Is physical space, the space of the natural world, continuous or discrete? If composed of space atoms, then discrete. The Weyl Tile argument (WTA), however, seems to show that physical space cannot be discrete or 'quantized' and therefore must be continuous. This is relevant to our ongoing debate about potential versus actual infinities. For if physical space cannot be discrete, then it must be at least compact (the lowest grade of continuity), where "A series is called compact when no two terms are consecutive, but between any two there are others." (Bertrand Russell, Our Knowledge of the External World, Norton, 1929, p. 144.) But if between any two points in space there are others, then there are infinitely many others, so that any line segment will be composed of an actual infinity of points.
But before we return to the question of actual infinities we need to get clear about the WTA itself. The nervus probandi lies in the following quotation from Hermann Weyl, Philosophy of Mathematics and Natural Science, Princeton UP, 1949, p. 43:
If a square is built up of miniature tiles, there are as many tiles along the diagonal as along the side; thus the diagonal should be equal in length to the side.
Take a gander at the chess board below. Consider the right triangle the sides of which are a1-a8 and a1-h1, and the hypotenuse of which is the diagonal a8-h1. The sides and the diagonal are each eight squares long. Count 'em and see. But this flies in the face of the theorem of Pythagoras. If the sides are each eight units in length, then the hypotenuse is equal to the square root of (82 + 82 =128), which is not 8, but the irrational 11.313 . . .
The question this curious fact raises is whether physical space can be quantized, i.e., whether there are space atoms. If so, space is discrete as opposed to continuous. It may help to bear in mind that the above array is a mere model in continuous space of discrete space. So it will do no good to object that if space atoms are squares, then the theorem of Pythagoras hold for them. Space atoms are not squares: they have no shape at all. But I am getting ahead of my story.
We need to define our terms. Space is discrete just in case every finite extended spatial region is composed of finitely many atomic spatial regions. That amounts to saying that every finite extended region of space is composed of finitely many space atoms, where 'atom,' as its etymology suggests, implies indivisibility. You cannot 'split' a space atom because such atoms are inherently 'unsplittable.' A space atom is thus an individual that has no proper parts: it is a mereological atom. A non-atomic region of space is then a mereological sum of space atoms. Note that every space atom, precisely because it is an atom, is an unextended region of space. It's an itty-bitty unextended bit of space itself, not of something in space. Space atoms are not in space; they compose space.
Now for the argument:
1) The theorem of Pythagoras is not true (or even approximately true) of discrete space.
2) The theorem of Pythagoras is true (or approximately true) of actual space. Therefore:
3) Actual space is not discrete.
To understand this argument, you have to understand that nothing rides on how small the tiles/squares are. Glance back at the chessboard. Consider the small right triangle in the bottom left corner of the board. Opposing sides and hypotenuse all have a length of two units. So it doesn't matter how small the space atoms are. No matter how small the squares, the hypotenuse remains equal in length to the other two sides.
You will be tempted to think of the array of tiles/squares against the backdrop of continuous Euclidean space for which the Pythagorean theorem holds. Thinking in this way, you will imagine that no matter how small you make the tiles, the diagonal will be longer than the sides. You have to resist this temptation to understand the 'Weyl tile' (vile tile?) argument. For if there are space atoms, then they have no shape and hence no different dimensions in different directions. As Wesley C. Salmon puts, "In discrete space, a space atom constitutes one unit, and that is all there is to it. It cannot be regarded as properly having a shape, for we cannot ascribe sizes to parts of it — it has no parts." (Space, Time, and Motion, U of Minnesota Press, 1980, p. 66)
I have found K. McDaniel, "Distance and Discrete Space," Synthese (2007) 155: 157-162, very helpful. He has an argument against the WTA which I may discuss in a subsequent post.
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