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{{DISPLAYTITLE:Epitext for Differential Logic and Dynamic Systems}}
 
{{DISPLAYTITLE:Epitext for Differential Logic and Dynamic Systems}}
 
 
__NOTOC__
 
__NOTOC__
  
{| width="100%"
+
{| style="height:36px; width:100%"
 
| align="left"  | ''Stand and unfold yourself.''
 
| align="left"  | ''Stand and unfold yourself.''
 
| align="right" | Hamlet: Francsico—1.1.2
 
| align="right" | Hamlet: Francsico—1.1.2
 
|}
 
|}
<br>
 
  
 
==Purpose==
 
==Purpose==
Line 15: Line 13:
 
==A Functional Conception of Propositional Calculus==
 
==A Functional Conception of Propositional Calculus==
  
{| align="right"
+
{| width="100%" cellpadding="0" cellspacing="0"
|
+
| width="40%" | &nbsp;
<p>Out of the dimness opposite equals advance . . . .<br>
+
| width="60%" |
 +
Out of the dimness opposite equals advance . . . .<br>
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br>
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br>
 
Always a knit of identity . . . . always distinction . . . .<br>
 
Always a knit of identity . . . . always distinction . . . .<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.
 
|-
 
|-
 +
| &nbsp;
 
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]
 
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]
 
|}
 
|}
{{-}}
 
  
 
===Qualitative Logic and Quantitative Analogy===
 
===Qualitative Logic and Quantitative Analogy===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Logical, however, is used in a third sense, which is at once more vital and more practical;  to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>[[John Dewey]], ''[[How We Think]]'', [Dew, 56]</p>
+
''Logical'', however, is used in a third sense, which is at once more vital and more practical;  to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]
 +
|}
  
 
===Philosophy of Notation : Formal Terms and Flexible Types===
 
===Philosophy of Notation : Formal Terms and Flexible Types===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking.  Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>
+
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking.  Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]
 +
|}
  
 
===Special Classes of Propositions===
 
===Special Classes of Propositions===
Line 48: Line 53:
 
===The Analogy Between Real and Boolean Types===
 
===The Analogy Between Real and Boolean Types===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Measurement consists in correlating our subject matter with the series of real numbers;  and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>
+
Measurement consists in correlating our subject matter with the series of real numbers;  and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]
 +
|}
  
 
===Theory of Control and Control of Theory===
 
===Theory of Control and Control of Theory===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>You will hardly know who I am or what I mean,<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
You will hardly know who I am or what I mean,<br>
 
But I shall be good health to you nevertheless,<br>
 
But I shall be good health to you nevertheless,<br>
And filter and fibre your blood.</p>
+
And filter and fibre your blood.
 
+
|-
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]
 +
|}
  
 
===Propositions as Types and Higher Order Types===
 
===Propositions as Types and Higher Order Types===
Line 68: Line 79:
 
===Reality at the Threshold of Logic===
 
===Reality at the Threshold of Logic===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>
+
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]
 +
|}
  
 
===Tables of Propositional Forms===
 
===Tables of Propositional Forms===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.  It provides explicit techniques for manipulating the most basic ingredients of discourse.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7-8]</p>
+
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.  It provides explicit techniques for manipulating the most basic ingredients of discourse.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]
 +
|}
  
 
==A Differential Extension of Propositional Calculus==
 
==A Differential Extension of Propositional Calculus==
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Fire over water:<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
Fire over water:<br>
 
The image of the condition before transition.<br>
 
The image of the condition before transition.<br>
 
Thus the superior man is careful<br>
 
Thus the superior man is careful<br>
 
In the differentiation of things,<br>
 
In the differentiation of things,<br>
So that each finds its place.</p>
+
So that each finds its place.
 
+
|-
<p>''I Ching'', Hexagram 64, [Wil, 249]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]
 +
|}
  
 
===Differential Propositions : The Qualitative Analogues of Differential Equations===
 
===Differential Propositions : The Qualitative Analogues of Differential Equations===
Line 98: Line 118:
 
===An Interlude on the Path===
 
===An Interlude on the Path===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>There would have been no beginnings:<br>
+
| width="4%"  | &nbsp;
instead, speech would proceed from me,<br>
+
| width="92%" |
while I stood in its path - a slender gap -<br>
+
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.
the point of its possible disappearance.</p>
+
| width="4%"  | &nbsp;
 
+
|-
<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>
+
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]
</blockquote>
+
|}
  
 
===The Extended Universe of Discourse===
 
===The Extended Universe of Discourse===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>
+
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]
 +
|}
  
 
===Intentional Propositions===
 
===Intentional Propositions===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Do you guess I have some intricate purpose?<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
Do you guess I have some intricate purpose?<br>
 
Well I have . . . . for the April rain has, and the mica on<br>
 
Well I have . . . . for the April rain has, and the mica on<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.
 
+
|-
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 45]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]
 +
|}
  
 
===Life on Easy Street===
 
===Life on Easy Street===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Failing to fetch me at first keep encouraged,<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
Failing to fetch me at first keep encouraged,<br>
 
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>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]
 +
|}
  
 
==Back to the Beginning : Some Exemplary Universes==
 
==Back to the Beginning : Some Exemplary Universes==
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>I would have preferred to be enveloped in words,<br>
+
| width="4%"  | &nbsp;
borne way beyond all possible beginnings.</p>
+
| width="92%" |
 
+
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.
<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>
+
| width="4%"  | &nbsp;
</blockquote>
+
|-
 +
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]
 +
|}
  
 
===A One-Dimensional Universe===
 
===A One-Dimensional Universe===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>There was never any more inception than there is now,<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
There was never any more inception than there is now,<br>
 
Nor any more youth or age than there is now;<br>
 
Nor any more youth or age than there is now;<br>
 
And will never be any more perfection than there is now,<br>
 
And will never be any more perfection than there is now,<br>
Nor any more heaven or hell than there is now.</p>
+
Nor any more heaven or hell than there is now.
 +
|-
 +
| &nbsp;
 +
| align="right" | &mdash; Walt Whitman, Leaves of Grass, [Whi, 28]
 +
|}
  
<p>Walt Whitman, Leaves of Grass, [Whi, 28]</p>
+
<br>
</blockquote>
 
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>The clock indicates the moment . . . . but what does<br>
+
| width="40%" | &nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?</p>
+
| width="60%" |
 
+
The clock indicates the moment . . . . but what does<br>
<p>Walt Whitman, 'Leaves of Grass', [Whi, 79]</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?
</blockquote>
+
|-
 +
| &nbsp;
 +
| align="right" | &mdash; Walt Whitman, 'Leaves of Grass', [Whi, 79]
 +
|}
  
 
===Example 1.  A Square Rigging===
 
===Example 1.  A Square Rigging===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Urge and urge and urge,<br>
+
| width="40%" | &nbsp;
Always the procreant urge of the world.</p>
+
| width="60%" |
 
+
Urge and urge and urge,<br>
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 28]</p>
+
Always the procreant urge of the world.
</blockquote>
+
|-
 +
| &nbsp;
 +
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]
 +
|}
  
 
===Back to the Feature===
 
===Back to the Feature===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>I guess it must be the flag of my disposition, out of hopeful<br>
+
| width="40%" | &nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.</p>
+
| width="60%" |
 
+
I guess it must be the flag of my disposition, out of hopeful<br>
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 31]</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.
</blockquote>
+
|-
 +
| &nbsp;
 +
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]
 +
|}
  
 
===Tacit Extensions===
 
===Tacit Extensions===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>
+
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]
 +
|}
  
 
===Example 2.  Drives and Their Vicissitudes===
 
===Example 2.  Drives and Their Vicissitudes===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>I open my scuttle at night and see the far-sprinkled systems,<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
I open my scuttle at night and see the far-sprinkled systems,<br>
 
And all I see, multiplied as high as I can cipher, edge but<br>
 
And all I see, multiplied as high as I can cipher, edge but<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.</p>
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.
 
+
|-
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 81]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]
 +
|}
  
 
==Transformations of Discourse==
 
==Transformations of Discourse==
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Robert Musil, ''The Man Without Qualities'', [Mus, 39]</p>
+
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]
 +
|}
  
 
===Foreshadowing Transformations : Extensions and Projections of Discourse===
 
===Foreshadowing Transformations : Extensions and Projections of Discourse===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]</p>
+
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
 +
|}
  
 
====Extension from 1 to 2 Dimensions====
 
====Extension from 1 to 2 Dimensions====
Line 220: Line 277:
 
===Thematization of Functions : And a Declaration of Independence for Variables===
 
===Thematization of Functions : And a Declaration of Independence for Variables===
  
<blockquote>
+
{| width="100%"
<p>And as imagination bodies forth<br>
+
| align="left"  |
The forms of things unknown, the poet's pen<br>
+
''And as imagination bodies forth''<br>
Turns them to shapes, and gives to airy nothing<br>
+
''The forms of things unknown, the poet's pen''<br>
A local habitation and a name.</p>
+
''Turns them to shapes, and gives to airy nothing''<br>
 
+
''A local habitation and a name.''
<p>''A Midsummer Night's Dream'', 5.1.18</p>
+
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
</blockquote>
+
|}
  
 
====Thematization : Venn Diagrams====
 
====Thematization : Venn Diagrams====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>The known universe has one complete lover and that is the greatest poet.<br>
+
| width="4%"  | &nbsp;
He consumes an eternal passion and is indifferent which chance happens<br>
+
| width="92%" |
and which possible contingency of fortune or misfortune and persuades<br>
+
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
daily and hourly his delicious pay.</p>
+
| width="4%"  | &nbsp;
 
+
|-
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 11-12]</p>
+
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]
</blockquote>
+
|}
  
 
====Thematization : Truth Tables====
 
====Thematization : Truth Tables====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>That which distorts honest shapes or which creates unearthly<br>
+
| width="4%"  | &nbsp;
beings or places or contingencies is a nuisance and a revolt.</p>
+
| width="92%" |
 
+
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 19]</p>
+
| width="4%"  | &nbsp;
</blockquote>
+
|-
 +
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]
 +
|}
  
 
===Propositional Transformations===
 
===Propositional Transformations===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>John Dewey, ''How We Think'', [Dew, 56-57]</p>
+
If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]
 +
|}
  
 
====Alias and Alibi Transformations====
 
====Alias and Alibi Transformations====
Line 261: Line 323:
 
====Transformations of General Type====
 
====Transformations of General Type====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Robert Musil, ''The Man Without Qualities'', [Mus, 34]</p>
+
''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred.  It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]
 +
|}
  
 
===Analytic Expansions : Operators and Functors===
 
===Analytic Expansions : Operators and Functors===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>Consider what effects that might ''conceivably''<br>
+
| width="4%"  | &nbsp;
have practical bearings you ''conceive'' the<br>
+
| width="92%" |
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>
+
| width="4%"  | &nbsp;
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>
+
|}
</blockquote>
 
  
 
====Operators on Propositions and Transformations====
 
====Operators on Propositions and Transformations====
Line 283: Line 347:
 
====Differential Analysis of Propositions and Transformations====
 
====Differential Analysis of Propositions and Transformations====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>The resultant metaphysical problem now is this:</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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?''
 
+
| width="4%"  | &nbsp;
<p>William James, ''Pragmatism'', [Jam, 43]</p>
+
|-
</blockquote>
+
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]
 +
|}
  
 
=====The Secant Operator : <font face=georgia>'''E'''</font>=====
 
=====The Secant Operator : <font face=georgia>'''E'''</font>=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]
 +
|}
  
 
=====The Radius Operator : <font face=georgia>'''e'''</font>=====
 
=====The Radius Operator : <font face=georgia>'''e'''</font>=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]
 +
|}
  
 
=====The Phantom of the Operators : '''&eta;'''=====
 
=====The Phantom of the Operators : '''&eta;'''=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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''!
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
 +
|}
  
 
=====The Chord Operator : <font face=georgia>'''D'''</font>=====
 
=====The Chord Operator : <font face=georgia>'''D'''</font>=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]
 +
|}
  
 
=====The Tangent Operator : <font face=georgia>'''T'''</font>=====
 
=====The Tangent Operator : <font face=georgia>'''T'''</font>=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]
 +
|}
  
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===
Line 335: Line 415:
 
====Analytic Expansion of Conjunction====
 
====Analytic Expansion of Conjunction====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
 +
| width="4%"  | &nbsp;
 +
| width="92%" |
 
<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>
 
+
| width="4%"  | &nbsp;
<p>Robert Musil, ''The Man Without Qualities'', [Mus, 118]</p>
+
|-
</blockquote>
+
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]
 +
|}
  
 
=====Tacit Extension of Conjunction=====
 
=====Tacit Extension of Conjunction=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>I teach straying from me, yet who can stray from me?<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
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>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]
 +
|}
  
 
=====Enlargement Map of Conjunction=====
 
=====Enlargement Map of Conjunction=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]
 +
|}
  
 
=====Digression : Reflection on Use and Mention=====
 
=====Digression : Reflection on Use and Mention=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 +
| width="92%" |
 +
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.
 +
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]
 +
|}
  
<p>John Dewey, ''How We Think'', [Dew, 57]</p>
+
<br>
</blockquote>
 
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]
 +
|}
  
 
=====Difference Map of Conjunction=====
 
=====Difference Map of Conjunction=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]
 +
|}
  
 
=====Differential of Conjunction=====
 
=====Differential of Conjunction=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 +
| width="92%" |
 +
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.
 +
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]
 +
|}
  
<p>Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]</p>
+
<br>
</blockquote>
 
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]
 +
|}
  
 
=====Remainder of Conjunction=====
 
=====Remainder of Conjunction=====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
 +
| width="40%" | &nbsp;
 +
| width="60%" |
 
<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>
Line 410: Line 520:
 
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>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]
 +
|}
  
 
=====Summary of Conjunction=====
 
=====Summary of Conjunction=====
Line 418: Line 529:
 
====Analytic Series : Coordinate Method====
 
====Analytic Series : Coordinate Method====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]
 +
|}
  
 
====Analytic Series : Recap====
 
====Analytic Series : Recap====
Line 428: Line 542:
 
====Terminological Interlude====
 
====Terminological Interlude====
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<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>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<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;
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]
 +
|}
  
 
====End of Perfunctory Chatter : Time to Roll the Clip!====
 
====End of Perfunctory Chatter : Time to Roll the Clip!====
Line 444: Line 561:
 
===Taking Aim at Higher Dimensional Targets===
 
===Taking Aim at Higher Dimensional Targets===
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>The past and present wilt . . . . I have filled them and<br>
+
| width="40%" | &nbsp;
 +
| width="60%" |
 +
The past and present wilt . . . . I have filled them and<br>
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br>
 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br>
And proceed to fill my next fold of the future.</p>
+
And proceed to fill my next fold of the future.
 
+
|-
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 87]</p>
+
| &nbsp;
</blockquote>
+
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]
 +
|}
  
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
Line 456: Line 576:
 
==Epilogue, Enchoiry, Exodus==
 
==Epilogue, Enchoiry, Exodus==
  
<blockquote>
+
{| width="100%" cellpadding="0" cellspacing="0"
<p>It is time to explain myself . . . . let us stand up.</p>
+
| width="4%"  | &nbsp;
 
+
| width="92%" |
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 79]</p>
+
It is time to explain myself . . . . let us stand up.
</blockquote>
+
| width="4%"  | &nbsp;
 +
|-
 +
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]
 +
|}

Latest revision as of 21:40, 2 July 2008


Stand and unfold yourself. Hamlet: Francsico—1.1.2

Purpose

Review and Transition

A Functional Conception of Propositional Calculus

 

Out of the dimness opposite equals advance . . . .
     Always substance and increase,
Always a knit of identity . . . . always distinction . . . .
     always a breed of life.

  — Walt Whitman, Leaves of Grass, [Whi, 28]

Qualitative Logic and Quantitative Analogy

 

Logical, however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.

 
— John Dewey, How We Think, [Dew, 56]

Philosophy of Notation : Formal Terms and Flexible Types

 

Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.

 
— W.V. Quine, Mathematical Logic, [Qui, 7]

Special Classes of Propositions

Basis Relativity and Type Ambiguity

The Analogy Between Real and Boolean Types

 

Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.

 
— W.V. Quine, Mathematical Logic, [Qui, 7]

Theory of Control and Control of Theory

 

You will hardly know who I am or what I mean,
But I shall be good health to you nevertheless,
And filter and fibre your blood.

  — Walt Whitman, Leaves of Grass, [Whi, 88]

Propositions as Types and Higher Order Types

Reality at the Threshold of Logic

 

But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.

 
— W.V. Quine, Mathematical Logic, [Qui, 7]

Tables of Propositional Forms

 

To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.

 
— W.V. Quine, Mathematical Logic, [Qui, 7–8]

A Differential Extension of Propositional Calculus

 

Fire over water:
The image of the condition before transition.
Thus the superior man is careful
In the differentiation of things,
So that each finds its place.

  I Ching, Hexagram 64, [Wil, 249]

Differential Propositions : The Qualitative Analogues of Differential Equations

An Interlude on the Path

 

There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.

 
— Michel Foucault, The Discourse on Language, [Fou, 215]

The Extended Universe of Discourse

 

At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.

 
— Michel Foucault, The Discourse on Language, [Fou, 215]

Intentional Propositions

 

Do you guess I have some intricate purpose?
Well I have . . . . for the April rain has, and the mica on
     the side of a rock has.

  — Walt Whitman, Leaves of Grass, [Whi, 45]

Life on Easy Street

 

Failing to fetch me at first keep encouraged,
Missing me one place search another,
I stop some where waiting for you

  — Walt Whitman, Leaves of Grass, [Whi, 88]

Back to the Beginning : Some Exemplary Universes

 

I would have preferred to be enveloped in words, borne way beyond all possible beginnings.

 
— Michel Foucault, The Discourse on Language, [Fou, 215]

A One-Dimensional Universe

 

There was never any more inception than there is now,
Nor any more youth or age than there is now;
And will never be any more perfection than there is now,
Nor any more heaven or hell than there is now.

  — Walt Whitman, Leaves of Grass, [Whi, 28]


 

The clock indicates the moment . . . . but what does
     eternity indicate?

  — Walt Whitman, 'Leaves of Grass', [Whi, 79]

Example 1. A Square Rigging

 

Urge and urge and urge,
Always the procreant urge of the world.

  — Walt Whitman, Leaves of Grass, [Whi, 28]

Back to the Feature

 

I guess it must be the flag of my disposition, out of hopeful
     green stuff woven.

  — Walt Whitman, Leaves of Grass, [Whi, 31]

Tacit Extensions

 

I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.

 
— Michel Foucault, The Discourse on Language, [Fou, 215]

Example 2. Drives and Their Vicissitudes

 

I open my scuttle at night and see the far-sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
     the rim of the farther systems.

  — Walt Whitman, Leaves of Grass, [Whi, 81]

Transformations of Discourse

 

It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.

 
— Robert Musil, The Man Without Qualities, [Mus, 39]

Foreshadowing Transformations : Extensions and Projections of Discourse

 

And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.

 
— Gaston Leroux, The Phantom of the Opera, [Ler, 126]

Extension from 1 to 2 Dimensions

Extension from 2 to 4 Dimensions

Thematization of Functions : And a Declaration of Independence for Variables

And as imagination bodies forth
The forms of things unknown, the poet's pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.

A Midsummer Night's Dream, 5.1.18

Thematization : Venn Diagrams

 

The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.

 
— Walt Whitman, Leaves of Grass, [Whi, 11–12]

Thematization : Truth Tables

 

That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.

 
— Walt Whitman, Leaves of Grass, [Whi, 19]

Propositional Transformations

 

If only the word 'artificial' were associated with the idea of art, or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that logical refers to artificial thought.

 
— John Dewey, How We Think, [Dew, 56–57]

Alias and Alibi Transformations

Transformations of General Type

 

Es ist passiert, "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.

 
— Robert Musil, The Man Without Qualities, [Mus, 34]

Analytic Expansions : Operators and Functors

 

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.

 
— C.S. Peirce, "The Maxim of Pragmatism", CP 5.438

Operators on Propositions and Transformations

Differential Analysis of Propositions and Transformations

 

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

 
— William James, Pragmatism, [Jam, 43]
The Secant Operator : E
 

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.

 
— William James, Pragmatism, [Jam, 46]
The Radius Operator : e
 

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.

 
— William James, Pragmatism, [Jam, 46]
The Phantom of the Operators : η
 

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!

 
— Gaston Leroux, The Phantom of the Opera, [Ler, 81]
The Chord Operator : D
 

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.

 
— William James, Pragmatism, [Jam, 45]
The Tangent Operator : T
 

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.

 
— William James, Pragmatism, [Jam, 300]

Transformations of Type B2B1

Analytic Expansion of Conjunction

 

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.

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

 
— Robert Musil, The Man Without Qualities, [Mus, 118]
Tacit Extension of Conjunction
 

I teach straying from me, yet who can stray from me?
I follow you whoever you are from the present hour;
My words itch at your ears till you understand them.

  — Walt Whitman, Leaves of Grass, [Whi, 83]
Enlargement Map of Conjunction
 

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.

 
— Robert Musil, The Man Without Qualities, [Mus, 62]
Digression : Reflection on Use and Mention
 

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.

 
— John Dewey, How We Think, [Dew, 57]


 

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.

 
— Robert Musil, The Man Without Qualities, [Mus, 130]
Difference Map of Conjunction
 

"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.

 
— Robert Musil, The Man Without Qualities, [Mus, 8]
Differential of Conjunction
 

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.

 
— Michel Foucault, The Archaeology of Knowledge, [Fou, 143]


 

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

 
— Robert Musil, The Man Without Qualities, [Mus, 144]
Remainder of Conjunction
 

I bequeath myself to the dirt to grow from the grass I love,
If you want me again look for me under your bootsoles.

You will hardly know who I am or what I mean,
But I shall be good health to you nevertheless,
And filter and fibre your blood.

Failing to fetch me at first keep encouraged,
Missing me one place search another,
I stop some where waiting for you

  — Walt Whitman, Leaves of Grass, [Whi, 88]
Summary of Conjunction

Analytic Series : Coordinate Method

 

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.

 
— Robert Musil, The Man Without Qualities, [Mus, 12]

Analytic Series : Recap

Terminological Interlude

 

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 …

 
— Gaston Leroux, The Phantom of the Opera, [Ler, 230]

End of Perfunctory Chatter : Time to Roll the Clip!

Operator Maps : Areal Views
Operator Maps : Box Views
Operator Diagrams for the Conjunction J = uv

Taking Aim at Higher Dimensional Targets

 

The past and present wilt . . . . I have filled them and
     emptied them,
And proceed to fill my next fold of the future.

  — Walt Whitman, Leaves of Grass, [Whi, 87]

Transformations of Type B2B2

Epilogue, Enchoiry, Exodus

 

It is time to explain myself . . . . let us stand up.

 
— Walt Whitman, Leaves of Grass, [Whi, 79]