From theism to causal finitism

Causal Finitism—the thesis that nothing can have an infinite causal history—implies that there is a first cause, and our best hypothesis for what a first cause would be is God. Thus:

1. If Causal Finitism is true, God exists.

But I think one can also argue in the other direction:

2. If God exists, Causal Finitism is true.

Aquinas wouldn’t like this since he thought that God could create a per accidens ordered backwards-infinite causal series.

In this post, I want to sketch an argument for (2). The form of the argument is this.

3. God cannot create a sequence of beings ..., A₋₃, A₋₂, A₋₁, A₀ where each being causes the next one.

4. If God cannot create such a sequence, such a sequence is impossible.

5. The best explanation of the impossibility of such a sequence is Causal Finitism.

Claim (4) comes from omnipotence. Claim (5) is I think the weakest part of the argument. Causal Finitism follows logically from the conjunction of two theses, one ruling out backwards-infinite causal chains and the other ruling out infinite causal cooperation (a precise statement and a proof is given in Chapter 2 of my Infinity book). But I am now coming to think that there is a not crazy view where one accepts the anti-chain part of Causal Finitism but not the anti-cooperation part. However, (a) the main cost of Causal Finitism come from the anti-chain part (the anti-chain part is what forces either a discrete time or a discrete causal reinterpretation of physics), (b) there are significant anti-paradox benefits to maintaining the anti-cooperation part, and (c) the theory may seem more unified in having both parts.

Now let’s move on to (3). Here is an argument. Say that an instance of causation is chancy provided that the outcome has a probability less than one.

6. If God can create a backwards-infinite causal sequence of beings, he can create a backwards-infinite chancy causal sequence of beings as the only thing in creation.

7. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings as the only thing in creation, then there is no creature x such that God determines x to exist.

8. Necessarily, if God creates, he acts in a way that determines that something other than God exists.

9. Necessarily, if God determines that something other than God exists then there is a creature x that God determines x to exist.

10. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings, then there is a creature x such that God determines x to exist. (8,9)

11. Hence, God cannot create a backwards-infinite chancy causal sequence of beings. (7,10)

12. Hence, God cannot create a backwards-infinite causal sequence of beings. (6,11)

The thought behind (6) is an intuition about modal uniformity. I think (6) is probably the most vulnerable part of the argument, but I don’t think it’s the one Aquinas would attack. What I think Aquinas would attack would most likely be (7). I will get to that shortly.

But first a few words about (8). In theory, it is possible to determine that something exists without determining any particular thing to exist. One can imagine a being with a chancy causal power such that if it waves a wand necessarily either a bunny or a pigeon is caused to exist, with the probability of the bunny being 1/2 and the probability of the pigeon being 1/2. But God is not like that. God’s will is essentially efficacious and not chancy. God can play dice with the universe, but only by creating dice. Thus, if God wanted to ensure there is a bunny or a pigeon without ensuring which specific one exists, he would have to create a random system that has chancy propensities for a bunny and for a pigeon and that must exercise one of the two propensities.

In fact, I think divine simplicity may imply this. For by divine simplicity, any two possible worlds that differ must differ in something outside God. Now consider a world w₁ where God determines a bunny to exist, and a world w₂ where God merely determines that a bunny or a pigeon exists and in fact a bunny is what comes about. There seems to be no difference outside God between these two worlds (one might wonder about the relation of being-created: could there be an relation of being-created-chancily and being-created-non-chancily? this seems fishy to me, and suggests a regress—how are the two relations differently related to God? and do we want to multiply such relations, saying there is such a thing as being-created-chancily-with-probability-0.7?). If both worlds are possible, by divine simplicity they must be the same, which is absurd. So at least one must be impsosible. And w₂ is a better candidate for that than w₁.

That still doesn’t establish (8). For I admitted that God can play dice if he creates dice. Thus, it seems that God could determine that something exists without determining where it’s A or B or C (say) by determining there to be dice that decide whether A or B or C are produced. But on this story, God still determines there to be dice, so there is an x—a die—that God determines to exist. I think a bit more could be said here, but as I said, I don’t think this is the main thing Aquinas would object to.

Back to (7). Why can’t God create a chancy backwards-infinite causal sequence while determining some item A_n in it to exist? Well, the sequence is chancy, so the probability that A_n − 1 causes A_n given that A_n − 1 exists is some p < 1. But, necessarily, if one creature causes another, it does so with divine cooperation (Aquinas will not disagree), and conversely if God cooperates with one creature to cause another, the one creature does cause the other. That the probability that God cooperates with A_n − 1 to cause A_n is equal to the probability that A_n − 1 causes A_n, because necessarily one thing happens if and only if the other does. Thus, the probability that God cooperates with A_n − 1 to cause A_n, given that A_n − 1 exists, is p. But p < 1, so it sure doesn’t look like a case of God determining A_n to exist!

But perhaps there is something like overdetermination, but between determination and chanciness (so not exactly over-determination). Perhaps God both determines A_n to exist and chancily cooperates with A_n − 1 to produce A_n. One problem with this hypothesis is with divine simplicity: it does not seem that there is any difference outside God between a world where God does both and God merely cooperates or concurs. But Aquinas may respond: “Yes, exactly. Necessarily, when one creature chancily causes another, God’s primary causation determines which specific outcome results. Thus there is no world where God merely cooperates.” So now the view is that whenever we have chancy causation, necessarily God determines the outcome. But suppose I chancily toss a coin, and it has chance 1/2 of heads and chance 1/2 of tails. Then on this view, I get heads if and only if God determines that I get heads. Hence the chance that God determines I get heads is 1/2. But it seems plausible that God’s determinations are not measured by numerical probabilities, and in any case that they are not measured by numerical probabilities coming from our world’s physics!

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

Metaphysical universism

Here’s a metaphysical view I haven’t seen: the fundamental obejcts (priority version) or the only objects (existence version) are universes, but there can be more than one of these. Call this metaphysical universism (as distinguished from Quisling’s philosophy).

If in fact there is only one universe, metaphysical universism extensionally coincides with monism. But even in that case, metaphysical universism is a different theory, because it has different modal implications. And if we live in a multiverse, metaphysical universism is extensionally different from monism, since monism says that the one fundamental (priority) or one and only (existence) entity is the multiverse as a whole, not the universes.

I can think of two main advantages of metaphysical universism over monism.

First, suppose there is only one universe. It is plausible that there could be another in addition to this one. Metaphysical universism embraces this possibility. Monism only says that The One could have been bigger so as to comprise two spatiotemporally disconnected regions.

Second, there is an old intuition that being and unity are connected. In a multiverse, monism violates this intuition, for in a multiverse it is the universes that have unity, not the multiverse. Indeed quantum entanglement arguments for monism in the context of a non-Everettian multiverse seem to me to point more towards metaphysical universism than monism.

On the other hand, monism has a significant advantage over metaphysical universism insofar as monism solves the problem of truthmakers of negative and universal claims by making The One be the truthmaker of all of them.

Of course, both theories are false.

The ethics of plant care

If someone devotes a significant part of their life to affectionately caring for plastic flowers, there is something wrong with them. Not so in the case of real, living plants. This points to me to the idea that life as such, and not just conscious life or the life of animals, is a valuable thing.

I don’t want to say that it’s always bad to affectionately care for artifacts. When the artifacts have an intimate and significant connection with human beings, as in the case of a chair that grandma made or a work of art, such affectionate care can make sense. But having an affectionate care for plants makes sense even in the absence of a connection to human beings.

What about microscopic forms of life? Can it make sense to fondly feed a bacterium? I think so, but I agree that the case is less clear.

Grim Toe-Cutters

Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.

Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.

This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.

Grim Reapers and logical impossibility

The main objection to the Grim Reaper paradox as an argument against infinite causal sequences is the Unsatisfiable Pair (UP) objection that notes that paradox sets up an impossible situation—and that’s why it’s impossible!

I’m exploring a response that distinguishes metaphysical and (narrowly) logical unsatisfiability. The Grim Reaper situation is not logically unsatisfiable. The UP objection (well, really, Unsatisfiable Quadruple) notes that the following cannot all be true:

1. For all n > 0, the nth reaper wakes up at 60/n minutes after 10 am and kills Fred if and only if Fred is alive.

2. Fred is alive at 10 am.

3. There are no possible causes of Fred’s death other than those described in (1).

4. There are no possible causes of Fred’s resurrection.

But all that’s needed to have these four claims hold is for each reaper to kill Fred and then have Fred causelessly come back to life before the next one kills him. And while I think causeless resurrections are metaphysically impossible, they are (narrowly) logically coherent.

In other words, for the UP objection to work, the unsatisfiability must be metaphysical, not merely narrowly logical. But this, I think, negatively affects the force of the UP objection. For instance, in my Infinity book I consider Grim Reapers with adjustable wake-up times, and I note that for some wake-up time settings (say, the nth reaper wakes up 60/n minutes before noon) there is no paradox, and I ask what metaphysical force prevents the wake-up time settings from being the paradoxical ones. Daniel Rubio in a review of the book responds (in the context of a parody) that “no metaphysical thesis is required to explain this impossibility; the fact that it would lead to a contradiction is enough.” But in fact a metaphysical thesis is required to explain the impossibility, since there is no contradiction (in the narrowly logical sense) in (1)–(4).

Perhaps this is not a big deal. After all the metaphysical thesis here, that causeless events are impossible, is one that I do accept. But nonetheless it is a metaphysical thesis, as such on par with causal finitism, and hence when we consider the explanation of the impossibility of the Grim Reaper story and the impossibility of various other of the causal paradoxes that I discuss, there is something appealing about seeing the case as nonetheless offering support for causal finitism, which explains all of them, while the thesis about causeless events being impossible does not.

Unreliable Grim Reapers

As usual, Fred is alive at 10 am, and there is an infinite sequence of Grim Reapers, where the nth has an alarm set for 60/n minutes after 10 am, and if the alarm goes off, it checks if Fred is dead, and swings its scythe at Fred if and only if Fred is alive. But here’s the twist. These Grim Reapers are unreliable killers. The probability that the nth Reaper’s swing would succeed in killing Fred is 1/n^p, where p is some positive real number, the same for each Reaper, and independently of all other relevant events.

Here’s the fun thing. It seems possible for Fred to survive the whole ordeal. All it takes is for every Grim Reaper to fail at killing Fred. Nothing absurd happens then. Moreover, it seems this isn’t the only way for absurdity to be avoided in this case. We could also suppose that the nth Reaper kills Fred, while Reapers n + 1, n + 2, … all fail.

Suppose we adopt what seems the best alternative to Causal Finitism, namely the Inconsistent Pair response to the original Grim Reaper paradox, which says that the reason the original paradox is impossible is simply because it embodies an Inconsistent set of propositions—some Reaper has to kill Fred and none can. If that’s what’s wrong with the original Grim Reaper paradox, then it seems we have to accept my Unreliable Reaper story as possible.

But things are a little bit more complicated. The only way to avoid paradox in the Unreliable Reaper story is if there is some n ≥ 0 such that all the Reapers starting with Reaper n + 1 fail. But now suppose that 0 < p ≤ 1. Then the event that all the Reapers starting with Reaper n + 1 fail is less than or equal to (1−1/(n+1)^p)(1−1/(n+2)^p)(1−1/(n+3)^p)... = 0 (this is because Σ_k 1/k^p = ∞ if p ≤ 1). Thus the probability that we have avoided paradox is 0. Hence, if we have to avoid paradox, a specific zero probability event—namely, the event of paradox-avoidance—has to happen (the probability of a countable disjunction of zero probability events is zero). But if it has to happen, it can’t be probability zero, but must be probability one!

Perhaps here we bring back the Inconsistent Pair response. We say that my Unreliable Reaper story is impossible if p ≤ 1, because if p ≤ 1, then a zero probability event has probability one, which is inconsistent. No such problem occurs if p > 1. Thus, on this version of the Inconsistent Pair response, my Unreliable Reaper story is impossible if the success probability of the nth Reaper is 1/n^p for p ≤ 1 but possible if p > 1. And that’s pretty counterintuitive.

On finitistic addition

By a finite alphabet encoding of a set X, such as the real numbers, I mean a one-to-one function ψ from X to countably infinite sequences s₀s₁... taken from some finite alphabet. For instance, standard decimal encoding, with a decision whether to have infinite sequences of trailing nines or not, is a finite alphabet encoding of the reals, with the alphabet consisting of ten digits, a decimal point and a sign. Write ψ_k(x) for the kth symbol in the encoding ψ(x) of x.

A function f from Rⁿ to R is finitistic with respect to a finite alphabet encoding ψ provided that there is a function h from the natural numbers to the natural numbers such that the value of ψ_k(f(x₁,...,x_n)) depends only on the first h(k) symbols in each of ψ(x₁), ..., ψ(x_n).

This concept is related to concepts in “real computation”, but I am not requiring that the finite dependences be all implemented by the same Turing machine.

Theorem: Let X be any infinite divisible commutative group. Then addition on X is not finitistic with respect to any finite alphabet encoding.

A divisible group X is one where for every x ∈ X there is a y such that ny = x. The real numbers under addition are divisible. So are the rationals. So is the set of all rotations in the plane.

This has a somewhat unhappy consequence for Information Processing Finitism. If reality encodes real numbers in a discrete way consistent with IPF, we should not expect each real number to have a uniquely specified encoding.

Proof of Theorem: Suppose addition is finitistic with respect to ψ. Let F be the algebra on X generated by the sets of the form {x : ψ_k(x) = α}. If addition is finitistic, then for any A ∈ F, there is a finite sequence of pairs (A₁,B₁), ..., (A_N,B_N) of sets in F such that

1. {(x,y) : x + y ∈ A} = ⋃_i(A_i×B_i).

Therefore:

2. x + y ∈ A if and only if x ∈ ⋃{A_i : y ∈ B_i}.

Thus:

3.  − y + A =  ∪ {A_i : y ∈ B_i}.

Now as y varies over the members of X, there are at most 2^N different sets generated by the right hand side. Thus,  − y + A can take on only finitely many values. Hence, A has only finitely many translates.

But this is impossible. Let Z be the set of x such that x + A = A. This is an additive subgroup of X. Note that x + Z = y + Z iff x − y ∈ Z iff (x−y) + A = A iff x + A = y + A. Thus, if there are only finitely many x + A, there are finitely many x + Z. Hence X/Z is a finite group. Let n be its order. Then n[x] = 0 for every coset [x] = x + Z in R/Z. For any x ∈ X choose y such that ny = x. Then n[y] = 0, and so [x] = 0, thus Z = X. It follows that A is invariant under every translation, so it must be either ⌀ or X. Hence |F| ≤ 2, which is absurd since F is infinite as X is infinite and ψ is one-to-one.

(I got the main idea for this proof from the answer here.)

Empirical mathematics

Suppose I want to figure out a good approximation to the eigenvalues of a certain Hamiltonian involving a moderately large number of Coulomb potentials. It could well be the case that the best way to do so is to synthesize a molecule with that Hamiltonian and then measure its spectrum. In other words, there are mathematical problems where our best solution to the problem uses scientific methods rather than mathematical proof.

Information Processing Finitism, Part II

In my previous post, I explored information processing finitism (IPF), the idea that nothing can essentially causally depend on an infinite amount of information about contingent things.

Since a real-valued parameter, such as mass or coordinate position, contains an infinite amount of information, a dynamics that fits with IPF needs some non-trivial work. One idea is to encode a real-valued parameter r as a countable sequence of more fundamental discrete parameters r₁, r₂, ... where r_i takes its value in some finite set R_i, and then hope that we can make the dynamics be such that each discrete parameter depends only on a finite number of discrete parameters at earlier times.

In the previous post, I noted that if we encode real numbers as Cauchy sequences of rationals with a certain prescribed convergence rate, then we can do something like this, at least for a toy dynamics involving continuous functions on between 0 and 1 inclusive. However, an unhappy feature of the Cauchy encoding is that it’s not unique: a given real number can have multiple Cauchy encodings. This means that on such an account of physical reality, physical reality has more information in it than is expressed in the real numbers that are observable—for the encodings are themselves a part of reality, and not just the real numbers they encode.

So I’ve been wondering if there is some clever encoding method where each real number, at least between 0 and 1, can be uniquely encoded as a countable sequence of discrete parameters such that for every continuous function f from [0,1] to [0,1], the value of each parameter discrete parameter corresponding to of f(x) depends only on a finite number of discrete parameters corresponding to x.

Sadly, the answer is negative. Here’s why.

Lemma. For any nonempty proper subset A of [0,1], there are uncountably many sets of the form f⁻¹[A] where f is a continuous function from [0,1] to [0,1].

Given the lemma, without loss of generality suppose all the parameters are binary. For the ith parameter, let B_i be the subset of [0,1] where the parameter equals 1. Let F be the algebra of subsets of [0,1] generated by the B_i. This is countable. Any information that can be encoded by a finite number of parameters corresponds to a member of F. Suppose that whether f(x) ∈ A for some A ∈ F depends on a finite number of parameters. Then there is a C ∈ F such that x ∈ C iff f(x) ∈ A. Thus, C = f⁻¹[A]. Thus, F is uncountable by the lemma, a contradiction.

Quick sketch of proof of lemma: The easier case is where either A or its complement is non-dense in [0,1]—then piecewise linear f will do the job. If A and its complement are dense, let (a_n) and (b_n) be a sequence decreasing to 0 such that both a_n and b_n are within 1/2^n + 2 of 1/2ⁿ, but a_n ∈ A and b_n ∉ A. Then for any set U of positive integers, there will be a strictly increasing continuous function f_U such that f_U(a_n) = a_n if n ∈ U and f_U(b_n) = a_n if n ∉ U. Note that f_U⁻¹[A] contains a_n if and only if n ∈ A and contains b_n if and only if n ∉ A. So for different sets U, f_U⁻¹[A] is different, so there are continuum-many sets of the form f_U⁻¹[A].

Information Processing Finitism

When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. I ended up largely abandoning that approach, partly because of my qualms about the concept of information and perhaps partly because of worries about physics that I will discuss below.

But I still think the alternative, which one might call information processing finitism, is something someone should work out in more detail.

• [IPF] Nothing with finite informational content can essentially causally depend on anything with infinite informational content.

Here, informational content is by definition contingent. The “essentially” excludes cases where finite informational content depends on a finite part of something with infinite informational content. How exactly the “essentially” is spelled out is one thing I am not clear on as yet.

The main difficulty with IPF is that our physics seems to violate it. The exact current temperature in Waco depends on the exact temperature, pressure and other facts around the world yesterday. Each of the latter facts involves infinite information—temperature is quantified with a real number, and a real number contains infinite information. Note that here IPF and CF may diverge. An advocate of CF can say that the exact current temperature in Waco depends on a finite number of past events such as “yesterday particle n has parameters P”, even if the parameters P involve real numbers that have infinite energy.

One way to escape this difficulty is to assume that our fundamental physics is actually discrete, and the real numbers in our equations are just an approximation. But I don’t want to stick my neck out so far.

Let’s see if we can make IPF work out with a continuous dynamics. We can suppose that metaphysically speaking, an entity’s having a real-valued parameter is constituted by the entity’s having an infinite sequence of discrete parameters, which parameters are more ontologically fundamental than the real-valued parameter.

For instance, by a one-to-one mapping we can assume our real number is strictly between zero and one, and then define it as an infinite decimal sequence 0.b₁b₂..., specified by an infinite sequence of digits. Unfortunately, then, we have some severe restrictions on what kind of dynamics we can have if we require that each digit of the output depend only on a finite number of digits of the input. For instance, multiplication by 3/4 cannot be defined, because to know whether f(x) starts with 0.24 or 0.25, you’d have to know whether x < 1/3 or x ≥ 1/3, and if the input is 0.333..., then you can’t tell from a finite number of digits which is the case. This kind of problem will occur with any other base.

It would be really nice to find some way of encoding a real number as an infinite sequence of discrete parameters each of which takes on a fixed finite range that escapes this kind of a problem. I am pretty sure this is impossible, but am too tired to prove it right now.

But there is another approach. We can have non-unique (many-to-one) encodings of reals. Here is one such approach, probably not the most natural one. Consider sequences of natural numbers n₁, n₂, ... such that for all k we have n_k ≤ 2k and there exists a real number x between 0 and 1 inclusive with the property that |x−n_k/2k| ≤ 1/k. Say that such a sequence encodes the real number x. In general, there will be more than one sequence encoding x by this rule.

Then if f is a function from [0,1] to [0,1], if we have a sequence n₁, n₂, ... encoding the real number x, to generate an acceptable kth term in a sequence encoding f(x), it suffices to know f(x) to within precision 1/2k, and if f is continuous, then we can do that by knowing a finite number of terms in a sequence encoding x (this is because every continuos function on [0,1] is uniformly continuous).

So any continuous dynamics from [0,1] to [0,1] can be handled in this way. The cost is that fundamental reality has degrees of freedom that are unimportant physically—for fundamental reality distinguishes between different sequences encoding the same real x, but the difference has no physical significance.

I don’t know if there is a way to do this with a unique encoding.

Mereology, plural quantification and free lunches

It is sometimes claimed that arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a new way of talking without any deep philosophical (or at least metaphysical) commitments.

I think this is false.

Consider this Axiom of Choice schema for mereology:

1. If for every x and y such that ϕ(x) and ϕ(y), either x = y or x and y don’t overlap, and if every x such that ϕ(x) has a part y such that ψ(y), then there is a z such that for every x such that ϕ(x), there is common part y of x and z such that ψ(y).

Or this Axiom of Choice schema for pluralities:

2. If for all xx and yy such that ϕ(xx) and ϕ(yy) either xx and yy are the same or have nothing in common, then there are zz that have exactly one thing in common with every xx such that ϕ(xx).

If arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a handy way of talking, then whether (1) or (2) is correct is just a verbal question.

But (1) and (2) respectively imply mereological and plural Banach-Tarski paradoxes:

3. If z is a solid ball made of points, then it has five pairwise non-overlapping parts, of which the first two can be rigidly moved to be pairwise non-overlapping and compose another ball of the same size as z, and the last three can likewise be so moved.

4. If the xx are the points of a solid ball, then there are aa, bb, cc, dd and ee which have nothing pairwise in common and such that together they make up xx, and there are rigid motions that allow one to move aa and bb into pluralities that have nothing in common but make up a solid ball of the same size as xx and to move cc, dd and ee into pluralities that have nothing in common and make up another solid ball of the same size.

Conversely, assuming ZF set theory is consistent, there is no way to prove (3) and (4) if we do not have some extension to the standard axioms of mereology or plurals like the Axiom of Choice. The reason is that we can model pluralities and mereological objects with sets of points in three-dimensional space, and either (3) or (4) in that setting will imply the Banach-Tarski paradox for sets, while the Banach-Tarski paradox for sets is known not to be provable from ZF set theory without Choice.

But whether (3) or (4) is true is not a purely verbal question.

One reason it’s not a purely verbal question is intuitive. Banach-Tarski is too paradoxical for it or its negation to be a purely verbal thing.

Another is a reason that I gave in a previous post with a similar argument. Whether the Banach-Tarski paradox holds for sets is not a purely verbal question. But assuming that the Axiom of Separation can take formulas involving mereological terminology or plural quantification, each of (3) and (4) implies the Banach-Tarski paradox for sets.