Tag Archives: Inquiry Into Inquiry

Unknown's avatar Jon Awbrey 2:40 pm on April 9, 2026
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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
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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
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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 9:00 am on November 11, 2025
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Survey of Pragmatic Semiotic Information • 9

This is a Survey of blog and wiki posts on a theory of information which grows out of pragmatic semiotic ideas.  All my projects are exploratory in character but this line of inquiry is more open‑ended than most.  The question is —

What is information and how does it impact the spectrum of activities answering to the name of inquiry?

Setting out on what would become his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions obscuring the issue and fixed on what he called the “laws of information” as the key to solving the puzzle.

The first hints of the Information Revolution in our understanding of scientific inquiry may be traced to Peirce’s lectures of 1865–1866 at Harvard University and the Lowell Institute.  There Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Fast forward to the present and I see the Big Question as follows.  Having gone through the exercise of comparing and contrasting Peirce’s theory of information, however much it yet remains in a rough‑hewn state, with Shannon’s paradigm so pervasively informing the ongoing revolution in our understanding and use of information, I have reason to believe Peirce’s idea is root and branch more general and has the potential, with due development, to resolve many mysteries still bedeviling our grasp of inference, information, and inquiry.

Inference, Information, Inquiry

Pragmatic Semiotic Information

Semiotics, Semiosis, Sign Relations

Sign Relations, Triadic Relations, Relation Theory

  • Blog Series • (1)
    • Discusssions • (1) • (2)

Excursions

Blog Dialogs

References

  • Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  Archive.  Journal.  Online (doc) (pdf).

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#abduction, #c-s-peirce, #communication, #control, #cybernetics, #deduction, #determination, #discovery, #doubt, #epistemology, #fixation-of-belief, #induction, #information, #information-comprehension-x-extension, #information-theory, #inquiry, #inquiry-driven-systems, #inquiry-into-inquiry, #interpretation, #invention, #knowledge, #learning-theory, #logic, #logic-of-relatives, #logic-of-science, #mathematics, #philosophy-of-science, #pragmatic-information, #probable-reasoning, #process-thinking, #relation-theory, #scientific-inquiry, #scientific-method, #semeiosis, #semiosis, #semiotic-information, #semiotics, #sign-relational-manifolds, #sign-relations, #surveys, #triadic-relations, #uncertainty, #visualization

Unknown's avatar Jon Awbrey 12:12 pm on November 4, 2025
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Survey of Cybernetics • 5

Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αρετης κυβερνητικης), do you perceive what must happen to him and his fellow sailors?

— Plato • Alcibiades • 135 A

This is a Survey of blog posts relating to Cybernetics.  It includes the selections from Ashby’s Introduction and the comment on them I’ve posted so far, plus two series of reflections on the governance of social systems in light of cybernetic and semiotic principles.

Anthem

Ashby’s Introduction to Cybernetics

  • Chapter 11 • Requisite Variety

Blog Series

  • Theory and Therapy of Representations • (1) • (2) • (3) • (4) • (5)

cc: FB | Inquiry Driven Systems • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#abduction, #c-s-peirce, #communication, #control, #cybernetics, #deduction, #determination, #discovery, #doubt, #epistemology, #fixation-of-belief, #induction, #information, #information-comprehension-x-extension, #information-theory, #inquiry, #inquiry-driven-systems, #inquiry-into-inquiry, #interpretation, #invention, #knowledge, #learning-theory, #logic, #logic-of-relatives, #logic-of-science, #mathematics, #peirce, #philosophy, #philosophy-of-science, #pragmatic-information, #probable-reasoning, #process-thinking, #relation-theory, #scientific-inquiry, #scientific-method, #semeiosis, #semiosis, #semiotic-information, #semiotics, #sign-relational-manifolds, #sign-relations, #surveys, #triadic-relations, #uncertainty

Unknown's avatar Jon Awbrey 12:08 pm on November 3, 2025
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Survey of Inquiry Driven Systems • 7

This is a Survey of work in progress on Inquiry Driven Systems, material I plan to refine toward a more compact and systematic treatment of the subject.

An inquiry driven system is a system having among its state variables some representing its state of information with respect to various questions of interest, for example, its own state and the states of potential object systems.  Thus it has a component of state tracing a trajectory though an information state space.

Anthem

Elements

Background

Blog Series

  • Pragmatic Cosmos • (1)
  • Reflection On Recursion • (1)

Blog Dialogs

  • Architectonics of Inquiry • (1)

Developments

Applications

  • Conceptual Barriers to Creating Integrative Universities
    (Abstract) (Online)
  • Interpretation as Action • The Risk of Inquiry
    (Journal) (doc) (pdf)
  • An Architecture for Inquiry • Building Computer Platforms for Discovery
    (Online)
  • Exploring Research Data Interactively • Theme One : A Program of Inquiry
    (Online)

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#abduction, #adaptive-systems, #analogy, #animata, #artificial-intelligence, #automated-research-tools, #c-s-peirce, #cognitive-science, #cybernetics, #deduction, #educational-systems-design, #educational-technology, #fixation-of-belief, #induction, #information-theory, #inquiry, #inquiry-driven-systems, #inquiry-into-inquiry, #intelligent-systems, #interpretation, #logic, #logic-of-science, #mathematics, #mental-models, #pragmatic-maxim, #semiotics, #sign-relations, #triadic-relations, #visualization