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{{DISPLAYTITLE:Dynamics And Logic}}
{{DISPLAYTITLE:Dynamics And Logic}}
==Note 1==
<pre>
<pre>
I am going to excerpt some of my previous explorations
I am going to excerpt some of my previous explorations
on differential logic and dynamic systems and bring them
on differential logic and dynamic systems and bring them
| ((dp) (dq)) |
| ((dp) (dq)) |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
==Note 2==
<pre>
We have just met with the fact that
We have just met with the fact that
the differential of the "and" is
the differential of the "and" is
as saying that dp implies dq, in other words, that there
as saying that dp implies dq, in other words, that there
is "no change in p without a change in q".
is "no change in p without a change in q".
</pre>
==Note 3==
<pre>
Given the proposition f<p, q> over the space X = !P! x !Q!,
Given the proposition f<p, q> over the space X = !P! x !Q!,
the (first order) "enlargement" of f is the proposition Ef
the (first order) "enlargement" of f is the proposition Ef
you happen to find yourself in the center cell and require
you happen to find yourself in the center cell and require
a complete and detailed description of ways to escape it.
a complete and detailed description of ways to escape it.
</pre>
==Note 4==
<pre>
Last time we computed what will variously be called
Last time we computed what will variously be called
the "difference map", the "difference proposition",
the "difference map", the "difference proposition",
to a point where the value of f<p, q> is different
to a point where the value of f<p, q> is different
from what it is where you started.
from what it is where you started.
</pre>
==Note 5==
<pre>
We have been studying the action of the difference operator D,
We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
also known as the "localization operator", on the proposition
any one of which may reveal some unsuspected aspect of its meaning.
any one of which may reveal some unsuspected aspect of its meaning.
We will encounter more and more of these variant readings as we go.
We will encounter more and more of these variant readings as we go.
</pre>
==Note 6==
<pre>
The enlargement operator E, also known as the "shift operator",
The enlargement operator E, also known as the "shift operator",
has many interesting and very useful properties in its own right,
has many interesting and very useful properties in its own right,
as telling us all the different ways to reach
as telling us all the different ways to reach
a model of f from any point of the universe X.
a model of f from any point of the universe X.
</pre>
==Note 7==
<pre>
To broaden our experience with simple examples, let us
To broaden our experience with simple examples, let us
now contemplate the sixteen functions of concrete type
now contemplate the sixteen functions of concrete type
| | | | | | |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
o---------o---------o---------o----------o------------------o----------o
</pre>
==Note 8==
<pre>
The next four Tables expand the expressions of Ef and Df
The next four Tables expand the expressions of Ef and Df
in two different ways, for each of the sixteen functions.
in two different ways, for each of the sixteen functions.
| | | | | | |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
</pre>
==Note 9==
<pre>
Table 9-a. Ef Expanded Over Differential Features
Table 9-a. Ef Expanded Over Differential Features
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | | | | | |
o------o------------o------------o------------o------------o------------o
o------o------------o------------o------------o------------o------------o
</pre>
==Note 10==
<pre>
If you think that I linger in the realm of logical difference calculus
If you think that I linger in the realm of logical difference calculus
out of sheer vacillation about getting down to the differential proper,
out of sheer vacillation about getting down to the differential proper,
three group elements fix 4 propositions each, and so we get:
three group elements fix 4 propositions each, and so we get:
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. -- Amazing!
Number of orbits = (4 + 4 + 4 + 16) / 4 = 7. -- Amazing!
</pre>
==Note 11==
<pre>
We have been contemplating functions of the type f : X -> B,
We have been contemplating functions of the type f : X -> B,
studying the action of the operators E and D on this family.
studying the action of the operators E and D on this family.
(u_1, ..., u_k) indicates the disjunctive region consisting of
(u_1, ..., u_k) indicates the disjunctive region consisting of
the cells that are "just next door" to the cell u_1 ... u_k.
the cells that are "just next door" to the cell u_1 ... u_k.
</pre>
==Note 12==
<pre>
| Consider what effects that might conceivably have
| Consider what effects that might conceivably have
| practical bearings you conceive the objects of your
| practical bearings you conceive the objects of your
More on the pragmatic maxim as a representation principle later.
More on the pragmatic maxim as a representation principle later.
</pre>
==Note 13==
<pre>
The above-mentioned fact about the regular representations
The above-mentioned fact about the regular representations
of a group is universally known as "Cayley's Theorem". It
of a group is universally known as "Cayley's Theorem". It
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
of lambda calculus, and I reckon you all know where that leads.
of lambda calculus, and I reckon you all know where that leads.
</pre>
==Note 14==
<pre>
The next few excursions in this series will provide
The next few excursions in this series will provide
a scenic tour of various ideas in group theory that
a scenic tour of various ideas in group theory that
I think that will serve to fix notation
I think that will serve to fix notation
and set up the remainder of the account.
and set up the remainder of the account.
</pre>
==Note 15==
<pre>
In Peirce's time, and even in some circles of mathematics today,
In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
the information indicated by the elementary relatives (i:j), as
This is consistent with the convention that Peirce uses in
This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
</pre>
==Note 16==
<pre>
We've been exploring the applications of a certain technique
We've been exploring the applications of a certain technique
for clarifying abstruse concepts, a rough-cut version of the
for clarifying abstruse concepts, a rough-cut version of the
and so the two representations have the very
and so the two representations have the very
same effects on each point of their bearing.
same effects on each point of their bearing.
</pre>
==Note 17==
<pre>
So long as we're in the neighborhood, we might as well take in
So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
some more of the sights, for instance, the smallest example of
(CP 4.227-323), with particular reference to the section
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).
that treats of "Trichotomic Mathematics" (CP 4.307-323).
</pre>
==Note 18==
<pre>
By way of collecting a short-term pay-off for all the work that we
By way of collecting a short-term pay-off for all the work that we
did on the regular representations of the Klein 4-group V_4, let us
did on the regular representations of the Klein 4-group V_4, let us
direction, but that will be an easy enough bridge to cross when we
direction, but that will be an easy enough bridge to cross when we
come to it.
come to it.
</pre>
==Note 19==
<pre>
To construct the regular representations of S_3,
To construct the regular representations of S_3,
we pick up from the data of its operation table,
we pick up from the data of its operation table,
though, of course, being representations of the
though, of course, being representations of the
same abstract group, they must be isomorphic.
same abstract group, they must be isomorphic.
</pre>
==Note 20==
<pre>
You may be wondering what happened to the announced subject
You may be wondering what happened to the announced subject
of "Dynamics And Logic". What occurred was a bit like this:
of "Dynamics And Logic". What occurred was a bit like this:
Biographical Data for Marius Sophus Lie (1842-1899):
Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
</pre>
==Note 21==
<pre>
We have seen a couple of groups, V_4 and S_3, represented in
We have seen a couple of groups, V_4 and S_3, represented in
several different ways, and we have seen each of these types
several different ways, and we have seen each of these types
| c:a | c:b | c:c |
| c:a | c:b | c:c |
o-----o-----o-----o
o-----o-----o-----o
</pre>
==Note 22==
<pre>
It would be good to summarize, in rough but intuitive terms,
It would be good to summarize, in rough but intuitive terms,
the outlook on differential logic that we have reached so far.
the outlook on differential logic that we have reached so far.
that need to be made from each point of X in order
that need to be made from each point of X in order
to feel a change in the felt value of the field pq.
to feel a change in the felt value of the field pq.
</pre>
==Note 23==
<pre>
I want to continue developing the basic tools of differential logic,
I want to continue developing the basic tools of differential logic,
which arose out of many years of thinking about the connections
which arose out of many years of thinking about the connections
o---------------------------------------------------------------------o
o---------------------------------------------------------------------o
Figure 23. Elements of a Cybernetic System
Figure 23. Elements of a Cybernetic System
</pre>
==Note 24==
<pre>
Now that we've introduced the field picture for thinking about
Now that we've introduced the field picture for thinking about
propositions and their analytic series, a very pleasing way of
propositions and their analytic series, a very pleasing way of
p q . (dp)(dq)
p q . (dp)(dq)
</pre>
==Note 25==
<pre>
Staying with the example pq : X -> B, Figure 25-1 shows
Staying with the example pq : X -> B, Figure 25-1 shows
the enlargement or shift map E[pq] : EX -> B in the same
the enlargement or shift map E[pq] : EX -> B in the same
(p)(q) . dp dq
(p)(q) . dp dq
</pre>
==Note 26==
<pre>
If we follow the classical line that singles out linear functions
If we follow the classical line that singles out linear functions
as ideals of simplicity, then we may complete the analytic series
as ideals of simplicity, then we may complete the analytic series
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
</pre>
</pre>
