Changes

==Transformations of Discourse==

==Transformations of Discourse==

<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>

 

<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.

| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]

In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another.  In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted.  Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another.  In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted.  Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

===Foreshadowing Transformations : Extensions and Projections of Discourse===

===Foreshadowing Transformations : Extensions and Projections of Discourse===

<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>

 

<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.

| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse.  An embedding of the general type [<font face="lucida calligraphy">X</font>]&nbsp;&rarr;&nbsp;[<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>.  When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse.  An embedding of the general type [<font face="lucida calligraphy">X</font>]&nbsp;&rarr;&nbsp;[<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>.  When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.

===Thematization of Functions : And a Declaration of Independence for Variables===

===Thematization of Functions : And a Declaration of Independence for Variables===

<p>And as imagination bodies forth<br>

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

In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized.  To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.

In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized.  To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.

====Thematization : Venn Diagrams====

====Thematization : Venn Diagrams====

<p>The known universe has one complete lover and that is the greatest poet.<br>

He consumes an eternal passion and is indifferent which chance happens<br>

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>

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 11-12]</p>

| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]

Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions.  The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression.  At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'',&nbsp;''v''].

Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions.  The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression.  At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'',&nbsp;''v''].

====Thematization : Truth Tables====

====Thematization : Truth Tables====

<p>That which distorts honest shapes or which creates unearthly<br>

beings or places or contingencies is a nuisance and a revolt.</p>

 

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>

| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]

Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.

Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.