A counterexample to Weak Supplementation

The Company axiom of mereology holds that an object cannot have only one proper part. This is a weaker version of the (Weak) Supplementation axiom which holds that if an object has a proper part, it has another proper part that doesn’t overlap the first.

Say that an object is simple at a time t provided that its instantaneous temporal part at t is simple.

Suppose we accept that:

1. fundamental particles have instantaneous temporal parts at every time at which they exist

2. fundamental particles are simple at every time at which they exist

3. there is no contingent identity.

Now, suppose x is a fundamental particle that comes into existence at time t₁ and persists until t₂ > t₁. Then it has an instantaneous temporal part y at t₁. Then y is a proper part of x: it is a part of x and distinct from x. Consider a world w that is just like ours up to and including t₁, but x comes to an end at t₁ (maybe time itself comes to an end at t₁, if one wants to be extreme). Then in w, y is still a part of x. And it must still be distinct from x, since identity cannot be contingent (this argument uses the Brouwer axiom).

Since x exists only at t₁ in w, any part it has at w is a part it has at t₁. The only candidates for such parts are x and y. Thus, in w, y is the only proper part of x. So, contrary to Company (and Supplementation) x has a proper part y and no other proper part.

I am inclined to deny (a). But I am also inclined to deny Company.

Avoiding temporal parts of elementary particles

It would be appealing to be able to hold on to all of the following:

1. Four-dimensionalism.

2. Elementary particles are simples.

3. There is only kind of parthood and it is timeless parthood.

4. Uniqueness of fusions: a plurality of parts composes at most one thing.

But (1)–(4) have a problem in cases where one object is transformed into another object made of the same elementary particles. For instance, perhaps, an oak tree dies and then an angel meticulously gathers together all the elementary particles the oak ever has and makes a pine out of them, which he shortly destroys before it can gain any new particles. Then the elementary particles of the oak seem to compose the pine, contrary to (4).

One common solution for four-dimensionalists is to deny (2). Elementary particles have temporal parts, and you can’t make the old temporal parts of the oak’s particles live again in the pine. But there are problems with this solution. First, you might believe in a patchwork principle which should allow the old temporal parts to get re-used again. Second, it is intuitive to think that elementary particles are parts of the oak. But on the temporal part solution, this violates the transitivity of parthood, since the elementary particles will have temporal parts that outlive the oak. Third, the temporal parts of particles seem to be just as physical as the particles, and you might think that it’s the job of physics and not metaphysics to tell us what physical objects there are, so positing the temporal parts steps on the physicist’s toes in a problematic way. Fourth, and I am not fully confident I understand all the ramifications here, we need some kind of primitive relation joining the temporal parts of the particle into a single particle, since otherwise we cannot distinguish the case where two electrons swap properties and positions (and thereby reverse the sign of the wavefunction) from the case where they don’t.

The second common solution is to deny (3), distinguishing parthood from an irreducible parthood-at-t, and then say that trees are merely composed-at-t from elementary particles. I find an irreducible parthood-at-t kind of mysterious, but perhaps it’s not too terrible.

I want to offer a different solution, with an unorthodox four-dimensionalist Aristotelianism. Like orthodox Aristotelianism, the unorthodox version introduces a further entity, a form. And now we deny that a tree is composed of the elementary particles. Instead, we say that a tree is composed of form and elementary particles. One minor unorthodox feature here is that we don’t distinguish the parthood of a form in a substance and the parthood of a particle in a substance: there is just one kind of parthood. The more unorthodox thing will be, however, that we allow elementary particles to outlive their substances. The resulting unorthodox four-dimensionalist Aristotelianism then allows one to accept all of (1)–(4), since the pine is no longer composed of parts that compose the oak, as the oak’s form is not a part of the pine.

But we still have to account for parthood-at-t. After all, it just is true that some electron e is a part of the oak at some but not other times. And this surely matters—it is needed to account for, say, the mass and shape of the oak at different times. How do we that? Well, we might suppose that even if in our unorthodox Aristotelianism particles can outlive their substances, they get something from the substance’s form, even if it’s not identity. Perhaps, for instance, they get their causal powers from the substance’s form. (We then still need to say something about unaffiliated particles—particles not inside a larger substance. Perhaps when a particle, considered as a bit of matter, gets expelled from a larger substance and becomes unaffiliated, it gains its own substantial form. It loses that form when it joins into a larger substance again. At any given time, it gets its causal powers from the substance’s form.) So we can say that e is a part of the oak at t if and only if e gets its causal powers from the oak’s form at t.

Aristotelianism and fundamental particles

A number of contemporary Aristotelians hold to the view that when a fundamental particle becomes or ceases to be a part of an organism, the particle perishes and is replaced by another. The reasoning is that the identity of parts comes from the whole substance, so parts are tied to their substances.

I’ve long inclined to this view, but I’ve also always found it rather hard to believe, feeling that a commitment to this view is a significant piece of evidence against Aristotelianism. I think I may now have found a way to reduce the force of this evidence.

Consider one of the main competitors to Aristotelianism, a non-Aristotelian four-dimensionalism with standard mereology that includes strong supplementation:

1. If y is not a part of x, then y has a part z that does not overlap x.

Together with antisymmetry (if x is a part of y and conversely, then x = y), it immediately follows that:

2. If everything that overlaps x also overlaps y and conversely, then x = y.

Now, suppose that we have a chair made of some fundamental particles. The planks from the chair are ripped off and reassembled into a model trebuchet, with no fundamental particles added or gained. Suppose the fundamental particles are simples. Then any z that overlaps the chair had better overlap at least one fundamental particle u of the chair (the Aristotelian will deny this: it might instead overlap the form) and since fundamental particles are simples it must have u as a part. But u is also a part of the trebuchet. Thus z overlaps the trebuchet, and so anything that overlaps the chair overlaps the trebuchet. And the converse follows by the same argument. Thus, the chair is the trebuchet, which is absurd.

Here is a standard solution to this: fundamental particles are not actually simples, because they have proper temporal parts, and temporal parts are parts. What are the true simples are the instantaneous slices of fundamental particles. Thus a z that overlaps the chair in a fundamental particle u need not overlap the trebuchet as the overlap can happen in disjoint temporal parts of u.

The main competitor to Aristotelianism, thus, has to suppose that fundamental particles are actually made up of their instantaneous slices. Now suppose the Aristotelian accepts this ontology of instantaneous slices of fundamental particles, but denies that there are fundamental particles composed of the slices. Problem solved! We don’t have the problem of fundamental particles persisting beyond the substances that they are parts of, because there are no fundamental particles, just instantaneous slices of fundamental particles.

Is there much cost to this? Granted, we have to deny that there are electrons and the like. But our non-Aristotelian four-dimensionalist mereologist either also denies that there are electrons or else has to construct the electrons out of electron slices, presumably by supposing some sort of a diachronic relation R that relates slices that are to count as part of the same electron. But if we have such a relation, then we can just paraphrase away talk of electrons into talk of maximal sets of electron-slices interrelated by R. If anything, we gain parsimony.

And if we cannot find such a diachronic relation that joins up electron-slices into electrons, then our non-Aristotelian four-dimensionalist has a serious problem, too.

Naturalism and perdurance

According to one version of naturalism, the only objects that have causal influence are objects posited by a completed science.

According to perdurantism, changing persisting objects have temporal parts which have the changing properties more fundamentally.

The changing properties of objects are causally efficacious. Thus, if perdurantism is true, the temporal parts have causal influence.

But the temporal parts of changing persisting objects are not among the objects posited by our current science. For instance, our current physics holds electrons and quarks to be fundamental, i.e., not made up of other objects studied by physics. The temporal parts of electrons and quarks are thus not studied by physics. Yet they have causal influence. If the completed physics is relevantly similar to our current physics in this regard, it won’t include temporal parts, either. And hence perdurantism posits causally efficacious entities that are unlikely to be posited by a complete science.

Thus perdurantism does not appear to fit with naturalism understood as above.

An argument against time travel or for temporal parts or for internal time

Start with this plausible claim:

1. If two objects are composed of the very same particles at the same time, then they have the same shape.

But now consider a statue of a horse that is reshaped into a statue of a tree and then time-travels back to sit besides the statue of the horse. Then the statue of the horse and the statue of the tree are composed of the very same particles at the same time, and yet they do not have the same shape.

I see three ways out of this paradox.

2. Deny the possibility of time travel.

3. Subscribe to temporal parts theory and modify (1) to speak of temporal parts of particles instead of particles.

4. Distinguish external and internal time, and qualify (1) to refer to internal time.

My preference is (4).