Tag Archives: Mathematics

Unknown's avatar Jon Awbrey 12:56 pm on December 16, 2025
Tags: , , , , , , Mathematics, , , , , ,   

Sign Relations • Signs and Inquiry

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

References

  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

Resources

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

#c-s-peirce, #connotation, #denotation, #inquiry, #logic, #logic-of-relatives, #mathematics, #relation-theory, #semiosis, #semiotic-equivalence-relations, #semiotics, #sign-relations, #triadic-relations

Unknown's avatar Jon Awbrey 9:36 am on December 14, 2025
Tags: , , , , , , Mathematics, , , , , ,   

Sign Relations • Anthesis

Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or semiosis, Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

Resources

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

#c-s-peirce, #connotation, #denotation, #inquiry, #logic, #logic-of-relatives, #mathematics, #relation-theory, #semiosis, #semiotic-equivalence-relations, #semiotics, #sign-relations, #triadic-relations

Unknown's avatar Jon Awbrey 7:16 am on November 20, 2025
Tags: Algorithms, , , , , , , , Data Structures, , , , , , , , , , Mathematics, , , , , ,   

Survey of Theme One Program • 7

This is a Survey of resources relating to the Theme One Program I worked on all through the 1980s.  The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning.  I had earlier developed separate programs for basic components of those tasks, in particular, two‑level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic.  Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate.  The long‑term vision is the implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Wiki Hub

Documentation

Blog Series

Blog Dialogs

Applications

References

  • Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
  • Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15.  Online.

cc: FB | Theme One Program • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#algorithms, #animata, #artificial-intelligence, #automated-research-tools, #boolean-functions, #c-s-peirce, #cactus-graphs, #constraint-satisfaction-problems, #data-structures, #differential-logic, #equational-inference, #formal-languages, #graph-theory, #inquiry-driven-systems, #laws-of-form, #learning-theory, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #painted-cacti, #propositional-calculus, #propositional-equation-reasoning-systems, #spencer-brown, #visualization

Unknown's avatar Jon Awbrey 1:00 pm on November 18, 2025
Tags: Abstraction, Ackermann, , , , Carnap, , , Hilbert, Hypostatic Abstraction, Kant, , Mathematics, Propositions As Types Analogy, , Saunders Mac Lane, , , Universals   

Survey of Precursors Of Category Theory • 6

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice.  A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background

Blog Series

Categories à la Peirce

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#abstraction, #ackermann, #analogy, #aristotle, #c-s-peirce, #carnap, #category-theory, #foundations-of-mathematics, #hilbert, #hypostatic-abstraction, #kant, #logic, #mathematics, #propositions-as-types-analogy, #relation-theory, #saunders-mac-lane, #semiotics, #type-theory, #universals

Unknown's avatar Jon Awbrey 11:30 am on November 15, 2025
Tags: , , , , , , Computational Complexity, , , , , , , , , Mathematics, , , , , Proof Theory, , , , Theorem Proving,   

Survey of Animated Logical Graphs • 8

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph‑theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Elements

Examples

Blog Series

  • Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)
  • Logical Graphs, Iconicity, Interpretation • (1) • (2)
  • Genus, Species, Pie Charts, Radio Buttons • (1)

Excursions

Applications

Anamnesis

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

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #computational-complexity, #constraint-satisfaction-problems, #differential-logic, #equational-inference, #graph-theory, #group-theory, #laws-of-form, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #model-theory, #painted-cacti, #peirce, #proof-theory, #propositional-calculus, #propositional-equation-reasoning-systems, #spencer-brown, #theorem-proving, #visualization

Unknown's avatar Jon Awbrey 2:00 pm on November 12, 2025
Tags: , , , , , Mathematics, , , , , , Triadicity,   

Survey of Semiotics, Semiosis, Sign Relations • 6

C.S. Peirce defines logic as “formal semiotic”, using formal to highlight the place of logic as a normative science, over and above the descriptive study of signs and their role in wider fields of play.  Understanding logic as Peirce understands it thus requires a companion study of semiotics, semiosis, and sign relations.

What follows is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

Elements

Sources

  • C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
  • C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

Topics

Blog Series

  • Peircean Semiotics and Triadic Sign Relations • (1) • (2) • (3)

Blog Dialogs

Excursions

  • Semiositis • (1)
  • Signspiel • (1)
  • Skiourosemiosis • (1)

References

  • 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, 269–284.  Abstract.  Online.
  • Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  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).
  • Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

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

#c-s-peirce, #icon-index-symbol, #inquiry, #logic, #logic-of-relatives, #mathematics, #relation-theory, #semiosis, #semiotics, #sign-relations, #triadic-relations, #triadicity, #visualization

Unknown's avatar Jon Awbrey 9:00 am on November 11, 2025
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , Mathematics, , , , , , , , , , , , , , , , ,   

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).

cc: FB | Semeiotics • Laws of Form • Mathstodon • Ontolog • 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, #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 11:48 am on November 9, 2025
Tags: Algebra, Algebra of Logic, , , Combinatorics, Discrete Mathematics, Duality, Dyadic Relations, , , , , , , , Mathematics, , , , Set Theory, , , ,   

Survey of Relation Theory • 9

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications.  This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relational Concepts

Relation Composition Relation Construction Relation Reduction
Relative Term Sign Relation Triadic Relation
Logic of Relatives Hypostatic Abstraction Continuous Predicate

Illustrations

Information‑Theoretic Perspective

  • Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)

Blog Series

Peirce’s 1870 “Logic of Relatives”

Peirce’s 1880 “Algebra of Logic” Chapter 3

Peirce’s 1885 “Algebra of Logic”

  • C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
  • C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

Resources

cc: FB | Relation Theory • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#algebra, #algebra-of-logic, #c-s-peirce, #category-theory, #combinatorics, #discrete-mathematics, #duality, #dyadic-relations, #formal-languages, #foundations-of-mathematics, #graph-theory, #group-theory, #logic, #logic-of-relatives, #logical-graphs, #mathematics, #model-theory, #relation-theory, #semiotics, #set-theory, #sign-relational-manifolds, #sign-relations, #triadic-relations, #type-theory

Unknown's avatar Jon Awbrey 11:20 am on November 7, 2025
Tags: , , , , , , , Change, , Differential Analytic Turing Automata, Differential Calculus, , Discrete Dynamics, , Frankl Conjecture, , Gradient Descent, , Hologrammautomaton, , Leibniz, , , Mathematics, , ,   

Survey of Differential Logic • 8

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

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.

Elements

Blog Series

Architectonics

Applications

Blog Dialogs

Explorations

cc: FB | Differential Logic • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#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, #inquiry-driven-systems, #leibniz, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #propositional-calculus, #visualization

Unknown's avatar Jon Awbrey 7:20 am on November 5, 2025
Tags: , Comprehension, Constraint, , , Extension, Form, Indication, , , , , Mathematics, , , , Structure, Systems Theory,   

Survey of Definition and Determination • 4

In the early 1990s, “in the middle of life’s journey” as the saying goes, I returned to grad school in a systems engineering program with the idea of taking a more systems-theoretic approach to my development of Peircean themes, from signs and scientific inquiry to logic and information theory.

Two of the first questions calling for fresh examination were the closely related concepts of definition and determination, not only as Peirce used them in his logic and semiotics but as researchers in areas as diverse as computer science, cybernetics, physics, and systems science would find themselves forced to reconsider the concepts in later years.  That led me to collect a sample of texts where Peirce and a few other writers discuss the issues of definition and determination.  There are copies of those selections at the following sites.

What follows is a Survey of blog and wiki posts on Definition and Determination, with a focus on the part they play in Peirce’s interlinked theories of signs, information, and inquiry.  In classical logical traditions the concepts of definition and determination are closely related and their bond acquires all the more force when we view the overarching concept of constraint from an information-theoretic point of view, as Peirce did beginning in the 1860s.

Blog Dialogs

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

#c-s-peirce, #comprehension, #constraint, #definition, #determination, #extension, #form, #indication, #information, #information-comprehension-x-extension, #inquiry-driven-systems, #logic, #mathematics, #scientific-method, #semiotics, #sign-relations, #structure, #systems-theory, #visualization

Unknown's avatar Jon Awbrey 12:12 pm on November 4, 2025
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , Mathematics, , , , , , , , , , , , , , , , , ,   

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
Tags: , Adaptive Systems, , , , , , Cognitive Science, , , Educational Systems Design, Educational Technology, , , , , , , Intelligent Systems, , , , Mathematics, Mental Models, , , , ,   

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)

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)

cc: FB | Inquiry Driven Systems • Laws of Form • Mathstodon • Academia.edu
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

Unknown's avatar Jon Awbrey 11:00 am on November 2, 2025
Tags: , , , , Dewey, , , , , , , Inference, , , , , , Mathematics, Morphism, Paradigmata, Paradigms, Pattern Recognition, , , , Pragmatism, , , , , , Syllogism, ,   

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 5

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through Charles S. Peirce.  Particular attention is paid to the way the inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Anthem

Blog Dialogs

Blog Series

Blog Surveys

OEIS Wiki

Ontolog Forum

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

#abduction, #aristotle, #c-s-peirce, #deduction, #dewey, #discovery, #doubt, #fixation-of-belief, #functional-logic, #icon-index-symbol, #induction, #inference, #information, #inquiry, #invention, #logic, #logic-of-science, #mathematics, #morphism, #paradigmata, #paradigms, #pattern-recognition, #peirce, #philosophy, #pragmatic-maxim, #pragmatism, #scientific-inquiry, #scientific-method, #semiotics, #sign-relations, #surveys, #syllogism, #triadic-relations, #visualization