A fun little argument against four-dimensionalism

1. Spinning a rigid object cannot affect its shape.

2. If four-dimensionalism is true, spinning a rigid object can affect its shape.

3. So, four-dimensionalism is not true.

The easiest way to see that 2 is true is to imagine that space is two-dimensional. Then if objects are considered to be extended in time, as the four-dimensionalist says, an object intuitively thought of as a rectangle that stays still is really a rectangular prism, while if that rectangle is spun by 90 degrees, it looks like a twisty thing.

I don’t think it’s too costly to deny 2. And perhaps one can make sense of some notion of internal shape that doesn’t change no matter how a rigid object moves around.

An argument against time travel or for temporal parts or for internal time

Start with this plausible claim:

1. If two objects are composed of the very same particles at the same time, then they have the same shape.

But now consider a statue of a horse that is reshaped into a statue of a tree and then time-travels back to sit besides the statue of the horse. Then the statue of the horse and the statue of the tree are composed of the very same particles at the same time, and yet they do not have the same shape.

I see three ways out of this paradox.

2. Deny the possibility of time travel.

3. Subscribe to temporal parts theory and modify (1) to speak of temporal parts of particles instead of particles.

4. Distinguish external and internal time, and qualify (1) to refer to internal time.

My preference is (4).

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.

Inductive evidence of the existence of non-spatial things

Think about other plausibly fundamental qualities beyond location and extension: thought, charge, mass, etc. For each one of these, there are things that have it and things that don’t have it. So we have some inductive reason to think that there are things that have location and things that don’t, things that have extension and things that don’t. Admittedly, the evidence is probably pretty weak.

Shape is not an intrinsic property

1. (Premise) If an object can change in shape without undergoing intrinsic change, shape is not an intrinsic property.
2. (Premise) If the diameter[note 1] of an object changes while its perimeter does not, the object changes in shape.
3. (Premise) An object can change in diameter but not in perimeter without undergoing intrinsic change.

The thought behind (2) is that the shape of an object determines the ratios of distances between parts.

Now I argue for (3). Imagine a giant hula-hoop, a light-year in diameter, without anything inside. Suppose that God creates a massive star in the middle. This distorts the spacetime manifold in the vicinity of the star, changing the distances between diametrically opposed points on the hula-hoop. But it will take half a year for the changes in the spacetime manifold to propagate to the hula-hoop. Thus the perimeter of the hula-hoop is unchanged for half a year. Furthermore, surely, the creation of a star half a light-year from any part of an object doesn't intrinsically change the object for at least half a year.

So, the hula-hoop (a) is intrinsically unchanged, (b) its perimeter is unchanged, and (c) its diameter is changed, which yields (3).

This is a modification of an argument in a paper of mine on the Eucharist.