a useful way to motivate the distinction between foundational and structural axioms. Consider any dispute over some axiom – say one mathematician says that in every ring there is an element such that multiplying it by any other element fixes the other element, while another mathematician denies that this is in general true. Consider another dispute – say one mathematician says that in the natural numbers, every element has a successor, while another denies that this is in general true. In the former case, it seems that there’s just a disagreement in terms of what we mean by “ring” – some people build in commutativity and identities, while others don’t. The disagreement is merely terminological. However, the ultrafinitist who denies the successor axiom isn’t engaged in a merely terminological dispute. We both intend to be talking about the counting process. We know what the other means by natural numbers, and say that they’re wrong in terms of what they claim. This dispute is not merely terminological.
The fact that there can be disputes like this, that aren’t about logical consequence and aren’t merely terminological, suggests that the axioms involved are in fact of this other “foundational” type. This doesn’t necessarily mean that they provide a foundation for the rest of mathematics, but just that they play a different sort of role. And this is the role that I am concerned with.