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.

Greg(21:01:17) :I certainly feel a difference between the two cases of disagreement you mention — but I’d like to know what

accounts for/ underwritesthat (felt) difference. (Is that what Chalmers’ paper is all about?)David Corfield(02:22:54) :I think the best way to catch a glimpse of a category theoretic take on the distinction you’re driving at is in the section ‘The Universal Structures’ from p.11 on in Steve Awodey’s Reply to Hellman.

Noah(04:57:19) :I’ve thought a little more about our brief discussion on this, and I’ve realized that although I don’t think that the axioms of set theory are structural, I do think that the axioms of the natural numbers are structural.

I don’t know whether this is constistant or sensical, but that’s what my intuition is.

Tanasije Gjorgoski(08:01:57) :Hi Kenny,

I know the subtitle of the blog says “A general distrust of strong metaphysical claims in mathematics and philosophy.”, but you seem to be pointing to the difference between math as part of (more general) metaphysics, and math as formalism.

Kenny(10:44:31) :Greg – Chalmers’ paper is about how we tell when things are merely terminological disputes, and what that means for philosophy. (I think it has a lot to do with his ideas on conceptual analysis.) I’ll have to look at it again to see what it might mean in this case. But I think what underlies the felt difference in this case is that rings and such are just defined structures, while the natural numbers and sets are things that we pick out more directly by ostension and then try to analyze.

Noah – that’s a consistent position to take, but I’m not entirely sure in what sense you’re going to mean that. Certainly the successor axiom has to be there, right? And we don’t want to count a structure that satisfies PA but also ~Con(PA) as the naturals – so we’re going to need to go beyond any recursive axiom system. But once we restrict to an isomorphism type, it certainly does seem plausible that there’s nothing in particular of that type that we mean.

Tanasije – that’s right. Mainly, I’m suspicious of lots of metaphysical claims, but that doesn’t mean that I don’t think they’re important and meaningful.

Aidan(16:38:44) :I’m not really seeing how this helps to motivate your project (or at least what I remember being your project, which may not be the same thing, but I’m hoping it’ll be close enough). The observation that there are substantial rather than merely verbal disputes about the acceptability of some of the axioms of mathematical theories doesn’t look like it sits too well with a picture on which the only role such axioms play is to bracket philosophical controversy. Indeed, it’s a simple-minded thought to be sure, but one might think your observation provides some good prima facie evidence that this cannot be the role these axioms play.

Kenny(17:48:13) :The fact that there are substantive disputes is just the fact that there is philosophical controversy to be bracketed. However, in ordinary practice, there are no disputes about these axioms, which is why they can be accepted. The few disputes that do arise in strange (ie, philosophical) cases, are substantive.

Alexandre Borovik(01:22:35) :99% of axiom systems appearing in everyday mathematical research have the status which is much below that of “foundational” or “structural” axioms: they are

disposable, just memos written by a mathematician to himself with the aim to fix the context of the argument and re-use the work already done. In most cases, such axioms end up in a waste basket without ever seeing light of publication. From a purely formal logical point of view, they have the same standing as at “structural” axioms.I believe that, for better understanding of the nature of mathematics, we should pay more attention to ephemera of everyday mathematical practice.

Aidan(11:08:37) :“The fact that there are substantive disputes is just the fact that there is philosophical controversy to be bracketed.”

But not the fact that there are substantives disputes *over the acceptability of particular axioms*, which is what I explicitly mentioned in my reply. On the face of it, that fact doesn’t look at all like it’s just the fact that there is controversy of the kind that might be bracketable by the mutual acceptance of some particular set of axioms for the theory. For this is by description substantive disagreement precisely over whether a particular axiom is acceptable.

Kenny(17:10:20) :Aidan – you’re right, I should be more careful. I think for the large part of the mathematical community, there is no dispute about the successor axiom, or just about any of the others, so they can accept these axioms to bracket further controversies. I suppose there’s a possibility that the positions questioning these axioms will come back into favor, and we’ll have to use a smaller set of axioms. For someone like Maddy, I don’t think there could possibly be a significant disagreement like this; for the foundationalist about axioms, the very possibility of this disagreement would get in the way of adopting the axioms. For me, we can adopt the axiom as long as we’re sure no one we work with is going to dispute it.

Alexandre – that’s definitely a very interesting idea, that doesn’t seem to be very much pursued yet.