Changes

====Analytic Expansion of Conjunction====

====Analytic Expansion of Conjunction====

<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of:  a soul.</p>

<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of:  a soul.</p>

<p>What is that?  It is easily defined negatively:  it is simply what curls up and hides when there is any mention of algebraic series.</p>

<p>What is that?  It is easily defined negatively:  it is simply what curls up and hides when there is any mention of algebraic series.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 118]</p>

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]

Figure&nbsp;35 pictures the form of conjunction ''J''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B''' as a transformation from the 2-dimensional universe [''u'',&nbsp;''v''] to the 1-dimensional universe [''x''].  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 ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B''' is being recast into the thematized role of a transformation ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''], where the new variable ''x'' takes the part of a thematic variable ¢(''J'').

Figure&nbsp;35 pictures the form of conjunction ''J''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B''' as a transformation from the 2-dimensional universe [''u'',&nbsp;''v''] to the 1-dimensional universe [''x''].  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 ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B''' is being recast into the thematized role of a transformation ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''], where the new variable ''x'' takes the part of a thematic variable ¢(''J'').

=====Tacit Extension of Conjunction=====

=====Tacit Extension of Conjunction=====

<p>I teach straying from me, yet who can stray from me?<br>

I teach straying from me, yet who can stray from me?<br>

I follow you whoever you are from the present hour;<br>

I follow you whoever you are from the present hour;<br>

My words itch at your ears till you understand them.</p>

My words itch at your ears till you understand them.

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 83]</p>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]

Earlier I defined the tacit extension operators <math>\epsilon</math>&nbsp;:&nbsp;''X''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''Y''^&nbsp;&bull; as maps embedding each proposition of a given universe ''X''^&nbsp;&bull; in a more generously given universe ''Y''^&nbsp;&bull; containing ''X''^&nbsp;&bull;.  Of immediate interest are the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, that locate each proposition of ''U''^&nbsp;&bull; in the enlarged context of E''U''^&nbsp;&bull;.  In its application to the propositional conjunction ''J''&nbsp;=&nbsp;''u''&nbsp;''v'' in [''u'',&nbsp;''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''].  The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table&nbsp;36, in effect, doing nothing more than conjoining a tautology of [d''u'',&nbsp;d''v''] to ''J'' in ''U''^&nbsp;&bull;.

Earlier I defined the tacit extension operators <math>\epsilon</math>&nbsp;:&nbsp;''X''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''Y''^&nbsp;&bull; as maps embedding each proposition of a given universe ''X''^&nbsp;&bull; in a more generously given universe ''Y''^&nbsp;&bull; containing ''X''^&nbsp;&bull;.  Of immediate interest are the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, that locate each proposition of ''U''^&nbsp;&bull; in the enlarged context of E''U''^&nbsp;&bull;.  In its application to the propositional conjunction ''J''&nbsp;=&nbsp;''u''&nbsp;''v'' in [''u'',&nbsp;''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''].  The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table&nbsp;36, in effect, doing nothing more than conjoining a tautology of [d''u'',&nbsp;d''v''] to ''J'' in ''U''^&nbsp;&bull;.

=====Enlargement Map of Conjunction=====

=====Enlargement Map of Conjunction=====

<p>No one could have established the existence of any details that might not just as well have existed in earlier times too;  but all the relations between things had shifted slightly.  Ideas that had once been of lean account grew fat.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 62]</p>

No one could have established the existence of any details that might not just as well have existed in earlier times too;  but all the relations between things had shifted slightly.  Ideas that had once been of lean account grew fat.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]

The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u''&nbsp;+&nbsp;d''u'' for ''u'' and ''v''&nbsp;+&nbsp;d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.

The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u''&nbsp;+&nbsp;d''u'' for ''u'' and ''v''&nbsp;+&nbsp;d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.

=====Digression : Reflection on Use and Mention=====

=====Digression : Reflection on Use and Mention=====

<p>Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.</p>

 

<p>John Dewey, ''How We Think'', [Dew, 57]</p>

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]

The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J''&nbsp;" to indicate the region ''J''^&ndash;1(1) and using "''J''&nbsp;" to indicate the function ''J''.  You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion.  But there seems to be no likelihood in practice that their interactions can be avoided.  If the name "''J''&nbsp;" is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J''&nbsp;" by transitivity a sign of the thing itself?  There are, of course, two answers to this question.  Not every act of signifying or referring need be transitive.  Not every warrant or guarantee or certificate is automatically transferable, indeed, not many.  Not every feature of a feature is a feature of the featuree.  Otherwise we have an inference like the following:  If a buffalo is white, and white is a color, then a buffalo is a color.  But a buffalo is not, only buff is.

The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J''&nbsp;" to indicate the region ''J''^&ndash;1(1) and using "''J''&nbsp;" to indicate the function ''J''.  You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion.  But there seems to be no likelihood in practice that their interactions can be avoided.  If the name "''J''&nbsp;" is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J''&nbsp;" by transitivity a sign of the thing itself?  There are, of course, two answers to this question.  Not every act of signifying or referring need be transitive.  Not every warrant or guarantee or certificate is automatically transferable, indeed, not many.  Not every feature of a feature is a feature of the featuree.  Otherwise we have an inference like the following:  If a buffalo is white, and white is a color, then a buffalo is a color.  But a buffalo is not, only buff is.

The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice.  The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using.  It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.

The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice.  The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using.  It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.

<p>The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things;  but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 130]</p>

The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things;  but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]

=====Difference Map of Conjunction=====

=====Difference Map of Conjunction=====

<p>"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders.  "In a tangle of forces like this it doesn't make a scrap of difference."  He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact.  And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there;  he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 8]</p>

"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders.  "In a tangle of forces like this it doesn't make a scrap of difference."  He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact.  And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there;  he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]

With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table&nbsp;41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'',&nbsp;''v''].

With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table&nbsp;41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'',&nbsp;''v''].

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

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

<p>By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities;  this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.</p>

 

<p>Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]</p>

By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities;  this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.

| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]

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 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.

<p>He had drifted into the very heart of the world.  From him to the distant beloved was as far as to the next tree.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 144]</p>

He had drifted into the very heart of the world.  From him to the distant beloved was as far as to the next tree.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]

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 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.

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

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

<p>I bequeath myself to the dirt to grow from the grass I love,<br>

<p>I bequeath myself to the dirt to grow from the grass I love,<br>

If you want me again look for me under your bootsoles.</p>

If you want me again look for me under your bootsoles.</p>

Missing me one place search another,<br>

Missing me one place search another,<br>

I stop some where waiting for you</p>

I stop some where waiting for you</p>

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]

Let us now recapitulate the story so far.  In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages.  First, we considered the equation E''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;D''J'', which was involved in the definition of D''J'' as the difference E''J''&nbsp;&ndash;&nbsp;<math>\epsilon</math>''J''.  Next, we contemplated the equation D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J''&nbsp;=&nbsp;D''J''&nbsp;&ndash;&nbsp;d''J''.  This remaining proposition r''J'' can be computed as shown in Table&nbsp;47.

Let us now recapitulate the story so far.  In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages.  First, we considered the equation E''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;D''J'', which was involved in the definition of D''J'' as the difference E''J''&nbsp;&ndash;&nbsp;<math>\epsilon</math>''J''.  Next, we contemplated the equation D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J''&nbsp;=&nbsp;D''J''&nbsp;&ndash;&nbsp;d''J''.  This remaining proposition r''J'' can be computed as shown in Table&nbsp;47.

====Analytic Series : Coordinate Method====

====Analytic Series : Coordinate Method====

<p>And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way.  So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.</p>

 

<p>Robert Musil, ''The Man Without Qualities'', [Mus, 12]</p>

And if he is told that something ''is'' the way it is, then he thinks:   Well, it could probably just as easily be some other way.  So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]

Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.

Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.

====Terminological Interlude====

====Terminological Interlude====

<p>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 …</p>

 

<p>Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]</p>

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 &hellip;

| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]

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 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.