Changes

* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.

* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.

For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X''&nbsp;=&nbsp;〈''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n''〉 <math>\cong</math> '''R'''^''n''. The coordinate

For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math>  The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''.  Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\operatorname{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:

system <font face=lucida calligraphy">X</font> = {''x''_''i''} is a set of maps ''x''_''i''&nbsp;:&nbsp;'''R'''_''n''&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections.  Given a "dataset" of points ''x'' in '''R'''_''n'', there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each ''i'' we choose an ''n''-ary relation ''L''_''i'' on '''R''', that is, a subset of '''R'''_''n'', and then we define the ''i''^th threshold map, or ''limen'' <u>''x''</u>_''i'' as follows:

: <u>''x''</u>_''i'' : '''R'''_''n'' &rarr; '''B''' such that:

: <u>''x''</u>_''i'' : '''R'''^''n'' &rarr; '''B''' such that:

: <u>''x''</u>_''i''(''x'') = 1 if ''x'' &isin; ''L''_''i'',

: <u>''x''</u>_''i''(''x'') = 1 if ''x'' &isin; ''L''_''i'',