Weyl's tile argument

In philosophy, the Weyl's tile argument (named after Hermann Weyl) is an argument against the notion that physical space is discrete, or composed of a number of finite sized units (or tiles).^[1] The argument purports to show a distance function approximating Pythagoras' theorem on a discrete space cannot be defined and, since the Pythagorean theorem has been confirmed to be approximately true in nature, physical space is not discrete.^[2][3][4][5] While academic debate continues, counterarguments have been proposed in the literature.^[6]

A demonstration of Weyl's argument proceeds by constructing a rectangular tiling of the plane representing a discrete space. A discretized triangle, n units tall and n units long, can be constructed on the tiling. The hypotenuse of the resulting triangle will be n tiles long. However, by the pythagorean theorem, a corresponding triangle in a continuous space—a triangle whose height and length are n -- will have a hypotenuse measuring n√2 units long. To show that the former result does not converge to the latter for arbitrary values of n, one can examine the percent difference between the two results:  ^{(n√2 - n)}⁄_n√2 = 1- ¹⁄_√2. Since n cancels out, the two results never converge, even in the limit of large n. The argument can be constructed for more general triangles, but, in each case, the result is the same. Thus, a discrete space does not even approximate the pythagorean theorem.

In response, Kris McDaniel has argued the Weyl Tile argument depends on accepting the Size Thesis, according to which the distance between two points is given by the number of tiles between the two points. However, as McDaniel points out, the size thesis is not accepted for continuous spaces. Thus, we might have reason not to accept the size thesis for discrete spaces.

Nonetheless, if a discrete space is constructed by a rectangular tiling of the plane and the Size Thesis is accepted, the Euclidean metric will be inappropriate for measuring distances on the resulting space. Instead, the so-called Hamming metric should be utilized. Computer scientists interested in the distance between two strings ^[7] and mathematical biologists interested in the distance between two genetic sequences employ the versions of the Hamming metric in each of their respective disciplines.^[8]

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