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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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</pre>
 
</pre>
   −
=Work Area 1=
+
=Materials from "Dif Log Dyn Sys" for Reuse Here=
   −
==Formal development==
+
Excerpts from "[[Differential Logic and Dynamic Systems]]"
   −
===Differential Propositions===
+
==A Functional Conception of Propositional Calculus==
   −
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in <math>[\mathcal{A}]</math> may change or move with respect to the features that are noted in the initial alphabet.
+
<blockquote>
 +
<p>Out of the dimness opposite equals advance . . . .<br>
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br>
 +
Always a knit of identity . . . . always distinction . . . .<br>
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.</p>
   −
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math>  (For all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)
+
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 28]</p>
 +
</blockquote>
   −
In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
+
In the general case, we start with a set of logical features {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world.  In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', &hellip;} or as meaningful words from a linguistic ''vocabulary'' of codes.  This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation.  In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space.  Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
   −
Proceeding as we did before with the base space <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:
+
The set of logical features {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>} provides a basis for generating an ''n''-dimensional ''universe of discourse'' that I denote as [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>].  It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉 and the set of propositions ''f''&nbsp;:&nbsp;〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉&nbsp;&rarr;&nbsp;'''B''' that are implicit with the ordinary picture of a venn diagram on ''n'' features.  Thus, we may regard the universe of discourse [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>] as an ordered pair having the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''), and we may abbreviate this last type designation as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or even more succinctly as ['''B'''<sup>''n''</sup>].  (Used this way, the angle brackets 〈&hellip;〉 are referred to as ''generator brackets''.)
   −
: <math>\operatorname{d}A = \prod_{i=1}^n \operatorname{d}A_i = \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>
+
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams.  Although it overworks the square brackets a bit, I also use either one of the equivalent notations [''n''] or '''''n''''' to denote the data type of a finite set on n elements.
   −
Here, <math>\operatorname{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\operatorname{d}\mathcal{A}_i = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \},</math> where <math>\overline{\operatorname{d}a_i}</math> is a symbol with the logical value of <math>\operatorname{not}\ \operatorname{d}a_i.</math>  Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} \cong  \{ 0, 1 \}.</math>  However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:
+
<font face="courier new">
−
 
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−
: <math>\mathbb{D} = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} = \{ \operatorname{same}, \operatorname{different} \} = \{ \operatorname{stay}, \operatorname{change} \} = \{ \operatorname{stop}, \operatorname{step} \}.</math>
+
|+ '''Table 2.  Fundamental Notations for Propositional Calculus'''
−
 
+
|- style="background:paleturquoise"
−
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
+
! Symbol
−
 
+
! Notation
−
===Extended Universe of Discourse===
+
! Description
−
 
+
! Type
−
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:
+
|-
−
 
+
| <font face="lucida calligraphy">A<font>
−
: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>
+
| {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
−
 
+
| Alphabet
−
This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:
+
| [''n''] = '''n'''
−
 
  −
:{| cellpadding=2
  −
| <math>\operatorname{E}A</math>
  −
| =
  −
| <math>A \times \operatorname{d}A</math>
   
|-
 
|-
−
| &nbsp;
+
| ''A''<sub>''i''</sub>
−
| =
+
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
−
| <math>\langle \operatorname{E}\mathcal{A} \rangle</math>
+
| Dimension ''i''
 +
| '''B'''
 
|-
 
|-
−
| &nbsp;
+
| ''A''
−
| =
+
|
−
| <math>\langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle</math>
+
〈<font face="lucida calligraphy">A</font>〉<br>
 +
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
 +
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
 +
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
 +
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub>
 +
|
 +
Set of cells,<br>
 +
coordinate tuples,<br>
 +
points, or vectors<br>
 +
in the universe<br>
 +
of discourse
 +
| '''B'''<sup>''n''</sup>
 
|-
 
|-
−
| &nbsp;
+
| ''A''*
−
| =
+
| (hom : ''A'' &rarr; '''B''')
−
| <math>\langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math>
+
| Linear functions
 +
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
 +
|-
 +
| ''A''^
 +
| (''A'' &rarr; '''B''')
 +
| Boolean functions
 +
| '''B'''<sup>''n''</sup> &rarr; '''B'''
 +
|-
 +
| ''A''<sup>&bull;</sup>
 +
|
 +
[<font face="lucida calligraphy">A</font>]<br>
 +
(''A'', ''A''^)<br>
 +
(''A'' +&rarr; '''B''')<br>
 +
(''A'', (''A'' &rarr; '''B'''))<br>
 +
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]
 +
|
 +
Universe of discourse<br>
 +
based on the features<br>
 +
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
 +
|
 +
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
 +
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
 +
['''B'''<sup>''n''</sup>]
 
|}
 
|}
 +
</font><br>
   −
thus giving <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
+
===Reality at the Threshold of Logic===
   −
Finally, the tangent universe <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\operatorname{E}\mathcal{A}:</math>
+
<blockquote>
 +
<p>But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.</p>
   −
: <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}] = [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n],</math>
+
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>
 +
</blockquote>
   −
thus giving the tangent universe <math>\operatorname{E}A^\circ</math> the type:
+
Table 5 accumulates an array of notation that I hope will not be too distracting.  Some of it is rarely needed, but has been filled in for the sake of completeness.  Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
−
 
  −
: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>
  −
 
  −
A proposition in the tangent universe <math>[\operatorname{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
  −
 
  −
With these constructions, to be specific, the differential extension <math>\operatorname{E}A</math> and the differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at one of the major subgoals of this study.  At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).
  −
 
  −
=Materials from "Dif Log Dyn Sys" for Reuse Here=
  −
 
  −
Excerpts from "[[Differential Logic and Dynamic Systems]]"
  −
 
  −
==A Functional Conception of Propositional Calculus==
  −
 
  −
<blockquote>
  −
<p>Out of the dimness opposite equals advance . . . .<br>
  −
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br>
  −
Always a knit of identity . . . . always distinction . . . .<br>
  −
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.</p>
  −
 
  −
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 28]</p>
  −
</blockquote>
  −
 
  −
In the general case, we start with a set of logical features {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world.  In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', &hellip;} or as meaningful words from a linguistic ''vocabulary'' of codes.  This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation.  In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space.  Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
  −
 
  −
The set of logical features {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>} provides a basis for generating an ''n''-dimensional ''universe of discourse'' that I denote as [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>].  It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉 and the set of propositions ''f''&nbsp;:&nbsp;〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉&nbsp;&rarr;&nbsp;'''B''' that are implicit with the ordinary picture of a venn diagram on ''n'' features.  Thus, we may regard the universe of discourse [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>] as an ordered pair having the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''), and we may abbreviate this last type designation as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or even more succinctly as ['''B'''<sup>''n''</sup>].  (Used this way, the angle brackets 〈&hellip;〉 are referred to as ''generator brackets''.)
  −
 
  −
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams.  Although it overworks the square brackets a bit, I also use either one of the equivalent notations [''n''] or '''''n''''' to denote the data type of a finite set on n elements.
      
<font face="courier new">
 
<font face="courier new">
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−
|+ '''Table 2.  Fundamental Notations for Propositional Calculus'''
+
|+ '''Table 5.  A Bridge Over Troubled Waters'''
 
|- style="background:paleturquoise"
 
|- style="background:paleturquoise"
−
! Symbol
+
! Linear Space
−
! Notation
+
! Liminal Space
−
! Description
+
! Logical Space
−
! Type
   
|-
 
|-
−
| <font face="lucida calligraphy">A<font>
+
|
−
| {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
+
<font face="lucida calligraphy">X</font><br>
−
| Alphabet
+
{''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>}<br>
−
| [''n''] = '''n'''
+
cardinality ''n''
 +
|
 +
<font face="lucida calligraphy"><u>X</u></font><br>
 +
{<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>}<br>
 +
cardinality ''n''
 +
|
 +
<font face="lucida calligraphy">A</font><br>
 +
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br>
 +
cardinality ''n''
 
|-
 
|-
−
| ''A''<sub>''i''</sub>
  −
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
  −
| Dimension ''i''
  −
| '''B'''
  −
|-
  −
| ''A''
   
|
 
|
−
〈<font face="lucida calligraphy">A</font>〉<br>
+
''X''<sub>''i''</sub><br>
−
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
+
〈''x''<sub>''i''</sub>〉<br>
−
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
+
isomorphic to '''K'''
−
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
+
|
−
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub>
+
<u>''X''</u><sub>''i''</sub><br>
 +
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
 +
isomorphic to '''B'''
 
|
 
|
−
Set of cells,<br>
+
''A''<sub>''i''</sub><br>
−
coordinate tuples,<br>
+
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
−
points, or vectors<br>
+
isomorphic to '''B'''
−
in the universe<br>
  −
of discourse
  −
| '''B'''<sup>''n''</sup>
  −
|-
  −
| ''A''*
  −
| (hom : ''A'' &rarr; '''B''')
  −
| Linear functions
  −
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
   
|-
 
|-
−
| ''A''^
  −
| (''A'' &rarr; '''B''')
  −
| Boolean functions
  −
| '''B'''<sup>''n''</sup> &rarr; '''B'''
  −
|-
  −
| ''A''<sup>&bull;</sup>
   
|
 
|
−
[<font face="lucida calligraphy">A</font>]<br>
+
''X''<br>
−
(''A'', ''A''^)<br>
+
〈<font face="lucida calligraphy">X</font>〉<br>
−
(''A'' +&rarr; '''B''')<br>
+
〈''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>〉<br>
−
(''A'', (''A'' &rarr; '''B'''))<br>
+
{‹''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>›}<br>
−
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]
+
''X''<sub>1</sub> &times; &hellip; &times; ''X''<sub>''n''</sub><br>
 +
&prod;<sub>''i''</sub> ''X''<sub>''i''</sub><br>
 +
isomorphic to '''K'''<sup>''n''</sup>
 
|
 
|
−
Universe of discourse<br>
+
<u>''X''</u><br>
−
based on the features<br>
+
〈<font face="lucida calligraphy"><u>X</u></font>〉<br>
−
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
+
〈<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>〉<br>
 +
{‹<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>›}<br>
 +
<u>''X''</u><sub>1</sub> &times; &hellip; &times; <u>''X''</u><sub>''n''</sub><br>
 +
&prod;<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
 +
isomorphic to '''B'''<sup>''n''</sup>
 
|
 
|
−
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
+
''A''<br>
−
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
+
〈<font face="lucida calligraphy">A</font>〉<br>
−
['''B'''<sup>''n''</sup>]
+
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
−
|}
+
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
−
</font><br>
+
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
−
 
+
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub><br>
−
===Reality at the Threshold of Logic===
+
isomorphic to '''B'''<sup>''n''</sup>
−
 
  −
<blockquote>
  −
<p>But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.</p>
  −
 
  −
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>
  −
</blockquote>
  −
 
  −
Table 5 accumulates an array of notation that I hope will not be too distracting.  Some of it is rarely needed, but has been filled in for the sake of completeness.  Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
  −
 
  −
<font face="courier new">
  −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
  −
|+ '''Table 5.  A Bridge Over Troubled Waters'''
  −
|- style="background:paleturquoise"
  −
! Linear Space
  −
! Liminal Space
  −
! Logical Space
   
|-
 
|-
 
|
 
|
−
<font face="lucida calligraphy">X</font><br>
+
''X''*<br>
−
{''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>}<br>
+
(hom : ''X'' &rarr; '''K''')<br>
−
cardinality ''n''
+
isomorphic to '''K'''<sup>''n''</sup>
 
|
 
|
−
<font face="lucida calligraphy"><u>X</u></font><br>
+
<u>''X''</u>*<br>
−
{<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>}<br>
+
(hom : <u>''X''</u> &rarr; '''B''')<br>
−
cardinality ''n''
+
isomorphic to '''B'''<sup>''n''</sup>
 
|
 
|
−
<font face="lucida calligraphy">A</font><br>
+
''A''*<br>
−
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br>
+
(hom : ''A'' &rarr; '''B''')<br>
−
cardinality ''n''
+
isomorphic to '''B'''<sup>''n''</sup>
 
|-
 
|-
 
|
 
|
−
''X''<sub>''i''</sub><br>
+
''X''^<br>
−
〈''x''<sub>''i''</sub>〉<br>
+
(''X'' &rarr; '''K''')<br>
−
isomorphic to '''K'''
+
isomorphic to:<br>
 +
('''K'''<sup>''n''</sup> &rarr; '''K''')
 
|
 
|
−
<u>''X''</u><sub>''i''</sub><br>
+
<u>''X''</u>^<br>
−
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
+
(<u>''X''</u> &rarr; '''B''')<br>
−
isomorphic to '''B'''
+
isomorphic to:<br>
 +
('''B'''<sup>''n''</sup> &rarr; '''B''')
 
|
 
|
−
''A''<sub>''i''</sub><br>
+
''A''^<br>
−
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
+
(''A'' &rarr; '''B''')<br>
−
isomorphic to '''B'''
+
isomorphic to:<br>
 +
('''B'''<sup>''n''</sup> &rarr; '''B''')
 
|-
 
|-
 
|
 
|
−
''X''<br>
+
''X''<sup>&bull;</sup><br>
−
〈<font face="lucida calligraphy">X</font>〉<br>
+
[<font face="lucida calligraphy">X</font>]<br>
−
〈''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>〉<br>
+
[''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>]<br>
−
{‹''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>›}<br>
+
(''X'', ''X''^)<br>
−
''X''<sub>1</sub> &times; &hellip; &times; ''X''<sub>''n''</sub><br>
+
(''X'' +&rarr; '''K''')<br>
−
&prod;<sub>''i''</sub> ''X''<sub>''i''</sub><br>
+
(''X'', (''X'' &rarr; '''K'''))<br>
−
isomorphic to '''K'''<sup>''n''</sup>
+
isomorphic to:<br>
 +
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> &rarr; '''K'''))<br>
 +
('''K'''<sup>''n''</sup> +&rarr; '''K''')<br>
 +
['''K'''<sup>''n''</sup>]
 
|
 
|
−
<u>''X''</u><br>
+
<u>''X''</u><sup>&bull;</sup><br>
−
〈<font face="lucida calligraphy"><u>X</u></font>〉<br>
+
[<font face="lucida calligraphy"><u>X</u></font>]<br>
−
〈<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>〉<br>
+
[<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>]<br>
−
{‹<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>›}<br>
+
(<u>''X''</u>, <u>''X''</u>^)<br>
−
<u>''X''</u><sub>1</sub> &times; &hellip; &times; <u>''X''</u><sub>''n''</sub><br>
+
(<u>''X''</u> +&rarr; '''B''')<br>
−
&prod;<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
+
(<u>''X''</u>, (<u>''X''</u> &rarr; '''B'''))<br>
−
isomorphic to '''B'''<sup>''n''</sup>
+
isomorphic to:<br>
 +
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
 +
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
 +
['''B'''<sup>''n''</sup>]
 
|
 
|
−
''A''<br>
+
''A''<sup>&bull;</sup><br>
−
〈<font face="lucida calligraphy">A</font>〉<br>
+
[<font face="lucida calligraphy">A</font>]<br>
−
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
+
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]<br>
−
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
+
(''A'', ''A''^)<br>
−
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
+
(''A'' +&rarr; '''B''')<br>
−
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub><br>
+
(''A'', (''A'' &rarr; '''B'''))<br>
−
isomorphic to '''B'''<sup>''n''</sup>
+
isomorphic to:<br>
−
|-
+
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
−
|
+
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
−
''X''*<br>
+
['''B'''<sup>''n''</sup>]
−
(hom : ''X'' &rarr; '''K''')<br>
+
|}
−
isomorphic to '''K'''<sup>''n''</sup>
+
</font><br>
−
|
+
 
−
<u>''X''</u>*<br>
+
The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''.  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.
−
(hom : <u>''X''</u> &rarr; '''B''')<br>
+
 
−
isomorphic to '''B'''<sup>''n''</sup>
+
I now proceed to explain these concepts in more detail.  The two most important ideas developed in the table are:
−
|
+
 
−
''A''*<br>
+
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.
−
(hom : ''A'' &rarr; '''B''')<br>
+
 
−
isomorphic to '''B'''<sup>''n''</sup>
+
* 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''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>〉 <math>\cong</math> '''R'''<sup>''n''</sup>.  The coordinate
−
''X''^<br>
+
system <font face=lucida calligraphy">X</font> = {''x''<sub>''i''</sub>} is a set of maps ''x''<sub>''i''</sub>&nbsp;:&nbsp;'''R'''<sub>''n''</sub>&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections.  Given a "dataset" of points ''x'' in '''R'''<sub>''n''</sub>, 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''<sub>''i''</sub> on '''R''', that is, a subset of '''R'''<sub>''n''</sub>, and then we define the ''i''<sup>th</sup> threshold map, or ''limen'' <u>''x''</u><sub>''i''</sub> as follows:
−
(''X'' &rarr; '''K''')<br>
+
 
−
isomorphic to:<br>
+
: <u>''x''</u><sub>''i''</sub> : '''R'''<sub>''n''</sub> &rarr; '''B''' such that:
−
('''K'''<sup>''n''</sup> &rarr; '''K''')
+
 
−
|
+
: <u>''x''</u><sub>''i''</sub>(''x'') = 1 if ''x'' &isin; ''L''<sub>''i''</sub>,
−
<u>''X''</u>^<br>
+
 
−
(<u>''X''</u> &rarr; '''B''')<br>
+
: <u>''x''</u><sub>''i''</sub>(''x'') = 0 if otherwise.
−
isomorphic to:<br>
+
 
−
('''B'''<sup>''n''</sup> &rarr; '''B''')
+
In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''.  Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values.  In each of these notations, the above definition could be expressed as follows:
−
|
+
 
−
''A''^<br>
+
: <u>''x''</u><sub>''i''</sub>(''x'') = <math>\chi (x \in L_i)</math> = <math>\lceil x \in L_i \rceil</math> = ''L''<sub>''i''</sub>(''x'').
−
(''A'' &rarr; '''B''')<br>
+
 
−
isomorphic to:<br>
+
Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''<sub>''i''</sub>} and the coordinate axes corresponding to {''x''<sub>''i''</sub>}, aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''<sub>''i''</sub> is bounded by some hyperplane that intersects the ''i''<sup>th</sup> axis at a unique threshold value ''r''<sub>''i''</sub> &isin; '''R'''.  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set ''L''<sub>''i''</sub> has points on the ''i''<sup>th</sup> axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''<sub>''i''</sub>,&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''<sub>''i''</sub> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
−
('''B'''<sup>''n''</sup> &rarr; '''B''')
+
 
−
|-
+
States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''<sup>''n''</sup> can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u><sub>''i''</sub>} as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''<sup>th</sup> threshold map.  This can
−
|
+
help to remind us that the ''threshold operator'' <u>&nbsp;</u>)<sub>''i''</sub> acts on ''x'' by setting up a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition <u>''x''</u><sub>''i''</sub> asserts that the representative point ''x'' resides ''above'' the ''i''<sup>th</sup> threshold.
−
''X''<sup>&bull;</sup><br>
+
 
−
[<font face="lucida calligraphy">X</font>]<br>
+
Primitive assertions of the form <u>''x''</u><sub>''i''</sub>(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''<sup>''n''</sup>, and
−
[''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>]<br>
+
a space of functions (regions, propositions), <u>''X''</u>^&nbsp;<math>\cong</math>&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''').  Together these form a new universe of discourse <u>''X''</u><sup>&nbsp;&bull;</sup> = [<font face="lucida calligraphy"><u>X</u></font>] of the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''')), which we may abbreviate as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or most succinctly as ['''B'''<sup>''n''</sup>].
−
(''X'', ''X''^)<br>
+
 
−
(''X'' +&rarr; '''K''')<br>
+
The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u><sub>''i''</sub>, and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.
−
(''X'', (''X'' &rarr; '''K'''))<br>
+
 
−
isomorphic to:<br>
+
Finally, let ''X''* denote the space of linear functions, (hom&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.
−
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> &rarr; '''K'''))<br>
  −
('''K'''<sup>''n''</sup> +&rarr; '''K''')<br>
  −
['''K'''<sup>''n''</sup>]
  −
|
  −
<u>''X''</u><sup>&bull;</sup><br>
  −
[<font face="lucida calligraphy"><u>X</u></font>]<br>
  −
[<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>]<br>
  −
(<u>''X''</u>, <u>''X''</u>^)<br>
  −
(<u>''X''</u> +&rarr; '''B''')<br>
  −
(<u>''X''</u>, (<u>''X''</u> &rarr; '''B'''))<br>
  −
isomorphic to:<br>
  −
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
  −
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
  −
['''B'''<sup>''n''</sup>]
  −
|
  −
''A''<sup>&bull;</sup><br>
  −
[<font face="lucida calligraphy">A</font>]<br>
  −
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]<br>
  −
(''A'', ''A''^)<br>
  −
(''A'' +&rarr; '''B''')<br>
  −
(''A'', (''A'' &rarr; '''B'''))<br>
  −
isomorphic to:<br>
  −
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
  −
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
  −
['''B'''<sup>''n''</sup>]
  −
|}
  −
</font><br>
     −
The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''.  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.)  The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.
+
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps.  The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets.  The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
   −
I now proceed to explain these concepts in more detail.  The two most important ideas developed in the table are:
+
==A Differential Extension of Propositional Calculus==
   −
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.
+
<blockquote>
−
 
+
<p>Fire over water:<br>
−
* 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 image of the condition before transition.<br>
−
 
  −
For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X''&nbsp;=&nbsp;〈''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>〉 <math>\cong</math> '''R'''<sup>''n''</sup>.  The coordinate
  −
system <font face=lucida calligraphy">X</font> = {''x''<sub>''i''</sub>} is a set of maps ''x''<sub>''i''</sub>&nbsp;:&nbsp;'''R'''<sub>''n''</sub>&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections.  Given a "dataset" of points ''x'' in '''R'''<sub>''n''</sub>, 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''<sub>''i''</sub> on '''R''', that is, a subset of '''R'''<sub>''n''</sub>, and then we define the ''i''<sup>th</sup> threshold map, or ''limen'' <u>''x''</u><sub>''i''</sub> as follows:
  −
 
  −
: <u>''x''</u><sub>''i''</sub> : '''R'''<sub>''n''</sub> &rarr; '''B''' such that:
  −
 
  −
: <u>''x''</u><sub>''i''</sub>(''x'') = 1 if ''x'' &isin; ''L''<sub>''i''</sub>,
  −
 
  −
: <u>''x''</u><sub>''i''</sub>(''x'') = 0 if otherwise.
  −
 
  −
In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''.  Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values.  In each of these notations, the above definition could be expressed as follows:
  −
 
  −
: <u>''x''</u><sub>''i''</sub>(''x'') = <math>\chi (x \in L_i)</math> = <math>\lceil x \in L_i \rceil</math> = ''L''<sub>''i''</sub>(''x'').
  −
 
  −
Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''<sub>''i''</sub>} and the coordinate axes corresponding to {''x''<sub>''i''</sub>}, aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''<sub>''i''</sub> is bounded by some hyperplane that intersects the ''i''<sup>th</sup> axis at a unique threshold value ''r''<sub>''i''</sub> &isin; '''R'''.  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set ''L''<sub>''i''</sub> has points on the ''i''<sup>th</sup> axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''<sub>''i''</sub>,&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''<sub>''i''</sub> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
  −
 
  −
States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''<sup>''n''</sup> can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u><sub>''i''</sub>} as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''<sup>th</sup> threshold map.  This can
  −
help to remind us that the ''threshold operator'' <u>&nbsp;</u>)<sub>''i''</sub> acts on ''x'' by setting up a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition <u>''x''</u><sub>''i''</sub> asserts that the representative point ''x'' resides ''above'' the ''i''<sup>th</sup> threshold.
  −
 
  −
Primitive assertions of the form <u>''x''</u><sub>''i''</sub>(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''<sup>''n''</sup>, and
  −
a space of functions (regions, propositions), <u>''X''</u>^&nbsp;<math>\cong</math>&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''').  Together these form a new universe of discourse <u>''X''</u><sup>&nbsp;&bull;</sup> = [<font face="lucida calligraphy"><u>X</u></font>] of the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''')), which we may abbreviate as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or most succinctly as ['''B'''<sup>''n''</sup>].
  −
 
  −
The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u><sub>''i''</sub>, and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.
  −
 
  −
Finally, let ''X''* denote the space of linear functions, (hom&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.
  −
 
  −
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps.  The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets.  The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
  −
 
  −
==A Differential Extension of Propositional Calculus==
  −
 
  −
<blockquote>
  −
<p>Fire over water:<br>
  −
The image of the condition before transition.<br>
   
Thus the superior man is careful<br>
 
Thus the superior man is careful<br>
 
In the differentiation of things,<br>
 
In the differentiation of things,<br>
Line 3,734: Line 3,672:  
| <math>\operatorname{d}y\!</math>
 
| <math>\operatorname{d}y\!</math>
 
|-
 
|-
−
| <math>f_{7}\!</math>
+
| <math>f_{7}\!</math>
−
| <math>(x y)\!</math>
+
| <math>(x y)\!</math>
−
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
+
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
−
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
+
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
+
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
−
|-
+
|-
−
| <math>f_{11}\!</math>
+
| <math>f_{11}\!</math>
−
| <math>(x (y))\!</math>
+
| <math>(x (y))\!</math>
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
+
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
−
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
+
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
−
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
+
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
−
|-
+
|-
−
| <math>f_{13}\!</math>
+
| <math>f_{13}\!</math>
−
| <math>((x) y)\!</math>
+
| <math>((x) y)\!</math>
−
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
+
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
+
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
−
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
+
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
−
|-
+
|-
−
| <math>f_{14}\!</math>
+
| <math>f_{14}\!</math>
−
| <math>((x)(y))\!</math>
+
| <math>((x)(y))\!</math>
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
+
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
−
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
+
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
+
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
−
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
−
|-
+
|-
−
| <math>f_{15}\!</math>
+
| <math>f_{15}\!</math>
−
| <math>((~))\!</math>
+
| <math>((~))\!</math>
−
| <math>((~))\!</math>
+
| <math>((~))\!</math>
−
| <math>((~))\!</math>
+
| <math>((~))\!</math>
−
| <math>((~))\!</math>
+
| <math>((~))\!</math>
−
| <math>((~))\!</math>
+
| <math>((~))\!</math>
−
|}<br>
+
|}<br>
−
 
+
 
−
=Work Area 2=
+
=Elegant Graveyard=
 +
 
 +
=Work Area=
 +
 
 +
==Formal development==
 +
 
 +
===Differential Propositions===
 +
 
 +
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in <math>[\mathcal{A}]</math> may change or move with respect to the features that are noted in the initial alphabet.
 +
 
 +
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> that is intended to be interpreted in the way just indicated.  It only remains to be understood that the precise interpretation of the symbols in <math>\operatorname{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math>  (For all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\operatorname{d}\mathcal{A}.</math>)
 +
 
 +
In the above terms, a typical tangent space of <math>A\!</math> at a point <math>x,\!</math> frequently denoted as <math>T_x(A),\!</math> can be characterized as having the generic construction <math>\operatorname{d}A = \langle \operatorname{d}\mathcal{A} \rangle = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math>  Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
 +
 
 +
Proceeding as we did before with the base space <math>A,\!</math> we can analyze the individual tangent space at a point of <math>A\!</math> as a product of distinct and independent factors:
 +
 
 +
: <math>\operatorname{d}A = \prod_{i=1}^n \operatorname{d}A_i = \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.</math>
 +
 
 +
Here, <math>\operatorname{d}\mathcal{A}_i</math> is an alphabet of two symbols, <math>\operatorname{d}\mathcal{A}_i = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \},</math> where <math>\overline{\operatorname{d}a_i}</math> is a symbol with the logical value of <math>\operatorname{not}\ \operatorname{d}a_i.</math>  Each component <math>\operatorname{d}A_i</math> has the type <math>\mathbb{B},</math> under the correspondence <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} \cong  \{ 0, 1 \}.</math>  However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D}, </math> whose intension may be indicated as follows:
 +
 
 +
: <math>\mathbb{D} = \{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \} = \{ \operatorname{same}, \operatorname{different} \} = \{ \operatorname{stay}, \operatorname{change} \} = \{ \operatorname{stop}, \operatorname{step} \}.</math>
 +
 
 +
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
 +
 
 +
===Extended Universe of Discourse===
 +
 
 +
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as:
 +
 
 +
: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math>
 +
 
 +
This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:
 +
 
 +
:{| cellpadding=2
 +
| <math>\operatorname{E}A</math>
 +
| =
 +
| <math>A \times \operatorname{d}A</math>
 +
|-
 +
| &nbsp;
 +
| =
 +
| <math>\langle \operatorname{E}\mathcal{A} \rangle</math>
 +
|-
 +
| &nbsp;
 +
| =
 +
| <math>\langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle</math>
 +
|-
 +
| &nbsp;
 +
| =
 +
| <math>\langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math>
 +
|}
 +
 
 +
thus giving <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
 +
 
 +
Finally, the tangent universe <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\operatorname{E}\mathcal{A}:</math>
 +
 
 +
: <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}] = [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n],</math>
 +
 
 +
thus giving the tangent universe <math>\operatorname{E}A^\circ</math> the type:
 +
 
 +
: <math>(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).</math>
 +
 
 +
A proposition in the tangent universe <math>[\operatorname{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
 +
 
 +
With these constructions, to be specific, the differential extension <math>\operatorname{E}A</math> and the differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at one of the major subgoals of this study.  At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 5).
    
==Orbit Table Template==
 
==Orbit Table Template==
Line 4,307: Line 4,307:  
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|}
 
<br>
 
<br>
−
  −
=Elegant Graveyard=
 
Jon Awbrey
12,080

edits

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