Quantum mechanics and points in space

There are two different pictures of space: space is made of points or space is made of regions. I will argue that quantum mechanics naturally suggests the latter view.

Consider a single-particle system. Its quantum state can be representated by a wavefunction Ψ, which is a square-integrable complex-value function. This seems to go nicely along with the idea that there is such a thing as space, and each point in space has the property of having such-and-such a value of the wavefunction Ψ. (And in a multi-particle system, tuples of points are related to a value.) But that misses a subtlety. While a square-integrable function Ψ represents a quantum state, any two square-integrable functions Ψ₁ and Ψ₂ that agree outside of a set of measure zero are taken to represent the same quantum state. The measure of a (set consisting of a) single point is going to be zero. So there is no a physical fact of the matter as to the value of the wavefunction is a particular point.

This seems to me to cohere a little better with a view on which space is built up out of extended regions rather than points. While there will be no fact of the matter as to what "the" value of the wavefunction is at a point x, for any extended (at least in the sense of being nonzero-measure) but bounded region R of space, say a ball or cube, there will be a fact about what the average value of the wavefunction over R is (functions that are square-integrable are locally integrable). Moreover, one can recover the value of the quantum state from the values of such averages over, say, all balls. (If space is potentially subdivisible but not actually subdivided, some of these balls will be potential regions of space, and the average of the wavefunction over such a ball may be a dispositional property--the average it would have if space were divided so as to have that ball as a region.)

This is not a knock-down argument against views on which space is made of points. One could say that space is made of points but deny that quantum states have values at single points. Or one could say that wavefunctions that differ on a set of zero measure represent different in principle empirically indistinguishable quantum states. The latter is, I think, unattractive. We should avoid positing an infinite number of empirically unobservable degrees of freedom in physics.

Gunky ontology and virtual points

Gunk is subdivisible into smaller parts, and these are subdivisible into yet smaller parts, and this happens ad infinitum, with no smallest indivisible parts or atoms.

But here is an interesting fact: One can introduce ersatz atoms or virtual points into a gunky ontology, given some plausible mereological axioms. Suppose that O is a gunky object. Then the set M(O) of the parts of O has a partial order ≤ where x≤y if and only if x is a part of y. Now we can say that an ersatz atom of O is any ultrafilter on O with respect to the ordering.

Thus, ersatz atoms are subsets U of M(O) such that:

1. U is a non-empty proper subset of M(O)
2. if x is in U then everything that has x as a part is also in U
3. if x and y are in U, then there is a z in U such that z≤x and z≤y
4. U is maximal: any larger subset satisfying (1)-(3) is all of M(O).

We can then say that an ersatz atom U is an ersatz part of x∈M(O) provided that x∈U.

To get the existence of ersatz atoms, we need some axioms of mereology in addition to the Axiom of Choice. Fortunately, pretty weak mereological axioms suffice:

5. parthood is a partial ordering
6. O has two parts x and y that do not overlap

As usual, two things are said to overlap provided that there is something that is a part of both.

In general, given any two parts x and y that do not overlap, there will be an ersatz atom U that is an ersatz part of x but not of y. Let's further assume the strong supplementation axiom that if y is not a part of x, then there is a z that is a part of y such that z does not overlap with x. Then whenever x≠y, there will be an ersatz atom that's an ersatz part of one but not of the other. Hence, we can identify every part of O with a set of ersatz atoms. However, given gunkiness, not every set of ersatz atoms corresponds to a part. In particular, singleton sets of ersatz atoms do not correspond to parts.

So the gunk theorist can talk as if objects were made out of atoms. Now, if we have a gunky ontology, then I think we should take the parts to be non-fundamental, and grounded in the wholes rather than the other way around on pain of a grounding regress. But if we allow non-fundamental parts in our ontology, then one may worry that the gunkiness of the ontology is merely verbal and non-substantive, dependent on the verbal decision not to talk of the ersatz atoms as real parts.