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, 09:35, 4 June 2009+ later notes
By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).
By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).
==Work Area==
==Note 20==
<pre>
By way of collecting a short-term pay-off for all the work --
not to mention the peirce-spiration -- that we sweated out
over the regular representations of the Klein 4-group V_4,
let us write out as quickly as possible in "relative form"
a minimal budget of representations of the symmetric group
on three letters, S_3 = Sym(3). After doing the usual bit
of compare and contrast among these divers representations,
we will have enough concrete material beneath our abstract
belts to tackle a few of the presently obscur'd details of
Peirce's early "Algebra + Logic" papers.
Table 1. Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| A B C | A B C | A B C | A B C | A B C | A B C |
| | | | | | |
| | | | | | | | | | | | | | | | | | | | | | | | |
| v v v | v v v | v v v | v v v | v v v | v v v |
| | | | | | |
| A B C | C A B | B C A | A C B | C B A | B A C |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
Writing this table in relative form generates
the following "natural representation" of S_3.
e = A:A + B:B + C:C
f = A:C + B:A + C:B
g = A:B + B:C + C:A
h = A:A + B:C + C:B
i = A:C + B:B + C:A
j = A:B + B:A + C:C
I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as X:Y
constitutes the turning of X into Y. It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.
</pre>
==Note 21==
<pre>
To construct the regular representations of S_3,
we pick up from the data of its operation table:
Table 1. Symmetric Group S_3
| ^
| e / \ e
| / \
| / e \
| f / \ / \ f
| / \ / \
| / f \ f \
| g / \ / \ / \ g
| / \ / \ / \
| / g \ g \ g \
| h / \ / \ / \ / \ h
| / \ / \ / \ / \
| / h \ e \ e \ h \
| i / \ / \ / \ / \ / \ i
| / \ / \ / \ / \ / \
| / i \ i \ f \ j \ i \
| j / \ / \ / \ / \ / \ / \ j
| / \ / \ / \ / \ / \ / \
| ( j \ j \ j \ i \ h \ j )
| \ / \ / \ / \ / \ / \ /
| \ / \ / \ / \ / \ / \ /
| \ h \ h \ e \ j \ i /
| \ / \ / \ / \ / \ /
| \ / \ / \ / \ / \ /
| \ i \ g \ f \ h /
| \ / \ / \ / \ /
| \ / \ / \ / \ /
| \ f \ e \ g /
| \ / \ / \ /
| \ / \ / \ /
| \ g \ f /
| \ / \ /
| \ / \ /
| \ e /
| \ /
| \ /
| v
Just by way of staying clear about what we are doing,
let's return to the recipe that we worked out before:
It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.
Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:
1. Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to x·y.
2. Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, >, with
the effect of producing the saturated
rheme <y, x> that evaluates to y·x.
In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into x·y, for y in G, all of which
is summarily notated as x = {(y : x·y) : y in G}.
The pairs (y : x·y) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin. This produces
the regular ante-representation of S_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:j + i:h + j:i
g = e:g + f:e + g:f + h:i + i:j + j:h
h = e:h + f:i + g:j + h:e + i:f + j:g
i = e:i + f:j + g:h + h:g + i:e + j:f
j = e:j + f:h + g:i + h:f + i:g + j:e
In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into y·x, for y in G, all of which
is summarily notated as x = {(y : y·x) : y in G}.
The pairs (y : y·x) can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin. This generates
the regular post-representation of S_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:i + i:j + j:h
g = e:g + f:e + g:f + h:j + i:h + j:i
h = e:h + f:j + g:i + h:e + i:g + j:f
i = e:i + f:h + g:j + h:f + i:e + j:g
j = e:j + f:i + g:h + h:g + i:f + j:e
If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.
</pre>
==Note 22==
<pre>
<pre>
| Consider what effects that might 'conceivably'
| the way of heaven and earth
| is to be long continued
| objects of your 'conception' to have. Then,
| in their operation
| your 'conception' of those effects is the
| without stopping
| whole of your 'conception' of the object.
|
|
| Charles Sanders Peirce,
| i ching, hexagram 32
You may be wondering what happened to the announced subject
of "Differential Logic", and if you think that we have been
taking a slight "excursion" -- to use my favorite euphemism
for "digression" -- my reply to the charge of a scenic rout
would need to be both "yes and no". What happened was this.
At the sign-post marked by Sigil 7, we made the observation
that the shift operators E_ij form a transformation group
that acts on the propositions of the form f : B^2 -> B.
Now group theory is a very attractive subject, but it
did not really have the effect of drawing us so far
off our initial course as you may at first think.
For one thing, groups, in particular, the groups
that have come to be named after the Norwegian
mathematician Marius Sophus Lie, have turned
out to be of critical utility in the solution
of differential equations. For another thing,
group operations afford us examples of triadic
relations that have been extremely well-studied
over the years, and this provides us with quite
a bit of guidance in the study of sign relations,
another class of triadic relations of significance
for logical studies, in our brief acquaintance with
which we have scarcely even started to break the ice.
Finally, I could hardly avoid taking up the connection
between group representations, a very generic class of
logical models, and the all-important pragmatic maxim.
Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
</pre>
==Note 23==
<pre>
| Bein' on the twenty-third of June,
| As I sat weaving all at my loom,
| Bein' on the twenty-third of June,
| As I sat weaving all at my loom,
| I heard a thrush, singing on yon bush,
| And the song she sang was The Jug of Punch.
We've seen a couple of groups, V_4 and S_3, represented in various ways, and
we've seen their representations presented in a variety of different manners.
Let us look at one other stylistic variant for presenting a representation
that is frequently seen, the so-called "matrix representation" of a group.
Recalling the manner of our acquaintance with the symmetric group S_3,
we began with the "bigraph" (bipartite graph) picture of its natural
representation as the set of all permutations or substitutions on
the set X = {A, B, C}.
Table 2. Permutations or Substitutions in Sym_{A, B, C}
Table 1. Permutations or Substitutions in Sym {A, B, C}
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
problem about writing:
Then we rewrote these permutations -- being functions f : X --> X
they can also be recognized as being 2-adic relations f c X x X --
in "relative form", in effect, in the manner to which Peirce would
have made us accostumed had he been given a relative half-a-chance:
e = A:A + B:B + C:C
f = A:C + B:A + C:B
g = A:B + B:C + C:A
h = A:A + B:C + C:B
i = A:C + B:B + C:A
j = A:B + B:A + C:C
These days one is much more likely to encounter the natural representation
of S_3 in the form of a "linear representation", that is, as a family of
linear transformations that map the elements of a suitable vector space
into each other, all of which would in turn usually be represented by
a set of matrices like these:
Table 2. Matrix Representations of Permutations in Sym(3)
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| e | f | g | h | i | j |
| | | | | | |
o=========o=========o=========o=========o=========o=========o
| | | | | | |
| 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 |
| 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 |
| 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
The key to the mysteries of these matrices is revealed by noting that their
coefficient entries are arrayed and overlayed on a place mat marked like so:
| A:A A:B A:C |
| B:A B:B B:C |
| C:A C:B C:C |
Of course, the place-settings of convenience at different symposia may vary.
</pre>
==Note 24==
<pre>
| In the beginning was the three-pointed star,
| One smile of light across the empty face;
| One bough of bone across the rooting air,
| The substance forked that marrowed the first sun;
| And, burning ciphers on the round of space,
| Heaven and hell mixed as they spun.
|
| Dylan Thomas, "In The Beginning", Verse 1
I'm afrayed that this thread is just bound to keep
encountering its manifold of tensuous distractions,
but I'd like to try and return now to the topic of
inquiry, espectrally viewed in differential aspect.
Here's one picture of how it begins,
one angle on the point of departure:
o-----------------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | | |
| | | |
| | Observation | |
| | | |
| | | |
| o--o----------o o----------o--o |
| / \ \ / / \ |
| / \ d_I ^ o ^ d_E / \ |
| / \ \/ \/ / \ |
| / \ /\ /\ / \ |
| / \ / @ \ / \ |
| o o--o---|---o--o o |
| | | | | | |
| | | v | | |
| | Expectation | d_O | Intention | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
From what we must assume was a state of "Unconscious Nirvana" (UN),
since we do not acutely become conscious until after we are exiled
from that garden of our blissful innocence, where our Expectations,
our Intentions, our Observations all subsist in a state of perfect
harmony, one with every barely perceived other, something intrudes
on that scene of paradise to knock us out of that blessed isle and
to trouble our countenance forever after at the retrospect thereof.
The least disturbance, it being provident and prudent both to take
that first up, will arise in just one of three ways, in accord with
the mode of discord that importunes on our equanimity, whether it is
Expectation, Intention, Observation that incipiently incites the riot,
departing as it will from congruence with the other two modes of being.
In short, we cross just one of the three lines that border on the center,
or perhaps it is better to say that the objective situation transits one
of the chordal bounds of harmony, for the moment marked as d_E, d_I, d_O
to note the fact one's Expectation, Intention, Observation, respectively,
is the mode that we duly indite as the one that's sounding the sour note.
A difference between Expectation and Observation is experienced
as a "Surprise", a phenomenon that cries out for an Explanation.
A discrepancy between Intention and Observation is experienced
as a "Problem", of the species that calls for a Plan of Action.
I can remember that I once thought up what I thought up an apt
name for a gap between Expectation and Intention, but I cannot
recall what it was, nor yet find the notes where I recorded it.
At any rate, the modes of experiencing a surprising phenomenon
or a problematic situation, as described just now, are already
complex modalities, and will need to be analyzed further if we
want to relate them to the minimal changes d_E, d_I, d_O. Let
me think about that for a little while and see what transpires.
</pre>
==Note 25==
<pre>
| In the beginning was the pale signature,
| Three-syllabled and starry as the smile;
| And after came the imprints on the water,
| Stamp of the minted face upon the moon;
| The blood that touched the crosstree and the grail
| Touched the first cloud and left a sign.
|
| Dylan Thomas, "In The Beginning", Verse 2
</pre>
==Note 26==
<pre>
| In the beginning was the mounting fire
| That set alight the weathers from a spark,
| A three-eyed, red-eyed spark, blunt as a flower;
| Life rose and spouted from the rolling seas,
| Burst in the roots, pumped from the earth and rock
| The secret oils that drive the grass.
|
| Dylan Thomas, "In The Beginning", Verse 3
</pre>
==Work Area==
<pre>
problem about writing
e = e:e + f:f + g:g + h:h
e = e:e + f:f + g:g + h:h
