Our thoughts live in natural and artificial languages the

way fish swim in natural and artificial bodies of water.

One of the lessons most strikingly impressed on me by my first year physics course and the mass of collateral reading I did at the time was to guard against the errors that arise from “projecting the properties and structures of any language or symbol system on the external world”. This was mentioned especially often in discussions of quantum mechanics — it was a common observation that our difficulties grasping wave-particle duality might be due to our prior conditioning to see the world through the lenses of our subject-predicate languages and logics. Soon after, I learned about the Sapir–Whorf hypothesis, and today I lump all these cautionary tales under the heading of GRAM (“Grammar Recycled As Metaphysics”).

Coping with collaboration, communication, context, integration, interoperability, perspective, purpose, and the reality of the information dimension demands a transition from conceptual environments bounded by dyadic relations to those informed by triadic relations, especially the variety of triadic sign relations employed by pragmatic semiotics.

Along the lines of my first post on this topic I am presently concerned with the logical and mathematical requirements of dealing with constraints but when it comes to the constraints involved in communicating across cultural and disciplinary barriers I could recommend a paper Susan Awbrey and I wrote for a conference devoted to those very issues.

- Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”,
*Second International Conference of the Journal ‘Organization’, Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century*, University of Massachusetts, Amherst, MA. Online.

- Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”,
*Organization : The Interdisciplinary Journal of Organization, Theory, and Society*8(2), Sage Publications, London, UK, 269–284. Abstract.

I remember Doug Medin from the year I was at Illinois and I recall a colloquium talk Frank Keil gave at Michigan State that intrigued me because he echoed ideas from Kant about the synthetic à priori, but I didn’t get a chance to ask him more about it.

As far as Fodor’s line goes, I’m generally sympathetic to faculty psychology, if only because it comports with the ways mathematicians and programmers analyze and synthesize functions and structures, but I find the faculties required to enable intelligence and inquiry interact with each other and mutually recur far too intricately to deserve the name *modules* in the strictest technical sense.

Still, if all we’re talking about is a native knack or a natural instinct for latching onto subsumptions wherever they may occur then I could go along with that for the sake of further argument.

I agree with previous comments that “subsumption” suffers from a surfeit of senses but here’s a couple of places where I found it natural to use “subsumes” or one of its synonyms, once in a logical sense and once in a grammatical sense.

There are reasons coming out of Peirce’s logic and also category theory for this usage but I’ll have to save that for another time.

]]>- JS:
- A binary relation is a set of ordered pairs of the elements of some other set.

That is the first definition I learned for binary relations.

Slightly more generally, a binary relation is a subset of a cartesian product of two sets, and In symbols, Of course and could be the same, but that’s not always the case.

I have long used the adjectives, *2-place*, *binary*, and *dyadic* pretty much interchangeably in application to relations but I developed a bias toward *dyadic* on account of computational contexts where *binary* is reserved for binary numerals.

Once again, partly due to computational exigencies, I would now regard this first definition as the *weak typing* version.

The *strong typing* definition of a -place relation includes the cartesian product as an essential part of its specification. This serves to harmonize the definition of a -place relation with the use of mathematical category theory in computer science.

When I get more time, I’ll go through the material I linked on relation theory in a slightly more leisurely manner …

Let me step back and talk about the research intention driving this work.

In a very real sense everything I’ve been doing along this line of inquiry for the last fifty years falls within the larger traditions of AI, A-Life, cybernetics, and systems theory that first got my attention in the late 1960s. Arbib, Ashby, McCulloch, Minsky and Papert, Wiener stand out among the early influences that whetted my appetite for computational and systems-theoretic approaches to inquiry. It’s fair to say the questions they asked, the hints and tools they provided are always on my mind even today.

A few references, among many others …

- Arbib, M.A.,
*Brains, Machines, and Mathematics*. 1st edition 1964. 2nd edition, Springer-Verlag, New York, NY, 1987. - Ashby, W.R.,
*An Introduction to Cybernetics*, Chapman and Hall, London, UK, 1956. Methuen and Company, London, UK, 1964. - McCulloch, W.S.,
*Embodiments of Mind*, MIT Press, Cambridge, MA, 1965. 3rd printing 1975. - Minsky, M., and Papert, S.,
*Perceptrons : An Introduction to Computational Geometry*, 1st edition 1969, 2nd printing 1972. Expanded edition, MIT Press, Cambridge, MA, 1988. - Wiener, N.,
*Cybernetics : or, Control and Communication in the Animal and the Machine*, 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.

Time scarce and scattered, will get to the rest later …

]]>*What I’m doing this summer …*

Eighteen years living in the same place and we blissfully forgot what it takes to pack up a house and find a new one. Thankfully most of the renovation work is done but it’s looking like it will be August before I get my head above water. Just bits and snatches of time till then …

The main functional test for me is getting a fully running version that works on the sorts of examples I stored at Google Drive:

There’s a sample of commenting I started — just barely started — at this place:

There’s a lot that could be done with the interface, especially making it more visual, displaying the graphical data structures, etc.

There’s hand-generated examples of “animated proofs” in the cactus graph variant of the CSP–GSB calculus at this place:

There’s discussion of those examples here:

- Logical Graphs • OEIS Wiki • InterSciWiki

And it would be wonderful to automate those eventually.

*That’s all for now …*

Just by way of clarifying and emphasizing a few points —

I use the word *relation* to mean a special type of mathematical object, namely, a designated subset included within a cartesian product of sets.

Whatever else this definition of a relation may have going for or against it, it does serve to pick out a class of formal structures that work in good stead as intermediary objects between the world of phenomena and our human capacity for coping with whatever reality may emanate in them. So that is mainly how I aim to use it here.

When I’m being careful, then, I’ll try to use words that maintain a distinction between objects, formal or otherwise, and the symbolic modifications of media we use to reference those objects.

For instance, I took some care with this statement from my last post:

The mathematical examples are typical of many in linguistic, logical, and mathematical contexts where we start out with compact, ready-made axioms, definitions, equations, expressions, formulas, predicates, or terms that denote the relations of interest.

For example, we might be discussing dyadic relative terms like “parent of —” or “square of —” and triadic relative terms like “giver of — to —” or “sum of — and —”.

I used “axioms, definitions, equations, expressions, formulas, predicates, terms” along with “dyadic relative terms” and “triadic relative terms” for various sorts of symbolic entities that serve to denote or describe formal objects of thought and discussion, while I tried to reserve “relations” for the objects themselves.

]]>I chose those examples of triadic relations to be as simple as possible without being completely trivial but they already exemplify many features we need to keep in mind in all the more complex cases as we use relational models of realistic phenomena and objective domains.

The mathematical examples are typical of many in linguistic, logical, and mathematical contexts where we start out with compact, ready-made axioms, definitions, equations, expressions, formulas, predicates, or terms that denote the relations of interest.

For example, we might be discussing dyadic relative terms like “parent of —” or “square of —” and triadic relative terms like “giver of — to —” or “sum of — and —”.

If we spend the majority of our time in contexts like that we may form the impression that all the relational concepts we’ll ever need can be requisitioned off-the-shelf from pre-fab stock, no assembly required.

That’s a pretty picture of our mental equipment. It may even be true if we cook the data long enough and fudge the meaning of pre-fab down to the level of amino acids or quarks or some other bosons on the bus.

As a practical matter, however, research pursued in experimental veins tends to push the envelope of pre-fab concepts into surprisingly novel realms of ideas.

I’ll discuss the examples of sign relations as I get more time …

]]>The middle ground between relations in general and the sign relations we need to do logic, inquiry, communication, and so on is occupied by triadic relations, also called ternary or 3-place relations.

Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic etc.

Here’s a primer with examples from mathematics and semiotics:

- Triadic Relations • InterSciWiki • Wikiversity

At the other end of the funnel, here’s an intro to relations in general, focusing on the discrete mathematical variety we find most useful in applications, for example, as background for relational data bases and empirical data.

- Relation Theory • InterSciWiki • Wikiversity