Thursday, February 1, 2024

Fusion and the Axiom of Choice

Assume classical mereology. Then for any formula that has a satisfier, there is a fusion of all of its satisfiers. More precisely, if ϕ is a formula with z not a free variable in ϕ, then the universal closure of the following under all free variables is true:

  1. xϕ → ∃zFϕ, x(z)

where Fϕ, x(z) says that z is a fusion of the satisfiers of ϕ with respect to the variable x (there is more than one account of what exactly the “fusion” is). This is the fusion axiom schema.

Stipulate that a region of physical space is a fusion of points.

Question: Is there a nonmeasurable region of (physical) space?

Assuming the language for formulas in our classical mereology is sufficiently rich, the answer is positive. For simplicity, suppose that physical space is Euclidean (the non-Euclidean case is handled by working in a small neighborhood which is diffeomorphic to a neighborhood of a Euclidean space). Let ψ be the isomorphism between the points of physical space and the mathematical space R3. Let ϕ be the formula ψ(x) ∈ y. Applying (1), we conclude that for any subset a of R3, there is a set of points of physical space that correspond to a under ψ. If we let a be one of the standard nonmeasurable subsets of R3, we get an affirmative answer to our question.

But now we have an interesting question:

  1. What grounding or explanatory relation is there between the existence of a nonmeasurable region of physical space and the existence of a nonmeasurable subset of mathematical space?

The two simplest options are that one is explanatorily prior to the other. Let’s explore these.

Suppose the existence of a nonmeasurable physical region depends on the existence of the nonmeasurable set. Well, it is a bit strange to think of a concrete object—a region of physical space—as partly grounded in the existence of a set. This doesn’t sound quite right to me.

What about the other way around? This challenges the fairly popular doctrine that complex things entities are a free lunch given simples. For if the existence of the nonmeasurable region is prior to the existence of an abstract set, it seems that we actually have quite a significant metaphysical “effect” of this complex object.

Moreover, if the existence of the nonmeasurable region is not grounded in the existence of nonmeasurable set, whether or not there is grounding running the other way, we have a difficult question of why there is in fact a nonmeasurable region. Without relying on nonmeasurable sets, it doesn’t seem we can get the nonmeasurable region out of the axioms of classical mereology. It seems we need some sort of a mereological Axiom of Choice. How exactly to formulate that is difficult to say, but one version that is enough for our purposes would be that given any formula ρ(x,y) that expresses a non-empty equivalence relation on the simples satisfying ϕ, there is an object z such that if ϕ(x) then there is exactly one simple x′ such that ρ(x,x′) and x is a part of z, and every object that meets z meets some simple satsifying ϕ.

But my intuition is that a mereological Axiom of Choice would badly violate the doctrine that complex objects are a free lunch. If all we had in the way of complex-object-forming axioms were reflexivity, transitivity and fusion, then it would not be crazy to say that complex objects are a fancy way of talking about simples. But the “indeterministic” nature of the Axiom of Choice does not, I think, allow one to say that.

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