Hello. You come to propose a conception of things coherent with naturalism. Great ! I stand for the opposite view ;-)
I actually never found a formulation of naturalism that seemed coherent, as it seems to me logically impossible, somehow already in principle, and then even more with quantum physics. So I am very curious when I see such a proposition announced ! For now most of the essays I reviewed here in support of such a view seemed to be amateur-level. I was full of hope for discussions to become at last serious and challenging, at the first sight of an essay with this purpose by a reputed physicist coming to the list.
One thing I was puzzled with when reading some naturalist views, is how they dismiss any idea of considering consciousness as fundamental, by calling this an "explanation by a mystery" and thus no explanation at all. Indeed it may look like this, in the sense that consciousness escapes all mathematical description. So if your condition to call something "non-mysterious" is to have a mathematical, deterministic description of it then indeed consciousness is "mysterious" in this sense. Which does not mean that noting can be said about it (as I did express some important features of consciousness for its connection with physics). However, on their side they claim to explain everything as "Nature". But what the heck do they mean by "nature", and, in lack of a clear definition for this kind of stuff and its working principles, how is an "explanation" of the world by an undefined "nature" assumed to be primary, be any less mysterious than the view taking consciousness as primary ?
I once saw an "argument" that if a miracle is real then by definition it must be part of nature because nature is "all what exists" so that nothing can be meaningfully called "supernatural". Then well, if "all what exists" is the definition of "nature" then it makes naturalism tautological, but no more informative. To be informative we need to specify what kind of stuff is "nature" supposed to be. It seems supposed to mean "physical stuff". Well if we were in the 19th century, and still with General Relativity, it could indeed look like there was such a thing as "physical stuff" that the universe could be made of. However, quantum physics broke that.
Namely, an important question I would have, is whether "nature" is supposed to be finitely or infinitely complex, or maybe just locally finitely complex, in case it could be considered locally (which you seem to reject as you seem to favor non-locality in interpretations of quantum physics). So for example if it is locally finitely complex but not locally causal then, finally, it is infinitely complex if the universe is infinite (in hope that the dependence of local stuff on the rest of the universe converges). Quantum physics makes the physical world locally finitely complex indeed. I consider consciousness infinitely complex. But if "nature" was physical and infinitely complex, how could it have definite causalities that depend on infinitely complex stuff ? Bohmian mechanics describes things as infinitely complex, but I suspect its laws to diverge when considered in their globality.
Here are points of interest I found in your article:
"The effectiveness of mathematics in physics is in [Platonism] mysterious because proponents of this view have failed to explain both how there could be such a correspondence and how we, as beings trapped in time bound physical reality, can have certain knowledge of the hypothesized separate realm of mathematical reality."
What failure to explain ???? I do not see the slightest problem here: it is a one-way dependence. Anything that exists must be coherent with itself, so that whenever we can discern mathematical structures somewhere, they have to be coherent with themselves, thus obey the laws of coherence which are the mathematical theorems. So it is "affected" by the mathematical world, but does not affect it in return (nothing can change the facts of what is coherent and what isn't). It is possible for mathematical structures to be more or less involved by contingent (non-mathematical) realities.
"if you believe that the ultimate goal of physics is to discover a mathematical object, O, which is in perfect correspondence with nature, such that every true fact about the universe, or its history, is isomorphic to a true fact about O, then you are also not a naturalist because you not only believe in the existence of something which is not part of nature, you believe that everything that is true about nature is explained by a true fact about something which exists apart from nature. You are instead a kind of mystic, believing in the prophetic power of the study of something which exists outside of time and apart from nature."
All right, so this means naturalism rejects any possibility to describe nature in mathematical terms. In this case, nature escapes any rigorous mathematical description and is therefore assumed to be fundamentally "mysterious". Like consciousness in my view.
You wrote: "Mathematics thus has no prophetic role in physics, which would allow us an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment". Then you "hypothesize two principles which we take to define temporal naturalism". Are these two principles not supposed to have any prophetic role in physics, that would allow you an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment? Because in the rest of your essay I did not find any big care to test these principles against experiment, or against the body of modern science which sums up so many experiments already done, in the sense of a possible challenge to the truth of your principles.
"All that exists is part of a single, causally connected universe. The universe and its history have no copies, and are not part of any ensemble."
Right. I would qualify the spiritual multiverse (where souls can migrate between universes) in these very terms, though the connections between parts (universes) can sometimes be poor.
"There is no other mode of existence, in particular neither a Platonic realm of mathematical objects nor an ensemble of possible worlds exist apart from the single universe." And why not ? You seem to have quite a faith in this negation.
"All that is real or true is such within a moment, which is one of a succession of moments"
You already multiply the modes of existence, between past, present and future existences, and where the time-status of the existence of any particular event... depends on time. So you admit multiple possible modes of existence, but you deny the possibility for still another mode of existence than these (the mathematical existence).
"The activity of time is a process by which novel events are generated out of a presently existing, thick set of present events. "
How thick is the set of present events, and how do you measure this thickness, both in space and time dimensions ? My view of the spiritual reality would be similar except that I take all past events as still presently existing and indestructible, and from which novel events are generated.
" we adopt a strong form of Einstein's principle of no unreciprocated action according to which there can be no entity A which plays a role in explaining an event B, that cannot itself be influenced by prior physical events."
That is quite an assumption, of trying to generalize a principle far beyond the form in which it was initially considered and justified by experiment ! But is it really just a plausible strengthening of a well-defined principle, or rather an endless multiplication of fanciful assumptions only superficially similar to the initially successful version ? Something like justifying philosophical relativism as "a strong form" of the Special Relativity principle.
Of course you cannot understand the possible relation between mathematical and physical realities if you exclude by principle the possibility of one-way influences, and by "satisfying explanation" you mean "explanation that agrees with this principle", assumed to have such a prophetic role in physics, that it allows you, in your own words, "an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment". By the way, how do you apply this principle to the dependence between past and future ? How can the past affect the future without being affected by it in return ?
You wrote " Among the things that violate a strict definition of naturalism are (...) absolute, timeless laws," yet you defend the view of "the singular universe" which seems to fit absolute timeless laws. It seems quite hard for these laws to vary inside the same universe, both theoretically (the formal rigidity of the physical laws that do not easily let coherent ways to glue together parts of space-time that do not obey the same laws) and as we did not see them vary, but it would be much easier between different universes. Don't you see it hard to reconcile both principles of uniqueness of the universe and contingency of the laws ?
I will write more remarks later.