Thursday, December 4, 2014

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:

  1. 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.

13 comments:

Joshua Rasmussen said...

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).

Alexander R Pruss said...

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.

Joshua Rasmussen said...

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?)

Joshua Rasmussen said...

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

IanS said...

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.

Heath White said...

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.

Unknown said...

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.

Unknown said...

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.

Alithea said...

Dr. Pruss

I have a hard time understanding this a bit. For example, if space is continuous and I move my hand closer to my chest, that would be a continuous change, so would there be continuous time? How does discrete space work on the Aristotelian view? I am confused. This is particularly important when talking about the Causal Principle and the Principle of Sufficient Reason. You make an argument in chapter 3 of your book The Principle of Sufficient Reason: A Reassessment that one can go from the CP to the PSR because there would still be skepticism about tiny infinite regresses. But this relies on there being able to be these tiny infinite regresses. I think what we want is a theory of time that gives us the possibility small regresses without causal finitism but is compatible with causal finitism.

Alexander R Pruss said...

Yeah, Causal Finitism rules out such "small" causal regresses. But that only makes the argument from CP to PSR work better.

Alithea said...

Dr. Pruss

I'm not really sure how causal finitism ruling out small regresses helps the PSR, but my concern is not with causal finitism ruling out small causal regresses. My main concern is discrete time ruling out small causal regresses. If one accepts causal finitism, then one already accepts a first cause. What really needs to be done is to have a theory of time that rules out small regresses with causal finitism, but keeps them without it. That way, we could have a situation where one is forced to accept a first cause either way. In your book, Infinity, Causation, and Paradox, you say that something undergoes an internal time change when it causes something else and comes into existence, among other things. Without causal finitism, it seems possible that an object appears, which was caused by an object that appeared half a second ago, and so on. Each object would have an instance of two internal times, it coming into existence, and it causing something else to come into existence. External time would be discrete, of course. But, if ruling out small causal regresses helps the PSR, I would like to know more. I'm still not sure about the continuous change in space I mentioned earlier. Maybe you could help me on that? Honestly, I could read a whole book on time by you, Dr. Pruss.

Alithea said...

I was thinking about this again. In Rob Koon's essay in the book Classical Theism, he argues that time is indiscrete for his argument from motion to work. So, this might have some tensions with that as well.

Heavenly Philosophy said...

It seems like the Grim Reaper Paradox supports a fixed discreteness. Imagine that there is only Grim Reaper, and the paradox happens when it is activated an infinite number of times in a Benardete sequence. Now, if time is Aristotelianly discrete, then the Grim Reaper will be able to activate in any of the times of the Benardete sequence. In fact, it will be able to activate in any finite combination of the Benardete sequence. But, it will not be able to activate at all of those points at once. So, it seems implausible that there are an infinite number of points at which it can activate in a finite interval.