C.S. Peirce and Category Theory • 7

Posted on July 1, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

HS:
I’d be very interested in the comments of people who know about Peirce on 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 …

Dear Henry,

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 …

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 3 Comments

C.S. Peirce and Category Theory • 6

Posted on June 30, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

HS:
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.
Dear Henry,

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.

Regards,

Jon

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 3 Comments

C.S. Peirce and Category Theory • 5

Posted on June 29, 2021 by Jon Awbrey

Re: C.S. Peirce and Category Theory • 2
Re: Category Theory • Henry Story • Avi Craimer • Henry Story

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.

Here’s an intro to Sign Relations from a Peircean point of view, still a bit “working on it” from my POV.

Here’s the skinny on the three main types of signs — Symbols, Icons, Indices — in Peirce’s theory of signs.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 3 Comments

C.S. Peirce and Category Theory • 4

Posted on June 28, 2021 by Jon Awbrey

Re: C.S. Peirce and Category Theory • 3
Re: Category Theory • Kyle Rivelli

Dear Kyle,

My Inquiry Into Inquiry blog has a Survey page where I collect blog and wiki resources on all the longer-running topics I write and dialogue about.  The following two collections bear on the close relationship, almost a kind of noun-verb or product-process duality, between signs and inquiry.

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.

\text{Information} = \text{Comprehension} \times \text{Extension}

I appear destined 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

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 3 Comments

C.S. Peirce and Category Theory • 3

Posted on June 27, 2021 by Jon Awbrey

Re: Category Theory • Kyle Rivelli

KR:
I really enjoyed the Diagrammatic Immanence book.
Gangle has another book that goes into more depth with Peirce:
Gianluca Caterina and Rocco Gangle (2016), Iconicity and Abduction.

Thanks, Kyle, I’ve been looking at this book for a while now, trying to decide if there’s anything in it I need to know badly enough to justify the purchase.

The connection between the types of inference (Abduction, Induction, Deduction) and the types of signs (Icons, Indices, Symbols) is a pivotal question in Peirce’s logic, occupying the interface between his theory of inquiry and his theory of signs.  It’s an issue I’ve done a lot of thinking, dialoguing, and blogging about.  I will dig up some links later but here is one for starters.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 4 Comments

C.S. Peirce and Category Theory • 2

Posted on June 24, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

HS:
This book Diagrammatic Immanence [preview]
has a whole chapter on Peirce and Category Theory.

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.

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.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 4 Comments

C.S. Peirce and Category Theory • 1

Posted on June 23, 2021 by Jon Awbrey

Cf: Category Theory • Jon Awbrey

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 …

Resources

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 3 Comments

Differential Logic • Discussion 7

Posted on June 20, 2021 by Jon Awbrey

Re: Differential Logic • 1
Re: FB | Pattern Languages for Systemic Transformation • Steve Kramer

SK:
Can differential logic be described using category theory?  To what other logical or mathematical modalities does differential logic relate?  Give an example.  Partial credit will be given.

Dear Steve,

The ultimate category-theoretic generalization of the functional derivative in calculus and the tangent vector in differential geometry is called a tangent functor.  Finding the proper logical analogue of a tangent functor is the main business of my essay on Differential Logic and Dynamic Systems.

But that’s a lot to take in at once so over the years I’ve written a number of easier pieces to work up to it more gradually.  The first intuitive inklings of the subject are provided by the following overture.

The following series provides a more systematic treatment of substantial issues.

Resources

cc: Cybernetics • Ontolog • Peirce (1) (2) (3) (4) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 8 Comments

Differential Logic • Discussion 6

Posted on June 19, 2021 by Jon Awbrey

Re: Differential Logic • 5
Re: Laws of Form • Lyle Anderson

JA:
The differential proposition \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))} may be read as saying “change p or change q or both”.  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.
LA:
Is this what is new:  “you happen to find yourself in the center cell [of a Venn diagram] and require a complete and detailed description of ways to escape it”?

Dear Lyle,

What’s improved, if not entirely new, is the development of appropriate logical analogues of differential calculus and differential geometry.  There has been work on applying the calculus of finite differences to propositions, but the traditional styles of syntax are so weighed down by conceptual clutter that the resulting formal systems hardly get off the ground before they become too unwieldy to stand.

That is where the formal elegance and practical efficiency of C.S. Peirce’s logical graphs and Spencer Brown’s graphical forms come to save the day.  That, I think, is new.  Or at least it was when I began to work on it.

Regards,

Jon (the Prisoner of Vennda, No More)

Resources

cc: Cybernetics • Ontolog • Peirce (1) (2) (3) (4) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 8 Comments

Differential Logic • Discussion 5

Posted on June 17, 2021 by Jon Awbrey

Re: Differential Logic • Discussion 4
Re: Laws of Form • Lyle Anderson

JA:
The differential proposition \mathrm{d}A is one we use to describe a change of state
(or a state of change) from A to \texttt{(} A \texttt{)} or the reverse.
LA:
Does this mean that if A is the proposition “The sky is blue”, then \mathrm{d}A would be the statement “The sky is not blue”?  Don’t you already have a notation for this in A and \texttt{(} A \texttt{)} \, ?  From where does “state” and “change of state” come in relation to a proposition?

Dear Lyle,

The differential variable \mathrm{d}A : X \to \mathbb{B} is a derivative variable, a qualitative analogue of a velocity vector in the quantitative realm.

Let’s say x \in \mathbb{R} is a real value giving the membrane potential in a particular segment of a nerve cell’s axon and A : \mathbb{R} \to \mathbb{B} is a categorical variable predicating whether the site is in the activated state, A(x) = 1, or not, A(x) = 0.  We observe the site at discrete intervals, a few milliseconds apart, and obtain the following data.

  • At time t_1 the site is in a resting state, A(x) = 0.
  • At time t_2 the site is in an active state, A(x) = 1.
  • At time t_3 the site is in a resting state, A(x) = 0.

On current information we have no way of predicting the state at time t_2 from the state at time t_1 but we know action potentials are inherently transient so we can fairly well guess the state of change at time t_2 is \mathrm{d}A = 1, in other words, about to be changing from A to \texttt{(} A \texttt{)}.  The site’s qualitative “position” and “velocity” at time t_2 can now be described by means of the compound proposition A ~ \mathrm{d}A.

Resources

cc: Cybernetics • Ontolog • Peirce (1) (2) (3) (4) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 8 Comments