It's one of the things I most admire about Simon Kochen, my co-author, that... in August 2006, we'd been talking about this... for years... Suddenly the scales fell away, that had been obscuring the thing, and I said... "We've proved if we have free will, so do the particles" and he... said "Yes... this means that my stuff with Ax is all nonsense, doesn't it?"
English mathematician (1937–2020)
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician, and Professor Emeritus of Mathematics at Princeton University in New Jersey. He was active in the theory of s, , number theory, and . He also made contributions to many branches of , most notably the invention of the with . Born and raised in , Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. He died of complications from COVID-19 at age 82.
From: Wikiquote (CC BY-SA 4.0)
Alternative Names:
John H. Conway
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JHC
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John Conway
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Conway
From Wikidata (CC0)
Let me phrase the free will theorem that Simon and I proved. ...[I]f we... have free will... then so do elementary particles have their... very small quantity of free will... to mean, our behavior is not a function of the past. ...[I]f some experimenters have free will ...then so do elementary particles... even the ones outside us...
[T]he best packings in up to eight dimensions belong to families <math>A_n, D_n </math> and <math>E_n</math>, and the corresponding s turn up in apparently unrelated areas... [I]n 24 dimensions the <math>\Lambda_{24}</math> has... connections with , s, and the Monster simple group... [O]ne day someone will write an article on "The Ubiquity of the Leech lattice." ...There are applications of... packings to number theory... [e.g.,] solving s, and to "the "... There are... applications of sphere packings... in digital communications... a typical question from... spread-spectrum communications for mobile radio... how many spheres of radius 0.25 can be packed in a sphere of radius 1 in 100-dimensional space? ...Two and three-d... packings... circles in a two-d... packing may represent s... in... a cable. Three-d... packings have applications in chemistry and physics... biology... antenna design... choosing directions for X-ray tomography... and... statistical analysis on spheres... n-dimensional packings may be used in... numerical evaluation of integrals... on the surface of a sphere in <math>\R^n</math> or in its interior. ...A related application ...n-dimensional search or approximation problems ...[I]n physics... dual theory and superstring theory... have involved the <math>E_8</math> and <math>\Lambda_{24}</math> lattices and the related Lorentzian lattices in dimensions 10 and 26...
[L]attice packing... has the properties that 0 is a center and... if there are spheres with centers <math>u</math> and <math>v</math> then there are spheres with centers <math>u + v</math> and <math>u - v</math>... [i.e.,] the sets of centers forms an . In crystallography these... are... called s... We can find... in general <math>n</math> centers <math>v_1,v_2, ...,v_n</math> for an n-dimensional lattice... such that the set of all centers consists of the sums <math>\sum k_i v_i</math> where <math>k_i</math> are s.
There has been a great deal of nonsense written... about the mysterious fourth dimension. ...4-dimensional space just consists of points with four coordinates instead of three (...similarly for any number of dimensions). ...[I]magine a telegraph ...over which numbers are ...sent in sets of four. Each set... is a point in 4-d... space.
The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space <math>\R^n</math> is... a string of real numbers<math>x = (x_1,x_2,x_3, ...,x_n)</math>.A sphere in <math>\R^n</math> with center <math>u = (u_1,u_2,u_3, ...,u_n)</math> and radius <math>\rho</math> consists of all points <math>x</math>... satisfying <math>(x_1-u_1)^2 + (x_2-u_2)^2+ ... +(x_n-u_n)^2 = \rho^2</math>. We can describe a sphere packing in <math>\R^n</math>... by specifying the centers <math>u</math> and the radius.
The classical... problem is... how densely a large number of identical spheres ([e.g.,] ball bearings...) can be packed together. ...[C]onsider an aircraft hangar... [A]bout one quarter of the space will not be used... One... arrangement... the face-centered cubic (or fcc) lattice... spheres occupy <math>\pi / \sqrt{18} = .7405...</math> of the total space.... the lattice packing has density <math>.7405...</math> . [H]pwever, there are partial packings that are denser than the face-centered cubic... over larger regions...
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