A first plausible thesis:

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

Here's a second plausible thesis:

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

*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.

## 8 comments:

But space is not discrete because:

1. If space is discrete, then there can't be extended objects.

2. There can be extended objects.

3. Therefore, space is not discrete.

On behalf of (1):

First, I assume the discreteness of space is not a contingent matter: if space *can* be continuous, then it is continuous. So can space be continuous? It can be if (i) there can be extended objects, and (ii) unextended simples are impossible. So why think unextended simples are impossible? Here is a reason that compels me. Suppose there could be an extended simple. Then it seems to me that there *could* be extended simples of any shape and size: it would be really weird of there could be a circular simple but not a triangular one, for example. Now with simples of any shape and size, it should be possible for one simple to bump into the "corner" or "top-half" of another... But the simplest and most straightforward account of "corners" and "top-halves" is that they are *parts* of the things they are corners/top-halves of. So that sort of interaction is not possible, So, extended simples are not possible. So, continuous space is possible if extended things are possible. Hence, premise 1 is true.

Premise 2 is plausible a priori (one might think).

I assume (ii) should read: *extended* simples are impossible.

But I don't see how it follows that space can be continuous from the claim that there can be extended objects and extended simples are impossible.

An extended object might occupy only two points.

Right, I meant extended simples are impossible. And I was thinking of extended *closed* objects. (You can't have two points *right* next to each other, can you?)

"closed" wasn't the right word, but you know what I mean ("connected").

On 2 of the OP: Maybe the world looks different at different scales. At a human scale we seem to see things moving and turning smoothly. But at some possible fundamental scale, who knows? Maybe the world is a giant cellular automaton.

For an analogy, think of a video game. You seem to see things moving, spinning and interacting. But at a scale of pixels and milliseconds the image is made up of discrete fixed pixels with discrete colour values that change at discete times.

Like IanS, I do not think intuitions drawn from macroscopic experience are of much use in addressing this question. The video game analogy occurred to me too. This criticism applies to the arguments for both (2) and (ii).

About the Aristotelian (?) idea that points themselves might move around: if they do, aren't they moving around IN some kind of second-order space? This just raises the same problem again.

Regarding point 1: Measuring at an arbitrary time the matter may become quantized (and discrete) but was probabilistically taking up a specific amount of area. Matter can exist in a probabilistic location as seen with electrons; within an electron shell no other matter can exist, which gives matter the appearance that it is occupying a discrete space that no other matter can violate. If this is the case of the rotating object, you would only ever know it's components' locations after measuring at a given time, which could then be quantized upon measuring, giving the appearance of being placed on a discrete grid system. When in motion, the object would appear "antialiased" so to speak, because of this probability principle. This has been proposed in the past with Planck scale being the unit of measurement, however this use of the Planck scale been disproved by measuring the resolution of moving light from gamma ray bursts (by 14 orders of magnitude, if memory serves)

I have been exhaustively looking at new papers coming out about discrete time just out of personal interest. Here's an article about time emerging from quantum entanglement:

https://medium.com/the-physics-arxiv-blog/quantum-experiment-shows-how-time-emerges-from-entanglement-d5d3dc850933

It seems like an interesting thought experiment, and I think it is possible that time and entanglement may have some relationship. I do not think that time doesn't exist at all (as implied by the article) simply because it is counter-intuitive. It may be that--like some properties of matter--time is only discrete when measured.

I forgot to make a closing statement to sum my point up. Ultimately I think that there is no way of knowing if space-time is discrete.

BUT, my personal view is that it is not discrete, location as a property of infinitesimal matter may be probabilistic until measured, and time may have infinite resolution but there is no way to observe or test that, as far as I know.

Interesting topic! I love it.

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