Sign Relational Manifolds

[Jon Awbrey]

Cf: Sign Relational Manifolds • 1
http://inquiryintoinquiry.com/2022/10/31/sign-relational-manifolds-1-2/

All,

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

I will have to wait for my present train of thought to stop at a station
before I can hop another but several recent discussions of geometry have
brought this subject back to mind and I thought it might serve to drop off
a few mail bags of related letters in anticipation of the next pass through
this junction.

Here are links to a set of excerpts from Murray G. Murphey (1961),
“The Development of Peirce's Philosophy”, discussing Peirce's reception
of Riemann's philosophy of geometry.

Manifolds of Signs
==================
https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#313
https://web.archive.org/web/20150206000400/http://stderr.org/pipermail/inquiry/2003-April/000313.html
https://web.archive.org/web/20150206000800/http://stderr.org/pipermail/inquiry/2003-April/000315.html
https://web.archive.org/web/20150206000818/http://stderr.org/pipermail/inquiry/2003-April/000316.html

Later developments of the manifold concept, looking to applications
on the one hand and theory on the other, are illustrated by excerpts
in the next two posts.

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • 2
http://inquiryintoinquiry.com/2022/11/01/sign-relational-manifolds-2-2/

All,

A taste of how manifolds are used in practice may be gleaned from
the set of excerpts linked below, from Doolin and Martin (1990),
“Introduction to Differential Geometry for Engineers”, which I used
in discussing differentiable manifolds with other participants in the
IEEE Standard Upper Ontology Working Group
( https://web.archive.org/web/20140512225349/http://suo.ieee.org/ ) .

Differential Geometry for Engineers
https://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04056
1. https://web.archive.org/web/20070302105328/http://suo.ieee.org/ontology/msg04056.html
•••
9. https://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04065.html

What brought the concept of a manifold to my mind in that context was
a set of problems associated with “perspectivity”, “relativity”, and
“interoperability” among multiple ontologies. To my way of thinking,
those are the very sorts of problems manifolds were invented to handle.

Reference
=========

Doolin, Brian F., and Martin, Clyde F. (1990), “Introduction to
Differential Geometry for Engineers”, Marcel Dekker, New York, NY.

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • 3
http://inquiryintoinquiry.com/2022/11/03/sign-relational-manifolds-3-2/

All,

I'm not sure when it was I first noticed the relationship between manifolds and
semiotics but I distinctly recall the passage in Serge Lang's “Differential and
Riemannian Manifolds” which brought the triadic character of tangent vectors into
high relief. I copied out a set of excerpts highlighting the point and shared it
with the Inquiry, Ontology, and Peirce lists.

Excerpts from Serge Lang, “Differential and Riemannian Manifolds”,
Springer‑Verlag, New York, NY, 1995.

Chapter 2. Manifolds
=====================

2.1. Atlases, Charts, Morphisms
===============================
https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#442
https://web.archive.org/web/20141220180402/http://stderr.org/pipermail/inquiry/2003-April/000442.html
•••
https://web.archive.org/web/20070309203913/http://stderr.org/pipermail/inquiry/2003-April/000447.html

2.2. Submanifolds, Immersions, Submersions
==========================================
https://web.archive.org/web/20141220174800/http://stderr.org/pipermail/inquiry/2003-May/thread.html#448
•••
https://web.archive.org/web/20061013220508/http://stderr.org/pipermail/inquiry/2003-May/000451.html
•••
https://web.archive.org/web/20061013220452/http://stderr.org/pipermail/inquiry/2003-May/000460.html

Using the concepts and terminology from Lang's text, I explained the
connection between manifold theory and semiotics in the following way.

Commentary Note
https://web.archive.org/web/20141220173001/http://stderr.org/pipermail/inquiry/2003-May/000454.html

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • 4
http://inquiryintoinquiry.com/2022/11/05/sign-relational-manifolds-4-2/

All,

Another set of notes I found on this theme strikes me as
getting to the point more quickly and though they read
a little rough in places I think it may be worth the
effort to fill out their general line of approach.

Representation Invariant Ontology
https://web.archive.org/web/20150302021003/http://stderr.org/pipermail/inquiry/2003-April/thread.html#439
https://web.archive.org/web/20141220180218/http://stderr.org/pipermail/inquiry/2003-April/000439.html
https://web.archive.org/web/20141220180220/http://stderr.org/pipermail/inquiry/2003-April/000440.html

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • 5
http://inquiryintoinquiry.com/2022/11/06/sign-relational-manifolds-5-2/

All,

Let me try to say in intuitive terms what I think is really going on here.

The problem we face is as old as the problem of other minds,
or intersubjectivity, or even commensurability, and it naturally
involves a whole slew of other old problems — reality and appearance,
or reality and representation, not to mention the one and the many.
One way to sum up the question might be “conditions on the possibility
of a mutually objective world”.

Working on what oftentimes seems like the tenuous assumption that there really
is a real world causing the impressions in my mind and the impressions in yours —
more generally speaking, that there really is a real world impressing itself in
systematic measures on every frame of reference — we find ourselves pressed to
give an account of the hypothetical unity beneath the manifest diversity — and
how it is possible to discover the former in the latter.

Manifold theory proposes one type of solution to that host of problems.

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • Discussion 1
http://inquiryintoinquiry.com/2022/11/07/sign-relational-manifolds-discussion-1-2/

All,

On one of the Facebook pages devoted to Semiotics
someone asked the following question:

• “What's at the End of a Chain of Interpretants?”
https://www.facebook.com/groups/373930009449106/permalink/854898248018944/

It's a variation on a question which comes up from time to time
and I gave a variation on the answer I have given now and again:

Semiotic manifolds, like physical and mathematical manifolds,
may be finite and bounded or infinite and unbounded but they
may also be finite and unbounded, having no boundary in the
topological sense. Thus unbounded semiosis does not imply
infinite semiosis.

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • Discussion 2
http://inquiryintoinquiry.com/2022/11/07/sign-relational-manifolds-discussion-2/

Re: FB | Paradoxology
https://www.facebook.com/groups/517592312405190/posts/1300714480759632
::: Alex Shkotin
https://www.facebook.com/groups/517592312405190/posts/1300714480759632?comment_id=1302559267241820

<QUOTE AS:>
Not on a narrow topic, but maybe you have a desire to answer.
Hypothesis. Any material something can be a sign.
Is it possible to give an example of something material that cannot be a sign?
</QUOTE>

Hi Alex,

Sign relations are mathematical relations we can use to model processes of
communication, learning, reasoning, just plain talking and thinking in general.
Anytime we can imagine a triadic relation where one thing, material or otherwise,
is related to a second thing in such a way that both refer to a third thing, and
that whole relationship is useful in modeling one of the above mentioned processes,
then we have a candidate which may be suitable for serving as a sign relation in the
pragmatic conception of the term.

Regards,

Jon

[Jon Awbrey]

Cf: Sign Relational Manifolds • Discussion 3
https://inquiryintoinquiry.com/2022/11/11/sign-relational-manifolds-discussion-3/

Re: FB | Paradoxology
https://www.facebook.com/groups/517592312405190/posts/1300714480759632

::: Alex Shkotin (
https://www.facebook.com/groups/517592312405190/posts/1300714480759632/?comment_id=1302697013894712&reply_comment_id=1302794400551640


<QUOTE AS:>
I see — “sign relation” is a special term for triadic relations
of some kind (with some properties); like this: thing in first
position and thing in second position must refer to the thing in
third position. Where “refer” is an unary partial function from
one thing to another. Am I on a right direction?

Hi Alex,

It is not uncommon in practice to find a sign s having many interpretant signs i
and many referent objects o. Generally speaking, then, we start out with a sign
relation L as a subset of a cartesian product L ⊆ O × S × I, where O, S, I are sets
called the “object domain”, “sign domain”, “interpretant sign domain”, respectively.
A definition of a sign relation — there are a few canonical ones we find useful in
practice — will specify what sort of constraint is involved in forming that subset.

Regards,

Jon