Re: Sign Relational Manifolds • Discussi on

[Correcting some typos.  Ignore previous post.]

Sign Relational Manifolds • Exposition

http://permalink.gmane.org/gmane.science.philosophy.peirce/8850
http://permalink.gmane.org/gmane.science.philosophy.peirce/8854
http://permalink.gmane.org/gmane.science.philosophy.peirce/8871

http://inquiryintoinquiry.com/2012/10/22/sign-relational-manifolds-1/
http://inquiryintoinquiry.com/2012/10/22/sign-relational-manifolds-2/
http://inquiryintoinquiry.com/2012/10/23/sign-relational-manifolds-3/

Sign Relational Manifolds • Discussion

JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/8861
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8868

JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/8862
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8872

IA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8866
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8875

JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/8873

JC: While I appreciate your responses to my question, it appears
     that you are headed down the Russell-Whitehead philosophy of
     mathematics as set theory + continuity + ...

JC: My question is simple.
     It appears that you are mis-interpreting the simplicity.

JC: If Riemann's concept of a manifold, and
     if Peirce's sign relations,
     are in close relationship,

JC: THEN simply state what it is that leads [you]
     to assert that such a close relationships exists.

JC: One sentence could be adequate.

Jerry & All,

One point of clarification.  I am using phrases like
“Riemann's concept of a manifold, as later developed”
in the way one might use “Galois' concept of a group”
or “Euclidean geometry” to refer to the entire subject
matter that follows on the work of its eponymous founder.
We all know that smudges many fine points of the actual
history, but it's the way subjects are described in the
literature, so that's my excuse, and I'm sticking to it.

That said, my sentence would be:

(X, E_i, E_j) = (O, S, I).

Of course the sentence is not self-contained — its meaning
recurs to a number of preliminary concepts and definitions,
which I referenced in a previous post and summarized here:

• http://stderr.org/pipermail/inquiry/2003-May/000454.html

On the manifold side, X is the space of objective interest,
where E_i is the codomain of the chart map q_i : U_i -> E_i,
where E_j is the codomain of the chart map q_j : U_j -> E_j,
and where (U_i, q_i) and (U_j, q_j) are any two charts in
the atlas of the manifold structure over X.

On the sign relation side, O is the object domain,
S is the sign domain, I is the interpretant domain
of a sign relation L that is contained as a subset
of the cartesian product O × S × I.

There is further detail and a figure in the note I linked above.

Regards,

Jon

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