Explanatory parsimony versus theoretical parsimony

Like the previous Ptolemaic model, Copernicus’ heliocentric model of the solar system required epicycles to predict the astronomical observations of the positions of the planets in the night sky. (The reason was that Copernicus’ orbits were circular instead of elliptical, and I suppose he needed epicycles to simulate the ellipticality.) Indeed, by Gingerich’s count, the Copernican system had more epicycles than the Ptolemaic.

What advantage did the Copernican theory have over the Ptolemaic, then?

The Copernican theory did not need epicycles to predict the qualitative movements of the planets in the night sky, with the famous retrograde motion where sometimes planets move from night to night in the opposite of their usual direction. Such qualitative features are nicely accounted for by the fact that the planets, including earth, are all orbiting the sun, and thus sometimes the earth overtakes a slower outer planet which thus appears to be moving backwards. On the other hand, Ptolemaic theory is both quantitatively and qualitatively wrong without epicycles: it needs epicycles to predict the existence of retrograde motion.

We think of the Copernican theory as superior due to its greater simplicity. But the theory as a whole is not simpler. However, it provides a simpler explanation of a puzzling or significant phenomenon.

This suggests an interesting theoretical virtue, which we might call not just parsimony, but explanatory parsimony, where the theory offers a particularly parsimonious explanation of one or more specific phenomena against a competitor, even if there happens to be little difference in the theories’ overall parsimony as needed for explaining all the phenomena, and even if there is less overall parsimony, as Gingerich claims was the case for Copernicus.

Take a more human case. We could explain the assassination of JFK in terms S₁, a theory about Oswald working alone, or we could explain it in terms of, S₂, where Oswald is working together with the CIA. The non-conspiracy explanation is more parsimonious: it involves two persons—Oswald and JFK—as opposed to many more. But notice that neither S₁ nor S₂ is capable of explaining all the humanly significant phenomena of 1963. To that end, both theories need to be enriched to include the activities of all of the 3.2 billion people of 1963. Let T₁ and T₂ be such respective enrichments of S₁ and S₂. These are vast theories.

Is T₁ more parsimonious than T₂? I doubt it. After all, the people at the CIA who, according to S₂, were working with Oswald were still doing something according to T₁, and indeed something that explains events of human significance. This activity is included in T₁.

Theory T₁ superior to T₂ not in terms of overall parsimony—or even the overall parsimony of the explanatorily fundamental posits—but in terms of how large a portion of (or approximation to?) the theory is needed to explain the specific events around the assassination of JFK. On T₁, a much smaller theory (with primary focus on two persons) S₁ is sufficient for the assassination, while on T₂, a much larger portion S₂ is needed.

If this is right, then we like focused explanations to be neater and smaller, even if the overall story is equally complex.

Here is an interesting application. Physicists have a strong preference for local theories, in the broad sense of the word according to which the behavior of a system depends only on what is happening near the system. But notice that in practice locality is often an approximation. Each system is affected to some degree by everything in its past light cone. To get a full explanation of the system, we need the whole past light cone. But we still have locality, because we get explanations of how the system works to a high degree of approximation in terms of how things are nearby, because all of the forces of physics drop off with distance. A preference for locality is thus a special case of a preference for explanatory parsimony: we want the approximate behavior of the system to have a more parsimonious explanation, even if the fully precise behavior needs a profligate one.

The general feature thus is this. It counts in favor of a theory if a puzzling or significant coarse-grained feature of our observations is explained in a more parsimonious way according to the theory, even if there is no parsimony advantage with respect to finer grained observations.

From a Bayesian point of view, this translates into a view on which we should have higher priors for theories that explain more of puzzling or significant phenomena with less.

Is the anthropocentrism in the notion of “puzzling or significant phenomena” a problem? Maybe not on my natural law epistemology: our epistemology is human epistemology.

I wonder if there is a philosophy of religion application. For instance, consider that theism provides a parsimonious explanation of the coarse-grained fact that there are rational animals: a perfect being has good reason to create rational animals given the objective value of such animals. But once we look at more fine-grained facts, such as that the rational animals have ten fingers and ten toes, it can be contended that we no longer have a more parsimonious explanation. Indeed, to explain all the details of humans, the theist is likely to need to invoke evolutionary mechanisms just as the naturalist does. Could the theist say that the more parsimonious explanation of the coarse-grained fact still counts for something?

I am also now thinking that this story gives the nominalist a good response to the Platonist in the context of yesterday’s post. The fact that a theory includes lots of fundamental nameable things is not very significant. What is significant is the ideology that needs to be brought to bear on particular explanations. When explaining our reaction to seeing grass, say, the Platonist needs to invoke not just the predicate “instantiates” but also the name “greenness”, while the nominalist only needs the predicate “is green”.

Platonism, nominalism and parsimony

Platonist: Platonism is ideologically way more parsimonious than nominalism. For Platonism has one fundamental predicate applying to objects: “instantiates”. Nominalism, on the other hand, has many: “is charged”, “has mass”, “is ____ units away from”, etc.

Nominalist: Platonism also has properties or relations such as charge, mass, or being ____ units away from.

Platonist: Yes, it has these properties among its objects, but there are no fundamental predicates corresponding to them. Platonism has extra fundamental objects, but that’s a question of ontological, not ideological parsimony.

Nominalist: But the Platonist has names for these objects: “charge”, “mass”, etc. And if fundamental predicates count against ideological parsimony, so do names of fundamental entities.

Platonist: If names of fundamental entities count against ideological parsimony, then either we have an objectionably anthropocentric notion of parsimony on which the parsimony of a theory depends on how many fundamental things we have in fact named, or else we are counting nameable fundamental things, and we have to say that a theory on which there are 10⁹⁰ fundamental particles is far less ideologically parsimonious than one on which there are 10⁸⁰ of them, since they’re all nameable. But it seems absurd to decide how many particles we have on grounds of ideological parsimony.

Nominalist: But Platonism has an additional categorial division, between properties (including non-monadic ones, if we want to be charitable and not count relations separately) and non-properties.

Platonist: Yes, but this division is not fundamental, since it can be entirely defined in terms of the Platonist’s one fundamental predicate, and logical and modal operators that nominalists also need: a property is an x such that possibly there are things that instantiate x. (Granted, this requires a sparse Platonism that doesn’t have properties like square circularity.)

Nominalist: But surely there is a difference in kind that explains why some things can be instantiated and others cannot.

Platonist: Sure, but not a difference in fundamental kind. Fundamentally, Platonism should be considered a one-category theory with one predicate, and lots and lots of names.

Personhood and supreme beauty

Start with the Socratic intuition that the most beautiful thing on earth is a human being living well. Extrapolating:

1. Persons living well are more beautiful than any impersonal entity.

Add a Platonic assumption:

2. Nothing is more beautiful than the Form of Beauty.

(Quick argument: Platonic Beauty is resemblance to the Form of Beauty, and nothing resembles x more than x itself does.)

3. There is a person who lives well.

Thus:

4. If the Form of Beauty is an impersonal entity, then there is a person who is more beautiful than the Form of Beauty. (1 and 3)

5. So, the Form of Beauty is a personal entity. (2 and 4)

6. The Form of Beauty is immaterial.

7. So, there is an immaterial personal entity that is supremely beautiful. (2, 6 and 7)

(A personal entity is one that is one or more persons.)

Substances as proper parts and constituent ontology

Here’s an argument against the claim that a human is sometimes (or always) a proper part of a larger whole such as a committee or an ecosystem, assuming constituent ontology.

1. An object is human iff humanity is a proper part of the object.

2. Proper parthood is transitive.

3. If a human x is a part of y, then x’s humanity is a proper part of y. (1 and 2)

4. So if a human is a proper part of something, that something is a human. (1 and 3)

5. No human is a proper part of a human.

6. So, no human is a proper part of anything. (4 and 5)

It seems that constituent ontologists have to have a pretty restricted theory of composition.

One way out is to distinguish “ontologically constituent parthood” from “integral parthood”, and say that an object is human iff humanity is an ontologically constituent part. Then the conclusion of the argument is going to be that no human is an ontologically constituent part of anything. But this requires multiplication of fundamental ideology.

There is another a way out of this. Let’s suppose that there are proper parts and immediate proper parts, with proper parthood being grounded in immediate proper parthood. Thus, perhaps, my fingers are immediate proper parts of my hand, my hand is an immediate proper part of my arm, and so on. Mediate parthood is then a chain (of length 0 for improper parts) of immediate parthood. There is no multiplication of fundamental ideology, since parthood is defined in terms of immediate proper parthood. Moreover, this is not unmotivated. Parthood seems intuitively hierarchical rather than flat. My big toe is a part of my leg because it is a part of my foot.

Now, notice that if we say this, we might hypothesize that an object’s essence is an immediate proper part of it. (Dialectical note: A constituent ontologist who accepts (1) and who buys the concept of immediate proper parthood has to agree with this hypothesis. For if my humanity were a mediate part of me, then I would have a proper part of which my humanity was a proper part, and by (1) that proper part would be human. But clearly no human is a proper part of me.)

Given that hypothesis, we can modify the constituent ontology claim (1) to:

7. An object is human iff humanity is an immediate proper part of the object.

And now humans can be parts of larger wholes, without infecting them with humanity.

Distinguishing eternalism and growing block at the end of time

Imagine that Alice lives in a classic growing block world, while Twin Alice lives in an eternalist world. Otherwise, the two worlds are exactly alike at all times. Moreover, in both worlds time started at noon, and ends at midnight.

Now let’s consider things from the point of view of both Alices right at the last moment of time, namely midnight.

In both cases, reality is a four-dimensional block, twelve hours long along the temporal axis. In both cases, there is a past but no future. However, the following would be true as asserted by Alice, but not as asserted by Twin Alice:

1. An hour ago, the block of reality was eleven hours long.

Question: What grounds the fact that (1) is true for Alice but not for Twin Alice? In other words, what grounds the fact that Alice inhabited a growing block world rather than an eternalist world?

This seems quite mysterious to me. And this seems to me to weaken the main arguments for why growing block is preferred to presentism and moving spotlight, namely that growing block gives you a better account of the grounds of claims about the past. But not in the case of (1). It seems that the growing blocker needs to ground (1) in one of the ways in which the presentist grounds claims about the past, say by positing a past-tensed fact at midnight of the block of reality having been eleven hours long an hour ago, or a past-tensed property, etc. And even so we have a puzzle.

My puzzle looks different depending on whether the eternalist world has A-eternalism or B-eternalism. If it has A-eternalism, then we can distinguish the two worlds as follows: the growing block world at the last moment of time does not have a spotlight anywhere, while the A-eternalist world does have a spotlight shining on midnight. But if that’s the answer we take, then there is a cost: there seems to be no difference between the growing block world at midnight and the B-eternalist world at midnight, and yet (1) is true in the growing block world and false in the B-eternalist world.

What did I mean by a classic growing block world? Well, in response to Merricks-style arguments that growing blockers can’t know that their thoughts are happening now, some have opted or a faded growing block, on which the past is missing some properties, like consciousness. On such a modified growing block theory, we have some hope of grounding the difference at the last moment of time between growing block and B-eternalism, because the B-eternalist world does not have a faded past. But now we can modify the story: suppose the B-eternalist world literally had a faded past (a past with no consciousness, say).

I think that if I was a growing blocker, I would be inclined to say something like this. The growing block theory is necessarily true in any world with time. What grounds the difference between being in a growing block world and an eternalist world is that an eternalist world doesn’t have time—it just has a fourth spatial dimension that enters differently into the laws of nature from the other three. Moreover, it is a necessary truth that at time t, time t is at leading edge of reality. This necessary truth does not need further grounding, but itself grounds (1) when combined with the fact that currently reality is twelve hours long.

I don’t like this invocation of an ungrounded necessary truth here. I wonder if it is much better than a Merricks-style presentism where truths about past facts are ungrounded (though admittedly contingent).

Is it wrong to assert falsehoods?

Years ago, when I was thinking hard about lying, I conjectured that perhaps we simply have a moral duty not to assert falsehoods. This way, we wouldn’t need to define lying for moral purposes–a very difficult task—because the wrongs in the vicinity of lying would all end up either being the assertion of a falsehood or the doing of something that one thinks is the assertion of a falsehood (and one should do neither wrong nor that which one thinks is a wrong). Of course, often, the assertion of a falsehood would be a non-culpable wrong, because one doesn’t know that one is asserting a falsehood.

I put that conjecture aside for a long time, and then just came back to it due to an email from a friend. But it now occurred to me that I have a pretty decisive argument against the idea that it’s wrong to assert falsehoods.

Consider a case where you know that someone has a false opinion on something, but you ask them their opinion on the topic anyway, for the sake of some minor good. There are many examples of such minor goods. You want to draw someone into an interesting conversation. You are asking a student a question on an oral exam where you know that the student will get it wrong but nonetheless you feel the student should have a chance at it like their peers. You’re on a panel and you know what your co-panelist thinks—and disagree with it—but you elicit their opinion so that the audience will know it, too.

However, in doing these things, you are leading another person to do wrong, if asserting falsehoods is always wrong. Yet these cases don’t seem like cases where you do wrong.

There are two possible subcases: you intend them to assert the thing you know to be false and you merely intend them to assert their opinion whatever it might be (while knowing that their opinion is in fact incorrect). In the first subcase, the action will be much more problematic if asserting falsehoods is always wrong. Granted, you are not intending that they do wrong qua wrong, but you are intending that they assert p when you know that asserting p is wrong. This is a pretty serious case of cooperation in evil. And this kind of case does happen, and apparently innocently so: perhaps you ask the opinion of a student in a class precisely because you think this student’s opinion is p, and you would like to have p represented in the discussion. In the second subcase, you intend only the expression of opinion as such, and you can make a reasonable Double Effect defense—but only if there is a proportionate good at stake. However, given that you know that this person’s expression of opinion will be a wrongdoing, that proportionate good would have to be fairly weighty. And yet we innocently elicit such opinions without “fairly weighty” reasons.

I guess if one thought that it’s wrong to assert a falsehood, instead of asking a first-order question like “Is it wrong to steal from the rich and give to the poor?” one might ask a second-order question like “Do you think it is wrong to steal from the rich and give to the poor?” For then perhaps the other person answers truly as long as they are sincere. But I don’t think we need to do such tricks. Plus, it’s not clear that a “Do you think …?” question is really a second-order question—it may just be a social form for asking the first-order question.

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.

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.