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A demonstration of Weyl's argument proceeds by constructing a
philosophy
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, Weyl's tile argument, introduced by Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
in 1949, is an argument against the notion that physical space is "discrete", as if composed of a number of finite sized units or tiles
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or ot ...
. The argument purports to show a distance function approximating Pythagoras' theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
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. Academic debate on the topic continues, with counterarguments proposed in the literature.
The argument
The tile argument appears in Weyl's 1949 book ''Philosophy of Mathematics and Natural Sciences'', where he writes:
A demonstration of Weyl's argument proceeds by constructing a square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.
Structure and properties
The square tili ...
of the plane representing a discrete space. A discretized triangle, units tall and units long, can be constructed on the tiling. The hypotenuse of the resulting triangle will be tiles long. However, by the Pythagorean theorem, a corresponding triangle in a continuous space—a triangle whose height and length are —will have a hypotenuse measuring units long. To show that the former result does not converge to the latter for arbitrary values of , one can examine the percent difference between the two results: Since cancels out, the two results never converge, even in the limit of large . 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.
Responses
In response, Kris McDaniel has argued the Weyl tile argument depends on accepting a "size thesis" which posits that 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.See also
*Digital physics
Digital physics is a speculative idea suggesting that the universe can be conceived of as a vast, digital computation device, or as the output of a deterministic or probabilistic computer program. The hypothesis that the universe is a digital com ...
* Discrete calculus
* Taxicab metric
* Causal sets
* Poisson point process
* Natura non facit saltus
''Natura non facit saltus'' Alexander Baumgarten, ''Metaphysics: A Critical Translation with Kant's Elucidations'', Translated and Edited by Courtney D. Fugate and John Hymers, Bloomsbury, 2013, "Preface of the Third Edition (1750)"p. 79 n. d " a ...
References
{{Reflist Philosophy of physics Philosophy of mathematics Discrete geometry Spacetime