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.

A problem with Special Relativity Theory for perdurantists

There seems to be a problem for the conjunction of Special Relativity and perdurantism.  Maybe this is a standard problem that has a standard solution? Let's say that being bent is an intrinsic property. Perdurantists of the sort I am interested in think that Socrates is bent at a time in virtue of an instantaneous temporal part of him being bent (I think the argument can be made to work with thin but not instantaneous parts, but it's a little more complicated). Therefore:

1. x is bent at t only if the temporal part of x at t is bent simpliciter.

The following also seems like something perdurantists should say:

2. x is bent simpliciter only if every temporal part of x is bent simpliciter.

Now, we need to add some premises about the interaction of Special Relativity and time.

3. There is a one-to-one correspondence between times and maximal spacelike hypersurfaces such that one exists at a time if and only if one at least partly occupies the corresponding hypersurface.

Given a time t, let H(t) be the corresponding maximal spacelike hypersurface. And if h is a maximal spacelike hypersurface, then let T(h) be the corresponding time. Write P(x,t) for the temporal part of x at t. Then:

4. P(x,t) is wholly contained within H(t) and if z is a spacetime point in H(t) and within x, then z is within P(x,t)

and, plausibly:

5. If a point within x is within a maximal spacelike hypersurface h, then P(x,T(h)) exists.

Now suppose we have Special Relativity, so we're in a Minkowski spacetime. Then:

6. For any point z in spacetime, there are three maximal spacelike hypersurfaces h₁, h₂ and h₃ whose intersection contains no points other than z.

Add this obvious premise:

7. No object wholly contained within a single spacetime point is bent simpliciter.

Finally, for a reductio, suppose:

8. x is an object that is bent at t.

Choose a point z within P(x,t) and choose three spacelike hypersurfaces h₁, h₂ and h₃ whose intersection contains z and only z (by 6). Now define the following sequence of objects, which exist by 4 and 5:

• x₁=P(x,t)
• x₂=P(x₁,T(h₁))
• x₃=P(x₂,T(h₂))
• x₄=P(x₃,T(h₃))

Observe that x₄ is wholly contained in the intersection of the three hypersurfaces h₁, h₂ and h₃, and hence:

9. x₄ is wholly at z.
10. It is not the case that x₄ is bent simpliciter.

Now:

11. x₁ is bent simpliciter. (By 1 and 8)
12. x₂ is bent simpliciter. (By 2 and 11)
13. x₃ is bent simpliciter. (By 2 and 12)
14. x₄ is bent simpliciter. (By 2 and 13)

Since 14 contradicts 10, we have a problem.  It seems the perdurantist cannot have any objects that are bent at any time in a Minkowski spacetime. This is a problem for the perdurantist. If I were a perdurantist, I'd deny 2, and maintain that an object can be bent simpliciter despite having temporal parts that are bent and temporal parts that are not bent. But I would not be comfortable with maintaining this. I would take this to increase the cost of perdurantism. What is ironic here is that it is often thought that endurantism is what has trouble with Relativity.