Changes

<br>

<br>

Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values.  This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of <math>f\!</math> and <math>g\!</math> to the 3-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math>  The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math>  The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math>  At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math>

Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values.  This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math>  The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math>  The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math>  At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math>

<br>

<br>

|}

|}

Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the 2-dimensional universe <math>[u, v]\!</math> to the 1-dimensional universe <math>[x].\!</math>  This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value.  Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math>

Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math>  This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value.  Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

* '''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 <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.

Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-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 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.

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.

<br>

<br>

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>

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 <math>1\!</math>-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>

<br>

<br>

|}

|}

In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y''] and the abstract type ''F''&nbsp;:&nbsp;['''B'''²]&nbsp;&rarr;&nbsp;['''B'''²]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:

In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math>  From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:

<br><font face="courier new">

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"

 

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"

|

|

{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"

<math>\begin{array}{ccccccl}

| align="left" | ''F''

F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],

| =

| ‹''f'', ''g''›

&& F_1 & = & f & : & [u, v] \to [x],

&& F_2 & = & g & : & [u, v] \to [y].

| :

\end{array}</math>

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>

| align="left" colspan="2" | where

| ''f''

| :

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>

| align="left" colspan="2" | and

| ''g''

| :

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''y''<nowiki>]</nowiki>

|}

|}

Then one tackles the separate components, now viewed as propositions ''F''_''i''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', one at a time.  At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation.  (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.)

 

Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time.  At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation.  (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)

Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''.  When we keep to transformations with a toll of 1, as ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''], we tend to get lazy about distinguishing a logical transformation from its component propositions.  However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.

Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''.  When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions.  However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.

Well, perhaps we can carry it a little further.  After all, the operator result W''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull; is a map of toll 2, and cannot be unfolded in one piece as a proposition.  But when a map has rank 1, like <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;''X''&nbsp;&sube;&nbsp;E''X'' or d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;d''X''&nbsp;&sube;&nbsp;E''X'', we naturally choose to concentrate on the 1-dimensional range of the operator result W''J'', ignoring the final difference in quality between the spaces ''X'' and d''X'', and view W''J'' as a proposition about E''U''.

Well, perhaps we can carry it a little further.  After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition.  But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math>

In this way, an initial ambivalence about the role of the operand ''J'' conveys a double duty to the result W''J''.  The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of W''J''.  This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results W''J'' as propositions or as transformations, indifferently.

In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.

But that's it, and no further.  Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion.  To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map ''F''&nbsp;:&nbsp;['''B'''²]&nbsp;&rarr;&nbsp;['''B'''²], and begin to pave the way, to some extent, for discussing any transformation of the form ''F''&nbsp;:&nbsp;['''B'''^''n'']&nbsp;&rarr;&nbsp;['''B'''^''k''].

But that's it, and no further.  Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion.  To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math>

<br>

<br>