Shape and parts

Alice is a two-dimensional object. Suppose Alice’s simple parts fill a round region of space. Then Alice is round, right?

Perhaps not! Imagine that Alice started out as an extended simple in the shape of a solid square and inside the space occupied by her there was an extended simple, Barbara, in the shape of a circle. (This requires there to be two things in the same place: that’s not a serious difficulty.) But now suppose that Alice metaphysically ingested Barbara, i.e., a parthood relation came into existence between Barbara and Alice, but without any other changes in Alice or Barbara.

Now Alice has one simple part, Barbara (or a descendant of Barbara, if objects “lose their identity” upon becoming parts—but for simplicity, I will just call that part Barbara), who is circular. So, Alice’s simple parts fill a circular region of space. But Alice is square: the total region occupied by her is a square. So, it is possible to have one’s simple parts fill a circular region of space without being circular.

It is tempting to say that Alice has two simple parts: a smaller circular one and a larger square one that encompasses the circular one. But that is mistaken. For where would the “larger square part” come from? Alice had no proper parts, being an extended simple, before ingesting Barbara, and the only part she acquired was Barbara.

Maybe the way to describe the story is this: Alice is square directly, in her own right. But she is circular in respect of her proper parts. Maybe Alice is the closest we can have to a square circle?
Here is another apparent possibility. Imagine that Alice started as an immaterial object with no shape. But she acquired a circular part, and came to be circular in respect of her proper parts. So, now, Alice is circular in respect of her proper parts, but has no shape directly, in her own right.

Once these distinctions have been made, we can ask this interesting question:

• Do we human beings have shape directly or merely in respect of our proper parts?

If the answer is “merely in respect of our proper parts”, that would suggest a view on which we are both immaterial and material, a kind of Hegelian synthesis of materialism and simple dualism.

Extended simples

There are at least two reasons to think we are simple:

1. It is difficult to explain how a non-simple thing can have a unity of consciousness.

2. There is David Barnett’s “pairs” argument.

But we are clearly extended.

So, we are extended simples.

So, there are extended simples.

(That said, while I am happy with the idea that we are extended simples, I am suspicious of both 1 and 2.)

Extended simples

1. It is possible to have a simple that exists at more than one time.

2. Four-dimensionalism is true.

3. So, temporally extended simples are possible. (By 1 and 2)

4. If four-dimensionalism is true, the time and space are metaphysically very similar.

5. So, probably, spatially extended simples are possible.

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.

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.

Partial location, quantum mechanics and Bohm

The following seems to be intuitively plausible:

1. If an object is wholly located in a region R but is not wholly located in a subregion S, then it is partially located in R−S.
2. If an object is partially located in a region R, then it has a part that is wholly located there.

The following also seems very plausible:

3. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is 1, then the particle is wholly located in the closure of R.
4. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is strictly less than one, then the particle is not wholly located in the interior of R.

But now we have a problem. Consider a fundamental point particle, Patty, and suppose that Patty's wavefunction is continuous and the integral of the modulus squared of the wavefunction over the closed unit cube is 1 while over the bottom half of the cube it is 1/2. Then by (3), Patty is wholly contained in the cube, and by (4), Patty is not wholly contained in the interior bottom half of the cube. By (1), Patty is partially located in the closed upper half cube. By (2), Patty has a part wholly located there. But Patty, being a fundamental particle, has only one part: Patty itself. So, Patty is wholly located in the closed upper half cube. But the integral of the modulus squared of the wavefunction over the closed upper half cube is 1−1/2=1/2, and so (4) is violated.

Given that scenarios like the Patty one are physically possible, we need to reject one of (1)-(4). I think (3) is integral to quantum mechanics, and (1) seems central to the concept of partial location. That leaves a choice between (2) and (4).

If we insist on (2) but drop (4), then we can actually generalize the argument to conclude that there is a point at which Patty is wholly located. Either there is exactly one such point--and that's the Bohmian interpretation--or else Patty is wholly multilocated, and probably the best reading of that scenario is that Patty is wholly multilocated at least throughout the interior of any region where the modulus squared of the normalized wavefunction has integral one.

So, all in all, we have three options:

• Bohm
• massive multilocation
• partial location without whole location of parts (denial of (2)).

This means that either we can argue from the denial of Bohm to a controversial metaphysical thesis: massive multilocation or partial location without whole location of parts, or we can argue from fairly plausible metaphysical theses, namely the denial of massive multilocation and the insistence that partial location is whole location of parts, to Bohm. It's interesting that this argument for Bohmian mechanics has nothing to do with the issues about determinism that have dominated the discussion of Bohm. (Indeed, this argument for Bohmian mechanics is compatible with deviant Bohmian accounts on which the dynamics is indeterministic. I am fond of those.)

I myself have independent motivations for embracing the denial of (2): I believe in extended simples.

Are elementary particles extended simples?

This argument is valid and every premise is plausible:

1. An elementary particle is located at every point where its wavefunction is non-zero.
2. An elementary particle is simple.
3. A simple located at every point of a region with non-zero volume is an extended simple.
4. Typical elementary particles have a wavefunction that is non-zero at every point of a region with non-zero volume.
5. So, typical elementary particles are extended simples.

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.