Sign Relational Manifolds


 

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


 

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


 

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


 

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


 

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


 

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


Lyle Anderson
 
Edited

This is a settled question in the Laws of Form:
The theme of this book is that a universe comes into being when a space is severed or
taken apart. The skin of a living organism cuts off an outside from an inside. So does the
circumference of a circle in a plane. By tracing the way we represent such a severance,
we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny,
the basic forms underlying linguistic, mathematical, physical, and biological science,
and can begin to see how the familiar laws of our own experience follow inexorably from
the original act of severance. The act is itself already remembered, even if unconsciously,
as our first attempt to distinguish different things in a world where, in the first place,
the boundaries can be drawn anywhere we please. At this stage the universe cannot be
distinguished from how we act upon it, and the world may seem like shifting sand
beneath our feet.
Although all forms, and thus all universes, are possible, and any particular form is
mutable, it becomes evident that the laws relating such forms are the same in any
universe. It is this sameness, the idea that we can find a reality independent of how the
universe actually appears, that lends such fascination to the study of mathematics. That
mathematics, in common with other art forms, can lead us beyond ordinary existence,
and can show us something of the structure in which all creation hangs together, is no
new idea. But mathematical texts generally begin the story somewhere in the middle,
leaving the reader to pick up the thread as best he can. Here the story is traced from
the beginning.
Unlike more superficial forms of expertise, mathematics is a way of saying less and
less about more and more. A mathematical text is thus not an end in itself, but a key to
a world beyond the compass of ordinary description.


 

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


Lyle Anderson
 

It seems to me that Pierce's sign is in fact Spencer-Brown's mark.  The distinction indicated by the mark is Pierce's object and they both share the observer.

Peirce’s Hegelianism, to which he increasingly admitted as he approached his most mature philosophy, is more difficult to understand than his Kantianism, partly because it is everywhere intimately tied to his entire late theory of signs (semeiotic) and sign use (semeiosis), as well as to his evolutionism and to his rather puzzling doctrine of mind. There are at least four major components of his Hegelian idealism. First, for Peirce the world of appearances, which he calls “the phaneron,” is a world consisting entirely of signs. Signs are qualities, relations, features, items, events, states, regularities, habits, laws, and so on that have meanings, significances, or interpretations. Second, a sign is one term in a threesome of terms that are indissolubly connected with each other by a crucial triadic relation that Peirce calls “the sign relation.” The sign itself (also called the representamen) is the term in the sign relation that is ordinarily said to represent or mean something. The other two terms in this relation are called the object and the interpretant. The object is what would ordinarily would be said to be the “thing” meant or signified or represented by the sign, what the sign is a sign of. The interpretant of a sign is said by Peirce to be that to which the sign represents the object. What exactly Peirce means by the interpretant is difficult to pin down. It is something like a mind, a mental act, a mental state, or a feature or quality of mind; at all events the interpretant is something ineliminably mental. Third, the interpretant of a sign, by virtue of the very definition Peirce gives of the sign-relation, must itself be a sign, and a sign moreover of the very same object that is (or: was) represented by the (original) sign. In effect, then, the interpretant is a second signifier of the object, only one that now has an overtly mental status. But, merely in being a sign of the original object, this second sign must itself have (Peirce uses the word “determine”) an interpretant, which then in turn is a new, third sign of the object, and again is one with an overtly mental status. And so on. Thus, if there is any sign at all of any object, then there is an infinite sequence of signs of that same object. So, everything in the phaneron, because it is a sign, begins an infinite sequence of mental interpretants of an object.
https://plato.stanford.edu/entries/peirce/

Now here is GSB's conclusion in the Laws of Form:

The value of a circumference to the space outside must be, therefore, the value of the mark, since the mark now distinguishes this space.
An observer, since he distinguishes the space he occupies, is also a mark.
In the experiments above, imagine the circles to be forms and their circumferences to be the distinctions shaping the spaces of these forms.
In this conception a distinction drawn in any space is a mark distinguishing the space. Equally and conversely, any mark in a space draws a distinction.
We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical.
Laws, 2011, pg 62-63

Now GSB makes the crucial addition of intention and calculation using intended steps.   The closest CSP comes is his concept of evolution in the phaneron.

But now, there is a fourth component of Peirce’s idealism: Peirce makes everything in the phaneron evolutionary. The whole system evolves. Three figures from the history of culture loomed exceedingly large in the intellectual development of Peirce and in the cultural atmosphere of the period in which Peirce was most active: Hegel in philosophy, Lyell in geology, and Darwin (along with Alfred Russel Wallace) in biology. These thinkers, of course, all have a single theme in common: evolution. Hegel described an evolution of ideas, Lyell an evolution of geological structures, and Darwin an evolution of biological species and varieties. Peirce absorbed it all. Peirce’s entire thinking, early on and later, is permeated with the evolutionary idea, which he extended generally, that is to say, beyond the confines of any particular subject matter. For Peirce, the entire universe and everything in it is an evolutionary product. Indeed, he conceived that even the most firmly entrenched of nature’s habits (for example, even those habits that are typically called “natural laws”) have themselves evolved, and accordingly can and should be subjects of philosophical and scientific inquiry. One can sensibly seek, in Peirce’s view, evolutionary explanations of the existence of particular natural laws. For Peirce, then, the entire phaneron (the world of appearances), as well as all the ongoing processes of its interpretation through mental significations, has evolved and is evolving.
IBID, Stanford

Of course, in keeping with the spirit of the age, this evolution is entirely accidental, all be it, subject to natural laws that one can, in principle, discover.  

Best regards,
Lyle


 

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