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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 12:25, 9 July 2008
, 12:25, 9 July 2008→Reality at the Threshold of Logic
===Reality at the Threshold of Logic===
===Reality at the Threshold of Logic===
====Original Format====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:ghostwhite"
! <math>\mbox{Linear Space}\!</math>
! <math>\mbox{Liminal Space}\!</math>
! <math>\mbox{Logical Space}\!</math>
|-
|
<math>\begin{matrix}
\mathcal{X} & = &
\{x_1, \ldots, x_n\} \\
\end{matrix}</math>
|
<math>\begin{matrix}
\underline\mathcal{X} & = &
\{\underline{x}_1, \ldots, \underline{x}_n\} \\
\end{matrix}</math>
|
<math>\begin{matrix}
\mathcal{A} & = &
\{a_1, \ldots, a_n\} \\
\end{matrix}</math>
|-
|
<math>\begin{matrix}
X_i & = &
\langle x_i \rangle \cong \mathbb{K} \\
\end{matrix}</math>
|
<math>\begin{matrix}
\underline{X}_i & = &
\{(\underline{x}_i), \underline{x}_i \} \cong \mathbb{B} \\
\end{matrix}</math>
|
<math>\begin{matrix}
A_i & = &
\{(a_i), a_i \} \cong \mathbb{B} \\
\end{matrix}</math>
|-
|
''X''<br>
〈<font face="lucida calligraphy">X</font>〉<br>
〈''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>〉<br>
{‹''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>›}<br>
''X''<sub>1</sub> × … × ''X''<sub>''n''</sub><br>
∏<sub>''i''</sub> ''X''<sub>''i''</sub><br>
isomorphic to '''K'''<sup>''n''</sup>
|
<u>''X''</u><br>
〈<font face="lucida calligraphy"><u>X</u></font>〉<br>
〈<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>〉<br>
{‹<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>›}<br>
<u>''X''</u><sub>1</sub> × … × <u>''X''</u><sub>''n''</sub><br>
∏<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
isomorphic to '''B'''<sup>''n''</sup>
|
''A''<br>
〈<font face="lucida calligraphy">A</font>〉<br>
〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉<br>
{‹''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>›}<br>
''A''<sub>1</sub> × … × ''A''<sub>''n''</sub><br>
∏<sub>''i''</sub> ''A''<sub>''i''</sub><br>
isomorphic to '''B'''<sup>''n''</sup>
|-
|
''X''*<br>
(hom : ''X'' → '''K''')<br>
isomorphic to '''K'''<sup>''n''</sup>
|
<u>''X''</u>*<br>
(hom : <u>''X''</u> → '''B''')<br>
isomorphic to '''B'''<sup>''n''</sup>
|
''A''*<br>
(hom : ''A'' → '''B''')<br>
isomorphic to '''B'''<sup>''n''</sup>
|-
|
''X''^<br>
(''X'' → '''K''')<br>
isomorphic to:<br>
('''K'''<sup>''n''</sup> → '''K''')
|
<u>''X''</u>^<br>
(<u>''X''</u> → '''B''')<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup> → '''B''')
|
''A''^<br>
(''A'' → '''B''')<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup> → '''B''')
|-
|
''X''<sup>•</sup><br>
[<font face="lucida calligraphy">X</font>]<br>
[''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>]<br>
(''X'', ''X''^)<br>
(''X'' +→ '''K''')<br>
(''X'', (''X'' → '''K'''))<br>
isomorphic to:<br>
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> → '''K'''))<br>
('''K'''<sup>''n''</sup> +→ '''K''')<br>
['''K'''<sup>''n''</sup>]
|
<u>''X''</u><sup>•</sup><br>
[<font face="lucida calligraphy"><u>X</u></font>]<br>
[<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>]<br>
(<u>''X''</u>, <u>''X''</u>^)<br>
(<u>''X''</u> +→ '''B''')<br>
(<u>''X''</u>, (<u>''X''</u> → '''B'''))<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
('''B'''<sup>''n''</sup> +→ '''B''')<br>
['''B'''<sup>''n''</sup>]
|
''A''<sup>•</sup><br>
[<font face="lucida calligraphy">A</font>]<br>
[''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]<br>
(''A'', ''A''^)<br>
(''A'' +→ '''B''')<br>
(''A'', (''A'' → '''B'''))<br>
isomorphic to:<br>
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
('''B'''<sup>''n''</sup> +→ '''B''')<br>
['''B'''<sup>''n''</sup>]
|}<br>
====Current Format====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"