Relations & Their Relatives : 13

The facts about relational reducibility are relatively easy to understand and I included links to relevant discussions in my earlier survey of relation theory.

The following article discusses relational reducibility and irreducibility in general terms and gives concrete examples of reducible and irreducible triadic relations of the sort we find in mathematics and semiotics, illustrating the two types of reducibility that usually come up in these settings.

These examples were introduced in the following articles on triadic relations and sign relations and I believe one could learn a lot from their careful consideration.

Relations & Their Relatives : 12

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.

Relations & Their Relatives : 11

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.

Relations & Their Relatives : 10

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.

My Thematics • 2

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!

| | 1 Comment

My Thematics • 1

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

| | 1 Comment

Relations & Their Relatives : 9

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:

• 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.