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Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it.  These mirrors were broken in parts.  Yes, they were marked and scratched;  they had been "starred", in spite of their solidity …

Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it.  These mirrors were broken in parts.  Yes, they were marked and scratched;  they had been “starred”, in spite of their solidity …

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At this point several issues of terminology have accrued enough substance to intrude on our discussion.  The remarks of this Section are intended to accomplish two goals.  First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate.  Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension.

At this point several issues of terminology have accrued enough substance to intrude on our discussion.  The remarks of this Subsection are intended to accomplish two goals.  First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate.  Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.

In talking about the actions of operators it is important to keep in mind the distinctions between the operators per se, their operands, and their results.  Furthermore, in working with composite forms of operators W = ‹W₁, …, W_''n''› , transformations F = ‹F₁, …, F_''n''› , and target domains ''X''^ • = [''x''₁, …, ''x''_''n''], we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components.  It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation.  Following the obvious paradigm would lead on to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time.  One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action.  I am following this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.

In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results.  Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components.  It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation.  Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time.  One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action.  We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.

* Scholium.  See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis, and for examples of their use in mechanics.  This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.

* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics.  This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.

Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have 1-dimensional ranges, we are free to shift between the native form of a proposition ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and the thematized form of a mapping ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;[''x''] without much trouble.  In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of the input and output domains of an operator than we otherwise might.  For example, in the preceding treatment of the example ''J'', and for each operator W in the set {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r}, both the operand ''J'' and the result W''J'' could be viewed in either one of two ways.  On the one hand, we could regard them as propositions ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;''B'', ignoring the qualitative distinction between the range [''x'']&nbsp;<math>\cong</math>&nbsp;'''B''' of <math>\epsilon</math>''J'' and the range [d''x'']&nbsp;<math>\cong</math>&nbsp;'''D''' of the other types of W''J''.  This is what we usually do when we content ourselves simply with coloring in regions of venn diagrams.  On the other hand, we could view these entities as maps ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;[''x'']&nbsp;=&nbsp;''X''^&nbsp;&bull; and <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;[''x'']&nbsp;&sube;&nbsp;E''X''^&nbsp;&bull; or W''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;[d''x'']&nbsp;&sube;&nbsp;E''X''^&nbsp;&bull;, in which case the qualitative characters of the output features are not allowed to go without saying, nor thus at the risk of being forgotten.

Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have 1-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble.  In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might.  For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways.  On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams.  On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.

At the beginning of this Division I recast the natural form of a proposition ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' into the thematic role of a transformation ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;[''x''], where ''x'' was a variable recruited to express the newly independent ¢(''J'').  However, in my computations and representations of operator actions I immediately lapsed back to viewing the results as native elements of the extended universe E''U''^&nbsp;&bull;, in other words, as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', where W ranged over the set {<math>\epsilon</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r}.  That is as it should be.  In fact, I have worked hard to devise a language that gives us all of these competing advantages, the flexibility to exchange terms and types that bear equal information value, and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.

At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be.  We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.

As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible.  For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology that I am using here, as applied to the case of ''J''&nbsp;=&nbsp;''uv''.  The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of an example, but to establish the general paradigm with enough solidity to bear the weight of abstraction that is coming on down the road.

As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible.  For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.

Table&nbsp;54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity.  Here, the operators <font face=georgia>'''W'''</font> in {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''d'''</font>,&nbsp;<font face=georgia>'''r'''</font>} and their components W in {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r} both have the same broad type <font face=georgia>'''W'''</font>,&nbsp;W&nbsp;:&nbsp;(''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;)&nbsp;&rarr;&nbsp;(E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;), as would be expected of operators that map transformations ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; to extended transformations <font face=georgia>'''W'''</font>''J'',&nbsp;W''J''&nbsp;:&nbsp;E''U^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;.

Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity.  Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math>

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Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''¹ or '''D'''¹), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;. As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column.  In one case, however, it is customary to depart from this scheme.  Because the phrase ''differential proposition'', applied to the result d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.

Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  For example, all the component operator maps <math>\mathrm{W}J\!</math> have 1-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math>  As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column.  In one case, however, it is customary to depart from this scheme.  Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math>

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====End of Perfunctory Chatter : Time to Roll the Clip!====

====End of Perfunctory Chatter : Time to Roll the Clip!====

Two steps remain to finish the analysis of ''J'' that I began so long ago.  First, I need to paste the accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <font face=georgia>'''W'''</font>''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;. This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b.  Finally, in Figures&nbsp;57-1 to 57-4 I put all the pieces together to construct the full operator diagrams for <font face=georgia>'''W'''</font>&nbsp;:&nbsp;''J''&nbsp;&rarr;&nbsp;<font face=georgia>'''W'''</font>''J''. There is a large amount of redundancy in these three series of figures.  At this early stage of exposition I thought that it would be better not to tax the reader's imagination, and to guarantee that the author, at least, has worked through the relevant exercises.  I hope the reader will excuse the flagrant use of space and try to view these snapshots as successive frames in the animation of logic that they are meant to become.

Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago.  First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math>  This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b.  Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math>  There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.

=====Operator Maps : Areal Views=====

=====Operator Maps : Areal Views=====

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]

<p><center><font size="+1">'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math>

=====Operator Maps : Box Views=====

=====Operator Maps : Box Views=====

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]

<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math>

=====Operator Diagrams for the Conjunction J = uv=====

=====Operator Diagrams for the Conjunction J = uv=====

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<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>

| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]

<p><center><font size="+1">'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math>

===Taking Aim at Higher Dimensional Targets===

===Taking Aim at Higher Dimensional Targets===