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Jon Awbrey (Jun 20 2021 at 20:56):I will use this space to post what comes to mind by way of Peirce and Category Theory.
Just to get the ball rolling (in good Sisyphean style) here's my blog of mostly Peirce-related discussion and thought.
Here's a historical perspective on the nature of signs and the conduct of inquiry in Aristotle, Peirce, Dewey, and a few other compatible thinkers.
More later …
Henry Story (Jun 20 2021 at 20:58):This book "Diagrammatic Immanence" has a whole chapter on Peirce and Category Theory https://twitter.com/bblfish/status/1406696563989155844
@KeithEPeterson_ Oh! That is a nice find. Diagrammatic Immanence a book from 2016 with chapters on: 1. Spinoza 2. Categories and Functors 3. Peirce 4. Presheaves 5. Deleuze 6. Adjunctions and Topoi... https://www.google.de/books/edition/Diagrammatic_Immanence/bjVYDwAAQBAJ?hl=de&gbpv=1&printsec=frontcover https://twitter.com/bblfish/status/1406696563989155844/photo/1
- The 🐟 BabelFish (@bblfish) Jon Awbrey (Jun 21 2021 at 19:06):There's a two-culture tension in the reception of Peirce these days. Maybe it's always been that way but it strikes me as more bifurcated today than any time since I began my Peirce studies 50+ years ago. Peirce for the logic-math-science researcher and Peirce for the humanities-literary-verbal stylist are almost immiscible types of thinkers. I find this especially irksome in the case of Peirce since I have felt from the beginning Peirce more than any other thinker gave us the framework and the tools we need to integrate the two-culture divide in society at large.
Avi Craimer (Jun 21 2021 at 19:20):Henry Story said:
This book "Diagrammatic Immanence" has a whole chapter on Peirce and Category Theory
I can't believe this books exists! Thank you for sharing. Talk about inter-disciplinary. I've recently become exposed to Deleuze in a completely different context, and I've been waiting for somebody in the humanities to take a serious look at the nature of diagrammatic reasoning as a fundamental question for philosophy.
Jon Awbrey (Jun 21 2021 at 19:52):The following paper touches on a number of related issues as they affect the education and research missions of universities.
Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract. Online.
Henry Story (Jun 21 2021 at 21:35):Avi Craimer said:
I've recently become exposed to Deleuze in a completely different context, and I've been waiting for somebody in the humanities to take a serious look at the nature of diagrammatic reasoning as a fundamental question for philosophy.
yes, I have also been waiting for a book like this. In fact now that you mention it, Deleuze did talk about diagrams (eg. searching, I just found this Deleuze and the Diagram) and gave them a central role. At the time when I listened to a few of his lectures, I knew too little about Category Theory, especially about diagrams, to know if there was a formal/serious relation, or if these words were just accidentally the same. But this book is certainly making the case for it being serious, and indeed that would also have been my guess for the simple reason that Mathematics in France has nearly a religious standing, so that it is hardly possible for a great philosopher to not be listening carefully to the breakthroughs happening in mathematics.
So that then makes me wonder if Deleuze's book on Folds could have been inspired by catamorphisms.... (I guess I should read some of them to find out).
Jon Awbrey (Jun 22 2021 at 12:18):Dear Avi, Henry,
Diagrams are a mixed bag, a complex and polymorphic species, in Peircean semiotics. All diagrams in common use, especially in mathematics, involve all three types of signs — Symbols, Icons, Indices — as interpreted by their user communities. There has been a tendency in recent years to overemphasize the iconic aspects of Peirce’s logical graphs, reading them a bit too much on the analogy of venn diagrams, but their real conceptual and computational power comes rather from their generic symbolic character.
Jon Awbrey (Jun 22 2021 at 12:45):Here's an intro to Sign Relations from a Peircean point of view, still a bit :working_on_it: from my POV.
Jon Awbrey (Jun 22 2021 at 21:40):Here's the skinny on the three main types of signs — Symbols, Icons, Indices — in Peirce's theory of signs.
Henry Story (Jun 23 2021 at 13:21):I'd be very interested in the comments of people who know about Peirce of the two chapters in the Book "Diagrammatic Immanence" I linked to above on "3. Peirce" and "4. Diagrams of Variation: Functor Categories and Presheaves".
The chapter on Presheaves has some good intutions on how to explain them that I recognise from studying them a year ago. At the end of that chapter the author @Rocco Gangle argues that Peirce's diagrams can be modelled in terms of Category Theory. I would have expected a long list of articles to follow to underwrite that claim. Perhaps this is all well known in Peirce or CT circles,...
Henry Story (Jun 23 2021 at 13:30):I guess one paper I really like and keep citing is Knowledge Representation in Bicategories of Relations which shows how regular and geometric logic can be modelled as something pretty close to sheaves, namely functors from bicategories of relations to Rel (the category of Sets and Relations). That does not give us modal logic, but all of first order logic. (And the article is presented in terms of string diagrams, but that should not affect the argument).
Henry Story (Jun 23 2021 at 13:32):Rocco also gives an explanation of the Peirce's tripartite relation there... I don't feel like I completely understand that yet.
Jon Awbrey (Jun 23 2021 at 13:45):Henry Story said:
I'd be very interested in the comments of people who know about Peirce of the two chapters in the Book "Diagrammatic Immanence" I linked to above on "3. Peirce" and "4. Diagrams of Variation: Functor Categories and Presheaves".
The chapter on Presheaves has some good intutions on how to explain them that I recognise from studying them a year ago. At the end of that chapter the author Rocco Gangle argues that Peirce's diagrams can be modelled in terms of Category Theory. I would have expected a long list of articles to follow to underwrite that claim. Perhaps this is all well known in Peirce or CT circles,...
I skimmed this when you first linked it. From the perspective of a long-time pursuer of Peirce it is very general and impressionistic but I'm in the process of taking another look and working up a few specific comments. As far as diagrams go my work for the last half-century has been focused on Peirce's logical graphs, the duality between their so-called entitative and existential interpretations for propositional logic, and their revamping along the lines of Spencer Brown's Laws of Form.
Jon Awbrey (Jun 23 2021 at 14:40):Just by way of explaining my personal orientation, when I first took up an interest in Peirce, as a freshperson in the late 60s, pretty much everyone in my circles of interest intersecting on Peirce intuited how radically different his drumming was from the mainstream dins of the day. These days I see a lot more Resistance Is Futile, You Will Be Assimilated spin on Peirce. So I'm finding it necessary to keep going back to the source and carefully pointing out exactly where the roads diverge.
Henry Story (Jun 23 2021 at 15:52):I'd love it of course if all of Peirce's graphs could be mapped to CT. That would help me integratet that work a lot faster. Or alternatively, if one could work out exactly where it could not be tied into CT, that would also be very helpful.
(Note that I am just zipping by here like Alice's Rabbit, always late on something to do, somewhere...)
Jon Awbrey (Jun 23 2021 at 16:20):Henry Story said:
I'd love it of course if all of Peirce's graphs could be mapped to CT. That would help me integrate that work a lot faster. Or alternatively, if one could work out exactly where it could not be tied into CT, that would also be very helpful.
The way I see it, Peirce’s work as a whole requires us to stand back from our current picture of category theory and adopt a more general perspective on the subject as we know it. That has not been a popular opinion in math circles and scarcely grasped in phil circles. It's on my big bucket list of Failures To Communicate but I haven't really tried all that hard lately so maybe I'll give it another go.
Henry Story (Jun 23 2021 at 16:46):I think you should team up with @David Corfield who is also complaining about how Philosophy has not caught up with Maths (at least analytic philosophy, whose founding father should have spurred a much deeper connection to CT. Or was it that the French with Grothendieck, somehow got hold of that first in philosophy?)
Jon Awbrey (Jun 23 2021 at 17:06):Here is the comment I made on another thread (in another stream?) placing Peirce in the chain of category brains from Aristotle to contemporary math cats.
• Precursors Of Category Theory
The thing to take away from all this is the common function of category markers as disambiguators of signs or unequivocators of terms. This function is the same through all the diverse systems of categories coming from different thinkers through history.
Jon Awbrey (Jun 23 2021 at 19:56):A sign relation is a triadic relation where is the object domain, is the sign domain, and is the interpretant sign domain. In many applications of great interest to me, is a formal language, usually a logical syntax or a programming language or a hybrid of both, is another system of signs, often identical or similar to but sometimes of a different sort, and is the universe of discourse which and denote, refer to, or at least aim at.
Henry Story (Jun 23 2021 at 22:33):Just found this talk in ACT2020 where Rocco Gangle is talking (with others) about "A Generic Figures Reconstruction of Peirce's Existential Graphs" https://youtu.be/j7Bp6_uiFaQ introduced by @John Baez
Jon Awbrey (Jun 23 2021 at 22:54):Thanks for the link. Here's a free-form intro and a slightly more formal introduction to the tack I've been developing since the late 60s when I started using Peirce's Alpha Graphs and Spencer Brown's Calculus of Indications as springboards for my work on theorem provers and propositional modelers.
🙞 Logical Graphs • Introduction
🙞 Logical Graphs • Formal Development
Jon Awbrey (Jun 23 2021 at 23:34):One of the first things I learned when I turned from scribbling alpha graphs on the backs of computer punch cards to hacking logical graphs on my trusty old TI-99/4A is that planar maps are a death-trap in terms of both conceptual elegance and computational efficiency — so I switched to the topological dual graphs. This is of course the same thing folks did in the map-coloring domain.
Well, have to break for dinner and a binge-worth of movies ...
Here's a Survey Page of blog and wiki resources.
• Survey of Animated Logical Graphs
Jon Awbrey (Jun 25 2021 at 04:54):Cf: C.S. Peirce and Category Theory • 2
Cf: Peirce List • John Sowa
FYI: John Sowa just posted the following review to the Peirce List.
• Diagrammatic Immanence : Category Theory and Philosophy
Henry Story (Jun 25 2021 at 05:04):Thanks. That is a nice and faire review, having read the book, and the little criticism is justified. I think it is a very good first step, that hopefully will be elaborated and developed on by others. For me, who knew very little on Spinoza, Peirce and Deleuze it will help me if and when I get to look at them more closely.
Jon Awbrey (Jun 25 2021 at 05:12):Henry Story said:
Thanks. That is a nice and faire review, having read the book, and the little criticism is justified. I think it is a very good first step, that hopefully will be elaborated by others.
Unfortunately the preview omits the chapter on Peirce — I'm limited to the little said about him in the Introduction, and at $132.00 that's probably all I'll have til Christmas :santa: :mother_christmas:
Jon Awbrey (Jun 25 2021 at 15:30):Very briefly, Spencer Brown's Revival — sounds like a great name for a band — augments Peirce's α-level logical graphs in two main ways:
Kyle Rivelli (Jun 26 2021 at 13:34):I really enjoyed the Diagrammatic Immanence book. Gangle has another book that goes into more depth with Peirce with his book, Iconicty and Abduction. https://www.amazon.com/Iconicity-Abduction-Philosophy-Epistemology-Rational/dp/3319442449
Jon Awbrey (Jun 26 2021 at 13:54):Kyle Rivelli said:
I really enjoyed the Diagrammatic Immanence book. Gangle has another book that goes into more depth with Peirce with his book, Iconicty and Abduction. https://www.amazon.com/Iconicity-Abduction-Philosophy-Epistemology-Rational/dp/3319442449
Thanks, Kyle, I've been looking at this book ...
The connection between the types of inference (Abduction, Induction, Deduction) and the types of signs (Icons, Indices, Symbols) is a major topic in Peirce's theory of inquiry and theory of signs. It's one I've done a lot of thinking, dialoguing, and blogging about. I will dig up some links later but here is one for starters.
Jon Awbrey (Jun 26 2021 at 14:56):I didn't like some of the things I wrote last week. Root canal early in the week and storm damage at the end, minor stuff but it's taking a longer series of iterations with several contractors than I ever would have thought possible — all in all too much distraction for due concentration. I cut the worst writs to a notepad and will work on making them more coherent down the line.
Jon Awbrey (Jun 27 2021 at 14:04):My Inquiry Into Inquiry blog has a Survey page where I collect blog posts and wiki resources on all the longer running topics I write and dialogue about. Here's a couple of collections most relevant to the close relationship, almost a kind of noun-verb or product-process duality, between signs and inquiry.
Jon Awbrey (Jun 27 2021 at 15:10):Especially relevant to the complex of connections Peirce suggests between the main types of signs (Icons, Indices, Symbols) and the main types of inference (Abduction, Induction, Deduction) are my study notes and blog series on Peirce's Laws of Information, the spirit of which is captured by the following formula.
I seem to revisit this subject every other summer or so. Here's an outline of the last time around.
Probably about due for another return …
Jon Awbrey (Jul 01 2021 at 14:18):@Henry Story said:
I'd be very interested in the comments of people who know about Peirce of the two chapters in the Book "Diagrammatic Immanence" I linked to above on "3. Peirce" and "4. Diagrams of Variation: Functor Categories and Presheaves".
The chapter on Presheaves has some good intuitions on how to explain them that I recognise from studying them a year ago. At the end of that chapter the author Rocco Gangle argues that Peirce's diagrams can be modelled in terms of Category Theory. I would have expected a long list of articles to follow to underwrite that claim. Perhaps this is all well known in Peirce or CT circles,...
Things are a little calmer in my neck of the woods at the moment so I’m paddling back up Peirce Bayou to clear up some of the points I missed during last week’s tempest and root canal. An hour’s expedition through Amazon’s creeks and tributaries finally turned up a pearl of not too great a price so far as Diagrammatic Immanence goes so I tumbled for a paperback edition to arrive in a couple of weeks but the purchase lets me read it on Kindle right away. So I’ll be perusing that …
Jon Awbrey (Jul 02 2021 at 13:30):@Henry Story
Cf: C.S. Peirce and Category Theory • 8
Re: Category Theory • Henry Story
Re: Laws of Form • Lyle Anderson
<QUOTE LA:>
As I am trying to get “frame sync” on this discussion, as the satellite communications people say, I am taking clues from the introduction to the listing for Gangle's Diagrammatic Immanence.
<QUOTE Edinburgh University Press>
A renewal of immanent metaphysics through diagrammatic methods and the tools of category theory. Spinoza, Peirce and Deleuze are, in different ways, philosophers of immanence. Rocco Gangle addresses the methodological questions raised by a commitment to immanence in terms of how diagrams may be used both as tools and as objects of philosophical investigation. He integrates insights from Spinozist metaphysics, Peircean semiotics and Deleuze's philosophy of difference in conjunction with the formal operations of category theory. Category theory reveals deep structural connections among logic, topology and a variety of different areas of mathematics, and it provides constructive and rigorous concepts for investigating how diagrams work.
</QUOTE></QUOTE>
This discussion keeps flashing me back to an unfinished syzygy from the mid '80s when I took a course on “applications of λ-calculus” with John Gray at Illinois examining the trio of combinators, computation, and cartesian closed categories, all hot topics of the day, and followed it up with a guided study on the connections to Peirce I had glimpsed at the time. I'll dig up some notes and get back to that. For the moment I'll focus on category theory in the light of Peirce. The lights of Spinoza and Deleuze I'll leave to observers who see better by them.
Henry Story (Jul 02 2021 at 13:40):yes, it's quite a feat to cover all three philosophers and do CT too.
But luckily one can criticise smaller pieces of the work :-)
Jon Awbrey (Sep 04 2021 at 14:54):(deleted)