Re: Sign Relational Manifolds • Discussi on

• 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/

JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8850
JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8854
JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/8861
JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/8862
IA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8866

IA: Jerry is quite right, that Riemann's concept of a manifold
     is a purely and exclusively a mathematical concept.

IA: For those with sufficient mathematical background and a reading ability in German,
     the best discussion of the history of manifolds remains Erhard Scholz's _Geschichte
     des Mannigfaltigkeitsbebriffs von Riemann bis Poincaré_ (Boston/Basel/Stuttgart:
     Birkhäuser, 1980) -- if you can still find a copy (unfortunately, it's out of print,
     and apparently has been difficult to find copies for quite some time).

IA: (Apropos of not much other than the mere mention on the list of differentiable
     manifolds, Jean van Heijenoort, my Doktorvater, did his Ph.D. with J.J. Stoker,
     at N.Y.U. in differential geometry, working on differentiable manifolds and the
     properties of convex sets, proving (1949) that if there is a support plane of
     a set through every boundary point of an open set, or of a closed set having
     interior points, then that set is convex. In his thesis, he proved it for
     2-dimensional manifold mapped into a 3-dimensional Euclidean space.  As it
     happened,unfortunately for van Heijenoort, A. D. Alexandrov proved and
     published the same theorem earlier that year. Van Heijenoort later (1952)
     was able to generalize his theorem to an (n-1)-dimensional space.)

Irving,

Thanks for the reference.  I will try to look it up sometime
when I find myself in the neighborhood of a bigger library.

I do not know what Jerry means by “a purely and exclusively a mathematical concept”,
so I cannot say whether I agree or disagree with his statement, or how it would bear
on what I wrote if I did.  All I have are several definitions of manifolds as they are
defined in standard sources, combined with my reminiscences of the more intuitive and
vivid explanations that my instructors and professors blessed me with over the years.

The definitions of a manifold that I have in mind all exhibit variations on a particular
genus of triadic relational structure, more or less “degenerate”, less or more “generic”
in triadic character, one which is recognizably akin to the structure of a sign relation.
I do not think this is purely coincidental, but that there are deep-lying reasons for it.

Regards,

Jon

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