Set Theory and Many Worlds

Round 1

Reviewer 1 Report

I am not an expert in quantum mechanics, and so there are some aspects of the presentation that I can't properly analyze, including the novelty of the view described here. However, it generally seems to be fairly clear.

There were two points where I thought a substantive edit would make sense.

The paragraph that currently goes from page 3 onto page 4 was a bit quick and not totally clear. I can't tell whether it is aiming to distinguish "all" and "each" in a way that some philosophical writing does, or if the point is slightly different. In any case, I think the discussion of the Principal Principle that occurs here could be given a bit more detail, and further clarified. I get the sense that when an event is certainly part of the whole set, but doesn't make up all of it, the suggestion is that the Principal Principle be modified in such a way that each subset gets only a fraction of the whole weight, rather than all of it. I got a bit lost about which terminology was being used for "objective probability" - I would have thought that the "objective probability" of some branch existing that sees the roll of 3 is 1, but the "subjective probability" is only 1/6 of the objective probability in this case, but that doesn't quite seem to be how the terminology is being used here.

The one other substantive question I had was in the discussion of spin on the bottom of p. 9 - and this one seems like it needs to be addressed for this paper to be publishable (unless it is based on a mistaken understanding, in which case it might still be relevant to clarify things to prevent this misunderstanding). I get the sense that what is being claimed here is that the state of an x-spin-up electron is being described as a set that contains elements that are z-spin-up and elements that are z-spin-down, and that this is supposed to solve some issues for spin. But it seems to me that a symmetric argument would say that z-spin-up electrons are themselves in states that are sets that contain elements that are x-spin-up and elements that are x-spin-down. But this would then seem to mean that an x-spin-up electron somehow contains elements (or elements of elements?) that are x-spin-down, which I would have thought should not occur. I must either be misunderstanding, or there must be a conceptual problem here.

I should also say that there were several places where there were either sentence fragments (e.g., a clause beginning "Because" that stands as its own sentence, rather than being attached to the previous sentence) or run-on sentence (i.e., two sentences linked together with a comma instead of a period). These weren't too confusing, but seem like they should be fixed.

Author Response

Please see attached file

Author Response File: Author Response.pdf

Reviewer 2 Report

The central thesis of this paper is that by adopting a set-theoretic approach to making sense of Everettian quantum mechanics under the fission program, it is possible to recover future-looking objective probabilities, even though it is the case that all possible outcomes of every process obtain. 

 

The author begins by acknowledging that the metaphysics associated with bare Everettian mechanics is not obvious. They suppose a fission program, and in this context they develop an account for future-directed objective probabilities. To this end, the author proposes a set-theoretic metaphysics that includes Everettian fission and microscopic local beables.

 

It is the case that there is no published satisfactory account of future-directed objective probability under the fission metaphysics of Everettian quantum mechanics. However, it is not obvious just how introducing this set-theoretic account accomplishes this. Throughout the paper, there are instances where a careful discussion of the various senses of probability would strengthen the account. With the inclusion of such a careful treatment of probability (in objective, subjective, epistemic, and ontic senses) such that it becomes clear just how the set-theoretic approach allows for future-directed objective probabilities in Everettian quantum mechanics, this paper could become an essential and important contribution to ongoing research on Everettian quantum mechanics. 

 

I now offer a handful of more specific points of feedback: 

 

• The author writes “Each of the six successors is a different observer seeing a different number so it’s logically impossible for them all to be the same observer as you. This demonstrates that a metaphysics of persistence needs to be invoked to make sense of Everettian fission even before considering uncertainty.” (3). It is not obvious why a metaphysics of persistence is necessary here; more explicit justification for this claim would be helpful.

• If we accept that any object is, at any given time t, simply a stage of its history, then in what sense are future and past counterparts the same as that object-stage at t? The author writes “A persisting object was its past temporal counterparts and will be its future temporal counterparts.” (3). What is the ontological status of the non-present stages on this account?

• Is Vaidman committed to this stage view of objects? Textual evidence for this commitment would be helpful. 

• Additional exposition of the notion of physical objects as self-membered singletons that are each identified with their hierarchy of unit sets would be helpful to the reader. 

• The author writes: “To make it fully intelligible, there has to be an account of how a well-informed observer can be uncertain as to which outcome they, and their entourage, will observer.” (5) In what sense is the observer distinct from their entourage? Is the entourage meant to be the set of all future counterparts of the observer?

• The author writes: ‘Uncertainty about the future is the cognitive state of assigning partial degrees of belief to multiple futures, whether those futures are thought of as alternative possibilities or coexistent actualities is an arbitrary choice.” (5) Is this really arbitrary? In one case, all possible outcomes are actualized, and so the probability that each possible outcome will obtain is 1. 

• The author writes: “To be certain that all outcomes will occur entails that 5 will occur. So we can be certain that 5 will occur whilst believing that the objective probability of 5 is 1/6.” Is 5 meant to be one of the available outcomes?

• What is the Unitary Interpretation of Mind?

• The author writes: “A free electron in our universe is a set of elemental electrons which are on different trajectories in the universes which are elements of our universe.” (7) It would be helpful to explicitly define what is meant by “elemental” here. 

• The author writes: “For the set-theoretic metaphysics the observer is situated in the set of interacting worlds; objects in the observer’s environment, including their body, are sets. The observer’s universe becomes a set of interacting universes.” (8) Is the observer meant to span the set of interacting universes? If this is the case, then in what sense does the set of instantiations of this observer across all the interacting universes constitute one being?

• The discussion of spin in section 4.1 would be made clearer with the inclusion of some equations. 

• The author writes: “an observer measuring an x-spin-up electron on the z axis will fission into to [sic] observers whose bodies are of equal measure, one observing an electron which is z-spin-up and the other observing an electron which is z-spin-down.” (9) It is not obvious just what it means to have two observers of equal measure.  This is central to providing an account of probability in the fission program. Is probability here meant to be a frequentist account, and does that necessitate branch-counting? How do we count branches?

• When the author writes “Emitted from a source, and collimated, the wavefunction propagates as a sphere with peaked amplitude in opposite directions” (10), it would be helpful to specify that the wavefunction propagates through configuration space. 

• When the author writes “When Alice makes her spin measurement on the x axis she has fissions into AliceUP and Alice_DOWN whose bodies occupy the local regions A_UP and A_DOWN respectively,” (10) is it the case that space itself fissions too, such that A_UP and A_DOWN are subsets of completely different spacetimes?

• The author writes: “​​Quantum processes induce the partitioning of those sets into macroscopically distinct subsets whose measures are the objective probabilities of outcomes” (12). Is this meant to give a future-looking account of objective probability in EQM, or is this a self-locating principle? Do agents really ask: Which doppelganger am I? Or, is it the case that they ask “Which doppelganger will I become?” If it is the latter, then it’s not obvious how to answer this, because all future doppelgangers are actual. 

• Is it the case that the author wishes to propose that future-directed objective epistemic probabilities are possible?

• The reader is left wondering what the author makes of Wallace and Deutch’s decision-theoretic program. Under this set-theoretic approach, might this help, particular with the goal of devising an account that is genuinely future directed?

 

A more general note:

• Typographical and grammatical errors abound. For instance:

• “There is as yet no agreed justification of the Principal Principle, it’s used by stochastic theorists simply because it seems self-evident.” (4) Two independent clauses are linked by only a comma here; consider changing to “There is as yet no agreed justification of the Principal Principle; it’s used by stochastic theorists simply because it seems self-evident.”

• “Whether the futures are understood as alternative possibilities or coexistent actualities is beside the point, uncertainty is the very same thing in both cases.” Two independent clauses are linked by only a comma here; consider changing to “Whether the futures are understood as alternative possibilities or coexistent actualities is beside the point, for uncertainty is the very same thing in both cases.” 

 

Author Response

Please see attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The author has tended to all of the feedback that I provided, and I feel confident that this is now publication-ready.