Dualism, humans and galaxies

Here is a mildly interesting thing I just noticed: given dualism, we cannot say that we are a part of the Milky Way galaxy. For galaxies, if they exist at all, are material objects that do not have souls as parts.

Against Monism

According to Monism:

• (M) Necessarily if there are any concrete physical objects, then there is a concrete physical fundamental object (“a cosmos”) that has all concrete physical objects as metaphysically dependent parts.

Here an object is fundamental just in case it is not metaphysically dependent. But Monism is difficult to reconcile with the Intrinsicness of Fundamentality:

• (IF) Necessarily, if x is a fundamental object, then any exact duplicate of x is fundamental.

For simplicity, let’s call concrete physical objects just “objects”, and let’s only talk of the concrete physical aspects of worlds, ignoring any spiritual or abstract aspects.

Now consider a world w₁ that consists of a single simple object (say, a particle) α. Let w₂ be a world consisting of an exact duplicate α′ of α as well as of one or more other simple objects. Then by (M), α′ is not fundamental in w₂, since it is dependent on w₂’s cosmos (which is not just α′, since w₂ has some other simple objects). But α is the cosmos of w₁, and hence is fundamental, and thus by (IF), α′ is a duplicate of a fundamental object, and hence fundamental.

I can think of one way out of this argument for the defender of (M), and this is to deny the weak supplementation axiom of mereology and say that in w₁, there are two objects: α and a cosmos c₁ which has exactly one proper part, namely α. This allows the monist to deny that α is fundamental in w₁. Many people will find the idea that you could have an object with exactly one proper part absurd. I am not one of them in general, but even I find it problematic when the object and the proper part are both purely physical objects.

Still, let’s consider this view. We still have a problem. For in w₁, there is an object, namely c₁, that has α as its only proper part. Now, suppose a world w₃ that contains a duplicate c₁′ of c₁, and hence a duplicate α′ of α that is a part of c₁′, as well as one or more additional simples. Then c₁′ has only one simple as a proper part, and hence is not the cosmos of w₃, and thus is not fundamental by (M), which contradicts (IF).

So, we cannot have a world w₃ as described. Why not? I think the best story is that a cosmos is a unique kind of organic whole that encompasses all of reality, and that exists in every world which has a (concrete physical) object, and nothing but the cosmos can be a duplicate of the cosmos.

But this story violates the following plausible Distinctness of Very Differents principle:

• (DVD) If x and y are organic wholes made of radically different kinds of particles and have radically different shape and causal structure, then x ≠ y.

But now consider a world consisting of a cloud of photon-like particles arranged in a two-dimensional sheet, and a world consisting of a cloud of electron-like particles arranged in a seven-dimensional torus. The cosmoses of the two worlds are made of radically different kinds of particles and have radically different shape and causal structure, so they are not identical.

The composition of a substance

Start with this plausible observation:

1. Any part of me either is an accident of me or has an accident.

For consider this: my corporeal parts all have accidents of size, shape, color, etc. And my non-corporeal parts are my soul or form, as well as my accidents. My soul has accidents: such as the accident of thinking about this or that. And my accidents are my accidents.

Now, add this plausible thesis:

2. Any accident of a part of me is identical with an accident of me.

Thus, my arm’s being tanned is identical with my being tanned-in-the-arm. Further:

3. An accident of a thing is a part of that thing.

Given 1-3, we conclude the following:

4. Any part of me has at least one accident of me as a part.

For suppose that x is a part of me. Then by (1), x is an accident of me or has an accident. If x is an accident of me, then x has an accident of me, namely x itself, as an improper part. If x has an accident y, then y is a part of x by (3) and identical with an accident of me by (2), so once again x has an accident of me as a part.

Now the standard definition of composition is:

5. The xs compose y if and only if every part z of y has a part in common with at least one of the xs.

It follows from (4) and (5) that:

6. I am composed of my accidents.

For every part of me has one of my accidents as a part by (4), and that accident is of course an improper part of one of my accidents.

But (6) seems really wrong!

Thomas Aquinas has a nice way out of (6). One of my parts is my esse, my act of being, and my esse has no proper parts, and no parts in common with any of my accidents. If Aquinas is right, then it seems (4) needs to be modified to:

7. Any part of me is either my esse or has at least one accident of me as a part.

Replacing (4) with (7) in the argument, we get:

8. I am composed of my esse and my accidents.

But that seems wrong, too. For the omission of form is really glaring.

One could get out of (8) if one supposed that my form has its own esse as a part of it. But that doesn’t seem right.

My own view is that (8) may actually be correct if we stipulate “compose” to be defined by (5). But what that points to is the idea that “compose” is not rightly defined by (5).

At least sometimes parts aren't prior to wholes

1. Efficient causes are explanatorily prior to their effects.

2. Circularity of explanatory priority is impossible.

3. I am the efficient cause of my teeth—I grew them!

4. Therefore, my teeth are not explanatorily prior to me. (1–3)

5. My teeth are parts of me.

6. Therefore, at least some parts are not explanatorily prior to the wholes. (4, 5)

Inclusive vs. proper parthood

Contemporary analytic philosophers seem to treat the “inclusive” concept of parthood, on which each object counts as an improper part of itself, as if it were more fundamental than the concept of proper parthood.

It seems to me that we should minimize the number of fundamental relations that all objects have to stand in. We are stuck with identity: every object is identical with itself. But anything beyond that we should avoid as much as we can.

Now, it is plausible that whatever parthood relation—inclusive parthood or proper parthood—is the more fundamental of the two is in fact a fundamental relation simpliciter. For it is unlikely that parthood can be defined in terms of something else. But if we should minimize the number of fundamental relations that all objects must stand in, then it is better to hold that proper parthood rather than inclusive parthood is a fundamental relation. For every object has to stand in inclusive parthood to itself. But it is quite possible to have objects that are not proper parts of anything else.

On this view, proper parthood will be a fundamental relation, and improper parthood is just the disjunction of proper parthood with identity.

Substances are not parts of substances

Here is a quick and simple argument for the Aristotelian axiom that substances are not parts of substances.

1. The parts of substances are at least partly grounded in the substances.

2. Substances are not even partly grounded in other things.

3. Therefore, substances are not proper parts of other substances.

I suppose (1) is probably just as controversial as (3).

A fun unsound argument for dualism

Here’s a fun argument for dualism.

1. What is a part of the body is a matter of social convention.

2. Persons are explanatorily prior to social conventions.

3. So, probably, persons are not bodies.

I think (2) is undeniable. And (1) is a not uncommon view among people thinking about prostheses, implants, transplants and the like.

That said, I think (1) is just false.

Two kinds of partial causation

It’s interesting that there are at least two significantly different kinds of partial causation. In both of the following cases it seems reasonable to say that x partially causes y:

1. x and z together cause y

2. x causes z and z is a part of y.

I.e., the partiality can be on either side of the causal relation. And one might even combine the two, no?

My previous post was about partial causation where the partiality was on the side of the cause, not the side of the effect.

An anti-Aristotelian argument for divine simplicity

The doctrine of divine simplicity fits comfortably with Aquinas’s Aristotelian framework. But it is interesting that anti-Aristotelianism also leads to divine simplicity.

1. The proper parts are more fundamental than the whole. (Mereological anti-Aristotelianism.)

2. Nothing is more fundamental than God.

3. So, God has no proper parts.

Of course, as an Aristotelian I reject 1, so while I accept the conclusion of this argument, I can’t use the argument myself.

Compositional and non-compositional trope theories

There are two kinds of trope theories: Those on which the tropes are parts of the particular object—call these “compositional” trope theories—and those on which the relation between the object and its tropes is not a whole-to-part relation. Compositional trope theories have an initial advantage over non-compositional ones: they have no need to introduce a new relation to join objects to their tropes.

But this is only an apparent advantage. Consider this old argument. Assume compositional trope theory. Suppose my toe is blue. Then its blueness trope is a part of the toe, which is in turn a part of me, and so the blueness trope is a part of me. Hence I am blue.

Of course, the compositionalist has an answer to this argument: there are two different kinds of parthood here. The toe is, as the medievals would say, an integral part of me. And the blueness trope is a non-integral part of the toe. Transitivity holds for integral parts. It may or may not hold for non-integral parts, but it certainly doesn’t hold across types of parthood: if y is an integral part of x and z is a non-integral part of y, it does not follow that y is any kind of part of x.

But notice now that the compositionalist has lost the main advantage over the non-compositionalist. The compositionalist’s initial advantage was not having to introduce a new kind of relation over and beyond the familiar composition relation. But the familiar composition relation was the one between wholes and integral parts, and our compositionalist now has to introduce a new relation over and beyond that. Granted, it is a new relation of the same type as the familiar one. But this actually makes the compositionalist’s theory more complicated. For now the compositionalist has two relations, integral composition and non-integral composition, plus a new relation type, composition. But the non-compositionalist need only have two relations, integral composition and the object-to-trope relation. These two relations don’t need to have a new relation type to fall under. In other words, the non-compositionalist has only one mystery in her theory—what is the object-to-trope relation—while the compositionalist has two mysteries—what is the object-to-trope relation and what is the type composition.

The same point applies more generally to compositional ontologies versus relational ontologies.

Self-colocation

Self-colocation is weird. An easy way to generate it is with time travel. You take a ghost or other aethereal object who time travels to meet his past self, and then walks into the space occupied by his past self--ghosts can walk into space occupied by themselves--so that he is exactly colocated with himself. If you don't like ghosts, time travel a photon--or any other boson--into the past and make it occupy the same place as itself. But time travel is controversial.

However, it occurs to me that one can get something a bit like self-colocation with an aethereal snake and no time travel. An aetherial snake can overlap itself. First, arrange the snake in spiral with two loops. Then gradually tighten the ring, so that the outer ring of the spiral overlaps the inner one, until the result looks like a single ring. Suppose that the snake exhibits no variation in cross-section. So we have a snake that is wound twice in the same volume of space. The whole snake occupies the same region as two proper parts of itself. [I'm not the only person in this room generating odd examples: Precisely as I write this, I hear our four-year-old remarking out of the blue that she wished she had two bodies, so she could be in two places at once. A minute or so later she is talking of twenty bodies.]

(The animation was generated with OpenSCAD using this simple code.)

So far it's not hard to describe this setup metaphysically: the whole overlaps two proper parts. But now imagine that our snake ghost is an extended simple. We can no longer say that the snake as a whole occupies the same region as a proper part of it does, as the snake no longer has any proper parts. But there seems to be a difference between the aethereal snake being wound twice around the loop and its being wound only once around it.

If we accept the possibility of aethereal objects that can self-overlap and extended simples, we need a way to describe the above situation. A nice way uses the concept of internal space and internal geometry. The snake's internal geometry does not change significantly as the spiral tightens. But the relationship between the internal space and the external space changes a lot, so that two different internal coordinates come to correspond to a each external coordinate. That's basically how my animation code works: there is an internal coordinate that ranges from 0 to 720 as one moves along the snake's centerline (backbone?), which is then converted to external xyz-coordinates. Initially, the map from the internal coordinate to the external one is one-to-one, but once things are completed, it becomes two-to-one (neglecting end effects).

The idea of internal and external space allows for many complex forms of self-intersection of extended simples. And all this is great for Aristotelians who are suspicious of parts of substances.

Two kinds of parthood?

I want to explore the thesis that every plurality has a fusion. Suppose that we live in a non-gunky materialist world where everything bottoms out in particles, with all particles being simple. Assume:

1. A fusion cannot gain or lose parts.

2. A fusion continues to exist if all its simple parts do.

3. Parthod is transitive.

4. I can gain and lose simple particles.

Now let F be the fusion of me, who I suppose am made of multiple particles, with some particle P₁ outside of me. Suppose now that the following happens: I, all my particles and P₁ all continue to exist, but a new particle P₂, distinct from P₁, additionally comes to be a part of me. Then by (2), F continues to exist, since all of F’s simple parts do. By (1), I continue to be a part of F. By (3), P₂ will be a part of F. But that violates (1).

(This is not a new argument—I vaguely remember seeing something like it.)

Maybe if we accept the universality of fusions, then the sense of “part” that goes along with fusions—the sense of part in (1), (2) and (3)—is different from the ordinary sense of “part”, as when I say that my kidneys are a part of me. Let’s talk of these as f-parts and o-parts. If we do that, then we can block the argument: P₂ comes to be an o-part of me in but one cannot infer that P₂ comes to be an f-part of F, since f-parthood is transitive and maybe o-parthood is transitive, but an o-part of an f-part need not be an f-part.

That doesn’t quite solve the problem. Let’s keep on elaborating the case. Suppose that I am a material object, and I eventually exchange all my particles, but these particles continue to exist outside of me. Then by (2), F continues to exist. Moreover, by (1) I continue to be an f-part of F. But interestingly, none of my particles are f-parts of F: for the particles I now have weren’t a part of F, since they were neither o- nor f-parts of me initially, and fusions don’t gain parts. Now suppose one of my particles were an f-part of me. Then by transitivity of f-parthood, that particle would be an f-part of F, which I argued it’s not. So none of my particles is an f-part of me.

This is very weird: I’m made of particles, but no particle is an f-part of me. It seems I am f-simple. (There are some alternatives, but they are also very weird.) But presumably if this is true after the exchange of particles, it’s true before—my f-simplicity status shouldn’t change in life. So I’m always f-simple, even though I have many proper o-parts, say my particles.

It’s now looking like f-parthood is very different from o-parthood, and I wonder if it’s a kind of parthood at all.

Fingers and other alleged body parts

Squeeze your fingers around something hard. It feels like you’re making an effort with your fingers. But you’re making an effort with muscles that are in your forearm rather than in your fingers—fingers have no muscles inside them.

Now, if I thought that bodies have proper parts, I would be inclined to think that my body’s parts are items delineated by natural boundaries, say, functional things like heart, lungs and fingers rather than arbitrary things like the fusion of my nose with my toes or even my lower half. But when we think about candidates for functional parts of the human body, it becomes really hard to see where the lines are to be drawn.

Fingers, for instance, don’t make it in. A typical finger has three segments, but the muscles to move these segments are, as we saw, far away from the finger. What is included in the finger, assuming it’s a real object? Presumably the tendons that move the segments had better be included. But these tendons extend through the wrist to the muscles. Looking at anatomical pictures online, they are continuous: they don’t have any special boundary at the base of the finger. Moreover, blood vessels would seem to have to form a part of the finger, but they too do not start at the base of the finger.

Perhaps the individual bones of the finger are naturally delineated parts? But bones only have delineated boundaries when dead. For instance, living bones have a nutrient artery and vein going into them, and again based on what I can see online (I know shockingly little about anatomy—until less than a year ago, I didn’t even know that fingers have no muscles in them), it doesn’t look like there is any special break-off point where the vessels enter the bone.

Perhaps there are some things that have delineated boundaries. Maybe cells do. Maybe the whole interconnected circularity system does. Maybe elementary particles qualify, too. But once we see that what are intuitively the paradigmatic parts of the body—things like fingers—are not in the ontology, we gain very little benefit vis-à-vis common sense by insisting that we do have proper parts, but they are things that require science to find. It seems better—because simpler—to say that in the correct ontology the body is a continuous simple thing with distributional properties (“pink-here and white-there”). We can then talk of the body’s systems: the circulatory system, the neural system, ten finger systems, etc. But these systems are not material parts. We can’t say where they begin an end. Rather they abstractions from the body’s modes of proper function: circulating, nerve-signaling, digital manipulating. We can talk about the rough locations of the systems by talking of where the properties that are central to the explanation of the system’s distinctive functioning lie.

Ontological grounding nihilism

Some people are attracted to nihilism about proper parthood: no entity has proper parts. I used to be rather attracted to that myself, but I am now finding that a different thesis fits better with my intuitions: no entity is (fully) grounded. Or to put it positively: only fundamental entities exist.

This has some of the same consequences that nihilism about proper parthood would. For instance, on nihilism about proper parthood, there are no artifacts, since if there were any, they'd have proper parts. But on nihilism about ontological grounding, we can also argue that there are no artifacts, since the existence of an artifact would be grounded in social and physical facts. Moreover, nihilism about ontological grounding implies nihilism about mereological sum: for the existence of a mereological sum would be grounded in the existence of its proper parts. However, nihilism about ontological grounding is compatible with some things having parts--but they have to be things that go beyond their parts, things whose existence is not grounded in the existence and relations of their parts.

Does the size of an organism matter morally?

One might with pull a small plant from one's garden with little thought. But one wouldn't do that to a full grown tree. Of course it's harder to pull out a tree, but that doesn't seem to be all that's going on. The tree seems more significant.

Part of that is that the tree has been growing for a longer time. Temporal size definitely seems to matter. We would think a lot harder about cutting down a tree that hundreds of years old rather than one that's five years old. (Interestingly, we tend to have the opposite judgment in the case of people: it is perfectly understandable when an older person lays down their life for a child. Maybe this is because people have an irreplaceability that plants do not.)

But what about pure spatial size? Does that matter? I once thought about this case. We kill insects for minor reasons. But would we do that if the insects were our size? I thought at the time that we would have more hesitation to kill the large insects for minor reasons (we might not hesitate on self defense), but that this was an irrational bias.

But I now think there might be a justification to thinking of spatially larger organisms as having more value. The larger organisms have more cells, and that makes for a complex system, just like a castle made of ten thousand Legos is more complex, other things being equal, than one made of a thousand.

In the case of people, I guess we will have a duty of justice to bracket reasons arising from the number of cells. So we shouldn't save the fatter person just because he has more cells.

But what about dogs, say. Is it really the case that if a Chihuahua and a Great Dane are drowning, other things being equal we should try to save the Great Dane?

Maybe the differences due to the number of cells are on a logarithmic scale, and hence are only significant given an order of magnitude difference? But a Great Dane is an order of magnitude heavier than a Chihuahua, and so I'd guess it has an order of magnitude more cells.

Maybe the moral difference requires several orders of magnitude? Or maybe it runs on a loglog scale?

Or maybe I'm barking up the wrong tree and spatial size doesn't matter morally at all.

If size doesn't matter morally at all, we have a nice argument that the parts of a substance are never substances. For if the parts of a substance are ever substances, the cells of a multicellular organism will surely qualify. But if the cells are substances, then they are living substances. But surely an order of magnitude difference in the number of living substances destroyed makes a moral difference.

A consideration against Weak Supplementation

The Axiom of Weak Supplementation (WS) says that if y is a proper part of x, then there is a part of x that doesn't overlap y. Standard arguments against WS adduce possible counterexamples. But I want to take a different tack. Proper parthood seems to be a primitive relation or a case of a primitive relation (the proper metaphysical component relation seems a good candidate; cf. here). Moreover, this relation does not involve any entities besides the two relata--it's not like the relationship of siblinghood, which holds between people who have a parent in common.

But if R is a primitive binary relation that does not involve any entities besides the two relata, then it is unlikely that the obtaining of R between two entities should non-trivially entail the existence of a third entity. (By "non-trivially", I want to rule out cases like this: everything trivially entails the existence of any necessary being; if mereological universalism is true, then the existence of any two entities trivially entails the existence of their sum.) But if WS is true, then existence of two entities in a proper parthood relationship non-trivially entails the existence of another part. Hence, WS is unlikely to be true.

A way of thinking about parthood

Take as primitive a "metaphysical component of" relation. Thus, accidents and essences are metaphysical components of their substances. I am interested in a family of accounts of parthood on which

1. x is a part of y if and only x is a metaphysical component of y and F(x) (and maybe G(x)).

In other words, to be a part just is to be a metaphysical component and to be the right sort of thing (and, maybe, for the thing you're a component of to be the right sort of thing). In other words, the relation of parthood would be defined in terms of the relation of being a metaphysical component (which, I suspect, is just the "mode of" or "accident or essence of" relationship) and something that isn't a relation between the part and whole.

Virtual parts, compressed files and divine ideas

Suppose I record some video to a file on my phone. The video on my phone then is made up of frames, say, 30 of them per second. The frames are parts of the video that it sure seems like we can quantify over. But what is a frame of the video? Well, it's natural to say: The file consists of a sequence of bits implemented as flash memory states arranged spatially in the flash memory of the phone (though not always in the "logical" order, because of wear-leveling and filesystem issues). A frame then would seem to be a subsequence of these flash memory states. But that is in general false. Video files are typically compressed. While some frames--the "key frames"--are stored as a whole as a discrete sequence of bits, typical frames are not stored as a whole. Instead, what is stored is basically a set of instructions on how to modify one or more other frames in order to get the current frame. For instance, if you are panning smoothly across a static object from left to right, the non-key frames will presumably say something like: "Take most of the previous frame, shift it over a little, and then add such and such pixels on the right." But we cannot identify the bits of an instruction like that with the video frame itself, because the video frame does not supervene on the instruction: it supervenes on the instruction and the previous frame.

Even a liberal materialist ontology with unrestricted composition that allows for fusions of arbitrary disconnected sequences of bit-encoding states, the parts of the video do not exist. Noentheless, we correctly (and truthfully) say in ordinary language that the video is made up of frames as parts.

This is a rather nice illustration, I think, of the Thomistic concept of virtual parts. Virtual parts are not fundamental ingredients in the ontology. Nonetheless it is correct (and truthful) to talk of them in ordinary language. There are other such illustrations in computing. For instance, images and sounds are compressed by algorithms that transform them from spatial or temporal sequential data to frequency data or wavelet coefficients. The "natural" parts of the image or sound (say, "the left half", or "the last third") will typically not correspond to a physical part of the device memory storage.

A more homely example is, I am pretty sure, the human visual system. I see an image composed of a variety of parts. There is a lit-up rectangular part (the laptop screen), which has a left half and a right half, and so on. But even without looking up any brain research, I am willing to bet quite a lot that the disjoint spatial parts of the visual image do not correspond to particular things in my brain: I do not have an array of pixels in the brain whose parts correspond to the parts of the image (I do not even have an array of pixels in the retina corresponding to the parts of the image, as the image is stitched together by the brain over time from a variety of images produced by eyes that are constantly moving across the image).

(Can the Platonist avoid talking of virtual parts, insisting that videos and pictures are abstract objects? But even if videos and pictures are abstract objects, I doubt that they have frames and subpictures as parts.)

One thing I would like to use this story for is divine ideas. God is fundamentally simple. But we can meaningfully and truthfully talk of a multiplicity of divine ideas, in much the way that we can talk of the parts of the visual image, which are all encoded in God's one idea of all possibility. And this grounds worlds, propositions and the like.

Branchy gunk

An object is gunky provided that all of its parts have proper parts. Gunk is usually considered a really outré possibility. I want to offer some examples of intuitively conceivable gunky objects to broaden the philosophical imagination. The examples are all predicated on an Aristotelian ontology that allows for parts but denies other aspects of classical mereology. The thought behind the Aristotelian ontology of parts is that the parts of a thing correspond to natural functionally delineated subsystems. My heart is a part of me, as is my left arm. But there is no such part of me as "the left half of my heart" or "me minus my left arm".

Example 1: An infinite tree in 3D.

Here's a plausible Aristotelian thought about trees. Suppose that we have a branch A that splits into sub-branches B and C. Then branch A is an object that has both B and C as parts. However, there is no such part as A minus (B plus C). I.e., there is no object that consists of the part of A before the split. For the naturally delineated subsystem is the whole branch, including sub-branches, rather than the part of the branch without the sub-branches. Now imagine a fractal tree-like structure where the branches split into sub-branches, and the sub-branches into sub-sub-branches, and so on ad infinitum. Suppose, further, that there are no smaller natural functionally delineated subsystems than branches, sub-branches, sub-sub-sub-branches, etc. (This differs from real-world trees, which are made of cells.) The result is gunky: each part of the structure is a branch at some level, and each branch itself gives rise to sub-branches.

Dynamically, the structure can be thought of as built out of extended simples. We start with a trunk (a zero-level branch) that grows gradually. Then the trunk splits into branches. As a result, the trunk ceases to be simple: it has two or more proper simple parts, namely the branches, but it is not just the sum of the branches. The branches initially are simples, but eventually split themselves. If each step takes half the time of the preceding, after a finite amount of time we have the full infinite gunky tree.

Example 2: A four-dimensional example.

Suppose a spatial simple can survive becoming non-simple.(This was a governing assumption in the dynamical story in Example 1.) Suppose there are no proper temporal parts. Now, imagine we have a simple A, which survives becoming a non-simple made of two simples B and C. Then repeat the process with each simple. Continue ad infinitum, but don't require the process to speed up in any way. At any finite time, there are only finitely many objects. But the whole four-dimensional thing is gunky: A is made of B and C, B is made of D and E, and so on.

Example 3: Aristotelian temporal parts of a spatially simple thing.

On the Aristotelian ontology of parts, there won't be arbitrary temporal parts: there won't be the temporal part of me from my third to my fourth year. However, there might be naturally delineated temporal parts, like my adult part. Now imagine a person who never dies, and every five years receives a PhD in another discipline. If PhD-in-discipline-X counts as a naturally delineated temporal part, then the person will have a sequence of temporal parts like: doctor of biology, doctor of physics, doctor of chemistry, etc. Moreover, if we list these parts in the correct order, it gets gunk-like. If her first PhD is in biology and the second is in physics and the third is in chemistry, then the doctor of chemistry will be a part of the doctor of physics which will be a part of the doctor of biology. Moreover, there might be no such part as not-a-doctor-of-biology or not-a-doctor-of-physics (by the same token as on the Aristotelian story, there is no such part as me-minus-my-left-arm). Now, suppose that the person in question is an angel and hence has no spatial parts, and that the person has no significant temporal divisions besides the acquiring of PhDs. Then the individual is gunky: each part has a proper part. And this is easy to imagine, as long as we aren't worried about temporally extended simples.

Final remark: I don't know if these conceivable things are metaphysically possible.

If only God is perfect, then God has no proper parts

This argument is valid:

1. Only God is perfect.
2. Every part of God is perfect.
3. So, every part of God is identical with God.
4. So, God has no proper parts.

Premise (2) seems obviously true. So, we learn from the argument that if only God is perfect, then God has no proper parts.