Space, time and discreteness

A first plausible thesis:

1. Space is discrete if and only if time is discrete.

This is very plausible in any picture like that given by modern physics where space and time, while not on par, are very closely related. It's also intuitive. For suppose space is continuous but time is discrete. Then a small enough object will jump over some intervening space when it moves during a unit of time, which seems strange. Conversely, if space is discrete but time is continuous, then objects always move jerkily: for a bit of time they stand still, then they instantaneously jump to a new point in space, and then they stand still for a little longer. That's strange, too.

Here's a second plausible thesis:

2. A small object can rotate by any small angle without its internal measurements changing much.

But now imagine that space is discrete, with the smallest distance between points being about α. Imagine an object occupying exactly two points x and y spaced out by approximately α. Now swivel that object slightly around the point x without changing internal measurements much. The other point in the object will either continue occupying y, in which case the object hasn't rotated (contrary to (2)), or will occupy some point very close to y (but there isn't any such point, since the minimum spacing is about α), or the object will come to occupy only one point (in which case its internal measurements will change much).

If the above argument isn't clear, just imagine a hexagonal or square grid, an object composed of two points on the grid, and think what it would be to rotate that object by a small angle (smaller than 90 degrees in the case of the square grid and smaller than 60 degrees for the hexagonal one).

So (2) gives us good reason to deny that space is discrete, and then (1) gives us good reason to deny that time is discrete.

But this was too quick. Both my arguments for (1) and my argument that (2) forbids space to be discrete made a crucial assumption, namely that space has a certain fixity to it that it is independent of the objects in space. For suppose that time is continuous but space is discrete. I said that it follows that objects move jerkily. Not so. For the points that the objects occupy could be moving with the objects! Thus an object could move smoothly because the spatial points in it could be moving. The discrete space, then, wouldn't be a regular grid. It would be a mess of points, which shift around as the objects they are in shift. (This doesn't affect the argument that we shouldn't say that space is continuous but time is discrete.)

The same flaw affects my argument based on (2). I was assuming that as I rotate the two-point object, the points x and y stay fixed. But what if points are defined by objects, and so the point y rotates with the object? Again, we wouldn't have a regular grid. We would have an irregular changing grid, where the real points are defined by the objects.

The resulting view of space would be, I think, a version of Aristotle's picture, where space is infinitely divisible but not actually infinitely divided. In the case of our two point object, there could be a point at distance α/10 from y, but there isn't, unless we rotate the object that defines the points.

In other words, Aristotle's account of space is the only discretist view of space that accommodates the intuition that objects can be rotated by small amounts without great distortion. That's pretty neat, I think.

What's the motivation for thinking this is the truth of the matter. Well, causal finitism gives one good reason to think that time is discrete (or at least discrete when we restrict ourselves to a local area of space). The implication from discrete time to discrete space in (1) survives my above criticism of the argument. So we have good reason to think space is discrete. And then the rotation argument yields a version of Aristotle's view.