Inquiry Into Inquiry

Re: Peirce’s Categories • 15

Understanding another person’s thought can be difficult.  Understanding the way another understands a third person’s thought, all the more so, even if that third person is not so formidable a thinker as Charles Sanders Peirce.  Measures of misunderstanding may be moderated if all thoughts and thinkers are guided by common objectives but the proof of the pudding is in the partaking, as they say.  So let’s step carefully and focus on the task of determining whether category theories, old and new, make good tools for understanding sign relations.

The interaction recorded in my last post continued as follows:

• Peirce List • RM • JA

RM:

I do not see how we can talk here about an operative relationship that would be a triad relationship.  It is not anything other than the composition of two morphisms and I do not ask for more.  3, 2, and 1 are the “place names” and “involves” are arrow names that I usually call alpha and beta.  Now if you think about the determinations in the sign, I have always assumed after much study of the 76 definitions, this idea that the composition of applications captures the presence in the mind of each of the elements of the sign, in such a way that they are themselves ipso facto connected by a triadic relationship.  There is a relationship of tricoexistence that is established as in this case evoked by Peirce:  “It predicates the genuinely Triadic relationship of tricoexistence, ” (CP 2.318, unfortunately there is a hole in my PDF of CP right after and I [left] my paper edition at the library of my university, inaccessible at the moment).

We have a mutual incomprehension?

JA:

I don’t often join the debates over sign classification so frequently animating the animadversions of the Peirce List.  As more the observer than the participant I see the same pattern over and over, with occasional hints but never any hue of resolution fast enough to last and satisfy every dyehard.

Situations of that sort are no novelty in philosophy, or politics, or even math and science on occasion.  And when they occur it is usually because the “place to stand” from which the subject appears in its proper light has yet to be reached by every viewer.

So I’ll back up a little and say how I see things from where I am.

Resources

• Precursors Of Category Theory
• Survey of Precursors Of Category Theory

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

Re: Peirce’s Categories • 15

Another point where the onrush of discussion and the impact of worldly distractions caused my train of thought to jump the track is here:

• Peirce List (1) (2) • RM • JA

RM:

First I note that the formulation “3ns involves 2ns, which involves 1ns” is very dangerous [as] it forgets that 2ns has its autonomy and 1ns too.  If you look at the podium [one] remains in the inner cylinder.  It seems to me that Peirce’s reproach to Hegel is:

“He has usually overlooked external Secondness, altogether.  In other words, he has committed the trifling oversight of forgetting that there is a real world with real actions and reactions.  Rather a serious oversight that.”

It is therefore important to prefer “3ns involves 2ns and 1ns, while 2ns involves 1ns” which preserves the autonomy of the Peircian categories so as not to encourage the idea of a possible peircean hegelianism.

JA:

I’ve been working on a comment about your first point but I’ll post it … when and if I manage to put it in respectable shape.  Just by way of a hint for now, the issue turns on whether we take involves or presupposes to be a dyadic relation and a transitive one at that, as we would if we pass from “3 involves 2” and “2 involves 1” to the conclusion “3 involves 1”.  That may be true for some concepts of involution or presupposition but I think the operative relation in this case is a thoroughly irreducible triadic relation, one whose properties do not reduce to the composition of two dyadic relations.

I think it will take a little more work to get clear about this.  I will go back to the draft remarks I was working on and see if I can bring them to bear on the question.

Resources

• Precursors Of Category Theory
• Survey of Precursors Of Category Theory

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

Re: Peirce’s Categories • 15

In These Uncertain Times, as people keep saying, it’s become even harder to concentrate than usual and I keep losing track of tricky points coming up in discussion which cry out for further discussion but then “human voices wake us, and we drown” or something … So let me go back to the list of loose ends I put together at the first of the month and try to address a few of them.

Here’s one juncture deserving of another look:

• JAS • JA • HR • JAS

JAS:

Every proposition is collective and copulative;  as I stated in a recent post, its dynamical object is “the entire universe” (CP 5.448n, EP 2:394, 1906), which is “the totality of all real objects” (CP 5.152, EP 2:209, 1903), while its immediate object is “the logical universe of discourse” (CP 2.323, EP 2:283, 1903).

This is a very important point, not the least because of the light it throws on a question John Corcoran raised on Facebook and elsewhere as to whether the logical universes of Peirce, or logicians in general, are conceived as referring to something like a holistic totality of existence or only a more limited universe of discourse relevant to a particular discussion.  I thought that significant enough to blog on it here:

• Semiotics, Semiosis, Sign Relations • 9

JA:

Incidentally … lack of care in distinguishing different objects of the same signs, in particular, immediate and ultimate objects and their corresponding universes or object domains, has been the source of many misunderstandings in scattered discussions on Facebook of late.

But then I added:

JA:

Another issue arising here has to do with the difference between the “dimensionality of a relation” and the “number of correlates”.  Signs may have any number of correlates in the object domain without requiring the dimensionality of the relevant sign relation to be greater than three.  This is one of the consequences of “triadic relation irreducibility”.

And that raised a number of further replies from HR and objections from JAS … which I’ll say more about when I get a chance.

Resources

• Precursors Of Category Theory
• Survey of Precursors Of Category Theory

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

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I’ve been too immersed in the Peirce List discussion of Robert Marty’s “Podium” paper to write much here — before I lose track of what I’ve been thinking the last several days I’ll need to ravel up my off-the-cuff remarks and pen them on the sleeves of this blog.

Re: Peirce List (1) (2) • Helmut Raulien (1) (2)

Helmut Raulien asked several questions about the composition, irreducibility, and reducibility of relations.  For background on relation composition as Peirce originally treated it, I referred him to Peirce’s 1870 Logic Of Relatives, especially the section titled “The Signs for Multiplication”, along with my commentary linked below.

• Peirce’s 1870 Logic Of Relatives

There is also this article:

• Relation Composition

For readers who want to skip to the chase for the quickest possible overview, the sorts of pictures floating through my head when I’m thinking about relational composition are the bipartite graph or “bigraph” pictures in the following section.

• Relation Composition • Graph-theoretic Picture

The ways in which relations are reducible or irreducible to simpler relations are covered in the following article.

• Relation Reduction

The following set of articles, in order of increasing generality, may be useful on these scores, providing background and concrete examples.

• Sign Relations
• Triadic Relations
• Relation Theory

Resources

• Precursors Of Category Theory
• Survey of Precursors Of Category Theory

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

Re: Peirce List • Robert Marty

RM:

I submit for your review this preprint which is awaiting publication:

The Podium of the Universal Categories of C.S. Peirce

Abstract

This article organizes Peirce’s universal categories and their degenerate forms from their presupposition relationships.  These relationships are formally clarified on the basis of Frege’s definition of presupposition.  They are visualized in a “podium” diagram.  With these forms, we then follow step by step the well-known and very often cited third Peirce Lowell Conference of 1903 (third draft) in which he sets out his entire method of analysis based on these categories.  The very strong congruence that is established between the podium and the text validates the importance, even the necessity, of taking into account these presuppositions in order to correctly understand Peirce’s phenomenology.

I would be very happy to read your comments.

There were numerous issues stemming from Robert Marty’s post and paper, some central and some tangential, which attracted my interest and which I hope I can get back to.  Seeing as how some of the earliest issues got a little lost in the flood of discussion that followed I thought I would take a moment to record a few threads for future follow up.

• Subthread 1 • RM • JAS • JAS • JAS • JA • HR • JAS • HR • HR • JAS
• Subthread 2 • JA • RM • JA
• Subthread 3 • GF • JA

I made a start at rehashing some of these questions on my blog:

• Semiotics, Semiosis, Sign Relations • (9)
• Peirce’s Categories • (13) • (14)
• Readings On Determination • Discussion • (4)

If I get inspiration and time enough, I may try to organize the issues and further comment on my blog.

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

Re: Peirce List • Robert Marty

RM:

What do you think of the presuppositions between the levels?
Do they make sense to you?

At this point I have mostly questions, which would take further research to answer, not to mention unpacking many books still in boxes from our move a year and a half ago, none of which I’m at liberty to do right now.  So, just off the cuff …

Presupposition is one of those words I tend to avoid, as it has too many uses at odds with each other.  There are at least the architectonic, causal, and logical meanings.  It it were only a matter of logic, I would say means   But usually people have something more pragmatic or rhetorical in mind than pure logic would require, something like enthymeme.

It’s also common for people to confound the implication order with the causal order whereas it’s more like the reverse of that.  In more complex settings we usually have the architectonic sense in mind, and that is what I sensed in the case of the normative sciences.  Viewed with regard to their bases, logic is a special case of ethics and ethics is a special case of aesthetics, but with regard to their level of oversight, aesthetics must submit to ethical control and ethics must submit to logical control.

Early on, Peirce used involution with the meaning it has in arithmetic or number theory, namely, exponentiation, where means   See the following passage and commentary.

• Peirce’s 1870 Logic Of Relatives • The Sign of Involution

As far as the boolean or propositional analogue goes, means the same thing as as one can tell by comparing the following two operation tables.

I haven’t looked into whether Peirce uses “involution” or “involvement” with that sense in his later writings.

Resources

• Peirce’s 1870 Logic Of Relatives
• Precursors Of Category Theory

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science