Inquiry Driven Systems • Discussion 4

Posted on July 20, 2021 by Jon Awbrey

Re: Category Theory • Eduardo Ochs

EO:
Do you have links on how to teach Logical Graphs to children (and to people like me!) and how to use them as a basis for learning Propositional Calculus and quantifiers?

Dear Eduardo,

There’s a lot of stuff I’ve put on the web over the last twenty years devoted to CSP and GSB and my own versions of Logical Graphs — I’m still working at organizing all that in a step-by-step tutorial fashion.  I’ll be doing more of that over time on a number of local streams and topics, e.g.

You might try sampling my Inquiry blog for the daily rushes and discussions or my OEIS user page and OEIS workspace to see if anything engages your interest.

Cheers,
Jon

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Inquiry Driven Systems • Laws of Form • Peirce List (1) (2) (3)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Learning Systems, Logic, Logical Graphs, Peirce, Semiotics, Sign Relations, Visualization | Tagged , , , , , , , , , , , , , , | 7 Comments

Inquiry Driven Systems • Discussion 3

Posted on July 19, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

HS:
Could one re-invent the whole curriculum from age 5 onwards
built on new [category theoretic] concepts?

If I were starting from scratch, and I’m always starting from scratch, I would ease my way up to the pons asinorum of logic and math using the types of logical graphs laid down by Peirce and Spencer Brown.  That is because I think it’s crucial to firm up propositional logic before taking on quantifiers and to grasp classical logic before intuitionistic.

The climb from “zeroth order logic” to first order logic is a lot more interesting and richer in adventure once you have a truly efficient calculus for propositional logic at the ready.  An approach to categories, combinators, etc. can then be made via the propositions as types analogy.  For the kiddies, Smullyan’s Mockingbird would be the primer of choice.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Inquiry Driven Systems • Laws of Form • Peirce List (1) (2) (3)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Learning Systems, Logic, Logical Graphs, Peirce, Semiotics, Sign Relations, Visualization | Tagged , , , , , , , , , , , , , , | 7 Comments

Inquiry Driven Systems • Discussion 2

Posted on July 18, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

Way back in the Summer of Love I met a girl who had just graduated in Chemistry and was thinking about grad school in Education, the hot new field of Instructional Media, we got to talking and dreamed up a vision of using media, at first just shapes in motion, to teach people math from scratch.  Long time passing, we got married, she did a dissertation — The Effect of the Hausdorff–Besicovitch Dimension of Figure Boundary Complexity on Hemispheric Functioning — studying the effects of fractal figure complexity on cognitive processing, Mandelbrot gave her permission to use a series of his figures and ranked them by fractal dimension for her, and I pursued an array of parallel lives in math, statistics, computing, philosophy, and psych.

Here is one of our later collaborations aimed toward integrating inquiry learning and information technology into education.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Inquiry Driven Systems • Laws of Form • Peirce List (1) (2) (3)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Learning Systems, Logic, Logical Graphs, Peirce, Semiotics, Sign Relations, Visualization | Tagged , , , , , , , , , , , , , , | 7 Comments

Differential Logic • Discussion 11

Posted on July 16, 2021 by Jon Awbrey

Re: Differential Logic • Discussion 9

Let’s look more closely at the “functor” from \mathbb{R} to \mathbb{B} and the connection it makes between real and boolean hierarchies of types.  There’s a detailed discussion of this analogy in the article and section linked below.

Assorted types of mathematical objects which turn up in practice often enough to earn themselves common names, along with their common isomorphisms, are shown in the following Table.

\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}

Analogy Between Real and Boolean Types

cc: Category Theory • Cybernetics • Ontolog • 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

Inquiry Driven Systems • Discussion 1

Posted on July 15, 2021 by Jon Awbrey

Re: Topos Lab • MathFoldr Project
Re: Category Theory • Valeria de Paiva

Dear Brendan and Valeria,

I’ve been a participant∫observer in web-ontology knowledge projects for a couple of decades and they always give far more attention to knowledge as a product than due reflection on the dynamics of inquiry required to develop our provisional knowledge.  Many such projects have come and gone with the winds of fashion and it’s my guess the lack of balance between process and product orientation is one of the reasons why.

So I’ve been working on that … here’s a few links to the model of knowledge development sketched in my work on Inquiry Driven Systems.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Inquiry Driven Systems • Laws of Form • Peirce List (1) (2) (3)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Learning Systems, Logic, Logical Graphs, Peirce, Semiotics, Sign Relations, Visualization | Tagged , , , , , , , , , , , , , , | 7 Comments

Differential Logic • Discussion 10

Posted on July 13, 2021 by Jon Awbrey

Re: Laws of Form • Lyle Anderson

Let’s say we’re observing a system at discrete intervals of time and testing whether its state satisfies or falsifies a given predicate or proposition p at each moment.  Then p and \mathrm{d}p are two state variables describing the time evolution of the system.  In logical conception p and \mathrm{d}p are independent variables, even if empirical discovery finds them bound by law.

What gives the differential variable \mathrm{d}p its meaning in relation to the ordinary variable p is not the conventional notation used here but a class of temporal inference rules, in the present example, the fourfold scheme of inference shown below.

Temporal Inference Rules

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
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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 9

Posted on July 12, 2021 by Jon Awbrey

Re: Laws of Form • Lyle Anderson

LA:
All I am asking is what is your definition of \mathrm{d}p in relation to p‌.  So far I have \mathrm{d}p is what one has to do to get from p to \texttt{(} p \texttt{)} or from \texttt{(} p \texttt{)} to p‌.  Is that all there is to it?  If that is the case, then what you are really dealing with is some flavor of Lattice Theory.

Dear Lyle,

Standing back for a moment to take in the Big Picture, what we’re doing here is taking all the things we would normally do in a “calculus of many variables” setting with spaces like:

\begin{matrix} \mathbb{R}, & \mathbb{R}^{j}, & \mathbb{R}^{j} \to \mathbb{R}, & \mathbb{R}^{j} \to \mathbb{R}^{k}, & \ldots \end{matrix}

and functoring that whole business over to \mathbb{B}, in other words, cranking the analogies as far as we can push them to spaces like:

\begin{matrix} \mathbb{B}, & \mathbb{B}^{j}, & \mathbb{B}^{j} \to \mathbb{B}, & \mathbb{B}^{j} \to \mathbb{B}^{k}, & \ldots \end{matrix}

A few analogies are bound to break in transit through the Real-Bool barrier, once familiar constructions morph into new-fangled configurations, and other distinctions collapse or “condense” as Spencer Brown called it.  Still enough structure gets preserved overall to reckon the result a kindred subject.

To be continued …

cc: Category Theory • Cybernetics • Ontolog • 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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 9 Comments

Differential Logic • Discussion 8

Posted on July 11, 2021 by Jon Awbrey

Re: Laws of Form • Lyle Anderson

A Reader inquired about the relationship between ordinary and differential boolean variables.  I thought it might help to explain how I first came to think about differential logic as a means of describing qualitative change.  The story goes a bit like this …

I wandered into this differential wonderland by following my nose through a budget of old readings on the calculus of finite differences.  It was a long time ago in a math library not too far away as far as space goes but no longer extant in time.  Boole himself wrote a book on the subject and corresponded with De Morgan about it.  I recall picking up the \mathrm{E} for enlargement operator somewhere in that mix.  It was a genuine epiphany.  All of which leads me to suspect the most accessible entry point may be the one I happened on first, documented in the Chapter on Linear Topics I linked at the end of the following post.

Maybe it will help to go through that …

Regards,

Jon

cc: Category Theory • Cybernetics • Ontolog • 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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 9 Comments

Animated Logical Graphs • 80

Posted on July 9, 2021 by Jon Awbrey

Re: Category Theory • Chad Nester

CN:
Re: Categorical Treatments of Existential Graphs
Cf: N. Haydon and P. Sobociński • Compositional Diagrammatic First-Order Logic

Thanks, Chad, for that extremely nice treatment of Peirce’s existential graphs at the β level, tantamount to predicate calculus or first order logic as we know it today.

The logic of relatives and the mathematics of relations appear in a different light from the perspective of Peirce’s own standpoint on logic, evolving as it does out of distinctive pragmatic and semiotic insights.  The reflections of Spencer Brown afford a few angles Peirce anticipated but in a glass, darkly.  And my own time tumbling recalcitrant calculi toward more ready tools for inquiry may add a few wrinkles, with luck to more than my own brow.  All that will develop as we go.

Resources

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
cc: Category Theory • Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
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Differential Logic • Comment 6

Posted on July 7, 2021 by Jon Awbrey

Re: Category Theory • Jon Awbrey

I opened a topic in the “logic” stream of “category theory zulipchat” to discuss differential logic in a category theoretic environment and began by linking to a few basic resources.

The topic on logical graphs introduced a style of graph‑theoretic syntax for propositional logic stemming from the work of Charles S. Peirce and G. Spencer Brown and touched on a generalization of Peirce’s and Spencer Brown’s tree‑like forms to what graph theorists know as cactus graphs or cacti.

Somewhat serendipitously, as it turns out, this cactus syntax is just the thing we need to develop differential propositional calculus, which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

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