## The Difference That Makes A Difference That Peirce Makes : 5

Re: Peirce List Discussion • Gary Richmond

When I think back to the conceptual changes my first university physics courses put me through, a single unifying theme emerges.  Relativity Theory and Quantum Mechanics had a way of making the observer an active participant in the action observed, having a local habitation, a frame of reference, and a bounded sphere of influence within the universe, no longer an outsider looking in.  As I soon discovered in my wanderings through the libraries and bookstores of my local habitation, this very theme was long ago prefigured in the corpus of C.S. Peirce’s work, most strikingly in his Logic of Relatives and Pragmatic Maxim, taken as a basis for his relational theories of information, inquiry, and signs.

It is more this level of underground conceptual revolution that comes to mind when I think of Peirce’s impact on the development of physical theory, needless to say science in general, more than any particular doctrines about continua, especially since continua posed no novelty to classical mechanics, indeed, if anything, were more catholic within its realm, while quantum mechanics introduced an irreducible aspect of discreteness to physics.

## The Difference That Makes A Difference That Peirce Makes : 4

Re: Peirce List Discussion • Mike Bergman

The mathematical perspectives and theories that made modern physics possible, perhaps even inevitable, were developed by many mathematicians, both abstract and applied, all throughout the 19th Century.  There was a definite sea change in the way scientists began to view the relationship between mathematical models and the physical world, passing from a monolithic concept to variational choices among multiple approaches, models, perspectives, and theories.

Charles Sanders Peirce was an astute observer and active participant in this transformation but it has always been difficult to trace his true impact on its course — so much of what he contributed operated underground, rhizome like, and without recognition.  But I think it’s fair to say that Peirce articulated the springs and catches of the workings of science better than any other reflective practitioner in his or subsequent times.  And I think the full import of his information-theoretic and pragmatic-semiotic approaches to scientific inquiry is a task for the future to work out.

## Icon Index Symbol • 17

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Our object being to clarify the relationships among icons, indices, and symbols, I believe the maximum benefit possible at this point is to be gained from studying the simple examples of triadic relations and sign relations discussed in the following places:

Once we get used to dealing with small examples like that we can move on to tackling more complex examples on the order of those we might encounter in realistic applications.

The sort of sign relation we normally encounter in practice will be a subset $L$ of a cartesian product $O \times S \times I,$ where the object, sign, and interpretant-sign domains all have infinitely many members in principle, though of course we tend to get by with finite samples at any given moment and it may even be possible to start small and build capacity over time.

All the objects we need to reference in a given application will go into the object domain $O$ and all the signs and interpretant-signs we need to denote these objects will go into the sign domain $S$ and the interpretant-sign domain $I.$

It may be useful to note at this point that there are such things as monadic projections:

$\begin{array}{lll} \mathrm{proj}_O & : & O \times S \times I \to O \\[4pt] \mathrm{proj}_S & : & O \times S \times I \to S \\[4pt] \mathrm{proj}_I & : & O \times S \times I \to I \end{array}$

For example, $\mathrm{proj}_O (L)$ gives the set of all elements in $O$ that actually occur as first correlates in $L,$ sometimes called the $O$-range.

There are interesting classes of relations that take place internal to the various domains.  For example, there are the syntactic relations or parsing relations that operate within the sign domain, relating complex signs to their component signs.

But that’s a topic a little ways down the road …

## Icon Index Symbol • 16

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • JASHR

Having lost my train of thought due to a week on the road, I would like to go back and pick up the thread at the following exchange:

JAS:
To be honest, given that the Sign relation is genuinely triadic, I have never fully understood why Peirce initially classified Signs on the basis of one correlate and two dyadic relations.
HR:
I have a guess about that:  I remember from a thread with Jon Awbrey about relation reduction something like the following:  A triadic relation is called irreducible, because it cannot compositionally be reduced to three dyadic relations.  Compositional reduction is the real kind of reduction.  But there is another kind of reduction, called projective (or projectional?) reduction, which is a kind of consolation prize for people who want to reduce.  It is possible for some triadic relations.

The course of discussion after that point left a great many of the original questions about icons, indices, and symbols unanswered, so I’d like to make another try at addressing them.  The relevant facts about triadic relations and relational reduction can be found at the following locations:

I introduced two examples of triadic relations from mathematics, two examples of sign relations from semiotics, and used them to illustrate the question of projective reducibility, in another way of putting it, whether the structure of a triadic relation can be reconstructed from the structures of three dyadic relations derived or “projected” from it.

• In the geometric picture of triadic relations, a dyadic projection is the shadow that a 3-dimensional body casts on one of the three coordinate planes.
• In terms of relational data tables, a dyadic projection is the result of deleting one of the three columns of the table and merging any duplicate rows.
• In Peircean terms, a projection is a type of “prescision” operation, abstracting a portion the structure from the original relation and ignoring the rest.

The question then is whether we keep or lose information in passing from a triadic relation to the collection of its dyadic projections.  If there is no loss of information then the triadic relation is said to be reducible to and reconstructible from its dyadic projections.  Otherwise it is said to be irreducible and irreconstructible in the same vein.

## Icon Index Symbol • 15

### Questions Concerning Certain Faculties Claimed For Signs

I put down the cup and turn to my mind.  It is up to my mind to find the truth.  But how?  What grave uncertainty, whenever the mind feels overtaken by itself;  when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it.  Seek?  Not only that:  create.  It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.

Re: Peirce List Discussion • TGJRJFS

That passage from Proust epitomizes for me one of the most distinctive features of the inquiry process, the fact that its object is a state of information that the inquisiturus, the agent of inquiry, may never have known before.

I have thought of inquiry and intelligence in terms of cybernetic or system-theoretic processes ever since my first encounters with the works of Arbib, Ashby, Bateson, McCulloch, Wiener, Young, and others during my undergrad years.  In the early 90s I returned to grad school in a Systems Engineering program with the idea that I might be able to string together many loose threads of unfinished business that continued to tug at my brain.  Here’s a bit I wrote at the outset of that project, that comes to mind in this context:

Prospects for Inquiry Driven Systems • Architecture of Inquiry

It is important to remember that knowledge is a different sort of goal from the run-of-the-mill setpoints that a system might have.  The typical goal is a state that a system has actually experienced many times before, like normal body temperature for a human being.  But a particular state of knowledge that an intelligent system moves toward may be a state it has never been through before.  The fundamental equivocation on this point expressed in Plato’s Meno, whether learning is functionally equivalent to remembering, was discussed above.  In spite of this quibble, it still seems necessary to regard states of knowledge as a distinctive class.  The reasons for this may lie in the fact that a useful definition of inquiry for human beings necessarily involves a whole community of inquiry.

## Icon Index Symbol • 14

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien
Re: Icon Index Symbol • (10)(11)(12)(13)

Let me sum up the main points of the above exchange before moving on.

Mathematics is useful in our present endeavor because it covers relations in general.  In addition — and multiplication, too — mathematics is chock full of well-studied examples of triadic relations.  When it comes to the job of analyzing sign relations and teasing out their relevant structures we could save ourselves a lot of trouble and trial and error by examining this record of prior art and adapting its methods to cover sign relations.

That brings us to the case of sign relations proper.  I think it’s clear that these types of triadic relations form our first stepping stones and also our first stumbling blocks in the inquiry into inquiry, and I think I gave some indications already of why that might be true.  I don’t know if I can do any better than that at this time, but I’ll think on it more after that all-essential secondness of caffeination.

## Icon Index Symbol • 13

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

HR:
I guess, that a difference between Peirce’s relation theory, and his semiotics and category theory, is, that the first is about all triadic relations, and the latter only about sign relations or representational relations (the special kind of triadic relations).

My guess is that Peirce’s category theory, when taken at its full promise and broadest historical perspective, will find its place in a line of inquiry extending from Aristotle’s Categories up through category theory in its present-day mathematical sense — and then beyond in certain directions, as guided by its more peircing insight into triadicity.  In this view, category theory, the logic of relatives, and the theory of relations all work in tandem toward the same object.

But it’s true, the initial focus and inciting application of all three converging operations — Peirce’s triple drill bit — must be to the matter of signs, information, and inquiry.  Our first imperative (!) is thus to interrogate (?) the indicative (.) faculty of signs.  Our experiences of confusion, comprehension, and communication place demands on us that only competent theories of inquiry and signs can bring to a close.