-To call space or extension non-continuous requires some sense of denying it altogether. We can only visualize discrete quanta of space as themselves as spacial things in space which is, *ex hypothesi, *not the case. We can only visualize a pixelized universe by making the pixels X meters across and/or with Y meters between them. “Discrete space” is of itself an oxymoron, and is a metaphor for something.

-The argument that space is discrete is really a claim that space itself is either illusory or emergent or that it arises within certain bounded conditions, e.g. at subluminal speeds and superatomic sizes.

-Space seems like the simplest hypothesis to allow for order and interaction between parts of a system. Non-applicability of continuity would mean a different way of accounting for order and interaction.

-Form in non-discrete space seems to be what accounts for unity in pixelization. Unity both over time and in time and throughout motion. Form is like whatever accounts for the lights of the outdoor news-ticker making one word or letter as they “move” across the bank of lights.

-Scotus on prime matter: “it is a contradiction for pure potency to exist in act.”

-On the spacial hypothesis, interaction is the simplest sense of eliminating space, the simplest way of allowing for non-interaction is space in between. Hence the scandal of action at a distance. It seems to contradict the very reason of having space in the first place.

-Discrete parts don’t interact. They are next to each other, or, if they are, they don’t push or pull. A figure going across a digital screen is not pushing something from one pixel to the next.

-“next to”, “immediate” and therefore “here” and “there” and other such terms are redefined. How are numbers next to each other?

-Discrete space would not allow for pushing or pulling or any recognizable sense of “force”.

-Discrete space would still me real or material – it need not be a theater of appearance.

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## Kristor said,

May 17, 2014 at 11:27 pm

You’re deleting the continuum and then adding it back in as the milieu within which discrete quanta of space are located. This way, you end up with *two* extensive domains: the extensive domain cobbled together as a reification of the causal relations between discrete quanta,and the extensive domain within which those quanta are located.

Prescind from the second step, and you’re OK. Let space and time be two metrics (among many) of causal importance in the relations of events, and you’re done.

E.g.: Say that the character of quantum A is particularly important in the constitution of quantum B. Say that B inherits or takes from A the same momentum that A has.

So A is particularly important in the constitution of B. One of the ways this importance is read out by other entities observing A and B is as spatial or temporal proximity of A to B. This way of reading out their causal relations is not exhaustive: A can be important to B in lots of different ways. A may not be spatially or temporally near to B, yet still be very important to B (as the assassination of Archduke Ferdinand is important to all of us today).

Back then to simple inheritance of momentum by B from A. Momentum, p, is mv, and v is defined spatially and temporally. Our customary way of thinking about v, and thus about p, is that they are both defined by reference to some inertial frame, in terms of which we can measure v, and thus derive p. But there is no necessity that we should think that way. The equation can be interpreted the other way with equal validity, so that we then would treat v, and thus the inertial frame, as cooking out of p.

The trick then is to think of the quanta, not as existing in some volume of actual physical space that pre-exists them, but as loci in a purely mathematical n-dimensional configuration space, or Platonic realm. If A is particularly important to B, then we would say that along at least one dimension of the configuration space, A and B are close to each other. But we would have to remember that temporal and spatial proximity are only two of the dimensions of the configuration space.

It seems we are forced to some such move, if it is correct to treat things as really discrete from each other. If they are not, then there’s really only one thing.

## Allen Hazen said,

May 18, 2014 at 10:20 pm

The argument of your first paragraph — that if discrete space is conceived on the analogy of pixels, with a volume of space containing some finite number of ordered pixels, the individual pixels themselves gain (by dividing the size of the containing region by their number) length, thereby reintroducing a continuous space — has been made by first-rate philosophers. Nonetheless, I think it is fallacious. The “width” of a pixel cane assigned, but it is a derivative notion. To draw the conclusion that there is an underlying continuous space in which each individual pixel is extended requires a further assumption: the assumption that the pixel is composed of smaller parts. Mathematically, of course, one can pretend it is: one can say that a pixel of “width” 2n located at point x (x being a co-ordinate, so chosen from a mathematical continuum of real numbers or pairs or triples of real numbers) occupies an interval (x-n,x+n), and so — mathematically — one can speak of the interval (x-n,x) as the “left half” of the pixel. But this is purely a mathematical fiction: speaking of the mathematical objects (intervals) used as coordinates AS IF they were the physical pixels themselves.

The claim that space is discrete amounts to the denial that talk of parts of pixels has any physical meaning. No physical facts (no observable facts, no facts postulated by well-founded theories) need anything smaller than whole pixels for their specification.

To borrow a Whiteheadian term, thinking of parts of pixels is a fallacy of misplaced concreteness: taking something that can meaningfully be affirmed of the mathematical abstraction (coordinate system) and assuming it is therefore to be affirmed of the physically real.

## Kristor said,

May 19, 2014 at 6:02 pm

Exactly. If a pixel or quantum of action is conceived as a point in a purely mathematical n-dimensional configuration space – so that within that space it has no extension – and we hang all its properties on it as coordinates of the configuration space, then among those coordinates may be numbered its extensions in physical space and time, its momenta, its charge, its charm, and so forth. We don’t say of a coordinate that if we halve it we will still be specifying the same locus in the configuration space that we were specifying before we performed the operation of division.

We then of course want to ask where the configuration space is “located.” But to ask that question is again to commit the fallacy of misplaced concreteness, and to add back in the physical space we had started out by deleting when we asked whether space is quantal. It would be more accurate perhaps to ask who is conceiving of the configuration space. One doesn’t ask where God is located, after all.