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: 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) which implies (C).

So, 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.

Psychological continuity and my first psychological state

Assume that human psychological states depend on brain states.

On psychological continuity theories of personal identity, if a time t is such that I did not have a psychological state at t, then I did not exist at t. Now consider the first time t₁ at which I have a psychological state (for simplicity, I am assuming that the set of times at which I have a psychological state is closed at the bottom; otherwise, the argument gets more complicated). Let ψ₁ be that psychological state. I think—this is perhaps a bit controversial, but very intuitive—that whether I have a psychological state at a given time does not depend on what happens after that time. But the brain states that psychological states depend on are, plausibly, unavoidably diachronic.

Since my having ψ₁ at t₁ cannot depend on how things are after t₁, it follows that it must depend on the brain state at and before t₁.

At first sight, this looks like an obvious contradiction for the psychological continuity theorist. Since I had no psychological states before t₁, I didn’t exist before t₁, and so I couldn’t have had any brain state before t₁.

But there is a way out. While I couldn’t exist before t₁, the brain that is now mine could have. What the psychological continuity theorist can say is that my psychological state at t₁ depends largely on the state that this brain had before I came into existence (together with the instantaneous state the brain has at t₁, when I do exist). This is somewhat counterintuitive.

Another way out, of course, is to drop the physicalism, and allow for neurally-ungrounded psychological states.

Parts and priority

Schaffer says that pluralists think that parts are ontologically prior to their wholes. But this seems wrong, even as restricted to bottom-up pluralists who think the fundamental realities are microphysical.

For consider that (assuming the materialism that bottom-up pluralists typically do) I am a part of the Milky Way Galaxy (both a part-at-a-time and an atemporal part). But it seems quite wrong to say that I am ontologically prior to the Milky Way. My fundamental particles may be ontologically prior to the Milky Way, but I am not.

Further, many bottom-up pluralists accept classical mereology. But now while it seems right, if I put myself in the mindset of a typical non-Aristotelian metaphysician, to say that my atoms, molecules, cells, and organs are prior to me, it seems wrong to think that “weird scattered parts” are prior to me. Imagine for instance ordering all my molecules from bottom to up, and then take the fusion of all the even-numbered molecules. That fusion exists and is a part of me by classical mereology, but it doesn’t seem like the sort of thing that’s prior to me. It seems to be something that emerges from my simples in parallel with me at best.

If this is right, then what are the priority relations between objects for bottom-up pluralists? For unrestricted compositionalists, I think the natural suggestion is this: The simples are prior to all the wholes made of them, and that’s the only priority relation there is. This has some counterintuitive results. For instance, it seems plausible that my molecules are prior to me, or that the individual people of France are prior to France. That’s probably a bullet that the unrestricted compositionalist should bite. For one of the primary arguments for unrestricted composition is worries about vagueness and continuous series, and as soon as we start inserting some additional priority relations, we are likely to get analogous worries. But it is an uncomfortable bullet to bite, because it undercuts many of the intuitive motivations for bottom-up views, such as that the Lego blocks are prior to the Lego sculpture.

A related point applies to top-down priority monistic views, at least assuming classical mereology. Maybe the Milky Way is prior to me, but it doesn’t seem right to say that the fusion of all people whose height in inches is even is prior to me (my height in inches is even). So the top-down priority monist seems to need a complicated story of when a whole is prior to its parts.

If we have highly restricted mereology, it becomes easier to say how the priority relation runs. If, say, van Inwagen is right and all we have are simples and organisms, then we can say that parts are always prior to wholes, and be done with it, as the only cases will be that simples are prior to cells and cells to multicellular organisms. Or we can reverse it and say that multicellural organisms are prior to their cells, cells are prior to their cells (while unaffiliated cells as well as unaffiliated simples are fundamental). In either case, the story is neat and simple.

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 circularity in psychological theories of personal identity

On psychological theories of personal identity, the diachronic identity of a person is defined in terms of psychological properties. But for any human person x who possesses a psychological property ψ at time t, x’s possession of ψ at t grounded both in how x is at t and how x was at times other than t (typically, at least times shortly before t). Thus, x’s possession of ψ presupposes x’s diachronic identity. I expect this is somewhere in the literature, but in any case it seems to me to be a serious problem.

Life, teleology, and the Holy Spirit

I suppose I think all substances have teleology. For as Rob Koons and others have noted, a causal power always has the immediate effect of its exercise as its telos, and substances seem to always have causal powers.

But I think we can make a distinction between having the immediate effect of one’s causal power’s exercise as a telos and having a further telos for the causal power. In the latter case, the substance is exercising the causal power as a means to an end—whether intentionally or not. This seems to happen for all living things. It is not merely the telos of a bacterium to seek food, but to seek food in order to ingest it. Thus the seeking behavior is ordered in the bacterium’s teleology to the ingesting, and the ingesting is presumably ordered to further goods of bacterial functioning. On the other hand, when an electron repels another electron, the repulsion is the only telos the electron is pursuing. Of course, someone else (God or an electrical engineer, say) may have a further purpose for that repulsion, but that doesn’t seem to be the electron’s telos.

It seems that we have a difference between living and non-living substances, then: non-living substances have only immediate teleology, while living substances have both immediate and mediate teleology.

What about God? Well, initially it seems easy to say that, of course, God engages in means-end reasoning in creation all the time (e.g., making fruit for primates like us to eat), and hence has mediate teleology. But if mediate teleology is essential to life, it is troubling to identify God’s mediate teleology with anything done within creation, since God does not need to create.

Does God in himself have mediate teleology? One might see a positive answer within a certain version of western trinitarian theology: The Father generates the Son in part in order that the Holy Spirit might proceed through the Son. Thus there is a mediate teleology within God’s trinitarian being.

If this is the right answer, then we have a way in which the filioque is needed for God to be alive, and in which the Holy Spirit is life-giver—as per the Nicene Creed—not just to creatures, but to the blessed Trinity.

Perhaps there are other trinitarian things one can invoke besides the filioque. The Father generates the Son in order that the Son might love him back. But in any case, it is difficult to see how there can be essential mediate teleology in God apart from something like the doctrine of the Trinity.

Do we thus have another argument for the multiplicity of persons in God?

1. God is simple.

2. God does not have to create.

3. Life essentially involves mediate teleology.

4. God is essentially alive.

5. If God is essentially alive and simple and does not have to create, the only way for him to have essential mediate teleology requires a multiplicity of hypostases.

All that said, I am not 100% convinced that we should think of mediate teleology as an essential feature of life.

Molinism and trials

A standard thing that theists have long said is that some difficult situations are “trials” where you have right and wrong options, and where you have an opportunity to grow closer to God through the right choice.

Suppose Molinism is true. Now suppose Curley the crooked politician has failed a trial by accepting a bribe. On Molinism, God knew that if Curley were to be offered a bribe in these circumstances, Curley would fail the trial. Now, presumably, God could also have tweaked the circumstances of the trial in different ways. For instance, he could have made sure that Curley had a headache at the time, or had noticed the beauty of God’s creation while walking to work through the city park, or had tripped and bruised his knee, etc. All these things would affect the psychological state that Curley was in while being bribed. Now, just as we would expect re-runs of Curley’s choice (with a memory wipe between re-runs) to result sometimes in the acceptance of the bribe and sometimes in its rejection in a way that looks random (as per van Inwagen’s rewind argument), likewise we would expect the results of Curley’s counterfactual choice to vary between the variant circumstances. In particular, we would expect some of the variant circumstances to be such that Curley would have passed the test.

Now, given that, we have a puzzle: Why didn’t God modify the circumstances in such a way? It seems like it would have been easy to do so without interfering in anyone’s freedom.

Of course, this is just a species of the problem of evil. But there is a difference between this and some other problems of evil. If we ask why God might have allowed someone to suffer, we can answer by saying that God might have planned to draw a greater good out of the suffering. But it is problematic to say that God planned to draw a greater good out of Curley’s sin. For to say that appears to imply that God intended Curley’s sin as a means to an end. But it is wrong to intend that someone sin, even as a means to much greater good.

What we need for theodicy here is a significant good achieved not by Curley’s sin, but by something independent of Curley’s sin. Let’s say that Curley would have rejected the bribe if he had a headache. Then God needs a significant reason not to have given Curley a headache, and a reason not dependent on the goods God would draw out of Curley’s sin. Since a headache is a minor bad, the good of avoiding a headache is not enough to do the job, it seems.

One solution is that God has good reason to minimize his interference in nature, even if the interference is really minor.

Another solution, and I both like it and yet feel it’s a bit of a cheat, is for the Molinist to embrace what I called the “Undefined” answer to the question of correlations between conditional of free will. On that answer, we cannot say that it is likely that in one of the variant circumstances Curley would have chosen differently.

Of course, the best answer is to abandon Molinism for mere foreknowledge.

An argument against the conjunction of perdurantism and physicalism

1. I think a thought.

2. For each thought I think derivatively, there is a unique entity that thinks it non-derivatively.

3. On standard perdurantism, all my thoughts are derivative, and derivative precisely from temporal parts thinking these thoughts.

4. Given physicalism, there is no unique temporal part thinking any of my thoughts non-derivatively.

5. So, at least one of standard perdurantism and physicalism is false.

There are two controversial premises: 2 and 4.

Regarding 2, there are two alternatives. Either there is an infinite regress of entities engaged in derivative thought behind my thought, or else my thought is derived from two or more entities non-derivatively thinking my thoughts. The infinite regress story is very implausible. And the two primary non-derivative thinkers story seems highly counterintuitive.

There are two arguments for 4. First, none of my thoughts are thought by an instantaneous temporal part: all my thoughts take time. But at least given physicalism, it seems obvious that the temporal boundaries of my thoughts are going to be sufficiently fuzzy that there won’t be a privileged temporal part for any of my thoughts—a temporal part that’s a picosecond longer or shorter would be just as good a non-derivative candidate for where I derive my thought from.

Second, temporal parts depend on reference frames, and I don’t have a privileged reference frame, so I don’t have a privileged temporal part non-derivatively thinking my thought.

A difference among relations

I want to get at a certain difference between relations like love and hate, on the one hand, and relations like causation and parthood, on the other.

One way to see the difference is that there are significant formal constraints on the distribution of causation and parthood, but no formal contraints on the distribution of love and hate. For instance, there cannot be cycles of causation and parthood must be transitive. But there are no such constraints on love and hatred: any combination of loves and hatreds is possible for ordinary human beings.

Another way to see what seems to be the same difference is this. Bracketing considerations of God’s goodness, God can just directly freely insert and remove instances of love and hatred within a bunch of people without changing anything else. But it seems that God can’t just directly freely insert and remove instances of causation and parthood within a bunch of entities. It seems he can’t, for instance, reverse the direction of causation between my father and myself, or will me to be a part of my heart.

I find it a very mysterious feature of causation and parthood that it has such formal constraints.

The lack of free recombinability for a relation is sometimes explained by the relation being non-fundamental. For instance, “darker than” must be transitive, but the reason is that its obtaining depends on more fundamental facts, such as how dark the two relata are, and this dependence induces the logical features of the relation. It could turn out that causation and parthood do depend on more fundamental things, though this does not seem very plausible. But at the same time, it seems pretty plausible that whatever more fundamental things causation and parthood depend on, these will include relations, and relations of a sort that have the non-recombinability features of causation and parthood.

Anyway, it seems to me that there is some fundamental difference between fundamental relations that have no formal constraints and fundamental relations that do have formal constraints, and I don’t know what accounts for it. I feel that this difference is a mere indicator of some deeper difference in kind. Causation and parthood “look” very different from love and hatred.

Causal well-foundedness

I have defended causal finitism at length, mainly (but not only) as a way of getting out of paradoxes.

I want to think about a weaker thesis than causal finitism today. Causal finitism says that there is no item that has infinitely many things causally prior to it.

A weaker thesis is that causal priority (which I will assume to be transitive) is causal well-foundedness, namely that any set S of items contains a minimal item—an item that has nothing in S causally prior to it. Assuming the Axiom of Choice (and its applicability to sets of “items”), this is equivalent to saying there are no backwards infinite regresses ... < A₋₃ < A₋₂ < A₋₁ < A₀, where < indicates causal priority.

The arrangement of the Grim Reapers appears not to be well-founded: each one’s actions respond to the actions of the one before. So causal well-foundedness seems to rule out that paradox just as well as causal finitism.

At the same time, just as causal finitism implies the existence of a first cause (and hence yields a kalaam argument), so does causal well-foundedness: just take the set of all items that are causally prior to something, and a minimal item will be a first cause.

So far, no differences.

But alas once we start looking at other paradoxes, differences arise. Thomson’s lamp is unaffected by causal well-foundedness: the story is well-founded. Similarly, it seems one can shuffle an infinite deck of cards using a well-founded supertask. Likewise, there are paradoxes arising from from infinitely many things affecting a single result without any chain, like the paradox where infinitely many people roll independent and fair dice without seeing the results, and an angel announces that all but finitely many got sixes.

So you get what you pay for. Causal well-foundedness is a weaker assumption than causal finitism, but it also rules out fewer paradoxes.

Beginnings and causation

Some people think that it’s way more obvious that a contingent thing that has a beginning has to have a cause than that a contingent that has no beginning has to have a case.

Here’s one reason to be suspicious of this intuition. Whether something contingent needs to have a cause should not depend on its orientation. But whether something has a beginning is a function of its orientation in spacetime. Take an object that has extends backward in time to infinity but forwards only finitely, and hence has no beginning. Consider this as a four-dimensional object. You can then rotate it so it extends forwards in time to infinity but only a finite distance backward. You’ve merely changed its orientation, but now the object has a beginning.

This is very much a four-dimensionalist argument, of course.

Schaffer and well-foundedness of priority

Jonathan Schaffer in his famous “Monism: The Priority of the Whole” paper assumes that priority relations are well-founded. The standard definition of well-foundedness in this context would say that every non-empty set has an element that has nothing prior to it in that set.

Imagine a world w with a continuously subdivided entity E made of pointlike simples (like a real line is made of points in standard mathematical constructions) that lies between the 0 and 1 coordinates on the x-axis, and suppose that for any real number a strictly between 0 and 1 there is the part E_a of E that is composed of all the simples with x-coordinate less than a. Assume further that either proper parts are always prior to wholes or wholes are always prior to proper parts.

In either case, we don’t have well-foundedness. For let C be the chain consisting of all the objects E_a for 0 < a < 1. Since E_a is a proper part of E_b if and only if a < b, C does not have a minimal element on either of the two priority relations (the part-before-whole and the whole-before-part one).

Schaffer’s justification for assuming well-foundedness is that without it, we could have a world with no basic entities. But notice that although w violates well-foundedness, it could still have basic entities. For it could be an atomic world—E itself is made of pointlike simples—in the case of the part-to-whole priority relation and it could be a priority monist world.

Perhaps Schaffer is misusing the term “well-founded”. He glosses it as “all priority chains terminate”. This is ambiguous between two readings. On a stronger reading, any priority chain terminates inside itself: there is an item in the chain that is prior to all the other items in the chain. Assuming the Axiom of Choice, Zorn’s Lemma shows that this stronger reading is equivalent to the standard definition of well-foundedness. But on the weaker reading, the idea would be that for any priority chain C, there is an entity that is prior to all the entities in C other than perhaps itself, but without the entity being required to be itself in the chain. This is equivalent to the priority ordering being the reverse of an inductive order.

On the weaker reading, assuming the Axiom of Choice, each non-basic item has something basic prior to it. This seems to me to be what Schaffer needs for his arguments.

That said, I think there is something to be said for the standard sense of “well-founded”, even if it is incompatible with the above world with the continuously subdivided entity E and whole-to-part or part-to-whole priority. Indeed, I am now thinking that well-foundedness is a really nice condition, but assuming there are continuously subdivided entities it does rule out both kinds of priority relations. But that should be fine. And in fact Schaffer could embrace the denial of whole-to-part and part-to-whole priority. In that paper, he only commits himself to integrated wholes being prior to their parts.

Still, here's one reason this matters. Plausibly, you could have infinite upward nesting: one integrated whole as part of another as part of another ad infinitum. Schaffer thinks that wholes are prior to parts, such such infinite upward nesting would violate would-foundedness in the standard mathematical sense. But it wouldn't violate the weaker condition, as long as the infinite cosmos as a whole was prior to all of the parts.

Ontology matters

Among things that metaphysicians wonder about is the question of what things referred to in ordinary language exist: Are there tables, chairs, people, shadows, holes, rocks, particles, etc.?

When metaphysicians discuss this question, others are apt to wonder about a different question: Why does anyone care? After all, the metaphysics will admit that whether there are chairs or not, there are particles (or there is a wavefunction) that’s chairwise arranged, and that’s what we really care about, no?

Here are two reasons to care, assuming theism. All the things that exist are either God or lovingly created by God. Thus, in finding out that a certain kind of non-divine thing exists, we have found out that God has lovingly created the thing. And to find out that God has lovingly created a thing surely matters. Furthermore, anything that exists is either God or its being is a participation in God. And to find out that there is a case of participation in God surely matters.

If time is necessary, it's discrete

Say that a total order is locally finite provided that between any two elements there are at most finitely many elements. A locally finite total order is discrete: each non-extreme element has an immediate predecessor and successor.

Here’s a valid argument:

1. If time is necessary, the order structure of time cannot be problematically arbitrary.

2. The only possible order structures for time that are not problematically arbitrary are locally finite.

3. Thus, if time is necessary, it has to be locally finite, and in particular discrete.

Premise (1) is motivated by my and Josh Rasmussen’s research project on which fundamental and necessary things cannot be problematically arbitrary. But if time is necessary, it is very likely to be fundamental.

Why accept (2)? I am thinking like this. Suppose there are two times between which there are infinitely many times. The size of that infinity of times is arbitrary—is it countable, is it the continuum, is it something in between (if there is anything in between), is it something even bigger?

So, if time is necessary, the order of time is locally finite.

Up to isomorphism, the following are the only possible locally finite total orders:

1. ...,  − 3,  − 2,  − 1, 0, 1, 2, 3, ...: no beginning or end

2. ...,  − 3,  − 2,  − 1, 0: end but no beginning

3. 0, 1, 2, 3...: beginning but no end

4. 0, 1, 2, 3, ..., n (for some finite n): beginning and end.

Now a complex thesis:

4. If time is necessary, then either (i) necessarily time has structure A or B, or (ii) necessarily time has structure C or D.

Here’s why. If there were possible worlds where time has structure A or B and other possible worlds where time has structure C or D, it would be problematically arbitrary whether time starts off with an infinite past or a finite past.

Can we go further? Maybe, but I get less confident. Still, it seems like it would be weird that time could end at some point, but had to have started. Thus:

5. If time does not have to go on endlessly, there need not have been time at all.

It follows that:

6. If time is necessary, then it necessarily has structure A or it necessarily has structure C.

In other words, if time is necessary, it must be like the integers (option A) or like the natural numbers (option C).

Can we narrow it down further? Well, causal finitism suggests but does not prove that options with no beginning are problematic. That would get one down to option C. But this is a weak argument. Option A also has the disadvantage that it leaves unexplained why events in time didn’t all happen a day earlier than they did.

In any case, there is another move we can make at this point. It follows from 6 that:

7. If time is necessary, all the specifics of the order structure of time are necessary.

I am personally inclined to add:

8. The specifics of the order structure of time are not necessary.

9. So, time is not necessary.

But my intuitions for 8 aren’t that strong, so I don’t think this is a very strong argument for the contingency of time.