I don't disagree too strongly with your characterization of my position, but I would modify it somewhat:

(1) It doesn't have to be physics per se unless one takes the position that all of natural science and all of human knowledge can ultimately be derived from physics. I don't take that position. It has to be derived from generalizations of observations of the natural world. Many of these, perhaps the majoirty, will be in the realm of physics just because we have defined physics to encompass the most fundamental and easily mathematizable aspects of the world, but not necessarily all. For example, if there are regularities in the way that societies behave that are not ultimately derivable from physics then these can form the basis of mathematics as well.

(2) Modulo the above, yes, but it is important to bear in mind that I intend a broad interpretation of the word "useful". If one theory is much more elegant and easy to think about than another, but their differences have no physical implications, then I would still call the first theory more useful than the second. For example, by this, I think I can justify that there may be a "correct" way of handling infinities in a mathematical theory, or at least ways that are more correct than others, even though this has no immediate physical consequences.

I admit that my view is logically compatible with yours. At least, I cannot rule out a very large mathematical multiverse containing all possibilities. However, my arguments are intended to at least undercut the reasons for thinking that this might be true. It seems to me that the main arguments for the maxiverse are simplicity (Occam's razor) and the ability to account for the role of mathematics in physics in a naturalistic way. I am not too impressed with Occam's razor arguments. It seems to me that, where it applies, its applicability can be derived from other more fundamental principles, and that there are areas where it doesn't apply, i.e. there are phenomena that are just complicated and messy. So, I'm not prepared to accept a hypothesis just because it is simple. There has to be more to it than that. For the naturalistic argument, I still think it is problematic to understand why our universe has the specific mathematical structure that it does, i.e. I don't think you are particularly successful in deflating the measure problem because it seems to me that there are still plenty of bizarre universes that are not incompatible with the continued existence of my consciousness. I also have trouble understanding why what *we* call mathematics should be what reality is fundamentally made of. I did emphasize mathematics as a social activity of finite beings after all.

My position is actually much closer to Sylvia Wenmackers' than to yours, although I don't agree with all her arguments for it. But basically we agree that reality is just what it is, we are in it, and our task is to figure out why the mathematics that we develop is at all relevant to it.

Apologies for not replying sooner.

I certainly did not intend my essay to provide a comprehensive analysis of all forms of mathematical enquiry. I was just trying to make it plausible that there is a process of mathematical abstraction that keeps enough of a tether to empirical reality to explain the later use of those theories in physics. I agree that there are other processes going on in the mathematical knowledge network at the same time. Once such process is problem solving. Another is the kind of free play with axiom changes that you mention. However, I don't think this play is entirely free. It is curious that mathematicians can usually agree on which axioms are the ones worth changing. For example, why drop the parallel postulate in geometry rather than some other? I think these choices have to do both with physical intutions, i.e. which changes are likely to lead to the most applicable theories, and with the internal structure of theories, e.g. which axioms changes are likely to leave me with a consistent theory that shares enough structure with its parent to be interesting. Both of these are to do with the structure of the knowledge network, so are ultimately constrained by empirical reality. I admit that this rough idea needs to be considerably fleshed out.

I note that in your essay you point out that the vast majority of mathematics is actually irrelevant to physics. However, instead of doing a total page count, I would be inclined to weight the pages by the number of links that the corresponding piece of knowledge has to other nodes in the knowledge network, as Google does in the Page Rank algorithm. We can admit that much of the published corpus of mathematics is "failed mathematics" in some sense, just as the majority of pages of theoretical physics produced today are probably irrelevant to reality. Using Page Rank is tantamount to defining "important mathematics" as "applicable mathematics", so I may be accused of circularity, but ultimately I think we agree that applicability is implicitly at least part of the definition of what we mean by a successful mathematical theory.

9 months later

Dear Matt:

I would find this meal more substantial if the distinguishing characteristics of mathematics as a formal system were mapped to what we know about physical reality. Playfully, thinking about realities in which mathematics would not apply might be revelatory. Let's consider a reality in which you had as many fingers as toes when you went to bed and woke up with half as many of the former. Or where a sheep arrived overnight (like flies on meat in the theory of spontaneous generation) - except that as you looked at it, it seemed to be more like a llama.

This guides me to a suggestion that the existence of an irreducible scale (quantization), fermion number conservation, and locality (charge cancellation and finite velocity in the propagation of effects) might all be critical to mapping mathematics to physics.

But I do like you idea that mathematics is an abstraction of physical reality. I see an explanation, in fact, of the observation that physics becomes more and more like mathematics every day. Physicists are reasoning about things that they cannot see. Their grasp of reality has become wholly Platonic. Therefore abstractions upon abstractions is all that they are left with. Rather like theologians arguing about angels dancing on the head of a pin, they have grasped the prop of the legitimacy of classical dynamics, extended it into the world of the unseen through quantum field theory, and then just gone off on a lark with group theory, discarding entirely the proposition that the complexity we observe might be explained by positing additional structure (as it always was in the past).

Brian