I will keep returning to my core values.
I will keep speaking from the center of my experience.
Disorder all around me? — What does it matter?
As long as there is order in my mind,
As long as my mind is in order,
I will start from there.
Jon Awbrey
7 August 2012
When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections. They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations. The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.
I’ve been working apace to format my old dissertation proposal on Inquiry Driven Systems for the web but I was reminded of this part when the subject of “signs about signs” came up recently on the Peirce List.
cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Inquiry Driven Systems • Structural Modeling • Systems Science
Angelina Suite • Duggan Place • Stratford • Ontario • 15 July 2012 • 5:24 am
Since the axiomatic method as it is now understood and practiced by mathematicians is the result of a long evolution in human thought, we shall precede our discussion of it by a brief description of some older uses of the term axiom. The modern usage of the term represents a high degree of maturity, and a better understanding of it may be achieved by some acquaintance with the course of its evolution.
If the reader has at hand a copy of an elementary plane geometry, of a type frequently used in high schools, he may find two groupings of fundamental assumptions, one entitled “Axioms,” the other entitled “Postulates.” The intent of this grouping may be explained by such accompanying remarks as: “An axiom is a self-evident truth.” “A postulate is a geometrical fact so simple and obvious that its validity may be assumed.” The “axioms” themselves may contain such statements as: “The whole is greater than any of its parts.” “The whole is the sum of its parts.” “Things equal to the same thing are equal to one another.” “Equals added to equals yield equals.” It will be noted that such geometric terms as “point” or “line” do not occur in these statements; in some sense the axioms are intended to transcend geometry — to be “universal truths.” In contrast, the “postulates” probably contain such statements as: “Through two distinct points one and only one straight line can be drawn.” “A line can be extended indefinitely.” “If L is a line and P is a point not on L, then through P there can be drawn one and only one line parallel to L.” (Some so-called “definitions” of terms usually precede these statements.)
This grouping into “axioms” and “postulates” has its roots in antiquity. Thus we find in Aristotle (384–321 B.C.) the following viewpoint: †
“Every demonstrative science must start from indemonstrable principles; otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. In (b) we have first the genus or subject-matter, the existence of which must be assumed.”
† As summarized by T.L. Heath, [The Thirteen Books of Euclid’s Elements, I, 119, Cambridge University Press, Cambridge, UK, 1908]. The reader is referred to this book for citations from Aristotle, Proclus, et al.
Re: Peirce List Discussions 2012 • (1) • (2) • (3) • (4)
Cf: Previous Discussions 2005–2006 • (A) • (B) • (C)
I will have to be off and on the internet for the next month or so, and won’t be able to keep up with the formal activities on the List. But I have been thinking a lot about the current state of discussion, along with the several bouts of past discussions on topics related to Peirce’s “Kaina Stoicheia” — I tend to call it that so as not to confuse it with the four volumes of The New Elements of Mathematics.
The editors of The Essential Peirce say Peirce’s “Kaina Stoicheia” was written “as a preface to an intended book on the foundations of mathematics”, but that much already requires our careful reflection in view of the way Peirce makes normative science, logic included, depend on the strife-born twins of mathematics and phenomenology. With that in mind, “foundations of mathematics” is a loose enough term that what the editors say may well be true, in one sense or another, but that sense is likely to be radically other than the meaning of thinkers who would reduce mathematics to deductive logic alone.
At any rate, that is a topic for another discussion.
What I’m noticing in my reflections on past and present discussions of these topics is the evident lack of a common language when it comes to the foundation of mathematics, in whatever sense we might have in mind. So I thought it might serve to collect a few notes on the subject from here and there among the canons and allied commentaries on foundations.
I am going to start with excerpts from the now-classic textbook by Raymond L. Wilder, Introduction to the Foundations of Mathematics. This was my first formal introduction to the subject, used in the course that Frank Harary taught at the University of Michigan back in the 1970s.
I gradually grow accustomed to the distinct possibility that there will always be different readings, and even divergent interpretations of Peirce’s writings. Some of that appears to be a two- or three-cultures issue — the readings that befit aesthetic, cultural, and literary aims often part ways with the readings that work best for logical, mathematical, and scientific ends. Partly this is due to the fact that applications to the humanities are soon over-whelmed with the vastly greater complexities of their theatres of operation, and so must be satisfied with very impressionistic and highly sketchy surveys of their realms.
It hasn’t always been this way with me, but most of the time these days I approach Peirce’s work from the standpoint of a practical mathematician focused on applications to empirical sciences, as luck would determine it, to the Odyssean no man’s land between qualitative and quantitative methods. That is far from how I started out, and there were many crises of mind and mood occasioned by the transits of my transdisciplinarity, but that is how it came to be at the present moment.
At any rate, what I find in Peirce are not antiques but tools toward the future.
When I returned to graduate school for the third time around, this time in systems engineering, I had in mind integrating my long‑standing projects investigating the dynamics of information, inquiry, learning, and reasoning, viewing each as a process whose trajectory evolves over time through the medium which gives it concrete embodiment, namely, a triadic sign relation.
Up until that time I don’t believe I’d ever given much thought to sign relations that had anything smaller than infinite domains of objects, signs, and interpretant signs. Countably infinite domains are what come natural in logic, since that is the norm for the formal languages it uses. Continuous domains come first to mind when turning to physical systems, despite the fact that systems with a discrete or quantized character often enter the fray.
So it came as a bit of a novelty to me when my advisor, following the motto of engineers the world over to “Keep It Simple, Stupid!” — affectionately known by the acronym KISS — asked me to construct the simplest non‑trivial finite example of a sign relation I could possibly come up with. The outcome of that exercise I wrote up in the following primer on sign relations.
As a “guide for the perplexed”, at least when it comes to semiotics, I’ll use this thread to collect a budget of resources I think have served to clarify the topic in the past.
By way of a first offering, let me recommend the following most excellent paper, which I can say with all due modesty in light of the fact all its excellence is due to my most excellent co‑author.
Choices at one level of freedom depend on enabling or prerequisite choices being available at more basic levels of freedom.
A situation of Pseudo-Choice is created when you offer people a choice at a high level of freedom without ensuring them equal access to the enabling choices.
It needs to be added that being put in situations of Pseudo-Choice tends to make human beings extremely angry. Intensely angry. And they always, eventually figure it out, no matter how long it takes.
So Watch Out For That …
One of the interesting things about the curse of our nation’s interesting times is the chance we have to observe how that triple threat — the War on Democracy, the War on Education, and the War on Science — work hand in hand in hand to wreak havoc on every core value of American society our parents and teachers impressed on us in what now seems like ancient days.
The inseparable bond between democratic government and public education is no doubt obvious to anyone whose mind and character have been nurtured by the lessons of progressive education — perhaps too obvious to understand how anyone could fail to see how each will die without the other.
At any rate, most of us can probably see how the war on democracy and the war on education are just two fronts in a larger campaign to nullify the core values our Founders labored to give birth on this Continent.
But the war on science? Or inquiry, knowledge, research, truth — however you want to put it? What is that about? Where does that come into the fray?
For one thing, think of the armory of double‑think‑tanks that constantly bombard the public with barrage on barrage of agenda‑driven reports, the host of which tanks operate in exact opposition to the way genuine researchers are trained to conduct historical and scientific research.
For another thing, the public is now so inundated by the rain of abuse on our university‑educated teachers that — unlike every other civilized country in the world — they forget the role that academic freedom plays in conveying the truth about realities not‑to‑be‑denied to the generations that will have to face those realities squarely and without the escape of wishful illusion.
So you can’t have a really good war on democratic education without a multi‑pronged assault on academic freedom, communication, information, inquiry, journalism, knowledge, research, science, and truth. Now can you?
Inquiry Into Inquiry 