Hello. This is the second essay I read after that of Lee Smolin, that tries to give a naturalistic philosophy of mathematics, as opposed to a Platonistic one. As I commented there, I have yet to see a coherent formulation of naturalism. The Stanford encyclopedia article you refer to on this point, admits it directly : "The term 'naturalism' has no very precise meaning in contemporary philosophy." It is remarkable to see that still after quite a long time that a majority of physicists and philosophers who care about metaphysics are trying to defend and develop naturalism, no clear formulation of its actual meaning could even reach a status of notability without being also loaded with big troubles (such as those of Bohmian mechanics); while I would explain the lack of well-known coherent formulation of idealism by the lack of serious tries by competent physicists and philosophers, a gap I care to fill by my essay.
The main difference I see between yours and Lee's exposition, is that Lee entered in specific details, especially about where he draws the line of existence between things. His claims are totally incoherent, both logically and with respect to existing knowledge in math and physics as I explained in my many comments there, however he has at least this merit, of taking that risk of making specific claims and thus expose himself to refutation. It seems you found a pretty good way to minimize the amount of criticism that you will get, by minimizing the quantity of claims contained in your essay. Indeed, most of the ideas there are already given in the abstract. You managed to approach the standard size of 9 pages not by adding up many ideas but rather by diluting the few you had. Anyway, by lack of effective matter to discuss from your essay, and also because I don't like to repeat myself, I invite you to read my comments to Lee's article (I'll still have more to write there), for the general remarks I made about this common topic between yours and his essay, of naturalism vs. Platonism, and for the specific analysis of his tries to effectively specify the claims of naturalism (I'm curious how would your options differ from his).
Now for your few specific claims :
You wrote: "Naturalism is the position that everything arises from natural properties and causes, i.e. supernatural or spiritual explanations are excluded. In particular, natural science is our best guide to what exists, so natural science should guide our theorizing about the nature of mathematical objects."
It is true that sciences of "natural" things (physics, biology) were extremely successful and these aspects of the world are a privileged field for scientific inquiry. However, like so many naturalist authors, you fail to distinguish between the scientific method and materialistic metaphysical prejudices, by using the ambiguous expression of "natural science" that may actually mean either the scientific method or the body of discovered knowledge in physics and biology, but where the adjective "natural" it contains can be irrationally played on in philosophical discussions to make it look as if the scientific method required to adopt such metaphysical prejudices. It doesn't.
"A naturalistic theory has no place for a dualistic mind that is independent of the structure of our brains. Therefore, if we have intuitive access to an abstract realm, our physical brains must interact with it in some way. Our best scientific theories contain no such interaction."
Our best scientific theories do not contain any explanation of anything psychological : what it may mean for a mind to understand something in general. Thus, there is no wonder why they do not explain either how we (I mean, some people) can understand mathematics in particular.
"The only external reality that our brains interact with is physical reality, via our sensory experience. Therefore, unless the platonist can give us an account of where the abstract mathematical realm actually is in physical reality, and how our brains interact with it, platonism falls afoul of naturalism."
The content of this argument looks like an absolute argument against platonism, to which I replied in comment to Lee's article. Strangely, for no reason I could see, the last phrase "platonism falls afoul of naturalism" looks as if it was only an argument relative to naturalism (thus void outside it). But then in this case, it implicitly seems to assume that a remark that "Naturalism contradicts X" should be taken as an argument against X, thus taking for granted that naturalism should be true. However I still beg for a rational argument for naturalism, in the name of which it would make sense to consider a view as being made less plausible just by its conflict with naturalism.
"The various attempts to reduce all of mathematics to logic or arithmetic reflect a desire [to] view mathematical knowledge as hanging hierarchically from a common foundation. However, the fact that mathematics now has multiple competing foundations, in terms of logic, set theory or category theory, indicates that something is wrong with this view." ; and in your comments: "What I mean is rather that which formal systems we decide to call mathematics, out of all the myriad of arbitrary axiom systems we might choose to lay down, is a matter of physics."
Well no. There is, for example, a clear absolute sense of which rules can form a valid and complete system of proofs for first-order logic, and satisfy the completeness theorem. It is clearly independent of physics. Now for the choice of an axiom system to best serve as the foundation of mathematics, I do not see such a competition between possibilities as you assume. Instead, as I explained in my work on the foundations of maths which I developed showing its intrinsic necessities independently of physics, I rather see a coherent whole of complementary parts of the foundations, with a dynamic articulation between them. Where there are alternative possibilities in competition for the same purpose (such as axiomatizations of set theory) I see a situation where in some aspects there is a preferred kind of formalization, and in other aspects such as the continuum hypothesis there is a remaining real diversity of acceptable systems corresponding to a real diversity of acceptable realities, that can often be understood as due to the real fact of ambiguity of the powerset of infinite sets.
"Firstly, in network language, the concept of a "theory of everything" corresponds to a network with one enormous hub, from which all other human knowledge hangs via links that mean "can be derived from". This represents a hierarchical view of knowledge, which seems unlikely to be true if the structure of human knowledge is generated by a social process. It is not impossible for a scale-free network to have a hierarchical structure like a branching tree, but it seems unlikely that the process of knowledge growth would lead uniquely to such a structure. It seems more likely that we will always have several competing large hubs..."
If things went as you describe here, we would not have got the amazing success of mathematics in physics we had, with an amazingly universal agreement on what is the right fundamental theory of gravitation on the one hand, of microscopic physics on the other hand, how amazingly well, each on their side, they indirectly explain so many things on almost all physical processes that could be tested. So I do not see your philosophy coherent with the state of science that could be observed.
"...and that some aspects of human experience, such as consciousness and why we experience a unique present moment of time, will be forever outside the scope of physics." What a nicely non-naturalistic claim ! ;-)
"I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics." Only under the assumption of naturalism. But, if, as I hold, naturalism is irrational, and the only reasonable way of understanding science is an idealistic one (more precisely a mind/mathematics dualism), then your conclusion fails.
Now for the general ideas : your account of mathematics and its success is much too vague, and fails to relate to the effective contents of how successful is mathematics for physics. You present an abstraction of explanation for an abstraction of a problem. But the difference between mathematical abstractions with their success in science (as abstractions of real problems), and the abstraction of your approach to the problem of the success of mathematics in physics, is that the success of mathematical abstractions depends on their care and success to keep effective logical articulations (as you say, "tether") with the real problems they are abstractions of, but your abstraction of approach to the issue of usefulness of mathematics, is a non-mathematical abstraction that fails at keeping a tether to the reality of this usefulness. If the success of mathematics in physics could be explained as simply as what you describe, there would have been no amazement at this success in the first place.
Now, of course, it would make little sense to only express this objection in the abstract: to justify it, I need to enter the specifics about how your approach fails to account for the specific success of mathematics. So here are a few specific details of how you fail.
First, you explain mathematics as "the study of regularities, within regularities, within ..., within regularities of the natural world". In this case, we have a hierarchy of different levels of abstractions, where some concepts are more abstract than others, as they are not directly natural objects but abstract generalities about wide ranges of natural objects, or generalities about generalities about natural objects. When abstract generalities are developed as abstract generalities, they are not themselves the natural objects. That structure of knowledge you describe can only be expected to be useful as a path of reasoning through which we might discover new natural objects by first going up to the abstraction and then down again from the abstraction to concepts of new objects that would be particular objects inside classes described by these abstractions: the new fundamental objects we discover should display this same dependence to mathematical abstract generalities as the objects of our experience do.
But this is not what we observe. Instead, what we observe is that what we discover as fundamental objects of physical reality are some very abstract mathematical objects themselves. And what I mean here as "abstract mathematical objects", does not look like any generality like a general description of regularities among a range of many particular objects in a naturalistic sense. You may make this confusion because some of the most famous mathematical concepts, such as category theory, precisely look like generalities describing a wide range of particular cases (it is a generality of generalities). But this is not the case for all highly abstract mathematical concepts, and this is not how it goes for the success of mathematical concepts used in physics. In short, "high abstraction" and "generality" are not synonyms.
And to explain how the concepts of "abstraction" and "generality" differ, I need to take a specific example. I would like you to consider the case of the Dirac equation. This is a particular case of equation of a particular object (electrons or other spin 1/2 massive particles), but nevertheless a very abstract one.
This equation describes the field of electronic presence as taking values in the space of bispinors of space-time. To say roughly, we can define this space as the sum of 2 spinor spaces (with conjugate types), where a spinor space is a 2-dimensional complex space E such that the space of hermitian forms on E is identified with the tangent space to this point of space-time. Namely, tangent vectors (x,y,z,t) to this point are identified with Hermitian forms on E with matrix
(t+x yв€'iz)
(y+iz tв€'x)
since the determinant of this matrix coincides with the relativistic invariant (t2в€'x2в€'y2в€'z2).
This is very abstract, but not any kind of "generality of things" like what category theory does by describing regularity classes of particular systems that may go down to objects that look "natural" in a naturalistic sense. Now what is amazing with the success of mathematics is that this spinor space E was found to be "what electrons (and other fermions) are actually made of". Yet its connection to space-time, as described above, is... quite abstract. Spinors are not "made of" space-time connections, since any spinor (element of E) would correspond to a light-like direction of space-time but any physical description by such a direction would fail to fix the phase of this spinor. Concretely, a big problem with a spinor is that its phase is reversed when you apply to it a rotation with angle 2pi.
So, unless you provide a naturalistic explanation of how an object can be reversed when applying a rotation with angle 2pi, and how such an amazing thing as the Dirac equation can be relevant to physics, I must consider that naturalism fails to account for the mathematical aspects of the physical reality as we observed.