Tag Archives: Visualization

Unknown's avatar Jon Awbrey 5:48 am on September 22, 2026
Tags: , , , , , , , , , , , , , , , , , , Visualization   

Objects, Models, Theories • 4

Re: Objects, Models, Theories • (1) • (2) • (3)

What are objects, models, theories, and how do they relate to one another?

Recurring questions about the relationship between objects, models, and theories take on a different aspect when viewed from the perspective of Peirce’s pragmatic semiotic, in other words, when cast within a framework of triadic sign relations.

In contemplating the array of questions which come to mind I always find it helpful to ruminate on the diagram shown below — I might even call it a mandala for its wealth of symbolic features and its aid in organizing the pro‑&‑con‑fusion of mental impressions.

Aristotle's Paradigm

Aristotle’s “Paradigm”

Here is the corresponding text from Aristotle and the context that leads on to Peirce’s viewpoint.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#adaptive-systems, #analogy, #biological-systems, #c-s-peirce, #information, #inquiry, #inquiry-driven-systems, #learning-theory, #logic, #logic-of-science, #mathematical-models, #mathematics, #mental-models, #model-theory, #pragmata, #semiotics, #sign-relations, #triadic-relations, #visualization

Unknown's avatar Jon Awbrey 8:48 am on September 15, 2026
Tags: , , , , , , , , , , , , , , , , , , Visualization   

Objects, Models, Theories • 3

Re: Objects, Models, Theories • (1) • (2)
Re: Peirce List • Tom Gollier

Here my task is to build bridges between several different classical and contemporary uses of the word model, so I don’t have the luxury of complete control over the words in play but have to start from the customary senses in the various communities of interpretation.  Of course I’m slyly working from a sign‑relational backdrop, but I have to be sleight‑handed about that and not hit people over the head with it.

You can probably guess I’m using object to cover sign‑relational objects, and theories are clearly syntacked together from complexes of sign‑relational signs, so all we have left to pin down is where the various kinds of model sit at the table set with the labels of Object, Sign, Interpretant.

In its theoretical sense, a model of a theory is anything the theory is true of, anything that satisfies the theory.  In that sense, a model is very like an object.  It is whatever the theory is talking about.  In the order of nature, indeed, models come before theories.  But there is another order, the order of art, and one may construct artificial models out of almost any stuff, even the stuff of signs.  So you see the kind of wiggle room we have to work with.

Things are easier outside of logic, in applied mathematics and the special sciences, where models are just things like analogues, icons, simulations, and similar representations of objects.  But that makes them objects serving as signs of other objects, and so you may find some semiotic subtlety lurking there.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#adaptive-systems, #analogy, #biological-systems, #c-s-peirce, #information, #inquiry, #inquiry-driven-systems, #learning-theory, #logic, #logic-of-science, #mathematical-models, #mathematics, #mental-models, #model-theory, #pragmata, #semiotics, #sign-relations, #triadic-relations, #visualization

Unknown's avatar Jon Awbrey 8:24 am on September 13, 2026
Tags: , , , , , , , , , , , , , , , , , , Visualization   

Objects, Models, Theories • 2

Re: Gödel’s Lost Letter • The Graph Of Math

GLL:
Kurt Gödel is said to have been a latecomer to appreciating the power of Model Theory.  He was of course the greatest architect of Proof Theory, which stands in contrast to Model Theory.  Model Theory concerns itself with what could be true, while Proof Theory deals with what can be proved.  The latter sounds more definite, but they are supplementary:  a statement is capable of being true somewhere precisely when its negation cannot be proved.  The question is, where is that somewhere?  And when?

What — if anything — is the common sense that connects the different senses of the word model, as it has been used over the years in logic, mathematics, and the special sciences?  It’s a problem I’ve been running into for several decades now and I think I can trace the roots of it going back as far as Aristotle’s treatment of analogy.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#adaptive-systems, #analogy, #biological-systems, #c-s-peirce, #information, #inquiry, #inquiry-driven-systems, #learning-theory, #logic, #logic-of-science, #mathematical-models, #mathematics, #mental-models, #model-theory, #pragmata, #semiotics, #sign-relations, #triadic-relations, #visualization

Unknown's avatar Jon Awbrey 12:32 pm on September 10, 2026
Tags: , , , , , , , , , , , , , , , , , , Visualization   

Objects, Models, Theories • 1

Happy Birthday, Charles Sanders Peirce❢ — September 10, 1839

I return once more to a recurring subject.

Re: Artem Kaznatcheev • Three Types of Mathematical Models

In speaking of models one tends to find denizens of different disciplines talking at cross purposes to one another.  Logicians use the word to describe what may be distinguished as logical models, saying a model is whatever satisfies a theory — anything a theory holds true of — and that is the sense used in the logical subject of model theory.

Almost everyone else uses the word to describe what may be called analogical models, analogues being things holding enough properties in common with other things that learning about Thing 2 (the analogue system) can teach us about Thing 1 (the object system).

It is actually quite easy to integrate those two senses of the word model into a coherent picture of the whole situation, namely, the triadic relationship among objects, analogues, and theories.  We’ll get into that further as the discussion proceeds.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#adaptive-systems, #analogy, #biological-systems, #c-s-peirce, #information, #inquiry, #inquiry-driven-systems, #learning-theory, #logic, #logic-of-science, #mathematical-models, #mathematics, #mental-models, #model-theory, #pragmata, #semiotics, #sign-relations, #triadic-relations, #visualization

Unknown's avatar Jon Awbrey 10:52 am on August 28, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflective Interpretive Frameworks • Incident 2

Re: Terence Tao • Modular Arithmetic Challenge

  • Can a neural network learn to do modular multiplication efficiently?

Incidental Reflection 1

There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

Incidental Reflection 2

The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling.  The latter method makes use of ideas from Grossberg’s competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

  • Theme One Program • Jets and Sharks • (1) • (2) • (3)

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 9:44 am on May 3, 2026
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Visualization   

Animated Logical Graphs • 2

Re: Peirce List • Jim Willgoose

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.  I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.

I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.

So I’ll try to keep focused on that.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#abstraction, #amphecks, #analogy, #animata, #automated-research-tools, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #deduction, #differential-logic, #duality, #form, #graph-theory, #iconicity, #laws-of-form, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #model-theory, #peirces-law, #praeclarum-theorema, #proof-theory, #propositional-calculus, #propositional-logic, #semiotics, #spencer-brown, #theorem-proving, #topology, #visualization

Unknown's avatar Jon Awbrey 10:10 am on April 30, 2026
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Visualization   

Animated Logical Graphs • 1

For Your Musement …

Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.

  • Proof Animations
    • Double Negation

      • Double Negation

    • Peirce’s Law

      • Peirce's Law

    • Praeclarum Theorema

      • Praeclarum Theorema

    • Two‑Thirds Majority Function

      • Two‑Thirds Majority Function

A full discussion of logical graphs can be found in the following article.

Additional Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#abstraction, #amphecks, #analogy, #animata, #automated-research-tools, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #deduction, #differential-logic, #duality, #form, #graph-theory, #iconicity, #laws-of-form, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #model-theory, #peirces-law, #praeclarum-theorema, #proof-theory, #propositional-calculus, #propositional-logic, #semiotics, #spencer-brown, #theorem-proving, #topology, #visualization

Unknown's avatar Jon Awbrey 4:36 pm on April 21, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

JM:
This is a very important and interesting topic.  I think you should consider the relationship to self‑reference, indeed are they really the same thing?

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed in the x‑ray vision of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.

And when we ask, “What is this, that we call an interpreter?”, the pragmatic theory of signs tells us we cannot tell when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 10:10 pm on April 18, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other.  It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Simple Recursion

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 8:30 am on April 14, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief.  Evidence of it can be found in the recursion diagram by examining the places where three paths meet.  On the descending side there is the point where three paths diverge.  On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

Simple Recursion

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

  • The three arrows from the initial node represent a function F : \mathbb{N} \to \mathbb{N} \times \mathbb{N} \times \mathbb{N} such that F(n) = ( p(n), n, f(n) ).
  • The three arrows at the penultimate node represent a function m : \mathbb{N} \times \mathbb{N} \to \mathbb{N} such that m(j, k) = jk.

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 2:40 pm on April 9, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflection On Recursion • 2

Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name ``f" occurring in the defined phrase ``f(n)" re‑occurs in the defining phrase ``m(n, f(p(n)))".

Simple Recursion

It needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat.  Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 5:30 pm on April 6, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflection On Recursion • 1

Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.  A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.

It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I’ve thought so far …

Comment 1

Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don’t find it in a given corpus I just take them at their word …

Comment 2

The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.

One way I’d call pragmatic recursion — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I’ll defer to more familiar language, calling it cognitive or conceptual recursion.

Comment 3

If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.

Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.

It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.

Comment 4

Recursion occurs more markedly in syntactic recursion, where the recursive process shows its character as such in the symbols of its syntactic expression.

A sense of the difference can be gained by looking at a case of ostensible syntactic recursion.  (How much substance backs the ostentation is a subject we’ll take up, maybe at length, but later …)

Consider the following diagram for the computation of a simple recursive function.

Simple Recursion

For example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j \cdot k.

Comment 5

Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.  But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.  So it’s fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it.  Then again, even if that’s all it does, is there anything of use to be learned from it?

Comment 6

The prevalence of recursion in mathematics arises from the architecture of mathematical systems.

Mathematical systems grow from a fourfold root.

  • Primitives are taken as initial terms.
  • Definitions expound ever more complex terms in relation to the primitives.
  • Axioms are taken as initial truths.
  • Theorems follow from the axioms by way of inference rules.

Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns.  And again, in computation, recursive programs follow the same patterns in action.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 8:36 am on March 28, 2026
Tags: , , , , , , , , , , , , , , , , Visualization   

Reflective Interpretive Frameworks • Incident 1

Re: William Waites • The Agent That Doesn’t Know Itself

WW:  ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that.  One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

  • Note.  I was looking for a word to describe a random encounter with something that jogs one’s memory of a recurring theme — incident plays into the reflection theme and looked worth trying for now.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Unknown's avatar Jon Awbrey 11:00 am on March 19, 2026
Tags: , , , , , , , , , , , , , , , , , , , , , , Visualization   

Differential Logic • Discussion 17

Re: Differential Logic • The Logic of Change and Difference
Re: Systems Science Working Group • Paola Di Maio

PDM:
Subject: Differential Logic —
A point of contact with AI Knowledge Representation

Dear Jon,

Thank you for keeping the bell tolling — your framing of differential logic as the logic of variation arrives at a propitious moment.

For the past year I have been working at the intersection of knowledge representation, non‑logical reasoning, and AI systems, partly through the W3C AI Knowledge Representation Community Group (which I chair) and partly through independent research.  One of the persistent problems we encounter is that classical propositional and first order logic, however powerful for static state description, cannot represent the dynamics of reasoning systems — what changes, how fast, under what perturbation.

Your formulation cuts right to it:  ordinary propositional calculus describes positions in logical space; differential propositional calculus describes movement through it.  The analogy to Leibniz–Newton augmenting Descartes marks a categorical shift.

This connects directly to work I have been developing on what I call the five‑corners framework, extending Nagarjuna’s “catuskoti” (the four‑cornered logic:  true, false, both, neither — with Graham Priest’s fifth corner as refusal of the frame) toward a relational and co‑evolutionary account of knowledge.  The catuskoti gives us positions;  your differential extension gives us the calculus of transitions between them.  The five corners are attractors;  differential logic describes the manifold on which the system moves.

I am attaching a recent research note —

  • “Beyond Formal Logic:  Non‑Logical Forms of Valid Reasoning and Their Implications for AI Knowledge Representation”.  Online.

It documents three classes of reasoning that produce valid outcomes yet resist formalization in FOL:  embodied ecological reasoning, somatic‑intuitive reasoning, and transrational insight.

I suspect your differential extension of propositional calculus may offer formal traction on at least the first two, precisely because it can represent how a reasoning agent’s truth‑value assignments shift as context changes.

I also noticed your reference to neural network activation states and competition constraints in relation to the boundary operator.

This is terrain I am actively exploring in connection with oscillatory network models and a citizen science project on anomalous luminous phenomena (where the signal is change, not static state).  I may have to write a paper on that.

Jotted down some thoughts —

With collegial regards,

Paola Di Maio
Chair, W3C AI Knowledge Representation Community Group
Research Lead, Center for Systems, Knowledge Representation and Neuroscience, Ronin Institute

Dear Paola,

Many thanks for your kind reply and comments.

I was getting ready to devote a blog post (or two or three) by way of responding to your very substantial comments and I see you addressed the Systems Science Working Group but your post did not make it through to the web interface.  Did you intend to post it there?  It would help if I had a list link in my response if you did so.  Otherwise, if it’s okay with you, I could just quote the whole of your remarks on my blog.  Please let me know what you prefer.

Regards,
Jon

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization

Unknown's avatar Jon Awbrey 9:00 pm on March 14, 2026
Tags: , , , , , , , , , , , , , , , , , , , , , , Visualization   

Differential Logic • The Logic of Change and Difference

Differential logic is the logic of variation — the logic of change and difference.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple case of a differential logical calculus is furnished by a differential propositional calculus, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #change, #cybernetics, #differential-calculus, #differential-logic, #discrete-dynamics, #equational-inference, #functional-logic, #gradient-descent, #graph-theory, #inquiry-driven-systems, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #time, #visualization