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==Back to the Beginning : Some Exemplary Universes==

==Back to the Beginning : Some Exemplary Universes==

<p>I would have preferred to be enveloped in words,<br>

borne way beyond all possible beginnings.</p>

 

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>

| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]

To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2.  In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.

To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2.  In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.

===A One-Dimensional Universe===

===A One-Dimensional Universe===

<p>There was never any more inception than there is now,<br>

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.

 

<p>Walt Whitman, Leaves of Grass, [Whi, 28]</p>

| align="right" | &mdash; Walt Whitman, Leaves of Grass, [Whi, 28]

Let <font face="lucida calligraphy">X</font> = {''x''₁} = {''A''} be an alphabet that represents one boolean variable or a single logical feature.  In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions.  At any rate, the basis element

Let <font face="lucida calligraphy">X</font> = {''x''₁} = {''A''} be an alphabet that represents one boolean variable or a single logical feature.  In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions.  At any rate, the basis element

It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time.  A time variable is actually a reference to a ''clock'', that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process.  This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others.  But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.

It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time.  A time variable is actually a reference to a ''clock'', that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process.  This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others.  But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.

<p>The clock indicates the moment . . . . but what does<br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?</p>

 

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?

| align="right" | &mdash; Walt Whitman, 'Leaves of Grass', [Whi, 79]

Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(d''A''),&nbsp;d''A''} are preserved or changed in the next instance.  In order to know this, we would have to determine d²''A'', and so on, pursuing an infinite regress.  Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d^''k''''A'' = 0 for all ''k'' greater than some fixed value ''M''.  Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.

Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(d''A''),&nbsp;d''A''} are preserved or changed in the next instance.  In order to know this, we would have to determine d²''A'', and so on, pursuing an infinite regress.  Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d^''k''''A'' = 0 for all ''k'' greater than some fixed value ''M''.  Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.

===Example 1.  A Square Rigging===

===Example 1.  A Square Rigging===

<p>Urge and urge and urge,<br>

Always the procreant urge of the world.</p>

 

Urge and urge and urge,<br>

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 28]</p>

Always the procreant urge of the world.

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]

By way of example, suppose that we are given the initial condition ''A''&nbsp;=&nbsp;d''A'' and the law d²''A''&nbsp;=&nbsp;(''A'').  Then, since "''A''&nbsp;=&nbsp;d''A''" &hArr; "''A''&nbsp;d''A'' or (''A'')(d''A'')", we may infer two possible trajectories, as displayed in Table 11.  In either of these cases, the state ''A''(d''A'')(d²''A'') is a stable attractor or a terminal condition for both starting points.

By way of example, suppose that we are given the initial condition ''A''&nbsp;=&nbsp;d''A'' and the law d²''A''&nbsp;=&nbsp;(''A'').  Then, since "''A''&nbsp;=&nbsp;d''A''" &hArr; "''A''&nbsp;d''A'' or (''A'')(d''A'')", we may infer two possible trajectories, as displayed in Table 11.  In either of these cases, the state ''A''(d''A'')(d²''A'') is a stable attractor or a terminal condition for both starting points.

===Back to the Feature===

===Back to the Feature===

<p>I guess it must be the flag of my disposition, out of hopeful<br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.</p>

 

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.

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]

Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>]&nbsp;=&nbsp;[''A'',&nbsp;d''A''].  Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''₁,&nbsp;d''x''₁} = {''A'',&nbsp;d''A''}, of cardinality 2^''n'' = 2, we generate the set of points, E''X'', of cardinality 2^2''n'' = 4, that bears the following chain of equivalent descriptions:

Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>]&nbsp;=&nbsp;[''A'',&nbsp;d''A''].  Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''₁,&nbsp;d''x''₁} = {''A'',&nbsp;d''A''}, of cardinality 2^''n'' = 2, we generate the set of points, E''X'', of cardinality 2^2''n'' = 4, that bears the following chain of equivalent descriptions:

===Tacit Extensions===

===Tacit Extensions===

<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>

 

<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.

| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]

Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \operatorname{E}X \to \mathbb{B},</math> even though they share the same logical expression.  Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives.  Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.

Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \operatorname{E}X \to \mathbb{B},</math> even though they share the same logical expression.  Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives.  Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.

===Example 2.  Drives and Their Vicissitudes===

===Example 2.  Drives and Their Vicissitudes===

<p>I open my scuttle at night and see the far-sprinkled systems,<br>

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>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]

Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.

Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.