Aquinas's embryology and the theory of relativity

Aquinas famously thinks that there is a succession of forms in utero, with first a vegetable form, then an animal form, and then a rational animal (human) form. No two of these forms are had simultaneously.

But if a three-dimensionally extended object that has form A comes to be a three-dimensionally extended object that has form B, with no other forms intervening, then, in every inertial reference frame except for at most one, there is a time at which some of the matter has form A and some of the matter has form B. So unless there is a privileged frame, Aquinas’s story doesn’t work.

In the following diagram, the slanted dashed line indicates a reference frame where some of the matter has form A (red) and some has B (blue).

Here is a variant that could work, but does not seem very plausible. We could imagine that when we have a transition from A to B, the matter of A, except at one point, passes to B through one or more other forms, indicated by the yellow portion of the diagram. These might be forms of mere particles, or they could be some special forms. In other words, A dies off into a point, with the dead matter acquiring transitional forms, and then B starts growing from the last point of A, incorporating the transitional forms. Where the A and B substances meet will be a point either of A or of B, but not of both. The narrowing of A and the growth of B happen at the speed or light or less.

On this variant, no inertial frame contains both A and B points. (If light moves at 45 degrees from horizontal in the diagram, then inertial frames correspond to lines like the dashed one making a less than 45 degree angle with the horizontal.)

But there is something rather weird going on here. Suppose that A is the vegetable form and B is the animal form (a similar argument will apply if A is the animal form and B the human form). Then close to the pointy meeting between A and B, the yellow stuff contains the vast majority of what biology would call “the embryo”, and a fairly well-developed one, since it’s on the cusp of becoming an animal. Yet the vast majority of that “embryo” is the yellow stuff—neither the vegetable nor the animal, but something else, maybe mere atoms. Indeed, once we get close enough to the meeting point, the yellow will materially function just like an embryo, since a tiny subatomic hole makes no difference to material functioning. This is very odd, and gives us reason to reject Aquinas’s story.

Of course, the main alternative to Aquinas’s story is that the gametes change into a human being. That faces some of the same difficulties. However, I think the difficulties are less if the gametes are not themselves a substance, but a plurality of substances, perhaps particle-substances. In the diagram below, the gamete-stuff is in yellow, and the blue indicates the human being. We still have the problem that early on most of what we have will need to be biologically very close to a functioning zygote, and yet it is in yellow, except for a small blue hole corresponding to where ensoulment is spreading out from a single point. But I think this is less problematic, because at this juncture the yellow stuff is something that is less obviously an organism. (Admittedly, the blue stuff is less obviously an organism when it is nearly a point. But what makes it an organism is that while its matter has little going for it, it’s got the right form.)

So, relativity makes it hard to hold on to Aquinas’s embryology. Which is a nice thing for pro-life Thomists who want to defend ensoulment at conception and hence deny Aquinas's embryology--or Catholic Thomists who find the doctrine of the Immaculate Conception incompatible with that embryology.

Maybe there is some way of getting out of this by using the considerations from yesterday’s post, though.

Pointy endpoints and internal space and time

In a series of recent posts, starting with this one, I argued that relativity theory gives us good reason to think that at the start and end of our lives we are pointlike in spatial extension. Crucial to these arguments was the idea that there is no privileged reference frame.

The conclusion is, of course, rather counterintuitive. While it is plausible that we start as a cell and that we shrink with old age, in neither case are we pointlike.

However, in some earlier posts, like this one, I explored the idea that substances may have internal space and internal time, in addition to the external spacetime of the universe.

Here is a suggestion that could save the intuition that we are not spartially pointlike at the beginning and end of life. We have internal space and time, and our internal space and time has an absolute simultaneity relation. With respect to internal time, at the first and last moments of our lives (I’m simplifying by assuming there are such moments), we have significant non-pointlike spatial extension. Of course, most external frames of reference do not agree with our internal simultaneity relation, and in most of them, we are pointlike at the beginning and end of our lives. But that’s fine: our intuition relates to our internal space and time.

The above suggestion is inspired by Rob Koons’ suggestion that our rest frame could be the privileged frame with respect to which we could have spatially non-pointlike endpoints of life. That suggestion doesn’t work for the technical reason, which Koons also noted, that there doesn’t seem to be a well-defined notion of a privileged frame of a squashy organism. (One might try to go with the rest frame of the center of mass. But an organism’s center of mass can move faster than light—e.g., in a fast amputation. And faster than light motion doesn’t define a reference frame.) But even if there isn’t a well-defined rest frame, there is a family of “approximate rest” frames that are intuitively close to what we would expect the rest frame to do. And then we could suppose that the internal simultaneity relation is pretty close to these approximate rest frames.

Note that we should not expect the internal simultaneity relations to align neatly between substances (except in the special case of substantial change where we might reasonably suppose that the old substances’s later internal simultaneity relations approximate the new substance’s earlier ones), and hence they do not define a global privileged simultaneity relation that would threaten relativity theory.

So now I think that my argument about spatially pointy endpoints of life gives us a choice between three options:

1. Accept spatially pointy endpoints.

2. Accept a global privileged reference frame.

3. Accept privileged reference frames specialized for individual substances.

In the last option, I think the best way is the way of internal space and time, but I suppose one could also think there are privileged external reference frames specialized for individual substances.

Accepting privileged reference frames specialized for individual substances could also help materialists deal with the problem of the unity of consciousness for a spatially extended brain.

Against the necessary unity of consciousness

Consider this unity of consciousness thesis:

1. Necessarily, if x at time t has phenomenal state A and phenomenal state B at time t, then x at t has a phenomenal state that includes both A and B.

(Note that (1) seems incompatible with time travel. But that is perhaps fixed by specifying that we are talking about internal time in (1).)

Christians have special reasons to be dubious of (1).

Start with this quick thought. Around 30 AD, the Logos was suffering pain on the cross and comprehensively experiencing his infinite divinity. But the Logos did not have a phenomenal state that subsumed these two phenomenal states. For orthodox Chalcedonian theology has it that the Logos had two minds: a divine mind and a human mind. Only in the divine mind can one have a mental state that subsumes the comprehensive experience of divinity. But no state subsuming suffering can be found in the divine mind.

This argument is suggestive but not conclusive. For we might say that it is not correct to say that the Logos has the divine experience around 30 AD, because God is outside of time.

But as we learn from Aquinas, once we accept that one incarnation of the Logos is possible, we should also accept that two simultaneous ones are possible. There are no further conceptual difficulties in two than in one, and if omnipotence allows for one, it should allow for two. But if the Logos had two simultaneous incarnations, then the Logos would have, in addition to the divine mind, two temporal creaturely minds. And if one of these minds houses unmitigated joy and the other sorrow, then none of the three minds of the Logos would house a state that includes both the joy and the sorrow, and hence the Logos would not have such a subsumptive state.

Further, if (1) is true, surely so is:

2. Necessarily, if x timelessly has phenomenal state A and phenomenal state B, then x timelessly has a phenomenal state that includes both A and B.

And now imagine that the timeless Logos engages in an analogue of the incarnation but as a timeless conscious being. Then the timeless Logos would have a divine phenomenal state and a human one which would not be unified.

Here is another thought against (1). Suppose that in heaven, Peter enjoys the beatific vision of God while enjoying Paul’s singing. A subsumptive phenomenal state that includes both the beatific vision and Paul’s singing would then be greater than either one of the included states. But no phenomenal state that Peter has is greater than Peter’s beatific vision.

There is a metaphysical version of this argument. The beatific vision has God directly as its content, rather than merely having a representation of God as its content. A state that included the beatific vision and Paul’s singing would have to have as its content God himself plus a representation of Paul’s singing. But there is no way to have a whole of which God is a proper part (Aquinas considers this to be a part of divine simplicity; but we might also think it follows from Anselmian theology—there cannot be anything greater than God).

Of course, even if (1) is false in general, a non-modal version may be true restricted to ordinary human phenomenal states, and we still will need an explanation of that fact. And it may be that some of the arguments people make from (1) against various materialist theories of consciousness would apply against the restricted thesis.

Lying in politics

Consider this reductio ad absurdum argument:

1. It is permissible to lie to achieve what one reasonably thinks to be practically necessary to save multiple innocent lives. (Assumption for reductio)

2. In typical elections to the highest political offices in a country, at least one candidate reasonably thinks that their winning the election is practically necessary to save multiple innocent lives.

3. In typical elections to the highest political offices in a country, at least one candidate is such that it would be permissible for them to lie to win the election.

4. But it would not be permissible for candidates for the highest political offices in a country to lie to win the election, except perhaps in atypical cases.

5. Contradiction!

6. So, it is false that it is permissible to lie to achieve what one reasonably takes to save innocent lives.

The thought behind (2) is that serious candidates tend to reasonably think that their policies would make a significant positive difference to the well-being of people. Given the tens of millions of people in a typical country, a fairly intelligent candidate will realize that this positive difference saves lives, by such factors as improving medical care, and decreasing stress, suicide and drug-abuse rates. And typical serious candidates are at least fairly intelligent. Additionally, in many countries abortion is relevant at election time, and these countries will often have candidates who reasonably think that abortion kills innocent people.

The “except perhaps in atypical cases” qualifier in (4) is to take care of the intuition that some people will have that lying is permitted to defeat someone with literally genocidal policies (which is fortunately an atypical case).

The above argument gives one reason to be dubious of the idea that it is permissible to lie to save lives. But I can also see an interesting answer. The most relevant kinds of lies of politicians would be lies to the public. But you might have this view: While it is permissible to lie once to save a life, it is not permissible to lie once in order to have a 0.1% chance of saving a life, nor to lie a thousand times to have a certainty of saving one life. For a lie is pretty bad, and too much lying outweighs the value of saving a life. Now, when you lie to a large group of people, you count as lying once to each member of the group. Thus, it would be wrong to lie to all the members of a population in order to save 0.1% of them or less. And in typical electoral cases, one would be unlikely to save more than 0.1% of the population!

I am not sure about this line of response. I am not sure the wickedness of a lie linearly multiplies with the number of people lied to. Imagine this. You are on the phone trying to dissuade a friend who has a large YouTube following from lying to their million followers. You see that you have a consideration they will think decisive, and are about to offer the consideration, but then you see a child drowning. If you jump in the water to save the child, you’ve lost your moment of influence and your friend will lie to a million. You should, typically, go and save the child. (Unless your friend’s lie would influence someone to commit murder.) So the disvalue of lying does not increase linearly with the size of the victim audience.

Some quantum semiholisms

I’ve been naively thinking about what a reductive physicalist quantum ontology that matches the Hilbert-space formalism in the Schroedinger picture might look like.

My first thought is something like this. “Space” is (the surface of) a sphere in a separable Hilbert space, with an inner product structure (perhaps derived from a more primitive linearity and metric structure using the polarization identity) and “the universe” is a point particle walking on that sphere.

But that description is missing crucial structure, because when described as above, all the points on the sphere are on par. Although the universe-particle was at a different location on the sphere 13 billion years ago than where it is now, there is nothing to distinguish these two points in the story, and hence nothing to ground the vast changes in the universe between then and now. What we need to do is to paint the sphere with additional structure.

There are multiple ways of having the additional structure. Here are two.

Option I. Introduce a number of additional causally impotent “point particles” living on the sphere but not moving around as “markers”, and define the rich intuitive structure of our universe from the inner-product relationships between the universe-particles and the marker-particles. Here are two variants on this option.

• (Ia): There are countably many point particles corresponding to basis vectors in some privileged countable Hilbert space basis, and these “marker-particles” are then located at a set of points on our sphere that form an orthonormal basis. For instance, if we “think of” the Hilbert space for a system of N particles as L²(R^3N), we might have a different static marker-particle for each 3N-dimensional Hermite polynomial.

• (Ib): There are uncountably many marker-particles, and they are located at a set of points of the sphere such that the closure of their span is the whole Hilbert space, but they are not orthogonal. For instance, in our N-particle case, we might think of each marker-particle as corresponding to a normalized indicator function of a subset of R^3N with non-zero Lebesgue measure, and require them to be located on our Hilbert space sphere in places which give them the “right” inner product relationships for normalized indicator functions.

Note that since what is physically significant are the inner products beween the positions of the marker-particles and the universe-particle, we need not think of the particles as having “absolute positions” on the sphere—we can have a “relationalist” version where all we need is the inner-product relationships between the particles (marker and universe). Or, if we want something more like the Heisenberg picture, we could suppose absolute positions, keep the universe particle static, and make the marker particles move. There are many variants.

Option II. We enrich the structure of our “space” (i.e., the surface of the Hilbert space sphere) by adding fundamental binary relations between points on that sphere that correspond to some privileged collection of operators (e.g., normalized projections onto subsets of R^3N with non-zero measure).

Anyway, here is an interesting feature of these two stories. On none of them do we have Schaffer-style holism. On Option I, we have an infinite number of fundamental “particles” in “space” (i.e., on our infinite-dimensional sphere), though only one of them is moving, and we may or may not have the “space” itself. On Option II, we have the two fundamental entities: the universe-particle and the sphere itself, with the universe-particle having merely positional structure, while the sphere has a complex operator structure.

We might call these stories semiholistic. Of course, there are fully holistic stories one can tell as well. But one doesn’t have to.

Fields and finetuning

Here is an interesting fine-tuning issue, inspired by a talk I heard from Brian Cutter at the 2023 ACPA meeting.

It seems likely that physical reality will involve one or more fields: objects that assign values to points in space (“ordinary” space or configuration space), which values then govern the evolution of the universe.

The fine-tuning issue is this. A plausible rearrangement principle should allow any mathematical assignment of values of the field to the points in space as metaphysically possible. But intuitively “most” such assignments result in a configuration that cannot meaningfully evolve according to our laws of nature. So we want to have an explanation of the fine-tuning—why are we so lucky as to have an assignment that plays nice with the laws of nature.

For a toy example, consider an electric field, which is a vector field E that generates a force F = qE on a particle of charge q. Intuitively, “most” vector fields will be nonmeasurable. But for a nonmeasurable electric field, we have no hope for a meaningful solution to the differential equations of motion. (OK, I’m ignoring the evolution of the field itself.)

For another example, suppose we think of the quantum wavefunction as a function over configuration space rather than as a vector in Hilbert space (though I prefer the latter formulation). If that function is nonmeasurable—and intuitively “most” are nonmeasurable—then we have no way to use quantum mechanics to predict the further evolution of this wavefunction. And if that function, while measurable, is not square integrable (I don’t know if there is a sense of “most” that applies here), then we have no way to use the Born rule to generate measurement predictions.

Metaphysical semiholism

For a while I’ve speculated that making ontological sense of quantum mechanics requires introducing a global entity into our ontology to ground the value of the wavefunction throughout the universe.

One alternative is to divide up the grounding task among the local entities (particles and/or Aristotelian substances). For instance, on a Bohmian story, one could divide up 3N-dimensional configuration space into N cells, one cell for each of the N particles, with each particle grounding the values of the wavefunction in its own cell. But it seems impossible to find a non-arbitrary way to divide up configuration space into such cells without massive overdetermination. (Perhaps the easiest way to think about the problem is to ask which particle gets to determine the value of the wavefunction in a small neighborhood of the current position in configuration space. They all intuitively have “equal rights” to it.)

It just seems neater to suppose a global entity to do the job.

A similar issue comes up in theories that require a global field, like an electromagnetic field or a gravitational field (even if these is to be identified with spacetime).

Here is another, rather different task for a global entity in an Aristotelian context. At many times in evolutionary history, new types of organisms have arisen, with new forms. For instance, from a dinosaur whose form did not require feathers, we got a dinosaur whose form did require feathers. Where did the new form come from? Or suppose that one day in the lab we synthesize something molecularily indistinguishable from a duck embryo. It is plausible to suppose that once it grows up, it will not only walk and quack like a duck, but it will be a duck. But where did it get its duck form from?

We could suppose that particles have a much more complex nature than the one that physics assigns to them, including the power to generate the forms of all possible organisms (or at least all possible non-personal organisms—there is at least theological reason to make that distinction). But it does not seem plausible to suppose that encoded in all the particles we have the forms of ducks, elephants, oak trees, and presumably a vast array of non-actual organisms. Also, it is somewhat difficult to see how the vast number of particles involved in the production of a duck embryo would “divide up” the task of producing a duck form. This is reminiscent of the problem of dividing up the wavefunction grounding among Bohmian particles.

I am now finding somewhat attractive the idea that a global entity carries the powers of producing a vast array of forms, so that if we synthesize something just like a duck embryo in the lab, the global entity makes it into a duck.

Of course, we could suppose the global entity to be God. But that may be too occasionalistic, and too much of a God-of-the-gaps solution. Moreover, we may want to be able to say that there is some kind of natural necessity in these productions of organisms.

We could suppose several global entities: a wavefunction, a spacetime, and a form-generator.

But we could also suppose them to be one entity that plays several roles. There are two main ways of doing this:

1. The global entity is the Universe, and all the local entities, like ducks and people and particles (if there are any), are parts of it or otherwise grounded in it. (This is Jonathan Schaffer’s holism.)

2. Local entities are ontologically independent of the global entity.

I rather like option (2). We might call this semi-holism.

But I don’t know if there is anything to be gained by supposing there to be one global entity rather than several.

Arguments for theism

How one met someone can significantly affect the shape of one’s relationship with them for years to come. Therefore, we can imagine that what reasons convinced one to believe in God can significantly affect the shape of one’s relationship with God. If this is right, and if God has a plan for a particular kind of relationship with a person, then God might have reason to keep the person from being convinced by some evidence for the existence of God, in order that the person might instead come to be convinced by reasons that will set the stage for the particular shape of relationship God wants.

In particular, it would not be surprising if some of the more abstract philosophical arguments for the existence of God, while perfectly fine as arguments, might not be fitting for leading a particular individual into a deeply interpersonal relationship. And if so, we could imagine that God could keep the person from being convinced by such arguments, in order that the person might come to belief in a different way.

But it all could depend on the person. We are broken in different ways, and God has different plans for us all.

A personalized argument from need

In the case of certain kinds of transformational experiences—such as meeting the love of your life—you have the realization that this is something you really needed for flourishing, but didn’t know it. Further, in many cases of these kinds of experiences, part of that realization is not just that you need the subjective experience for flourishing, but that what you need for reality the flourishing. For instance, if it turns out that your experience of meeting the love of your life was a hallucination, it would be a part of your realization not just that you need an experience of meeting the love of your life, but that you need the reality.

Now I think this kind of realization is a part of the life of many theists in connection with the experience of a relationship with God: they come to realize that they have always needed both this experience and the reality of it. But then the theist has this argument:

1. What is needed for the flourishing of a human being is in principle possible.

2. A relationship with God is in principle possible only if God exists.

3. A relationship with God is needed for my flourishing.

4. So, God exists.

An interesting thing about this argument is that for a number of people, the full realization of the truth of premise (3) only comes about once they believe in God. The argument thus has a circularity of sorts: it works best for those who already believe the conclusion. This is an innocent circularity: the relationship with God, of which belief in God is a part, makes available significant evidence for that belief.

This is a bit like Kierkegaard’s “argumentum Spiritus Sancti” which is only available to those who believe. It sounds paradoxical, but I do not think it is actually all that paradoxical. Imagine you have a friend who is accused of some crime, but refuses to show you evidence of their innocence unless you believe in their innocence first. Then, your belief is needed for you to have the evidence, but the evidence can be perfectly genuine and unparadoxical.

Even more on pointy beginnings

In a recent post I argued that in Aristotelian substantial change, given special relativity, the resultant substance starts as basically a point—it is arbitrarily small.

I think the argument doesn’t actually require much in the way of Aristotelian assumptions, but works for any caused extended substance, or at least any ordinary one.

Suppose that a substance B is caused to exist by A (which might be a single entity or a plurality) and initially (at least) at distinct points in spacetime. There is a reference frame according to which one of these points is earlier than the other (this is true in all frames if the points are timelike-related and some frames if they are spacelike-related). Let F be a frame where this happens, and let z₁ be the F-earlier and z₂ the F-later point. From now on, work in F.

Let t_i be the time of z_i. Now B is at least partly present arbitrarily close to z₁, and hence arbitrarily close to t₁, and since t₁ < t₂, it follows that B already existed before t₂. Therefore, any causal influence of A sufficiently close to time t₂ is irrelevant to B’s existence. In fact, B wasn’t even partly caused by A to exist at times close to t₂, since it had already existed for a while before this. And this contradicts our assumption that A caused B at both z₁ and z₂.

A crucial assumption here is that nothing that happens later than a time t is relevant to whether a substance B exists at t.

What if there is backwards causation? If so, then this argument fails. But even if there is backwards causation, it is rare and extraordinary. It is still true that in ordinary cases, substances are caused to exist at a single point.

What if B is uncaused? Again, the argument fails. But even if there are uncaused extended substances, they are not the norm. So, again, the argument still works in ordinary cases.

The fundamentality of souls

Some dualists say that the soul is a fundamental entity.

I think we’re not in a position to think that. Compare this. We have no reason to think electrons are not elementary particles. They certainly aren’t made of any of the other particles we know of, so they are, we might say, “relatively elementary” with respect to the particles we know. But we would not be very surprised if electrons turned out to be made of other particles.

Similarly, we have good reason to think the soul is not grounded in any of the other things we know of (matter, accidents, etc.) But we should not be really surprised if a finer-grained analysis would reveal the soul to have a grounding structure beyond our current knowledge. We should be cautious and say the soul is “relatively fundamental” with respect to the entities we know.

Another argument that we likely start small

In a number of recent posts, I argued that mid-sized objects like ourselves start microscopic. All my arguments so far relied on relativity. Here is one that doesn’t.

1. Biological entities are unlikely to have a perfectly flat macroscopic geometrical face (biological things tend to be rounded, rough, pointy, but not perfectly flat).

2. We are four-dimensional.

3. We are biological entities.

4. If we don’t start microscopic and we are four-dimensional, then we have a perfectly flat macroscopic geometrical face at our temporal beginning.

5. So, probably, we start microscopic.

Why the restriction to macroscopic faces? Two reasons. First, if space is discrete and grid-like, then it may be that all objects have perfectly flat sides at the grid-spacing level. Second, if we are made of point particles, then our geometry likely includes perfectly flat triangles between three outer point particles.

Relationship without belief

Consider this fairly standard version of the argument from hiddenness:

1. If God exists, he produces everything that is necessary for a personal relationship with every nonresisting person.

2. Belief in the existence of x is necessary for a personal relationship with x.

3. So, if God exists, every nonresisting person comes to believe in God.

4. Some nonresisting person does not come to believe in God.

5. So, God does not exist.

I noticed today that (2) is just plain false. My example is a skeptic about other minds. You can take seriously the hypothesis that you are the only real person around, seriously enough that you do not believe the hypothesis false, and still have a personal relationship with other people. Surely Unger, in his phase of believing that people don’t exist, had personal relationships with them!

A perhaps even better counterexample to (2) was given by one of my students. You can have a long-standing Internet-based personal relationship while taking seriously the possibility that the other person doesn’t exist (e.g., maybe you are interacting with a chatbot).

This observation doesn’t destroy the hiddenness argument. One might, for instance, replace (2) with:

6. A personal relationship with x is incompatible with consistent disbelief in the existence of x

and then replace (4) with:

7. Some nonresisting persons end up consistently disbelieving in God (e.g., due to their reasonable evaluation of the problem of evil, or due to low priors for theism).

But now (7) is less plausible than (4). One might well think that the evidence against theism is insufficiently strong to make it possible for a nonresister to disbelieve in God.

Alternately, one might replace the deductive hiddenness argument with a probabilistic one by noting that it’s a lot harder to have a personal relationship without belief in the other person, and it’s unlikely that a loving God would make it this hard. I think that’s not a very strong argument, but it is an option for the defender of hiddenness.

Relativistic Aristotelian beginnings

From purely geometrical facts, it follows that every spatially extended entity is arbitrarily small at its beginning and at its end in almost every reference frame.

A stronger result is possible in the special case of the beginnings of substances in simple Aristotelian substantial change. In simple Aristotelian substantial change, substance A wholly changes into a new substance B by having all of the terminal matter of A be the proximate matter of B without any temporal gap. I claim that then substance B comes into existence at a single point in every reference frame (i.e., the temporal bottom of B fits into a light cone).

For suppose that in some frame F, substance B comes into existence at two F-simultaneous and distinct spacetime points z₁ and z₂ (these could be points at which there is still A but arbitrarily close to points of B or these could be points at which there is B but that are arbitrarily close to points of A). Then there is another frame F′ at which z₁ is earlier than z₂. Let t′_i be the time of z_i in F′. Because of how the terminal matter of A is the proximate matter of B, there is matter of A arbitrarily close to z₂. Hence, arbitrarily close to time t′₂, we will have A still existing. However, arbitrarily close to time t′₁ we will have B already existing. Since t′₁ < t′₂, it follows that according to F, we will have A still existing after B has already come to exist. So the cause is partly later than the effect, which is absurd.

Maybe there is a way around this in more complex cases where multiple substances result in one new substance. I am not sure.

But here is another way to see the pointiness of the beginnings of substances if one accepts the Aristotelian idea that substances are individuated by their initial matter and we take matter to be infinitely subdivisible. Suppose in frame F, substance B begins at distinct and simultaneous points z₁ and z₂. Let F′ as before be a frame where z₁ is earlier than z₂. Then according to frame F, substance B already exists before its matter close to z₂ exists (I am assuming matter is infinitely subdivisible, and in this case a relevant division happened). So its matter close to z₂ cannot be essential to its individuation. And there is a third frame, F″, where z₁ is later than z₂, and so the matter close to z₁ cannot be essential to B’s individuation. It follows that none of the matter can be essential to the individuation of the substance if the substance starts at two or more places at once. Thus, a substance must start at a single point.

What happens in substantial change then? It seems that if we are to preserve relativity, we have to say that the new substance comes into existence at a single point z out of one or more preceding substances. If matter individuates (which I am dubious of), then the matter immediately around z is what does the individuating. The substance’s form then spreads out from z, perhaps incorporating more and more of the stuff around z, at the speed of light or less.

Of course, all these problems disappear if we allow for faster-than-light causation in substantial change. But that should be a last resort.

Gratuitous evil and compensation

In the last couple of decades, the most prominent argument from evil is base on the idea that God couldn’t allow a gratuitous evil. Here is one way to define a gratuitous evil, paraphrasing Rowe:

1. E is gratuitous if and only if there is no greater or equal good G that is only obtainable by God if God permits E or something equal or worse.

(For simplicity, I am taking the prevention of an evil as itself a good.)

By (1), if E is any earthly evil, it’s too easy to show that for all we know E is not gratuitous.

To see this, observe that for all we know, the persons suffering from E are compensated in an afterlife by God bestowing on them a greater good G₀. Now let G be the good of receiving G₀ in compensation from God for suffering E. Then G is a good, and is at least as good as G₀ itself. But G₀ is greater in magnitude than E. Thus, G is greater than E. Moreover, God’s permitting E is a necessary condition for God’s compensating someone for suffering E, since God cannot compensate someone for something that didn’t happen. Therefore, G is obtainable by God only if God permits E.

So we don’t want to go for (1), since it’s too easy to find goods that are only obtainable given E if there is an afterlife.

The problem with this example is that while it is impossible for God to give G₀ as compensation for E without E, God can give G₀ gratuitously, and that seems to be just as good as giving it as compensation. The atheist who wants to argue for the existence of God should modify (1) as follows:

2. E is gratuitous if and only if there is no greater or equal good G such that something at least as good as G is only obtainable by God if God permits E or something equal or worse.

But given (2), it is implausible to say that God couldn’t allow a gratuitous evil. Consider a case where Alice makes a slightly mean joke at Bob’s expense. She then repents, and asks Bob for forgiveness, who forgives. It is easy to imagine that the value of the repentance and forgiveness is greater than the disvalue of the joke, but just as the joke was a minor evil, the repentance and forgiveness are minor goods. It seems intuitively clear that the case of Alice’s slightly mean joke does not require any theodicy beyond what has just been said. Yet Alice’s joke appears to be a gratuitous evil according to (2).

For the goods in the previous paragraph are all minor goods. But God could create a major good without permitting any evil at all. For instance, God could create an infinite number of happy mathematicians who eternally enjoy the search for truth and who do not have the freedom to choose between good and evil, but must do good. That infinity of happy mathematicians would be a major good. But surely any minor good is less than any major good. Thus, if we let G be the minor goods of repentance and forgiveness in the previous paragraph, then something greater than G—namely, the infinite number of happy mathematicians—is obtainable by God without any evil at all. And hence the story in the previous paragraph isn’t enough to provide a theodicy, if (2) is the right account of gratuitous evils.

Maybe the solution is a very strong doctrine of incommensurability, on which the good of the infinitely many mathematicians is incommensurable with the goods of repentance and forgiveness even when the latter two are minor. But given such a strong doctrine of incommensurability, we can go back to my original compensation story. The problem I saw with that original story is that God could give G₀ gratuitously and not as compensation, and I said that that would be at least as good. But given a strong doctrine of incommensurability, giving G₀ gratuitously will be incommensurable with giving G₀ as compensation for E. And, plausibly, any good that isn’t and instance of compensating for E or something at least as bad will be incommensurable with the good of compensating for E with G₀. Thus, mere compensation will suffice for theodicy.

I am not comfortable with saying that mere compensation suffices for theodicy. But there is something to this idea.