More on minimal diachronic fusions and temporal parts

This is yet another attempt to answer the question of the conditions under which in an atomic world with classical mereology we have unrestricted minimal diachronic fusions, with neater assumptions (and in particular without atomism).

Say that an object is instantaneous provided that it exists at exactly one time.

Assumptions:

1. Classical (atemporal) Mereology.

2. Parthood-at-t is reflexive and transitive, and x is a part at t of y only if x and y both exist at t.

3. SO: If x and y exist at t, but x is not a part of y at t, then x has some part at t that does not overlap y at t.

4. I-Parthood Bridge: If x is instantaneous, then x is a part of y if and only if x is a part of y at the time at which x exists.

5. I-Temporality: Every part of an instantaneous object exists at a time.

Say that a function is admissible provided that it is a non-empty function from times to sets of objects such that all the objects in f(t) exist at t. Define minimal diachronic fusions as in Sider’s paper. Say that a t-temporal part of x is a minimal diachronic fusion of the function f(t′) = x defined only for t′ = t.

Theorem: Given Assumptions 1-5, the following are equivalent:

1. Every admissible function f has a minimal diachronic fusion.

2. For every object x and every time t at which x exists, x has a t-temporal part.

3. For every object x and every time t at which x exists, there is an instantaneous part y of x that exists at t.

Notes: Since Sider in his paper explicitly accepts 1–3. I don’t love 3. It’s hard to deny 4.

I am actually a bit worried about 5, because I am worried about the possibility of objects that have both a temporal and atemporal mode of existence, and such an object could be instantaneous (i.e., its temporal mode of existence could be at only one time) while yet having a part that is atemporal. I think I might be able to handle the case of objects with a temporal and atemporal mode of existence by strengthening the concept of a minimal diachronic fusion and a temporal part to forbid the object from having an atemporal mode of existence.

In any case, the result shows that in Sider’s setting, the crucial question is whether every object has an instantaneous part at every time at which it exists.

Proof of Theorem: Clearly (A) implies (B).

Assume (B). Let y be a t-temporal part of x. This exists at t and only at t, so it’s instantaneous. It is, further, a fusion-at-t of x itself. If we can show that y is a part of x at t, then by I-Parthood Bridge, we will have y a part of x, and thus we will have (C). Suppose y is not a part of x at t. Then by SO there is a z at t that is a part of y and does not overlap x. But this contradicts fusion-at-t.

It remains to show that (C) implies (B). Assume (C). Let S be the set of all x for which there is a t and a y in f(t) such that x is an instantaneous part of y existing at t. Let z be the (atemporal) fusion of S. We need to prove that z is a minimal diachronic fusion of f.

This requires us to show that for any time t in the domain of f, z is a fusion-at-t of f(t), and that z exists ony at the times in the domain of f.

Suppose y ∈ f(t). We must show that y is a part of z at t. Suppose it’s not. Then by SO, y has a part x at t that does not overlap z at t. By (C), let u be an instantaneous part of x that exists at t. Then u ∈ S, and so u is a part of z. By I-Parthood Bridge, u is a part of z at t and u is a part of x at t. Hence x overlaps z at t, a contradiction.

Next, we must show that everything that overlaps z at t also overlaps some y ∈ f(t) at t. Suppose x overlaps z at t. Let u be a part at t of both x and z. Let v be an instantaneous part of u that exists at t. Then v is a part of u at t by I-Parthood Bridge. Thus, v is a part of z at t by transitivity. Hence, v is a part of z by I-Parthood Bridge.

So, v overlaps some w ∈ S. Let s be a common instantaneous part of v and w (the existence of s follows from (C) and I-Temporality). By I-Parthood Bridge (using the fact that a part at a time exists at that time), s exists at t, since v exists only at t. Thus s is an instantaneous part, existing at t, of some y ∈ f(t) (since s is a part of w, and w is a part of such a y). Hence s is a part of y at t by I-Parthood Bridge. Moreover, s is a part of v at t by I-Parthood Bridge, and hence by transitivity it is a part of u at t, and likewise of x at t. Thus, z overlaps y at t.

We have shown that z is a fusion-at-t of f(t).

It remains to show that z exists only at times in the domain of f. Suppose z exists at a time t. Let x be an instaneous part of z existing at t by (C). Then x overlaps some member y of S. Any member y of S is an instantaneous object existing at some time t′ in the domain of f. Now, x and y will have to have a common instantaneous part (again using (C) and I-Temporality). That instaneous part must exist at precisely t as well as at precisely t′ by I-Parthood Bridge. Thus, t = t′, and so t is in the domain of f.

Classical mereology, atomism, temporal parts, and unrestricted diachronic fusions

[This post is obsolete. A much better version is here.]

In this post, I aim to answer the question of the conditions under which in an atomic world with classical mereology we have unrestricted minimal diachronic fusions, as Sider claims we always do. I will need to make some additional plausible assumptions.

My assumptions:

1. Classical (atemporal) Mereology.

2. Parthood-at-t is reflexive and transitive, and x is a part at t of y only if x and y both exist at t.

3. Atomism: Every object has a simple part.

4. Parthood Bridge A (from temporal to atemporal parthood): If y is a part of x at all the times at which y exists, then y is a part of x.

5. (Weak) Parthood Bridge B (from atemporal to temporal parthood): If y is a part of x while y exists at exactly one time and x also exists at that time, then y is a part of x at that time.

6. Fusion Bridge: If z is a fusion of the set S of objects, then z exists at a time t if and only if at least one of the objects in S exists at t.

7. Temporal Fusion Parthood: If x is a fusion-at-t of U and y is a fusion-at-t of V while U is a subset of V, then x is a part of y at t.

Note that, assuming (2), Temporal Fusion Parthood follows from Sider’s condition:

8. PO: If x and y exist at t, but x is not a part of y at t, then x has some part at t that does not overlap y at t.

For suppose in the setting of Temporal Fusion Parthood we don’t have x a part of y at t. Then there is a part w of x at t that does not overlap y. We must then have w overlapping at t some member z of U by the fusion-at-t condition, and hence w must overlap y at t (since y is a fusion of a superset of U), a contradiction.

Further, note that (1)–(4) imply:

9. Weak Temporality: If there are two objects, every object exists at at least one time.

For suppose y exists at no time. Then y is a part of x at every time at which y exists, for any x. Hence there is an object that is a part of every object, which violates Classical Mereology if there are two objects (e.g., use fusion and supplementation).

Say that an admissible function is a function f defined on some non-empty domain consisting of times such that f(t) is a non-empty set of objects existing at t. A minimal diachronic fusion of f is an object that exists at all and only times in the domain of f which is the fusion-at-t of f(t) for all such times in the domain.

Say that an object is temporally extended provided it exists at two or more times.

Theorem: Under the above assumptions, every admissible function has a minimal diachronic fusion if and only if there are no temporally extended simples.

Corollary: Under the above assumptions, the following are equivalent:

1. For any non-empty set T of times and any object x, there is a T-temporal part of x

2. For any time t and any object x, there is a {t}-temporal part of x

3. There are no temporally extended simples.

Here, a T-temporal part of x is a minimal diachronic fusion of f_T, x where f_T, x has domain T and f_T, x(t) = x for all x. The Theorem shows (iii) implies (i), and (i) trivially implies (ii). To see that (ii) implies (iii), suppose x is a temporally extended simple, and let t be one of the times at which x exists. A t-temporal part of x would have to exist at t and only at t, and would be a part of x at t, and hence would be a part of x by Parthood Bridge B, which would violate the simplicity of x.

Now, a philosophical comment. These results shows that given classical mereology, atomism, and some plausible assumptions, the crucial question for Sider’s argument is whether there are temporally extended simples. But this question side-steps all the stuff in his paper about vagueness and continuity. Sider’s continuity argument does nothing to establish that there are no temporally extended simples.

And a technical comment: I’d like to find a good way to weaken Fusion Bridge.

Proof of Theorem: Note that the Theorem is trivial if there is only one object, so we can assume every object exists at a time by Weak Temporality.

Suppose x is a temporally extended simple. Let f have as its domain a non-empty proper subset S of the times at which x exists, and let f(t) = x for all x. Suppose y is a minimal diachronic fusion of f. Then by Parthood Bridge A, y is a part of x. Since x is simple, y = x. But since y exists at a proper subset of the times at which x exists, this is a contradiction.

Conversely, suppose there are no temporally extended simples. Given an object y that exists at a time t, I claim that y is a fusion-at-t of all of y’s simple parts that exist at t.

For by Classical Mereology and Atomism, y is a fusion of all of y’s simple parts. By Fusion Bridge, at least one of y’s simple parts exists at t since y exists at t. Suppose z is one of these simple parts. Then z exists at t and only at t as there are no temporally extended simples. Hence, by Parthood Bridge B, z is a part of y at t. Next suppose that z and y overlap-at-t. Let x be a common part at t. Then x is a fusion of its simple parts by Classical Mereology and Atomism, and hence has a simple part w that exists at t by Fusion Bridge since x exists at t. Then w is a part of y at t by Parthood Bridge B (since w exists only at t). Then w is a part of y at t as well as of z at t (by transitivity of parthood-at-t), and so z overlaps w at t, while w is a simple part of y that exists at t.

Now suppose f is admissible. Let S be the set of all simples x such that there is a t and a y ∈ f(t) such that (a) x exists at t and (b) x is a part of y. By Classical Mereology, let z be the fusion of S.

I claim that z is minimal diachronic fusion of f. First, note that z exists at all and only the times in the domain of f. Let t be a time in the domain of f. Fix any object y in f(t). By Classical Mereology and Atomism, y is a fusion of simples. By Fusion Bridge, at least one of these simples, say x, exists at t. Then x is in S. Hence z exists at t by Fusion Bridge. Conversely, suppose that t is a time not in the domain of f, and suppose z exists at t. By Fusion Bridge, one of the simples in S exists at t. But each of the simples in S exists at a time in the domain of f. Thus, that simple must be temporally extended, and we assumed there aren’t any such.

Hence, we have shown that z at exists at all and only the times in the domain of f. It remains to show that at any time t in the domain of f, z is a fusion-at-t of f(t). Fix any y that is in f(t). We must show that y is a part of z at t. Now, z is a fusion-at-t of its simple parts existing at t, and likewise y is a fusion-at-t of its simple parts existing at t. But any simple part of y that exists at t is a simple part of z by the construction of z, so by Temporal Fusion Parthood, y is a part at t of z.

Next, suppose y overlaps z at t. Thus, y and z have a common part x at t. Then x is a fusion of its simple parts by Classical Mereology and Atomism, and hence there is a simple part w of x that exists at t by Fusion Bridge. Then w is a part of z. Since z is the fusion of S, there must be some member of S that overlaps w, and since w is simple, w is a member of S. Hence w is a simple part existing at t′ of some u in f(t′). Since w is not temporally extended, t = t′, and so w is a simple part existing at t of some u in f(t). By Parthood Bridge B, w is a part of u at t, and likewise w is a part at t of x. Hence w is a part at t of y by transitivity of parthood-at-t, and so y and u overlap at t, which completes our proof.

A counterexample to Sider's argument for temporal parts

Sider famously argues for the existence of arbitrary temporal parts by arguing for the existence of arbitrary minimal diachronic fusions. A minimal diachronic fusion of a function f from times to non-empty sets of objects such that all the objects in f(t) exist at t is an object x that exists at all and only the times in the domain of f, and at any t in that domain is a fusion-at-t of all the objects in f(t). Here, an object x is a fusion-at-t of the objects in a set S provided that every member of S is a part of x at t, and every part of x at t overlaps at t some member of S.

The argument is a continuity argument starting with the “continuous series” assumption:

• if not every function f as above has a minimal diachronic fusion, there will be a continuous series of cases starting with one where there is a minimal diachronic fusion and ending where there isn’t one.

Then Sider argues that this results in an implausible abrupt cut-off in whether there is a minimal diachronic fusion.

But the continuous series assumption need not be true. Consider a world consisting of a single temporally extended simple, x, that exists at all times from 0 to 2. In this world, u is a part of v at t if and only if t is in the interval from 0 to 2 and u = v = x.

Now, let f₀ have an interval from 0 to 1 as its domain, and let f₀(t) = x for t in that interval. Then there is no minimal diachronic fusion of f₀, because the only object there is is x, and x exists at times outside the interval from 0 to 1. Next, let f₁ have the interval from 0 to 2 as its domain, and let f₁(t) = x for all x. Then x is a minimal diachronic fusion of f₁.

Is there a continuous series of cases between the two? The best candidate is to consider the functions f_u which has as its domain the interval from 0 to 1 + u, with f_u(t) = x for all x. It seems like this is a continuous series of cases. But it’s not. For in a continuous series of cases, nearby cases are very similar. But in this case, nearby cases are not very similar: for f_1 − δ for small positive δ differs very significantly from f₁, because the domain of f₁ has the special property that it includes all the times at which any of the objects in the range of f₁ exist, while the domain of f_1 − δ does not have that special property.

More generally, consider four-dimensional atomic worlds with temporally extended simples and classical atemporal mereology. The objects of these worlds are fusions of all non-empty sets of these temporally extended simples. A fusion of simples exists at all the times at which any of its simples do. Suppose further that x is a part of y at t provided that x exists at t, y exists at t, and all the simples that exist at t and are parts of x are parts of y. It is easy to verify that this parthood-at-t relation satisfies all the formal conditions in Sider’s footnote 10.

On this story, there will be minimal diachronic fusions only for functions f whose domain is a union of the sets of existence-times for simples. For instance, if all the simples come in and out of being at integer times, there will be minimal diachronic fusions only for functions f whose domain is a union of intervals of time with integer values.

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).