My old advisor Peter Koellner, in a paper on pluralism in mathematics, raises the kind of interesting point (based on a simple observation) that I'm often disappointed didn't occur to me already. The main issue Koellner is considering is the problem of selection of mathematical theories, analogous to the problem of selection in physics. For physics, the problem is to select from among classes of empirically equivalent theories; for math, it is to select from among various interpretability degrees, i.e. equivalence classes under the relation of mutual interpretability. (Don't worry about the precise definition of that for now.) The point I have in mind is an aside Koellner makes.

He writes:

The structure of the hierarchy of interpretability is more disorderly than one might expect---it forms a distributive lattice that is neither linearly ordered nor well-founded. This is shown via the construction of non-standard theories via coding techniques. Remarkably, however, when one restricts to the natural theories that occur in mathematical practice the theories are well-behaved---they are well-ordered under interpretability. (28)

(NB: The single best source I know for such material is Lindström's Aspects of Incompleteness, where he investigates the structure of interpretability degrees for theories containing arithmetic.) Koellner has in mind the hierarchy of large cardinal axioms---e.g. inaccessibles, Mahlos, measurables, Woodins, and so on---for extending . Similarly, in the context of second-order arithmetic instead of set theory, there are the extensive results in reverse mathematics showing that many theorems of ordinary mathematics fall directly into the well-ordered progression of subsystems , , , , - and so on. (Simpson's Subsystems of Second Order Arithmetic is the standard reference.)

As Koellner points out, there is some imprecision in the notion of what counts as a "large cardinal" axiom. But setting that aside, we can just talk of the finitely many instances of axioms that currently go by the name. Or in the context of we can talk of the finitely many subsystems in the list above. The fact is that "natural" theories tend to fall into the corresponding well-orderings of interpretability. But now there is the notion which would be interesting to examine and perhaps try to make somewhat precise, that of a natural theory.

Let's focus for the moment on subsystems of , taking as a base theory. We know that theorems from countable algebra and separable analysis tend to fall into the hierarchy of subsystems listed above, that is, along one particular chain from to in the lattice of interpretability degrees. Here are a couple of questions I would ask; they are rather vague, and I do not know what form satisfactory answers would take.

Question 1: What is so special about this particular chain? Why does ordinary mathematics lie on it? (How could we characterize this in a precise manner?)

Question 2: Would ordinary mathematics be different (and in what ways) if it, say, lived on a different chain? Could one argue that it necessarily resides where it does, in some interesting sense of the word?

Again, these are pretty vague, but I could imagine rigorous logico-mathematical work that would inform a consideration of these philosophical questions and others like them.

In sum, while I've long been aware of the fact that "natural" theories tend to fall along the well-ordering of large cardinal axioms (or, within , the well-studied subsystems), and also of the rather wild structure of the lattice of interpretability degrees, for some reason I never thought about examining the inherent tension there. I agree with Koellner that this is "a mystery that calls for clarification" (28).