Notes on the Foundations of Mathematics • 1

Posted on July 5, 2012 by Jon Awbrey

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.

Posted in C.S. Peirce, Foundations of Mathematics, Kaina Stoicheia, Logic, Mathematics, Semiotics | Tagged , , , , , | Leave a comment

Approaching Peirce

Posted on June 30, 2012 by Jon Awbrey

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.

Posted in Anthem, Inquiry, Logic, Mathematics, Peirce, Science, Semiotics | Tagged , , , , , , | 2 Comments

Peircean Semiotics and Triadic Sign Relations • 2

Posted on June 27, 2012 by Jon Awbrey

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.

Posted in C.S. Peirce, Inquiry, Logic, Logic of Relatives, Relation Theory, Semiotics, Sign Relations | Tagged , , , , , , | 3 Comments

Peircean Semiotics and Triadic Sign Relations • 1

Posted on June 27, 2012 by Jon Awbrey

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.

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52. Archive. Journal.
    Online (doc) (pdf).
Posted in C.S. Peirce, Inquiry, Logic, Logic of Relatives, Relation Theory, Semiotics, Sign Relations | Tagged , , , , , , | 3 Comments

Pseudo-Choice

Posted on June 26, 2012 by Jon Awbrey

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 …

Posted in Commerce, Democracy, Education, Governance, Rhetoric, The Big Picture | Tagged , , , , , | 3 Comments

The Place Where Three Wars Meet

Posted on June 21, 2012 by Jon Awbrey

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?

Posted in Democracy, Education, Governance, Information, Inquiry, Science | Tagged , , , , , | 8 Comments

How Did It Come To This?

Posted on June 20, 2012 by Jon Awbrey

What brought us to this pass is the simple fact that too many people who take the most good from operating in a civilized, democratic, educated society have chosen not to pay for the goods they receive.

These people harbor the delusion that they created all those goods out of their own miraculous selves and thus bear no obligation to pay anything forward to the next harvest, much less toward the next generation.

The common wealth that makes their uncommon wealth possible — that is a concept beyond their grasp, and so they live like parasites, destroying the host that gives them a home.

These people abhor the idea of spending money on anything they can neither totally consume nor totally control. And so the idea of paying taxes to support a civilized, democratic, educated society, with no more say than one equal vote in what the common good will be — that is utter anathema to self-created deities such as these.

That is how it came to this.

The usual thing, worshiping false gods.

Posted in Democracy, Education, Governance, The Big Picture | Tagged , , , | 1 Comment

The Unmet Challenge of Peirce’s Work

Posted on June 15, 2012 by Jon Awbrey

NB. I am posting these incipient thoughts as a promissory note, in hopes of nudging myself to develop the theme as time goes by.

The Unrealized Potential of Peirce’s Thought

One of my main philosophical and practical concerns for many years now has rested, in its restless way, with the potential contribution of Charles Sanders Peirce to our understanding of inquiry. If I were starting a new project today, instead of trying to dig my way out from under a mountain of unfinished business, it would get a title like “The Unmet Challenge of Peirce’s Work” or “The Unrealized Potential of Peirce’s Thought”. My feeling is that only a small fraction of Peirce’s potential contribution to our understanding has yet been realized and that something critical has been lost in the years since he lived. Consequently, my concern is less with the thinkers who came after him than with the clues their work provides to what was found and what was lost.

It has long been my experience that we cannot grasp the full import of Peirce’s work from the shadows that are cast on the analytic, atomistic, logistic, reductive, syntactic plane. I prefer looking at the work of the intervening years from Peirce’s conceptual perspective, instead of the other way around. I think that affords a much clearer view of things.

Posted in Anthem, Inquiry, Peirce | Tagged , , | 7 Comments

C.S. Peirce • Of Triadic Being

Posted on June 14, 2012 by Jon Awbrey

Selection from C.S. Peirce, “Some Amazing Mazes, Fourth Curiosity” (c. 1909)

Of triadic Being the multitude of forms is so terrific that I have usually shrunk from the task of enumerating them; and for the present purpose such an enumeration would be worse than superfluous: it would be a great inconvenience. In another paper, I intend to give the formal definition of a sign, which I have worked out by arduous and long labour. I will omit the explanation of it here.

Suffice it to say that a sign endeavors to represent, in part at least, an Object, which is therefore in a sense the cause, or determinant, of the sign even if the sign represents its object falsely. But to say that it represents its Object implies that it affects a mind, and so affects it as, in some respect, to determine in that mind something that is mediately due to the Object. That determination of which the immediate cause, or determinant, is the Sign, and of which the mediate cause is the Object may be termed the Interpretant.

C.S. Peirce, Collected Papers, CP 6.347

Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Volume 6 : Scientific Metaphysics, 1935.

Posted in Logic, Logic of Relatives, Mathematics, Peirce, References, Relation Theory, Semiotics, Sources | Tagged , , , , , , , | 3 Comments

Definition and Determination • 8

Posted on June 13, 2012 by Jon Awbrey

Re: Peirce List • Jim Willgoose (1) (2)

The most general meaning of “formal” is “concerned with form”,
but the Latin “forma” can mean “beauty” in addition to “form”,
so perhaps a normative “goodness of form” enters at this root.

The Latin word “norma” literally means a “carpenter’s square”.
The Greek “gnomon” is a sundial pointer taking a similar form.
The most general meaning of “normative” is “having to do with
what a person ought to do”, but a pragmatic interpretation of
ethical imperatives tends to treat that as “having to do with
what a person ought to do in order to achieve a given object”,
so another formula might be “relating to the good that befits
a being of our kind, what must be done in order to bring that
good into being, and how to tell the signs that show the way”.

Defining logic as formal or normative semiotic differentiates
logic from other species of semiotic under the general theory
of signs, leaving a niche open for descriptive semiotic, just
to mention the obvious branch. This brings us to the question:

How does a concern with form, or goodness of form, along with
the question of what is required to achieve an object, modify
our perspective on sign relations in a way that duly marks it
as a logical point of view?

cc: Inquiry List • Peirce List (1) (2) (3)

Posted in C.S. Peirce, Definition, Determination, Inquiry, Logic, Mathematics, Peirce, Phenomenology, Semiotics | Tagged , , , , , , , , | 9 Comments