In philosophy, the Weyl's tile argument, introduced by Hermann Weyl 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.[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] Academic debate on the topic continues, with counterarguments proposed in the literature.[5][6][7]
We approximate the diagonal with vertical and horizontal edges. No matter how large n is, the lengths don't match.
A demonstration of Weyl's argument proceeds by constructing a square 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 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:
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 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.[5]
weyl, tile, argument, philosophy, introduced, hermann, weyl, 1949, argument, against, notion, that, physical, space, discrete, composed, number, finite, sized, units, tiles, argument, purports, show, distance, function, approximating, pythagoras, theorem, disc. In philosophy the Weyl s tile argument introduced by Hermann Weyl 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 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 Academic debate on the topic continues with counterarguments proposed in the literature 5 6 7 We approximate the diagonal with vertical and horizontal edges No matter how large n is the lengths don t match A demonstration of Weyl s argument proceeds by constructing a square 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 displaystyle n sqrt 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 displaystyle frac n sqrt 2 n n sqrt 2 1 frac 1 sqrt 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 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 5 See also EditDigital physics Discrete calculus Taxicab metric Causal sets Poisson point processReferences Edit Weyl Hermann 1949 Philosophy of Mathematics and Natural Sciences Princeton University Press Hagar Amit 2014 Discrete or Continuous The Quest for Fundamental Length in Modern Physics Cambridge University Press ISBN 978 1107062801 Cohen S Marc Atomism faculty washington edu Retrieved 2015 05 02 Fritz Tobias June 2013 Velocity polytopes of periodic graphs and a no go theorem for digital physics Discrete Mathematics 313 12 1289 1301 arXiv 1109 1963 Bibcode 2011arXiv1109 1963F doi 10 1016 j disc 2013 02 010 S2CID 15066745 a b McDaniel K 2007 Distance and Discrete Space Synthese 155 1 157 162 doi 10 1007 s11229 005 5034 7 ISSN 0039 7857 JSTOR 27653481 S2CID 8768211 Van Bendegem Jean Paul 2019 09 12 Finitism in Geometry In Zalta Edward N ed Stanford Encyclopedia of Philosophy Chen Lu August 2021 Intrinsic local distances a mixed solution to Weyl s tile argument Synthese 198 8 7533 7552 doi 10 1007 s11229 020 02531 4 ISSN 0039 7857 S2CID 210135018 This philosophy related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Weyl 27s tile argument amp oldid 1131453333, wikipedia, wiki, book, books, library,
