## Peirce’s Categories • 3

Recent travels and other travails (dental work) have scattered my thoughts to the four winds, so let me just document a few bits from my current state of mind in case I can get back to it someday.

Here is the figure I drew to illustrate a recurring theme from Peirce.

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

Here are a few additional resources that I find useful by way of establishing a foothold on these shores.

## Peirce’s Categories • 2

According to Peirce, it is logic that draws on both mathematics and phenomenology.

At any rate, Peirce takes the distinctive position that normative science, which includes logic, “rests largely on” phenomenology and mathematics.  Unless there is a case to be made for a practical difference between drawing on and resting on, as those phrases are intended in the present setting, I would have to say they mean the same thing.

I discussed the relationship among these sciences in a previous post and drew the following figure to illustrate it.

Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.

— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)

The following post contains a longer excerpt from Peirce’s Classification of the Sciences.

## Peirce’s Categories • 1

Just from my experience, the best first approach to questions of firstness, secondness, thirdness, and so on is to regard k-ness as the property that all k-adic relations possess in common.  There is more to say once this first point is appreciated but it is critical to begin from this understanding.

It is best to view k-adic relations as whole sets of k-tuples rather than fixating on single k-tuples at a time since all the most relevant properties of relations are “holistic” properties of whole sets or whole systems that are not reducible to properties of their individual elements.

A k-adic relation and its converses, numbering k! possibilities in all, each bears essentially the same information about the relation of its domains.  This means that fixating on a particular ordering will tend to distract us with inessential features of a particular presentation rather than highlighting the essential properties of the relation in view.

Peirce demonstrates in several places that he appreciates the significance of these facts.

Just my k bits …

## What Makes An Object? • 2

Re: Peirce List Discussison • (1)(2)

Visual metaphors and perceptual analogies can be instructive — they make for most of my personal favorites — but in logic, mathematics, and science our interest extends through the abductive spectrum, from percept formation to where it shades off to concept formation to where it takes off in theory formation.

Objects in logic and semiotics are any objects of discussion or thought — atoms and atomic clocks, bubbles and bubble chambers, clouds and cohorts, determinants and deuterium, electrons and ellipses, galaxies and ganglia, photons and positrons, quarks and question marks, …, you get the picture.

If we take care that our signs make sense, and by this “sense” mean to say they have objects that are logically consistent, then we pass from the realm of mere semiotics, where literary clutches will dilletate till the twelfth of never on the Madness Of Prince Hamlet (MOPH1) or the Method Of Prince Hamlet (MOPH0), or the taste of Organic Martian Potatoes (OMPs), and leaving all that till the twelfth of never we enter the realm of formal or normative semiotics that we know as logic.

## What Makes An Object? • 1

What makes an object is a perennial question.

I can remember my physics professors bringing it up in a really big way when I was still just a freshman in college.  They always cautioned us then about extrapolating our everyday intuitions about everyday objects beyond their native realms.

Anyone who has been graced or grazed by a modicum of process thinking, say Whitehead or Bucky Fuller, is aware of the trade-off between process thinking and product thinking that rules our descriptions of every domain of phenomena, but in a retrograde time like the one we are currently experiencing it takes a mighty effort to recollect the way that hidebound objects are precipitated from more primal processes.

Here’s an old post I happened on that may apply here:

## Forgetfulness Of Purpose • 8

Re: Peirce List Discussion • (1)(2)

Just to review, we were looking at Ashby’s example of a regulation game as given below.

$\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}$

I observed that this gives us a triadic relation $G_1 \subseteq D \times R \times O$ whose triples are listed next.

$\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}$

Sungchul Ji raised a question about the irreducibility of $G_1$ as a triadic relation, suggesting that it would imply a degree of “communication” or “transfer of information” between $D$ and $R$ as constrained by the relation $G_1.$

For the moment I remain puzzled how we ought to measure various orders and quantities of information that affect the play of regulation games.  That question evidently depends on the kinds of structures that are key to the regulation task.  The quickest way to gain information on those scores is probably to read a little further in Ashby’s text, so I will set a goal of doing that.

On the other hand, the irreducibility of $G_1$ as a triadic relation can be investigated more directly, without considering the information question, so I will try that tack first.

### Reference

• Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.

## Forgetfulness Of Purpose • 7

I invited readers to consider Ashby’s example of a regulation game as a triadic relation $G_1 \subseteq D \times R \times O$ whose triples $(d, r, o)$ are given by either one of the following tables.

$\begin{array}{cc|ccc} \multicolumn{5}{c}{\text{Table 11/3/1}} \\[4pt] & & & R & \\ & & \alpha & \beta & \gamma \\ \hline & 1 & b & a & c \\ D & 2 & a & c & b \\ & 3 & c & b & a \end{array}$

$\begin{matrix} D & R & O \\ \hline 1 & \alpha & b \\ 1 & \beta & a \\ 1 & \gamma & c \\ 2 & \alpha & a \\ 2 & \beta & c \\ 2 & \gamma & b \\ 3 & \alpha & c \\ 3 & \beta & b \\ 3 & \gamma & a \end{matrix}$

Sungchul Ji asked a rather good question about the degree of “communication” or “transfer of information” between D and R within $G_1$ and its bearing on the irreducibility of $G_1$ as a triadic relation.  Given the light that concrete examples can throw on abstract questions, I thought it worth the effort to work through a detailed answer and I began as follows:

Ashby cast this example at a high level of abstraction in order to isolate the “bare bones” structure of the situation but we know the intended interpretation makes D the source of disturbances arising from the environment and R the regulator who has to counteract the disturbances in order to maintain the goal.

With regard to the question of communication or transfer of information between D and R, it is clear that R has to make different choices depending on what D does, so there is a degree of information transmission between D and R to that extent.  In effect, R is trying to cancel the variety that D creates.

Careful discussions of irreducibility will require us to work from explicit definitions and here it turns out there are several different notions of irreducibility that are often confounded in common use.

Looking to the case at hand, $G_1 \subseteq D \times R \times O$ is a triadic relation consisting of $9$ triples in the larger set $D \times R \times O$ that consists of $3 \times 3 \times 3 = 27$ triples.

We could reasonably use the ratio $3/9 = 1/3$ as one measure of constraint, determination, information, selection, or “law” involved in carving $G_1$ from the uncarved block $D \times R \times O.$

If we prefer an additive measure, we could extract the exponents from the ratio $3^1/3^3$ and use their difference as a measure of the information it takes to select $G_1$ from $D \times R \times O.$  Peirce pulled this very trick in some manuscripts I saw one time, getting a simple version of our modern day logarithmic measure of information.

To be continued …

### Reference

• Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.
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