Changes

===Analytic Expansions : Operators and Functors===

===Analytic Expansions : Operators and Functors===

<p>Consider what effects that might ''conceivably''<br>

have practical bearings you ''conceive'' the<br>

objects of your ''conception'' to have.  Then,<br>

Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have.  Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.

your ''conception'' of those effects is the<br>

whole of your ''conception'' of the object.</p>

 

| align="right" colspan="3" | &mdash; C.S. Peirce, "The Maxim of Pragmatism", CP 5.438

<p>C.S. Peirce, "The Maxim of Pragmatism", CP 5.438</p>

Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.

Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.

====Differential Analysis of Propositions and Transformations====

====Differential Analysis of Propositions and Transformations====

<p>The resultant metaphysical problem now is this:</p>

 

<p>''Does the man go round the squirrel or not?''</p>

he resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''

 

<p>William James, ''Pragmatism'', [Jam, 43]</p>

| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]

The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''.  The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents.  After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.

The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''.  The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents.  After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.

=====The Secant Operator : <font face=georgia>'''E'''</font>=====

=====The Secant Operator : <font face=georgia>'''E'''</font>=====

<p>Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce:  that conduct is for us its sole significance.</p>

 

<p>William James, ''Pragmatism'', [Jam, 46]</p>

Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce:  that conduct is for us its sole significance.

| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]

Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study.  From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis.  In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type.  The secant operator has the component description <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›, and its active ingredient E is known as the ''enlargement operator''.  (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'')&nbsp;=&nbsp;''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)

Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study.  From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis.  In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type.  The secant operator has the component description <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›, and its active ingredient E is known as the ''enlargement operator''.  (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'')&nbsp;=&nbsp;''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)

=====The Radius Operator : <font face=georgia>'''e'''</font>=====

=====The Radius Operator : <font face=georgia>'''e'''</font>=====

<p>And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.</p>

 

<p>William James, ''Pragmatism'', [Jam, 46]</p>

And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.

| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]

The operator identified as <font face=georgia>'''d'''</font>⁰ in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context.  Construed in terms of its broadest components, <font face=georgia>'''d'''</font>⁰ is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>".  Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›&nbsp;=&nbsp;<font face=georgia>'''d'''</font>⁰ as the ''radius operator''.  The operation that is intended by all of these forms is defined by the equation:

The operator identified as <font face=georgia>'''d'''</font>⁰ in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context.  Construed in terms of its broadest components, <font face=georgia>'''d'''</font>⁰ is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>".  Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›&nbsp;=&nbsp;<font face=georgia>'''d'''</font>⁰ as the ''radius operator''.  The operation that is intended by all of these forms is defined by the equation:

=====The Phantom of the Operators : '''&eta;'''=====

=====The Phantom of the Operators : '''&eta;'''=====

<p>I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!</p>

 

<p>Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]</p>

I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!

| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]

I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect.  In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.

I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect.  In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.

=====The Chord Operator : <font face=georgia>'''D'''</font>=====

=====The Chord Operator : <font face=georgia>'''D'''</font>=====

<p>What difference would it practically make to any one if this notion rather than that notion were true?  If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.</p>

 

<p>William James, ''Pragmatism'', [Jam, 45]</p>

What difference would it practically make to any one if this notion rather than that notion were true?  If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.

| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]

Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding.  It may appear once as a record:  a relic or revenant that reprises the reminders of an earlier stage of development.  Or it may appear always as a resource:  a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage.  And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding.  It may appear once as a record:  a relic or revenant that reprises the reminders of an earlier stage of development.  Or it may appear always as a resource:  a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage.  And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

=====The Tangent Operator : <font face=georgia>'''T'''</font>=====

=====The Tangent Operator : <font face=georgia>'''T'''</font>=====

<p>They take part in scenes of whose significance they have no inkling.  They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken.  So we are tangent to the wider life of things.</p>

 

<p>William James, ''Pragmatism'', [Jam, 300]</p>

They take part in scenes of whose significance they have no inkling.  They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken.  So we are tangent to the wider life of things.

| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]

The operator tagged as <font face=georgia>'''d'''</font>¹ in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'', and is usually denoted in this text as <font face=georgia>'''d'''</font> or <font face=georgia>'''T'''</font>.  Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a ''tangent functor''.  According to the custom adopted here, we dissect it as <font face=georgia>'''T'''</font>&nbsp;=&nbsp;<font face=georgia>'''d'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›, where d is the operator that yields the first order differential d''F'' when applied to a transformation ''F'', and whose name is legion.

The operator tagged as <font face=georgia>'''d'''</font>¹ in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'', and is usually denoted in this text as <font face=georgia>'''d'''</font> or <font face=georgia>'''T'''</font>.  Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a ''tangent functor''.  According to the custom adopted here, we dissect it as <font face=georgia>'''T'''</font>&nbsp;=&nbsp;<font face=georgia>'''d'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›, where d is the operator that yields the first order differential d''F'' when applied to a transformation ''F'', and whose name is legion.