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.)

An argument that organisms aren't wholly material

Here’s an argument for non-materialism about all organisms.

Start with these premises:

1. Organisms exist.

2. If organisms exist, it is possible to have two organisms that have all their matter in common.

3. Antisymmetry of parthood: Two distinct things can’t each be a part of the other.

4. Strong supplementation: If y isn’t a part of x, then y has a part that doesn’t overlap x.

Now suppose that x and y are two organisms that have all their matter in common. By antisymmetry of parthood, they can’t both be mutual parts of each other. Swapping labels if necessary, suppose y isn’t a part of x. Then by strong supplementation, y has a part z that doesn’t overlap x. But any material part of y is a material part of x by (2). Thus, y has a part that isn’t a material part. Hence:

5. It is possible to have an organism with a non-material part.

But this is weak in two ways. First, it doesn’t show that any actual organism has a non-material part. Second, even if we showed that, that wouldn’t show that all organisms have non-material parts. Moreover, I have given no argument for premise 2.

I will fix all these problems together. In a recent post, I gave a plausible construction of a world where there is a dog-like organism and a cat-like organism with all their matter in common. The same construction would give the following conclusion: there is a world where there is a mushroom-like organism and a grass-like organism thathave all their matter in common. Plugging that into the above argument yields:

5. Possibly a mushroom-like or a grass-like organism has a non-material part.

The likeness here is functional: the way of life of the organism is like that of mushroom or a grass plant. Then add:

6. If possibly a mushroom-like organism has a non-material part, all organisms have non-material parts.

7. If possibly a grass-like organism has a non-material part, all organisms have non-material parts.

From 5-7, we get:

8. All organisms have non-material parts.

Why believe 6 and 7? Here’s a thought. You might think that conscious animals have to have souls or something like that. But if a mushroom- or grass-like organism can have a non-material part, probably it actually does, and if it has a non-material part, probably all organisms do.

There is a lot in this argument that doesn’t personally convince me. In fact, I am skeptical of premises 2 and 4, and my main reason for accepting 6 and 7 is that I already accept anti-materialism about all organisms. But I think the argument has some force.

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.

What has form?

On the question of what has a substantial form, I have tended to think something similar to van Inwagen’s answer to the question of what wholes there are. Namely, I assign form to:

1. organisms, and

2. fundamental objects in physics that are good candidates for being substances.

Regarding 2, if the correct physics is particle-based (which I doubt, in light of the apparent possibility of the world being in a superposition of states with different numbers of particles), these will be particles, or at least those particles that aren’t part of an organism. If the correct physics is field-based, the substances in physics will be fields (or maybe just one field-like object, namely “the global wavefunction”).

A lot of Aristotelians have substances, with forms, that are intermediate between (1) and (2), such as hydrogen atoms or water molecules or chunks of iron, and maybe astronomical objects like stars or galaxies. While I don’t have a knock-down argument against such substances, I also don’t see any reason to posit them.

My reasons for positing form for organisms and fundamental physical objects are quite different. For organisms, the reasons are largely normative. Parrots and oak trees can flourish or languish; they have ends and proper functions. In the case of humans, the normativity extends much further. Furthermore, we need well-defined boundaries for organisms for ethical reasons—there is reason not to harm an organism, especially but not only a human one—and there need to be well-defined persistence conditions for humans for moral responsibility. Something needs to ground all this. And the best candidate is form.

It is a central commitment of Aristotelianism that all of physical reality is grounded in physical substances and their accidents. But it is false that all of physical reality is grounded in organisms. There was a time when the physical universe had no organisms. So we need other substances. The fundamental objects of physics are the best candidates. They are active and have very clear kind-boundaries. The electromagnetic field is a different kind of thing from the gravitational field (which is just spacetime, according to Einstein). Photons are clearly different from electrons. (Though if it turns out that particle number is indeterminate, then particles won’t be the fundamental objects of physics.)

Granted, it is not obvious (and somewhat counterintuitive) that organisms have well-defined kind-boundaries and identity conditions. And it is not obvious (and somewhat counterintuitive) that fundamental physical objects have norms. But here I just take these to be consequences of the theory. Organisms have well-defined kind-boundaries and identity conditions, but we don’t know where they lie. Fundamental physical objects have normative properties, but I suspect they are perfect instances of their kind, and always do exactly what they should (C. S. Lewis says something like that in Mere Christianity).

Neither of my two reasons applies much to objects like atoms, molecules, chunks of stuff, or astronomical objects. There is no strong independent reason to suppose that they have normative properties in their own right, and their boundaries are, if not quite as fuzzy as those of organisms, pretty fuzzy. How far apart do I get to move a hydrogen atom from two oxygen atoms before I destroy a water molecule? How many sodium and chloride ions do I add to water to change it from water with impurities to a salt solution? (I suppose the concept of impurity pulls in the direction of thinking there are normative properties. But here is a reason to think this is mistaken. If impure water is languishing, then we have reason to distill water independently of any practical benefit to any organism, just for the sake of the water itself. That seems absurd.)

That the reasons don’t apply doesn’t show that there aren’t other reasons to posit substantial forms for these other candidates. But I don’t see such reasons. And so we can apply Ockham’s razor.

Video: Three Mysteries of the Concrete

Alexander Pruss, “Three mysteries of the concrete: Causation, mind and normativity”, Christian Philosophy 2022, online, Cracow, Poland, September, 2022.

The afterlife of humans and animals

I’ve been thinking a bit about the afterlife for non-human animals. The first thought is that there is a relevant difference between human and non-human animals in terms of flourishing. There is something deeply incomplete about the eighty or so years a human lives. The incompleteness of our earthly life is a qualitative incompleteness: it is not just that we have not had enough pieces of cake or run enough miles. Typically, whole areas of virtue are missing, and our understanding of the world is woefully incomplete, so that one of the most important things one learns is how little one knows. The story of the life is clearly unfinished, even if life has gone as well as it is reasonable to expect, and flourishing has not been achieved. Not so for non-human animals. When things have gone as well as it is reasonable to expect, the animal has lived, played and reproduced, and the story is complete.

If we think of the form of an entity as specifying the proper shape of its life, we have good reason to think that the human form specifies the proper shape of life as eternal, or at least much longer than earthly life. But there is little reason to think that form of an animal’s life specifies the length of life as significantly longer than the typical observed life-span of in its species.

If we accept the thesis which I call “Aristotelian optimism”, namely that things tend to fulfill their form or nature, we have good reason to think there is more to human life than our earthly life, but not so for non-human animals. In the case of humans, this line of argument should worry typical atheistic Aristotelian ethicists, because it would push them to reject Aristotelian optimism, which I think is central to ensuring knowledge of the forms in Aristotle’s system.

By the way, there may be an exception in the above argument for animals whose flourishing consists in relationships with humans. For there its flourishing might be incomplete if it cannot be a companion to the human over its infinite life-span. So there is some reason to think that species that are domesticated for human companionship, like dogs and to a lesser extent cats and horses (where companionship is less central to flourishing), might have an afterlife.

A Thomistic argument for the possibility of an afterlife for animals

1. Accidents are more intimately dependent on substance than substantial forms on matter.

2. If (1) is true and God can make accidents survive without the substance, then God can make forms survive without matter.

3. If God can make forms survive without matter, then God can ensure life after death for animals by making their forms survive and restoring their matter.

4. God can make accidents survive without the substance.

5. So, God can ensure life after death for animals.

The most controversial claim here is (4), but that follows from the Thomistic account of the transsubstantiation.

Of course, there is a great gap between the possibility of an afterlife for an animal and its actuality. And the above argument works just as well for plants and fungi.

Formas aristotélicas y leyes de la naturaleza

Rafael Abuchedid kindly translated my piece on Aristotelian forms and laws of nature into Spanish.

A metaphysical argument for survivalism

Corruptionist Thomists think that after death and before the resurrection, our souls exist in a disembodied state and have mental states, but we do not exist. For we are not our souls. Survivalist Thomists think we continue to exist between death and the resurrection. They agree that we are not our souls, but tend to think that in the disembodied we have our souls as proper parts.

Here is a metaphysical argument against corruptionism and for survivalism.

1. An accident that has a subject is a part of that subject.

2. There are mental state accidents in the disembodied state.

3. All mental state accidents in the disembodied state have a subject.

4. The soul does not have accidents as parts.

5. Therefore, the mental state accidents in the disembodied state have something other than the soul as their subject.

6. The only two candidates for a subject of mental state accidents are the soul and the person.

7. Therefore, the mental state accidents in the disembodied state have the person as their subject.

8. Therefore, the person exists in the disembodied state.

(This argument is a way of turning Jeremy Skrzypek’s accident-based defense of survivalism into a positive argument for survivalism. Maybe Skrzypek has already done this, too.)

The argument is slightly complicated by the fact that Thomists accept the possibility of subjectless accidents existing miraculously (in the Eucharist). Nonetheless, I do not know of any Thomists who think the disembodied state is such a miracle. Given that Thomists generally think that the survival of the soul after death is not itself miraculous, they are unlikely to require the miracle of subjectless accidents in that case, and hence will accept premise 3.

Premise 2 is common ground between survivalists and corruptionists, as both agree that there is suffering in hell and purgatory and joy in heaven even in the disembodied state.

I think the controversial premises are 1 and 4. I myself am inclined to deny the conjunction of the two premises (even though I think survivalism is true for other reasons).

Premise 1 is a core assumption of compositional metaphysics, and compositional metaphysics is one of the main attractions of Thomism.

One reason to accept premise 4 is that the soul is the form of the human being, and one of the main tasks for forms in Aristotelian metaphysics is to unify complex objects. But if forms are themselves complex, then they are also in need of unification, and we are off on a regress. So forms should be simple, and in particular should not have accidents as parts.

Another reason to accept 4 is that if the soul or form has mental state accidents as parts, it becomes very mysterious what else the form is made of besides these accidents. Perhaps there is the esse or act of being. But it seems wrong to think of the form as made of accidents and esse. (I myself reject the idea that objects are “made of” their parts. But the intuition is a common one.)

Progress on Norms, Natures and God

In the fall, I opened a github repository for my in-progress Norms, Natures and God book manuscript, but all I had was a table of contents. I’ve finally started to regularly contribute text to the book. You can monitor my progress here, and you’re welcome to submit suggestions and bug reports via the Issues page.

Don’t count on the repository being available permanently: it will disappear when it’s time to submit to a publisher. (My preferred way to write books is to write them and then submit the whole draft to a publisher.)

Wilde lectures now online

Videos of my 2019 Wilde Lectures in Natural Theology and Comparative Religion at Oxford’s Oriel College are now available online.

Here are the slides:

• Beauty
• Scepticism
• Forms I
• Forms II
• The finite

Book in Progress: Norms, Natures and God

I have begun work with a working title of Norms, Natures and God, which should be a book on how positing Aristotelian natures solves problems in ethics (normative and meta), epistemology, semantics, metaphysics and mind, but also how, especially after Darwin, to be an intellectually satisfied Aristotelian one must be a theist. The central ideas for this were in my Wilde Lectures.

There is a github repository for the project with a PDF that will slowly grow (as of this post, it only has a table of contents) as I write. I welcome comments: the best way to submit them is to click on "Issues" and just open a bug report. :-)

The repository will disappear once the text is ready for submission to a publisher.

An Aristotelian account of proper parthood (for integral parts)

Here it is: x is a proper part of y iff x is informed by a form that informs y and x's being informed by that form is derivative from y's being informed by it.

Emotions and naturalism

On occasion, I’ve heard undergraduates suggest that naturalism faces a problem with emotions. They feel that a mere computational system would not have emotional states.

One might take this to be a special case of the problem of qualia, and I think it has some plausibility there. It is indeed hard to see how an emotionless Mary would know what it’s like to be scared or in love. Is it harder than in the case of ordinary sensory qualia, like that of red? I don’t know.

But I think it’s more interesting to take it to be a special case of the problem of intentionality or content. Emotions are at least partly constituted by intentional (quasi?) perceptual states with normative content: to be scared involves perceiving reality as containing something potentially bad for one and being in love involves perceiving someone as wonderful in some respects.

The standard materialist story about the content of perceptual states is causal: a perception of red represents an object as reflecting or emitting light roughly of a certain wavelength range because the perception is typically triggered by objects doing this. But on standard naturalist stories do not have room for normative properties to play a causal role. Post-Aristotelian scientific explanations are thought not to invoke normative features.

There is, of course, nothing here to worry an Aristotelian naturalist who believes that objects have natures that are both normative and causally explanatory.

Over the past year, I’ve been coming to appreciate the explanatory power of the Aristotelian story on which the very same thing grounds normativity and provides a causal explanation.

Tropes of tropes

Suppose that x is F if and only if x has a trope of Fness as a part of it.

Here is a cute little problem. Suppose Jim is hurting and has a trope of pain, call it Pin. But Pin is an improper part of Pin. Thus, Pin has a trope of pain—namely itself—as a part of it, and hence Pin is hurting. Thus, wherever someone is hurting, there is something else hurting, too, namely their pain.

The standard move against “two many thinkers” moves is to say that one of them is thinking derivatively. But if we do that, then it looks like the fact that Jim is hurting is more likely to be derivative than the fact that Pin is hurting. For Jim hurts in virtue of having Pin as a part of it, while Pin hurts in virtue of having itself as a part of it, which seems a non-derivative way of hurting. But it seems wrong to say that Jim is hurting merely derivatively, so the real subject of the pain is Pin.

An easy solution is to say that x is F if and only if x has a trope of Fness as a proper part of it.

But this leads to an ugly regress. A trope is a trope, so it must have a trope of tropeness as a proper part of it. The trope of tropeness is also a trope, so it must then have another trope of tropeness as a proper part and so on. (This isn’t a problem if you allow improper parthood, as then you can arrest the regress: the trope of tropeness has itself as an improper part, and that’s it.)

One can, of course, solve the problem by saying that the trope theory only applies to substances: a substance x is F if and only if x has a trope of Fness as a proper part of it, while on the other hand, tropes can have attributes without these attributes being connected with the tropes having tropes. But that seems ad hoc.

As a believer in Aristotelian accidents and forms, which are both basically tropes, I need to face the problem, too. I have two ways out. First, maybe all tropes are causal powers. Then we can say that if “is F” predicates a power, then x is F if and only if x has a trope of Fness as a proper part. But for attribution of non-powers, we have a different story.

Second, maybe the relation between objects and their tropes is not parthood, but some other primitive relation. Some things stand in that relation to themselves (maybe, a trope of tropeness stands in that relation to itself) and others do not (Pin is not so related to itself). This multiplies primitive relations, but only if the relation of parthood is a primitive relation in the system.

A Traveling Forms Interpretation of Quantum Mechanics

Paper is here.

Abstract: The Traveling Minds interpretation of Quantum Mechanics is a no-collapse interpretation on which the wavefunction evolves deterministically like in the Everett branching multiple-worlds interpretation. As in the Many Minds interpretation, minds navigate the Everett branching structure following the probabilities given by the Born rule. However, in the Traveling Minds interpretation (a variant by Squires and Barrett of the single-mind interpretation), the minds are guaranteed to all travel together--they are always found in the same branch.

The Traveling Forms interpretation extends the Traveling Minds interpretation in an Aristotelian way by having forms of non-minded macroscopic entities that have forms, such as plants, lower animals, bacteria and planets, travel along the branching structure together with the minds. As a result, while there is deterministic wavefunction-based physics in the branches without minds, non-reducible higher-level structures like life are found only in the branch with minds.

From unrestricted composition to unrestricted caninity

According to unrestricted composition (UC) for any plurality of things there is a whole that is exactly composed of them. Sider offers a continuity argument for UC. Here's a vivid formulation. Let the Ps be the particles in the even-numbered books on one of my bookshelves. If UC is false, then in the actual world the Ps will be a paradigm case of something that doesn't compose a whole. But there is a world where the Ps compose a dog. And between these two worlds there is a continuous sequence of worlds where the Ps gradually migrate from their every-second-book positioning to their canine positioning. It is absurd to think that suddenly somewhere in this continuous sequence the particles come to compose something. So, Sider concludes, they compose something all along, even in the actual world.

But to a hylomorphist, the argument as I've put it simply fails. There is no world where the Ps compose a dog, since a dog—or any other complex entity—is not composed of matter, but of matter and form. The argument can, however, be reformulated. Say that the Ps materially compose an F provided that the Ps are material and together with some form compose an F. Then the argument gets off the ground. In the actual world, the Ps do not materially compose anything while in the final world they materially compose something. Where along the line do they come to materially compose something?

Now, however, the story is underdescribed. For we have failed to say in which worlds in the sequence there is a substantial form of the dog informing the Ps. Facts about substantial forms should not be assumed to supervene on facts about the arrangement of the particles. There could be zombie dogs that are nothing but heaps of particles looking like a dog. In other words, it's a contingent matter whether a certain kind of arrangement of particles materially composes something—if there is a form informing them, then they compose and if not, not.

Of course, there is a question of explanation: Why is there no form informing the Ps in the actual world but there is one in the the non-zombie dog worlds? But the answers aren't particularly troublesome. Maybe the laws of nature explain that. Maybe God just decides when to create forms and make them inform particles.

However, there is a final move that Sider can make. Instead of asking in which worlds the Ps (materially) compose something, he could ask which arrangements of particles are such that something could be materially composed of the particles in that arrangement. Of course the dog-like arrangement is like that. And the even-numbered-book arrangement is not. So where is the transition in the continuous deformation of the even-numbered-book arrangement into the dog-like arrangement?

This is an interesting question for the hylomorphist. It is closely to the question of what forms there could be (cf. the discussion here and in the Murphy book referenced in the comments there). The hylomorphist could take an unrestricted view. There is a sufficiently wide variety of possible forms and defects that any possible arrangement of matter is compatible with being informed by some form—perhaps defectively. There could be a possible world where something looking just like our even-numbered-book arrangement is a highly defective (it doesn't grow or reproduce) plant.

Nonetheless, there is a remaining problem. While the even-numbered-book arrangement may be apt for materially composing a defective plant, it's surely inapt for materially composing a dog. So there will seem to be a discontinuous transition between those arrangements that can and those that cannot materially compose a dog. One answer here is that "dog" is vague. This doesn't fit with traditional Aristotelian views, though, on which all dogs have an exactly similar form, and so one could meaningfully ask about the range of arrangements that could be informed by a form that's exactly like that. But perhaps the Aristotelian can yield some ground here. Another answer would be unrestricted canine composition: any material arrangement could materially compose a dog, albeit a highly defective one. I am somewhat drawn to this strange view. Yet is it that strange? I think I can imagine a dog continuously deforming into the even-numbered-book arrangement but where rather than dying the dog comes to be more and more defective. I am dualist enough that I can even imagine the dog being conscious throughout the process.

Are parts modes?

There are two variations on Aristotelian ontology. On the sparser version there are substances and their modes (accidents and essences). On the more bloated version there are substances, modes and (proper) parts. I want to argue that the more bloated version should be reduced to the sparser one.

Parts in an Aristotelian ontology are unlike the parts of typical contemporary ontologies. They are not substances, but rather they are objects that depend on the substance they are parts of. At least normally when a part, say a finger, comes to be detached from the substance it is a part of, it ceases to exist—a detached finger is a finger in name only, as Aristotle insists.

This makes the parts of Aristotelian ontology mode-like in their dependence on the whole. Ockham's razor then suggests that rather than supposing three fundamental categories—substances, mode and parts—we will do better to posit that a part is just a kind of mode. Thus, I really do have a heart, but my heart is just much a mode or accident of me—my cardiacality—as my knowing English is. Both my heart and my knowledge of English confer on me certain causal powers and causal liabilities (knowing English makes me liable to having my feelings hurt by uncomplimentary assertions in English!)

This is not an elimination of parts. Some of my accidents are parts and others are not. Which ones? I do not know. Maybe those accidents that occupy space are parts and those accidents that do not are not. My knowing English doesn't occupy space, while my cardiacality is somewhat vaguely but really located located in space.

Perhaps we need a finer distinction, though. Consider the strength of my arm. This isn't a part of me, but it seems to be located in my arm. I suggest that we distinguish between three ways that a mode can get a location. It can (a) inherit a location from a subject, or (b) it can inherit a location from its own modes, or (c) it can be located in its own right. I suggest that a mode is a part if and only if it has a location of type (b) or (c). The strength of my arm inherits its location from its subject—my arm—and hence is not a part. (It's important to the full development of this ontology that modes can nest. Thus, my arm is a mode of me, and the strength of that arm is a mode of this mode. Both I and my arm are subjects of the strength of the arm.)

I think the distinction between type (b) and type (c) parts is worth thinking about. Maybe matter, that mysterious ingredient in Aristotelian ontology, can be identified with type (c) parts?

The vagueness argument against restricted compositionality

Lewis and Sider have argued that if restricted compositionality is true—some but not all pluralities of two or more objects compose a whole—then there will be cases where it's vague how many objects there are. For instance, imagine two universes, A and B, each with the same finite set of n particles with the same intrinsic properties. But in A, the particles are neatly arranged into galaxies, trees, tables, buildings, etc. And in B there is just a blooming buzzing confusion. If restricted compositionality holds, then, assuming there are no immaterial objects, universe B has exactly n or at most n+1 objects—it's just too messy to have any cases of composition, except perhaps for the universe as a whole (that's why it might be n+1 rather than n). But A is much like our universe, and so we would expect lots of cases of composition, and hence the number of objects will be a lot more than n+1, say n+m for some large m. However, we can now imagine a continuous sequence of universes ranging from A to B, differing continuously in how the particles are arranged. As we move that continuous sequence, the number of objects will have to change from no more than n+m to n+1. But it is incredible that the object count should sharply change due to a very tiny shift in particle positions. Instead, the object count will at times be vague. But how many objects there are is a matter of which sentences using universal quantification, conjunction, negation and identity are true. But quantification, conjunction, negation and identity are not vague. So we have vagueness where we cannot have vagueness.

There may be some technical problems with the argument as I formulated it, given the assumption of no immaterial objects. Maybe we can't do without immaterial entities like God or numbers. One could reformulate the argument to restrict the counting to material entities, but "material" might actually be a vague term. Perhaps the best thing to do is to assume that these universes have no immaterial contingent entities, and then just count contingent entities. Contingency shouldn't be a vague matter, after all. The Aristotelian may balk at this. For it may well be that a necessary condition for a bunch of material entities to compose a whole that they have a form, and forms are immaterial but contingent. Maybe, though, "form" is not vague, and so we can just count the contingent non-forms.

But talking of forms suggests a more serious difficulty. If there are Aristotelian forms, then how many material objects there are may well not supervene on how material objects are spatiotemporally arranged and what intrinsic properties they have. For objects to come to compose a whole, there must come into existence a form. There is nothing absurd about there being sharp laws of nature specifying under which precise conditions a form comes into existence. There is no need for the laws of nature to be continuous (and the possibility of fundamental discreteness is empirically still open). Or perhaps God decides on a case-by-case basis whether to create a form. Then there is no vagueness as to how many material objects there are: the number of material objects equals the number of forms of material objects that in fact inform some matter (the souls of the resurrected are forms of material objects but temporarily fail to inform any matter). Of course in transitional cases we won't have much confidence whether some objects compose a whole, but that's just because we are unable to see forms except through their normal effects.

Plato might have been a "nominalist"

I was reading the SEP entry on nominalism by Rodriguez-Pereyra. Rodriguez-Pereyra sees nominalism as basically the rejection of causally inert non-spatiotemporal entities. If so, then Plato might have been a nominalist. It seems that Plato did not think the Form of the Good was causally inert--it caused the good arrangement of things in the universe. I don't know if Plato generalized from that case, but he might well have--he might have taken all of the Forms to be capable of causing things to be like them. So, for all I know, Plato was a nominalist.

And Leibniz might have been was a nominalist despite going on and on about abstract objects, because he thought of them as ideas guiding God's deliberation, and hence perhaps we should say that on his view they had a causal role in creation.

This isn't a big deal. Rodriguez-Pereyra's account nicely captures a rejection of modern forms of Platonist.

I wonder, too, whether a belief in Newtonian space is compatible with nominalism by this definition. Newtonian space seems to be causally inert (perhaps unlike the Riemannian manifold of General Relativity). And it may be a category mistake to say that space is spatiotemporal. Though maybe it's fine to say that space is spatiotemporal in some trivial sense.