Inquiry Into Inquiry

Mathematical Method • Discussion 5

Posted on August 10, 2020 by Jon Awbrey

Dear Paul,

“How We Think” is a topic for the descriptive science of psychology, and its ways are legion beyond definitive or exhaustive description.

“How We Ought To Think” if we wish to succeed at specified purposes is a topic for the normative science of logic, lumping together for the moment the evolving varieties of informal, formal, mathematical, and technologically augmented methods.

They’re all good questions and I see no reason not to pursue them all, aside from the limitations of our brief lives, but we have to keep the spectrum of different aims sorted.

John Dewey wrote the book How We Think in 1910. Peirce had earlier summed up his “non-psychological conception of logic” in the pithy motto “Logic has nothing to do with how we think” and this led some scholars to suspect Dewey’s title was aimed as a poke in Peirce’s ribs. But the book itself is a How-To guide devoted to improving our capacity for learning and reasoning, what we’d call today instruction in critical thinking.

All that is prologue to Vannevar Bush’s 1945 article, “As We May Think”, projecting the ways technology may amplify our capacity for inquiry going forward into the future. I think this is where we came in …

Reference

Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Posted in Abstraction, C.S. Peirce, Essentialism, Hypostatic Abstraction, Logic, Mathematics, Metaphysics, Method, Nominalism, Ockham, Ockham's Razor, Peirce, Pragmatic Maxim, Pragmatism, Realism, Semiotics, Theory | Tagged Abstraction, C.S. Peirce, Essentialism, Hypostatic Abstraction, Logic, Mathematics, Metaphysics, Method, Nominalism, Ockham, Ockham's Razor, Peirce, Pragmatic Maxim, Pragmatism, Realism, Semiotics, Theory | Leave a comment

Mathematical Method • Discussion 4

Posted on August 8, 2020 by Jon Awbrey

Re: Peirce List • Helmut Raulien

Dear Helmut,

It’s one of the occupational hazards of the classifying mind that one can start out consciously characterizing aspects of real situations and end up unwittingly thinking one’s gotten everything under the sun sorted into mutually exclusive bins.

Once the idols of compartmentality and the illusions of autonomous abstraction get their hold on our minds it is almost impossible to reconstitute or synthesize what we’ve torn asunder, if only in our own minds. The ounce of prevention here is always keeping in mind that from which all abstractions are abstracted — living experience.

Reference

Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Mathematical Method • Discussion 3

Posted on August 7, 2020 by Jon Awbrey

Re: Peirce List • John Sowa • Auke van Breemen • Jon Awbrey • Gary Fuhrman

Dear Gary,

Auke van Breemen wrote:

AvB:: It seems to come down to: never consider the textual production of a scientist only in itself, but also look at the reality the text tries to explain.

I took this as an admirably succinct statement of the difference between (1) scriptural hermeneutics — I’d call it “corpus hermeneutics” except for the risk of confusion with Corpus Hermeneticum — and (2) scientific interpretation, that is, any development of interpretant texts in relation to an independent object domain with the aim of forming true descriptions or gaining knowledge of that domain.

What I wrote in response to Auke was this:

JA:: Exactly!; We interpret texts
in relation to
the object in view.

All I did there was mention the three roles in a sign relation. We take in texts or whole bodies of work as signs of an object domain and we form interpretive texts as signs of the same domain. For my part, I interpret Peirce’s work as signs of an object world, one with respect to which other writers, artists, signifactors of all sorts have generated signs worthy of our interpretation.

I wouldn’t take the words “in view” too literally. I just as easily could have said “at hand” or “in mind” but I went with “object in view” on account of the fondness one of my old teachers had for Dewey’s signature “end in view”. As far as indicial signs are concerned, we’re all embroiled in concernful situations all the time, making our selves the initial signs of those pragmata, from which we derive all the remainder.

Peirce’s distinction between theorematic and corollarial reasoning has come up before. From what I recall of previous discussions, we should not read the word “theorematic” in too reductive or purely deductive a sense. Years ago it was something of a commonplace, even outside Peircean circles, to call attention to the etymology of “theorem” as having an observational, even “visionary” sense, cognate with “theatre”, and some would even point to the sacred origins of theatre, though maybe that’s a bridge too far …

As far as the iconic aspects of mathematics go, or even our knowledge representations in general, they are nice when we can get them, but I’m careful not to stress them too far — it’s too easy to “fall victim to a picture”, in Wittgenstein’s phrase, or succumb to the short-sightedness of Russell’s isomorphism theory of knowledge. Icons are specializations of symbols and thus fall short of symbols’ full potential. There is more to science than serving as a mirror of nature.

Reference

Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Mathematical Method • Discussion 2

Posted on August 5, 2020 by Jon Awbrey

Re: Peirce List • Auke van Breemen

AvB:: It seems to come down to: never consider the textual production of a scientist only in itself, but also look at the reality the text tries to explain.

Dear Auke,

Exactly!

We interpret texts
in relation to
the object in view.

Reference

Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.

ccc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Mathematical Method • Discussion 1

Posted on August 4, 2020 by Jon Awbrey

Re: Peirce List • John Sowa

Dear John,

Thanks for the notice of Carolyn Eisele’s article — it’s always worth reading what she has to say. We’ve had discussions of Peirce’s distinction between theorematic and corollarial reasoning before and I know there’s a respectable amount of literature out there about it.

The subject has curiously enough come up just recently in discussions on Facebook and Academia.edu, mostly on account of points brought up by John Corcoran. It’s also related to a number of discussions I’ve had over the years about the difference between “insight” proofs and “routine” proofs, partly in connection with theorem proving apps and Peirce’s logical graphs.

Usually these discussions take off into the stratosphere of high‑sounding blue‑skying about Gödel incompleteness and all that — but I want to keep my focus on more nuts and bolts issues at the moment and I’ll try to avoid going off on those planes.

Reference

Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341. Online. PDF.

cc: Cybernetics • Ontolog • Peirce List (1) (2) (3) • Structural Modeling • Systems Science

Differential Analytic Turing Automata • Discussion 2

Posted on July 31, 2020 by Jon Awbrey

Re: Scott Aaronson • The Busy Beaver Frontier

Dear Scott,

This discussion inspired me to go back and look at some of the work I did in the late 80s when I was trying to understand Cook’s Theorem. One of the programs I wrote to explore the integration of sequential learning and propositional reasoning had a propositional calculus module based on an extension of C.S. Peirce’s logical graphs, so I used that syntax to write out the clauses for finite approximations to Turing machines, taking the 4-state parity machine from Herbert S. Wilf’s Algorithms and Complexity as an object example. It was 1989 and all I had was a 289 PC with 600K heap, but I did manage to emulate a parity machine capable of 1 bit of computation. Here’s a link to an exposition of that.

Differential Analytic Turing Automata • Part 1 • Part 2

It may be quicker to skip to Part 2 and refer to Part 1 only as needed.

I’ll work up the case of a 2-state Busy Beaver when I get a chance.
I always learned a lot just from looking at the propositional form.

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

Posted in Algorithms, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Differential Analytic Turing Automata, Differential Logic, Logic, Logical Graphs, Peirce, Propositional Calculus, Turing Machines | Tagged Algorithms, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Differential Analytic Turing Automata, Differential Logic, Logic, Logical Graphs, Peirce, Propositional Calculus, Turing Machines | Leave a comment

Riffs and Rotes • 5

Posted on July 23, 2020 by Jon Awbrey

Re: Scott Aaronson • The Busy Beaver Frontier

All my favorite integer sequences, some very fast growing, spring from the “lambda point” where graph theory, logic, and number theory meet. My fascination with them goes back to a time when I was playing around with Gödel numbers of graph-theoretic structures and thinking about computational complexity. I’m busy as a beaver with other business at the moment so I’ll leave it now with just a few links to chew on till whenever.

Resources

Riffs and Rotes
The On-Line Encyclopedia of Integer Sequences
• A007097 • A050924 • A109300 • A111788 • A111791
Differential Analytic Turing Automata • Part 1 • Part 2

Posted in Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | Tagged Algebra, Combinatorics, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Riffs and Rotes | 2 Comments

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9

Posted on July 19, 2020 by Jon Awbrey

Re: FB | Systems Sciences • Kenneth Lloyd

Dear Kenneth,

Mulling over recent discussions put me in a pensive frame of mind and my thoughts led me back to my first encounter with category theory. I came across the term while reading and I didn’t fully understand it. But I distinctly remember a short time later catching up with my math TA — it was on the path by the tennis courts behind Spartan Stadium — and asking him about it.

The instruction I received that day was roughly along the following lines.

“Actually . . . we’re already doing a little category theory, without quite calling it that. Think about the different types of spaces we’ve been discussing in class, the real line $\mathbb{R},$ the various dimensions of real-value spaces, $\mathbb{R}^n, \mathbb{R}^m,$ and so on, along with the various types of mappings between those spaces. There are mappings from the real line $\mathbb{R}$ into an $n$ -dimensional space $\mathbb{R}^n$ — we think of those as curves, paths, or trajectories. There are mappings from the plane $\mathbb{R}^2$ to values in $\mathbb{R}$ — we picture those as potential surfaces over the plane. More generally, there are mappings from an $n$ -dimensional space $\mathbb{R}^n$ to values in $\mathbb{R}$ — we think of those as scalar fields over $\mathbb{R}^n$ — say, the temperature at each point of an $n$ -dimensional volume. There are mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$ and mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$ where $n$ and $m$ are different, all of which we call transformations or vector fields, depending on the use we have in mind.”

All that was pretty familiar to me, though I had to admire the panoramic sweep of his survey, so my mind’s eye naturally supplied all the arrows for the maps he rolled out. A curve $\gamma$ through an $n$ -dimensional space would be typed as a function $\gamma : \mathbb{R} \to \mathbb{R}^n,$ where the functional domain $\mathbb{R}$ would ordinarily be regarded as a time dimension. A mapping $\alpha$ from the plane to a real value would be typed as a function $\alpha : \mathbb{R}^2 \to \mathbb{R},$ where we might be thinking of $\alpha(x, y)$ as the altitude of a topographic map above each point $(x, y)$ of the plane. A scalar field $\beta$ defined on an $n$ -dimensional space would be typed as a function $\beta : \mathbb{R}^n \to \mathbb{R},$ where $\beta(x_1, \ldots, x_n)$ is something like the pressure, the temperature, or the value of some other dependent variable at each point $(x_1, \ldots, x_n)$ of the $n$ -dimensional volume. And rounding out the story, if only the basement and ground floor of a towering abstraction still under construction, we come to the general case of a mapping $f$ from an $n$ -dimensional space to an $m$ -dimensional space, typed as a function $f : \mathbb{R}^n \to \mathbb{R}^m.$

To be continued …

Resources

Differential Propositional Calculus

Part 1 • Part 2 • Appendices • References

Differential Logic • Part 1 • Part 2 • Part 3

Differential Logic and Dynamic Systems

Part 1 • Part 2 • Part 3 • Part 4 • Part 5
Appendices • References

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

Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Hill Climbing, Hologrammautomaton, Information Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Systems, Visualization | Tagged Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Hill Climbing, Hologrammautomaton, Information Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Systems, Visualization | 8 Comments

Differential Logic, Dynamic Systems, Tangent Functors • Comment 1

Posted on July 16, 2020 by Jon Awbrey

Re: Differential Logic, Dynamic Systems, Tangent Functors • 1

Seeing as how quasi-neural models and the recurring issues of symbolic vs. connectionist paradigms have come round again, I thought I might revisit work I began initially in that context, investigating logical, qualitative, and symbolic analogues of systems studied by McClelland, Rumelhart, and the Parallel Distributed Processing Group, and especially Stephen Grossberg’s competition-cooperation models.

⁂

People interested in category theory as applied to systems may wish to check out the following article, reporting work I carried out while engaged in a systems engineering program at Oakland University.

The problem addressed is a longstanding one, namely, building bridges to negotiate the gap between qualitative and quantitative descriptions of complex phenomena, like those we meet in analyzing and engineering systems, especially intelligent systems endowed with a capacity for processing information and acquiring knowledge of objective reality.

One way the problem arises has to do with describing change in logical, qualitative, and symbolic terms, long before we grasp the reality beneath the appearances firmly enough to cast it in measured, quantitative, real-number form.

Development on the quantitative shore got no further than a Sisyphean beachhead until the invention of differential calculus by Leibniz and Newton, after which things advanced by leaps and bounds. And there’s our clue what we need to do on the qualitative shore, namely, develop the missing logical analogue of differential calculus.

With that preamble …

Differential Logic and Dynamic Systems

This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

The reading continues at Differential Logic and Dynamic Systems

Resources

Differential Propositional Calculus • Part 1 • Part 2
Differential Logic • Part 1 • Part 2 • Part 3
Differential Logic and Dynamic Systems
• Part 1 • Part 2 • Part 3 • Part 4 • Part 5

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

Posted in Amphecks, Animata, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Information & Control, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged Amphecks, Animata, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Information & Control, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | 7 Comments

Category Theory • Comment 1

Posted on July 14, 2020 by Jon Awbrey

I’m deep in the middle of upgrading my intro to sign relations and I am determined to stick to it this time but there will be a phase when it’s critical to bring category theory to bear on the development. I had a nagging sense we had been discussing category theory in a related connection just recently but when I went back through my records it turned out this was way back in late 2018. (I was a bit occupied with moving our household and lost track of many loose threads.) At any rate, I’ll just post a few links here as reminders of topics to pick up later.

Survey of Precursors Of Category Theory
Differential Logic, Dynamic Systems, Tangent Functors • (1)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Systems Science • GST Conceptual Framework Development
• My Comment • GST, DST, MST

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

Posted in Abstraction, C.S. Peirce, Category Theory, Differential Logic, Graph Theory, Group Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Categories, Research Technology, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Tagged Abstraction, C.S. Peirce, Category Theory, Differential Logic, Graph Theory, Group Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Categories, Research Technology, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations | Leave a comment

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Inquiry Into Inquiry

Mathematical Method • Discussion 5

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Mathematical Method • Discussion 4

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Mathematical Method • Discussion 3

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Mathematical Method • Discussion 2

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Mathematical Method • Discussion 1

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Differential Analytic Turing Automata • Discussion 2

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9

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Differential Logic, Dynamic Systems, Tangent Functors • Comment 1

Differential Logic and Dynamic Systems

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