Relations & Their Relatives • Discussion 9

Posted on July 2, 2015 by Jon Awbrey

Re: Peirce List Discussion • Jeffrey Brian Downard

In viewing the structures of relation spaces, even the smallest dyadic cases we’ve been exploring so far, no one need feel nonplussed at the lack of obviousness in this domain.  Anyone who spends much time doing mathematics will discover how far from being that advertised brand of purely à priori, non-empirical, non-observational, non-nitty-gritty practice it really is.  This is especially true of combinatorics, where a would-be theorist for the lack of a good theory about a species of combinatorial creatures will proceed like a seventeenth century naturalist, collecting specimens of combinatorial fauna and flora until their natures impress themselves on a thus-prepared mind.  Just as I’ve been doing here.

Posted in C.S. Peirce, Combinatorics, Dyadic Relations, Graph Theory, Group Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Tertium Quid, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , , , | 12 Comments

Relations & Their Relatives • Discussion 8

Posted on June 30, 2015 by Jon Awbrey

Re: Peirce List Discussion • Jeffrey Brian Downard

In discussing the “combinatorial explosion” of dyadic relations that takes off in passing from a universe of two elements to a universe of three elements, I made the following observation:

Looking back from the ascent we see that the two-point universe \{ \mathrm{I}, \mathrm{J} \} manifests a type of formal degeneracy (loss of generality) compared with the three-point universe \{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.  This is due to the circumstance that the number of “diagonal” pairs, those of the form \mathrm{A\!:\!A}, equals the number of “off-diagonal” pairs, those of the form \mathrm{A\!:\!B}, so the two-point case exhibits symmetries that will be broken as soon as one adds another element to the universe.

There are two types of symmetry that we might be talking about in this setting and it behooves us to keep them distinctly in mind:

  1. There is the symmetry exhibited by pairs of the form \mathrm{A\!:\!A} versus the asymmetry exhibited by pairs of the form \mathrm{A\!:\!B}.
  2. There is the number of pairs of the form \mathrm{A\!:\!A} versus the number of pairs of the form \mathrm{A\!:\!B} and whether those numbers are equal or not.

The type of symmetry (“sameness in measure”) motivating the above observation is the second type, where the number of pairs on the diagonal is equal to the number of pairs off the diagonal.  That is the symmetry that will be broken when we pass from the 2-point universe to the 3-point universe.

Posted in Combinatorics, Graph Theory, Group Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Tertium Quid, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , | 13 Comments

Relations & Their Relatives • Discussion 7

Posted on June 28, 2015 by Jon Awbrey

Re: Peirce List Discussion • Jim Willgoose

Here is the series of blog posts on Chapter 3 (The Logic of Relatives) from Peirce’s 1880 “Algebra of Logic” up to the point where I left off on May Day.

Up to this point we are still dealing mainly with dyadic relations and as interesting as those may be, especially to a graph theorist, the level of complexity it takes for the first signs of semiosis to get up and running does not come into play until we reach the playing field of triadic relations.

Posted in Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Tertium Quid, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , | 13 Comments

My Thematics • 2

Posted on June 28, 2015 by Jon Awbrey

Communication is so much harder
Than mere invention or discovery.

Will they have in mind what I have in mind?
Will I find the signs?   Will I have the time?

There is so much shadow there must be light!

Posted in Anthem, Anthematics, Communication, Discovery, Inquiry, Invention, Mantra, Mathematics, Meditation, Morpheus, Morphism, Mythematics, Nostalgebra, Nostalgia, Reflection, Semeiosis, Semiosis, Semiotics | Tagged , , , , , , , , , , , , , , , , , | 1 Comment

My Thematics • 1

Posted on June 26, 2015 by Jon Awbrey

I miss the days I’d spend my days
And nights dreaming mathematics

Posted in Anthem, Anthematics, Mathematics, Morpheus, Morphism, Mythematics, Nostalgebra, Nostalgia | Tagged , , , , , , , | 1 Comment

Relations & Their Relatives • Discussion 6

Posted on June 19, 2015 by Jon Awbrey

Re: Peirce List Discussion • Helmut Raulien

In discussing Peirce’s concept of a triadic sign relation as existing among objects, signs, and interpretant signs the question arises whether any of the classes so related are classes by themselves, that is, whether there is necessarily anything distinctive about the being of an object, the being of a sign, or the being of an interpretant sign.

Maybe I can clear up a few points about the relational standpoint by resorting to a familiar case of a triadic relation, one I’m guessing we all mastered early in our schooling, namely, the one involved in the operation of subtraction, x - y = z.  When I was in school we learned a set of quaint terms for the numbers x, y, z in the relation and I wasn’t sure they still taught such things so I checked the web and found a page that described the terms just as I remembered them:

☞ Maths Is Fun • Subtraction

  • The number x is called the minuend.
  • The number y is called the subtrahend.
  • The number z is called the difference.

So we come to the questions:

  • Are minuends a class by themselves?
  • Are subtrahends a class by themselves?
  • Are differences a class by themselves?

To answer these questions we need to observe the distinction between relational roles and absolute essences (inherent qualities, ontological substances, or permanent properties).

If our notion of number is generous enough to include negative numbers then any number can appear in any one of the three places, so minuend, subtrahend, and difference are relational roles and not absolute essences.  We can tell this because it follows from the definition of the subtraction operation.

When it comes time to ask the same questions of objects, signs, and interpretant signs then any hope of a definitive answer must come from the definition of a sign relation we’ve chosen to fit our subject matter.

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Tertium Quid, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , | 11 Comments

Information Resistance • Ω

Posted on June 10, 2015 by Jon Awbrey

The hardest thing to understand about information is people’s resistance to it.

Posted in Anamnesis, Information, Information Resistance, Inquiry, Meditation, Reflection, Resistance | Tagged , , , , , , | 3 Comments

Signs Of Signs • 4

Posted on May 31, 2015 by Jon Awbrey

Re: Michael Harris • Language About Language

But then inevitably I find myself wondering whether a proof assistant, or even a formal system, can make the distinction between “technical” and “fundamental” questions.  There seems to be no logical distinction.  The formalist answer might involve algorithmic complexity, but I don’t think that sheds any useful light on the question.  The materialist answer (often? usually?) amounts to just‑so stories involving Darwin, and lions on the savannah, and maybe an elephant, or at least a mammoth.  I don’t find these very satisfying either and would prefer to find something in between, and I would feel vindicated if it could be proved (in I don’t know what formal system) that the capacity to make such a distinction entails appreciation of music.

Peirce proposed a distinction between corollarial and theorematic reasoning in mathematics which strikes me as similar to the distinction Michael Harris seeks between technical and fundamental questions.

I can’t say I have a lot of insight into how the distinction might be drawn but I recall a number of traditions pointing to the etymology of theorem as having to do with the observation of objects and practices whose depth of detail always escapes full accounting by any number of partial views.

On the subject of music, all I have is the following incidental —

🙞 Riffs and Rotes

Perhaps it takes a number theorist to appreciate it …

Posted in Aesthetics, C.S. Peirce, Category Theory, Coherentism, Communication, Connotation, Form, Formal Languages, Foundations of Mathematics, Higher Order Propositions, Illusion, Inquiry, Inquiry Into Inquiry, Interpretation, Interpretive Frameworks, Logic, Mathematics, Objective Frameworks, Objectivism, Pragmatic Semiotic Information, Pragmatics, Pragmatism, Recursion, Reflection, Semantics, Semiotics, Sign Relations, Syntax, Translation, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 3 Comments

Signs Of Signs • 3

Posted on May 27, 2015 by Jon Awbrey

Re: Michael Harris • Language About Language

And if we don’t [keep our stories straight], who puts us away?

One’s answer, or at least one’s initial response to that question will turn on how one feels about formal realities.  As I understand it, reality is that which persists in thumping us on the head until we get what it’s trying to tell us.  Are there formal realities, forms which drive us in that way?

Discussions like those tend to begin by supposing we can form a distinction between external and internal.  That is a formal hypothesis, not yet born out as a formal reality.  Are there formal realities which drive us to recognize them, to pick them out of a crowd of formal possibilities?

Posted in Aesthetics, C.S. Peirce, Category Theory, Coherentism, Communication, Connotation, Form, Formal Languages, Foundations of Mathematics, Higher Order Propositions, Illusion, Inquiry, Inquiry Into Inquiry, Interpretation, Interpretive Frameworks, Logic, Mathematics, Objective Frameworks, Objectivism, Pragmatic Semiotic Information, Pragmatics, Pragmatism, Recursion, Reflection, Semantics, Semiotics, Sign Relations, Syntax, Translation, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Signs Of Signs • 2

Posted on May 27, 2015 by Jon Awbrey

Re: Michael Harris • Language About Language

I compared mathematics to a “consensual hallucination,” like virtual reality, and I continue to believe that the aim is to get (consensually) to the point where that hallucination is a second nature.

I think that’s called coherentism, normally contrasted with or complementary to objectivism.  It’s the philosophy of a gang of co‑conspirators who think, “We’ll get off scot‑free so long as we all keep our stories straight.”

Posted in Aesthetics, C.S. Peirce, Category Theory, Coherentism, Communication, Connotation, Form, Formal Languages, Foundations of Mathematics, Higher Order Propositions, Illusion, Inquiry, Inquiry Into Inquiry, Interpretation, Interpretive Frameworks, Logic, Mathematics, Objective Frameworks, Objectivism, Pragmatic Semiotic Information, Pragmatics, Pragmatism, Recursion, Reflection, Semantics, Semiotics, Sign Relations, Syntax, Translation, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment