Problems with saying that our universe is physically closed (reformulating Kant's antinomies)

Question

Initial caveat: some misapprehension seems to have arisen over my reference to physical sets. But in this, I am trying to follow the language of modern topology, which seems to be applied everywhere without hesitation in physics (as with quantum field theory). I should also mention Augenstein[96] (c.f. Svozil[99]), which I saw linked to over on the MathOverflow (in a post asking about connections between set theory and physics). —Now, I suppose there is a problem with my question being posed on the PhilosophySE if it just so happens that the more helpful responses will have to come from people who are well-versed in both philosophy and physics; I optimistically assume there are respondents meeting those criteria, however (migrating my question to the PhysicsSE would probably trigger a "fringe science" VTC, I pessimistically expect).

Without further ado... For the sake of specific argument, let strong physicalism be the thesis that the universe is physically closed. (Weak physicalism would be more like methodological naturalism, or then an epistemological thesis, which I have no strong objections to (AKA instead of "all possible objects are physical" we would just say "all strongly knowable objects are physical," although as to what "strong knowledge" would be, well...).) Now closure is a matter of quantification via predication, i.e. quantifying over satisfiers of predicates, whereby we quantify over a closed set of satisfiers, and since we are speaking of a universe, then we are speaking of universal closure. It seems then that:

1. If the universe is the set of all physical objects, then if this set is well-founded, then the universe is not itself a physical object. So it would be (A) a physical non-object, (B) a non-physical object, or (C) a non-physical non-object.
2. If the universe is the set of all physical objects and is a physical object itself, then it is not a well-founded set.
3. If the universe is the set at the head of an infinitely descending membership chain, then the regress of physical objects doesn't terminate in smallest physical objects, so subatomic particles would not be "indivisible" after all and there would never be a limit to particle decomposition.

None of these options seems unproblematic. (1) ends up with hyperphysical objects, physical hyperobjects, or hyperphysical hyperobjects, any of which seem perhaps fancifully named (or outright outlandish); mystics and theists might be happy with such a conclusion, but not so much would physicalists, I suppose.^☆ (2) would mean that the objective world has circular phenomena, in fact is a circular phenomenon "as a whole," which raises the specter of allowing circularity among the parts too, and I know there are those who would be upset with this.^☆☆ (3) would irritate believers in the absolutely small, and might raise the issue of hyperstuff too (though the predicate inversion function on hypersets works a little differently compared to the one for well-founded sets).

Can Kant's antinomies be reformulated as tension/conflict between not just two, but three, lines of reasoning (per antinomy)?

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^☆Denying that the universe is an object seems equivalent to denying that the universe is predicatively closed in the first place (the default concept of an object is that which predicates are true of, so if the universe is not an object, then the universe is not a boundary of predication, but is open, and then physicalism is defeated at least insofar as we can't epistemically rule out the existence of hyperphysical predicates and objects beyond the physical domain).

^☆☆Closed timelike curves, for example, though, seem admissible among physicists.

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First corollary: or put the issue in terms of physical laws. If the set of all physical laws is well-founded, then this set does not itself represent a physical law, so it represents (A) a physical non-law, (B) a non-physical law, or (C) a non-physical non-law. Or if this set is an element of itself, then the quintessential law(s) of the entire universe will be derived by circular reasoning. Or if there is a highest physical law that is not well-founded, then there will be no fundamental physical laws and nuclear physics will prove no more fundamental than chemistry turned out to be. —Or consider temporal sets: if there is a well-founded set of all temporal sets, then this set is not itself temporal; or time encloses itself; or time goes backwards forever without a beginning.

Second corollary: the case of causal sets, and so the Third Antinomy, seems fraught with peculiarities. (I would qualify that phrase "causal sets" as "generically causal sets," to avoid bringing in the details of the actual causal-set program in the metatheory of quantum gravity.) To wit:

1. A well-founded set of all causes would not be a cause, and therefore would have no effect. A well-founded set of all effects would not be an effect, and therefore would have no cause.
2. A circular set of all causes would have some effect, and a circular set of all effects would have some cause. I'm not sure the universal generically causal set would be a contributing cause, directly, to every effect, but maybe it would be the cause of the universal generic effect-set specifically.
3. Hyperfounded generically causal/effected sets might pertain to backwards causation/quantum retrocausality, but for now I'm kind of just guessing about that.

Answer 1

So first caveat, my understanding of Kant is fairly sketchy, but I believe his diagnosis of the antinomies is something to the effect that our conceptual scheme has to come apart from the structure of the world as such, and we're theorising about the world, not trying to determine its absolute for-all-time metaphysical truth.

With this in mind, I think for Kant, your 1) would be a first attempt at a resolution of the antinomy, rather than the phrasing of a component of it. Sets are separate things from the noumenal world, as they're phenomenal, and that's not a problem for Kant - it's just the answer he's settled on. So in one sense, this might just be a case of one person’s Modus Ponens being another’s Modus Tollens.

Of course we might now ask whether the set of Phenomena might be a well-founded one, and with that in mind we might dig a bit deeper into 2), because I don't necessarily accept a counterpart line that "The universal set" being ill-founded means that sets aren't physical in the sense of being reducible qua phenomenal theory to physical things. There is an obvious rejoinder from the modern set theorist - even in this sense, the universal class is not a set.

I understand this to be a semantic position - the class, concept or property theory is the semantic ground into which the scaffolding of mathematical set theory is built, and the axioms of set theory within it aren’t necessarily adjoined but rather theories of things that we take to already be true of the phenomenal world the axioms describe. So if we maintain something like a commitment to everything being physical then the same thing would be the case of the physical universe - that it does not exist. What exists is everything, not “the everything set”.

The puzzle then of course is the rehabilitation of the semantics of mathematics in light of the other modern antinomies, but the pessimism about this is very much overstated - grounding the rules of the proof game in the practice of maths as needed for science has a lot of traction (see e.g. Feferman (92).)

Answer 2

As a possible critique of physicalism, thinking about whether the set of all sets of things is included in a set of things -- seems like a fairly abstract and ultimately irrelevant question, rather than central to the validity/invalidity of physicalism.

Physics operates off the Locke/Popper indirect realism model, where we perceive things, and we come up with models, but physicalism is an assertion about what is REAL, not about our models. Logic problems with our models, do not lead to problems for physicalism.

Explicitly, set theory is a math model. It looks like you have identified a logic fault per classical logic, if you assume set theory applies perfectly to our universe. This is not a problem for our universe, as this problem occurs when ever you apply set theory to a set of sets.

• The problem you have identified is a problem for set theory PLUS the assumption of classical logic. This is only one of many problems that classical logic has been found to struggle with, and these have been the reason that many alternative logics have been developed. The current near-consensus among logicians is that, like math is infinite, there are also an infinitude of logics. Taht set theory does not mesh perfectly with one particular logic -- is just an interesting observation. Both set theory, and classical logic, can still be used separately as tools.
• It is also a problem for the presumption that set theory applies fully to our world. However, given the infinity of maths and of logics, this is not a surprising discovery.

There ARE possible refutations for physicalism related to its closure assumption, but they do not appear within this abstract logic question. I will try to note the empirical questions that closure raises for physicalism.

1. Physics is not itself causally determined. See Deterministic or stochastic universe?. An underdetermined universe is not logically closed.
2. No physics problem is ever treated as necessarily closed by physicists. Every space, and every postulated actual or thought problem within our universe will have influences from outside that problem. There can be no fully closed space within a universe. The only possibility is an entire universe being closed. And cosmologists don't treat our universe as closed, basically all cosmology models trat our universe as a whole as open -- from the spontaneous particle generation within "empty" space for the Steady State model, thru the spontaneous breakdown of a background quantum field in the quantum instability model, thru an infinity of outside universes spawning new baby universes in the Cosmic Landscape -- theoretical cosmology is pretty much is universal in rejecting causal closure.
3. Hempel's Dilemma points out that a field of study in science is intrinsically open -- none of what physicalists want to exclude -- Gods, ghosts, and souls -- can be definitively excluded from methodological naturalism.
4. Physicalism, to exclude the non-physical, must show that physics encompasses everything in our universe. This is untrue, as physics -- a science -- relies upon philosophy to pre-establish the methodology of science and the validity of the knowledge it arrives at. This philosophic epistemology is explicitly NOT physics, hence physicalism is self-refuted by the definition of physics.
5. In addition to philosophy not being physics, the consensus of philosophers of science (this is important, as the method of empiricism relies upon expert consensus to settle when a conclusion is valid) is that the other sciences are not reducible to physics, and that there are other fields of subject that are non-sciences that produce valid knowledge. So -- physics is not only not the only way to gain valid knowledge of our world, but there are multiple subject areas that physics is useless to understand.
6. There are major areas of lack of clarity in any physicalist worldview. The only way to accommodate point 5 within physicalism is to postulate a Strong emergence principle, but emergence is currently so ill-defined/characterized as to be a nearly blank placeholder. And causation -- has not had a valid definition since Newton refuted the contact/push theory of causation.
7. To date, no physicalist theories of consciousness can solve the hard problem of why we are conscious (including emergence theories, despite having a nearly "blank chip" to use to resort to), nor physicalist theories of abstract objects explain why abstractions appear to exist.

There are other avenues to very plausibly critique physicalism but set theory does not appear to be a particularly fruitful avenue.