Re: Sign Relational Manifolds • Discussi on

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

SE: http://permalink.gmane.org/gmane.science.philosophy.peirce/8874

SE: I share Jerry's confusion and would ask that you include
     definitions of “mathematics” and “sign relations” so that
     we can be sure.

SE: For example, we might define “mathematics” as “the science
     that draws necessary conclusions” — which is the definition
     that Peirce adhered to (and which came from his father).  We
     might say that, in a similar vein, that “Peirce's sign relations”
     are the necessary conclusions drawn from a study of apprehension.
     This would be consistent with the view of Benjamin and Charles
     Peirce, i.e., that Logic rests upon the (abduction of) necessary
     conclusions of mathematics (not vice versa, as posited by Logicism).

SE: So, when Jerry says that Riemann's manifold is an exclusively
     mathematical concept, he is simply appealing to the necessity
     that geometry, in general, is a purely mathematical concept —
     a position with which I agree.

SE: The relationship with Peirce's “sign relations” is that these too
     (I am confident that Peirce will argue) are necessary conclusions —
     but this is a general relation that allows the construction of
     “mathematical logic” and would not be drawn from Riemann manifolds
     in particular.

SE: Now, as I know Jon explores some of the same space as I, differentiable manifolds
     of one kind or another are important for the mathematical characterization of
     sign systems — so this may be the concept that Jon has in mind, upon which
     the confusion rests.  As a consequence we might argue the inverse — that
     a certain kind of differentiable manifold is the necessary conclusion
     drawn from a study of sign relations and biophysical evidence.  But
     I see not reason to prefer Riemann's formulation over others.

Steven,

I wrote as follows:

JA: Riemann's concept of a manifold, as later developed,
     has a close relationship to Peirce's sign relations.

A compulsion to parallel sentence structure impressed on me
by the English teachers of my impressionable youth forced me
to rewrite my statement as follows on my blog:

JA: Riemann’s concept of a manifold, especially as later developed,
     bears a close relationship to Peirce’s concept of a sign relation.

At any rate, my statement was designed to accord Riemann credit
for his part in delivering the manifold concept, modulo perhaps
the distinct possibility that its true paternity is due to Kant,
but my parenthetical clause was not purely incidental, which is
why I then proceeded to cite a few examples of the developments
I had in mind.

Jerry then remarked as follows:

JC: Riemann's concept of a manifold is a purely
     and exclusively a mathematical concept.

I simply don't know what Jerry means by “a purely and exclusively
a mathematical concept” or how it bears on what I wrote.  Does he
mean “pure" as opposed to “applied”, as one uses those adjectives
in mathematics?  That kind of “pure” does not apply to the notion
of a manifold.  What does the “exclusively” mean?  I cannot think
of any definition of mathematics that would exclude the treatment
of sign relations under that head.  So I think the burden remains
on Jerry to draw the line he prefers between the mathematical and
the non-mathematical.  I'm guessing it would be different strokes
for different folks if it comes to that, but I can't see anything
in Peirce that'd block a mathematical inquiry into sign relations.

For my part, I mentioned manifolds, and I pointed to the definitions
that I had in mind.  I mentioned sign relations, and for that let me
refer again to the same two variants of Peirce's NEM definition of a
sign that I have recommended every time the question has arisen over
the last dozen years or so on this list.

The two NEM variants are quoted here:

C.S. Peirce • On the Definition of Logic
• http://inquiryintoinquiry.com/2012/06/01/c-s-peirce-%E2%80%A2-on-the-definition-of-logic/

There is additional discussion here:

MyWikiBiz • Sign Relation
• http://mywikibiz.com/Sign_relation#Definition

Awbrey and Awbrey • Interpretation as Action : The Risk of Inquiry
• http://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry

Regards,

Jon

-- 

academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache