Dear Arnold,
I believe L 224, the letter Peirce wrote to William James on 1909 Feb 26 is
exceedingly important here. In print you find it in volume III/2, p.836
ff. of
"The New Elements of Mathematics", ed.: Carolyn Eisele.
What is important is the fact that general transitivity has a property that
would in Boolean algebra amount to the "Idempotency Law", i.e. AA=A. You
can see
this clearly on page 838. It would be very tedious to try to cite things
here in
an email text, so I can only indicate what is meant.
Since the idempotency expresses identity, we can see here quite clearly why
Peirce in connection with the Existential Graphs spoke of "ter-identity"
(as in
"line of teridentity"). But this very insight in fact goes back to the 1867
paper "On the Natural Classification of Arguments" (see CP 2.461 ff) where
Peirce writes:
"There is, however, an intention in which these substitutions are
inferential.
For, although the passage from holding for true a fact expressed in the
form "No
A is B", to holding its converse, is not an inference, because, these facts
being identical, the relation between them is not a fact; yet the passage
from
one of these forms taken merely as having "some" meaning, but not this or
that
meaning, to another, since these forms are not identical and their logical
relation is a fact, is an inference. The distinction may be expressed by
saying
that they are not inferences, but substitutions having the "form" of
inferences."(CP 2.496)
And this then lead to the "On a New List of Categories" in 1867.
I think we should address this as "diagonalization", since in L 224, after
exploring what is meant by "Logical Analysis of a Concept and a Real
Definition
of it" (p.844) and addressing Johann Benedict Listing's "Barycentral
Calculus"
[Sorry, it is impossible to go through all this mathematics; but these
things,
as e.g. the Barycentral Calculus is not so much important here in itself,
anyway] Peirce writes (on page 854) concerning the "theory that concepts
are
combined in a form substantially like that of the combination of points in
the
barycentric calculus":
"I may remark, that it is in barycentral composition that forces are
combined
according to the principle of the parallelogram of forces".
I remind you, what Peirce here still is talking about is general
transitivity
and what he found in 1867! (you'll see that later in the text of L 224/see
below)
Next (poor William James:-)) he gives his proof of what today is known in
mathematics as the Peirce Theorem. It says that "Every linear associative
algebra is isomorphic to a matrix algebra." (compare e.g. Birkhoff/
MacLane,
Modern Algebra, p.398; by the way interesting to compare this to the
theorem
that "Any abstract group is isomorphic with a group of transformations" on
page
139 of the same book;-)).
In the proof Peirce gives of his theorem, it is not the isomorphism that
is
interesting. By way of proof analysis it is interesting to note how Peirce
arranges these (A:A), (A:B) etc forms, which are part of sums, in a square
and
again compare this to Cantor's ¨€¨Diagonalverfahren¨€œ, secondly it is
interesting
to note the distinction Peirce makes on page 858 between ���substitution��� and
¨€¨replacement¨€œ and it is interesting to note that the last formula on page
858
somehow reminds one strongly of the form of the general transitive
relation back
at the beginning of L 224 and in CP 3.523f. (here then the formula on top
of the
page 332 in the CP.)
Sorry, if Peirce could do this with poor William James, then I here simply
do
the same with poor you :-)
I could even "top" it by saying that what is important here, as in the
Bernoulli
series, is not the concrete system of numbers or abstract algebraic
"units", but
the relation beween multiplication and addition or "distributivity" and
"collectivity" (just to coin two new terms to be thrown away immediately
after
use!).
By the way, Jacob Bernoulli didn't derive his "Bernoulli numbers"
algebraically.
That probably would have been utterly hopeless. He SAW A PATTERN in the
series
he was exploring. With his very eyes. That's historical fact and not a
myth, as
it seems. Interesting curiosity, isn't it?
What I want to say: the conception of probability and its logical
foundation is
not very far here.
Sorry again, I don't say this here to appear "intelligent" or something,
or to
impress anybody with my "superior personality". I simply don't know a
better way
to say it and perhaps it is better so to say it, instead that it is lost
for
further use. In other words: this might be a good field for a research
project,
Arnold;-)
Again, analysing Peirce's proof, keep categorial structure in mind!
Cheers,
Thomas.
On Wed, 15 Mar 2006 08:32:12 +0100, Arnold Shepperson
<[EMAIL PROTECTED]> wrote:
Thomas
TR: Thomas Riese
AS: Arnold Shepperson
TR: Peirce is exactly interested in the relation between isomorphous
forms.
His primary relation is the general form of transitivity.
TR: The difference has far reaching, profound implications.
AS: I agree with you on this. Contemporary work that exploits
transitivity
with which I am familiar includes the work on welfarist economics by
Arrow
and Sen, amongst others. Lister James Wible (are you there, James?) has
noted the very important role transitivity plays in Peirce, especially in
the way this relates to the economy of research. I would venture to
suggest
(subject to the better sense of those on the list who have greater
experince
with the MSS than I have) that the notion of a Sign contains the concept
of
a transitive function, making a very strong case for what Thomas has
said on
this subject. Other transitive functions in Peirce can be found in Vols
III
and IV of the CP (see especially 3.562).
Cheers
Arnold Shepperson
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