Changes

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Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.

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The difference map D''J'' can also be given a ''dispositional'' interpretation.  First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''.  Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below.  Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'',&nbsp;E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other.  In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.

The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation.  First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below.  Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other.  In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.

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

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

|

<math>\begin{array}{lllll}

| width="6%"  | <math>\epsilon</math>''J''

\boldsymbol\varepsilon J

| width="47%" | = {Dispositions from ''J''  to ''J''  }

& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}

| width="47%" | + {Dispositions from ''J''  to (''J'') }

& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}

| &nbsp;

| width="6%"  | E''J''

& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}

| width="47%" | = {Dispositions from ''J''  to ''J'' }

| width="47%" | + {Dispositions from (''J'') to ''J'' }

\mathrm{D}J

|-

& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}

| &nbsp;

& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}

|-

|}

| width="47%" | = (<math>\epsilon</math>''J'', E''J'')

 

| width="47%" | &nbsp;

|-

 

| &nbsp;

Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math>

 

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

| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]

|-

|-

| width="6%" | D''J''

| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math>

| width="47%" | = {Dispositions from  ''J''  to (''J'') }

| width="47%" | + {Dispositions from (''J'') to  ''J''  }

|}

|}

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Figures&nbsp;44-a through 44-d illustrate the difference proposition D''J''.

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

| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]

<p><center><font size="+1">'''Figure 44-b. Difference Map of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]

<p><center><font size="+1">'''Figure 44-d. Difference Map of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math>

=====Differential of Conjunction=====

=====Differential of Conjunction=====

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Finally, at long last, the differential proposition d''J'' can be gleaned from the difference proposition D''J'' by ranging over the cells of [''u'',&nbsp;''v''] and picking out the linear proposition of [d''u'',&nbsp;d''v''] that is "closest" to the portion of D''J'' that touches on each point.  The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position.  There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.

Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point.  The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position.  There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.

{| width="100%" cellpadding="0" cellspacing="0"

{| width="100%" cellpadding="0" cellspacing="0"

|}

|}

Let us venture a guess about where these developments might be heading.  From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis.  Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a self-corrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.

Let us venture a guess as to where these developments might be heading.  From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis.  Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.

Awaiting that determination, I proceed with what seems like the obvious course, and compute d''J'' according to the pattern in Table&nbsp;45.

Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.

 

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"

{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math>

|+ Table 45. Computation of d''J''

|

|

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

<math>\begin{array}{cllllllll}

| width="6%"  | D''J''

\mathrm{D}J

| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))

& = &

| width="23%" | + ''u'' (''v'')(d''u'') d''v''

u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}

| width="23%" | + (''u'') ''v'' d''u'' (d''v'')

& + &

| width="23%" | + (''u'')(''v'') d''u'' d''v''

u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v

& + &

\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}

\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}

& = &

& + &

\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0

 

 

Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects.  This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math>

 

|-

|-

| width="6%" | &rArr;

| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math>

 

 

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

| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]

|-

|-

| width="6%" | d''J''

| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math>

| width="23%" | + ''u'' (''v'') d''v''

| width="23%" | + (''u'')(''v'') <math>\cdot</math> 0

|}

|}

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<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]

<p><center><font size="+1">'''Figure 46-b. Differential of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math>

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<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]

<p><center><font size="+1">'''Figure 46-d. Differential of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math>

=====Remainder of Conjunction=====

=====Remainder of Conjunction=====