Triadic Relations, Intentions, Fuzzy Subsets • 1

Posted on November 27, 2012 by Jon Awbrey

Re: Foundations Of Math Discussion • Lotfi Zadeh

Way back during my first foundational crisis (1967–1972), I had been willing to consider almost any alternatives to the usual set theories, so I can remember looking at early accounts of fuzzy set theory.  There was in addition a link to certain issues that came up in my studies of C.S. Peirce, especially the idea that many dyadic relations we use in logic, mathematics, and semantics — as a rule being functions that assign meanings and values to symbols and expressions — are better understood if taken in the context of triadic relations that serve to complete and generalize them.

My line of thought went a bit like this:

Consider a fuzzy set as a triadic relation of the form x ∈r S among an element x, a degree of membership r, and a set S.

Ask yourself:  Where do these assigned degrees of membership come from?  Imagine that they come from averaging the results of many judges making binary {0, 1} = {Out, In} = {∉, ∈} decisions.

Now consider the more fundamental triadic relation from which this data is derived, the relation of the form x ∈j S that exists among an element x, an interpreter (judge, observer, user) j, and a set S.

That formulates fuzzy sets in a way that links up with many Peircean themes.

Posted in C.S. Peirce, Fuzzy Logic, Fuzzy Sets, Intentional Contexts, Intentional Objects, Intentionality, Intentions, Logic, Logic of Relatives, Lotfi Zadeh, Mathematics, Peirce, Relation Theory, Semiotics, Triadic Relations | Tagged , , , , , , , , , , , , , , | 1 Comment

“Is it possible to advance philosophy today?”

Posted on November 25, 2012 by Jon Awbrey

Re: Stephen Rose

Is it possible to advance philosophy today?  To do so, one would have to use terms that appear to have evolved in different disciplines to a point where dialog is almost impossible, even when desired.

I suppose that would depend on one’s definition of philosophy, one’s definition of advance, and one’s definition of possible.  Now there’s a fearsome threesome if ever I saw one!

I believe it is possible to advance the state of logic, mathematics, and science, not just in theory but also in practice.  And Peirce’s hints, ideas, and methods open up so many directions of exploration that I constantly wonder at the fact that all of them remain barely touched even today.  If part of philosophy’s task is to bring about critical reflection on the state of logic, mathematics, and science, among other things, then I can’t help but to charge it with falling down on the job for failing to nudge these other arts further along.

Posted in Logic, Mathematics, Peirce, Philosophy, Pragmatism, Science | Tagged , , , , , | 4 Comments

Abduction, Deduction, Induction, Analogy, Inquiry • 2

Posted on November 21, 2012 by Jon Awbrey

Re: Peirce List • Kirsti Määttänen

Inference from particulars to particulars is also called analogy.

Peirce gave a fair account of the logic behind statistical inference, as used in the research sciences from before his time to the present day.  That logic depends on a fair amount of probability theory, which Boole had already begun to cast as a generalization of classical logic, but a lot of the underlying logic of inquiry is already visible in the infrastructure at the level of propositional and predicate logic.  And that much of the basic structure was more or less roughly outlined by Aristotle himself.

Here’s a set of notes on the role of abduction, deduction, induction, and analogy in inquiry, as seen in the works of Aristotle and Peirce.

More discussion of these topics can be found on the following pages.

cc: Peirce List (1) (2) (3) (4) (5) (6)

Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | Tagged , , , , , , , , , , , , , , , | 6 Comments

That Aristotling Town

Posted on November 1, 2012 by Jon Awbrey

The man’s reputation for dualing exceeds him.
It’s a mode more the eyebeam of the beholden.
Western wayfarers will claim him their founder,
But they founder on the way his meta*physick
Straddles the narrow straits of their harbor.

Jon Awbrey
30 Oct 2012

Posted in Aristotle, Verse | Tagged , | Leave a comment

Sign Relational Manifolds • 5

Posted on November 1, 2012 by Jon Awbrey

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.

Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Interoperability, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | Leave a comment

Sign Relational Manifolds • 4

Posted on October 30, 2012 by Jon Awbrey

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.

Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Interoperability, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | 4 Comments

Sign Relational Manifolds • 3

Posted on October 23, 2012 by Jon Awbrey

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

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

Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Intentionality, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | Leave a comment

Sign Relational Manifolds • 2

Posted on October 22, 2012 by Jon Awbrey

A sense of how manifolds are applied 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.

What brought the concept of a manifold to 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.
Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Interoperability, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | Leave a comment

Sign Relational Manifolds • 1

Posted on October 22, 2012 by Jon Awbrey

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 is 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.

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.

Posted in C.S. Peirce, Cybernetics, Differential Geometry, Differential Logic, Geometry, Interoperability, Logic, Manifolds, Mathematics, Riemann, Semiotics, Sign Relational Manifolds, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , | 2 Comments

Landfail

Posted on October 18, 2012 by Jon Awbrey

Adrift on the oceans of memory
I touch on an isle of solidity
And try to plant my feet —
But only push off back to sea.

Posted in Verse | Tagged | 2 Comments