Signs Of Signs • 3

Re: Michael HarrisLanguage About Language

And if we don’t, who puts us away?

One’s answer, or at least one’s initial response to that question will turn on how one feels about formal realities.  As I understand it, reality is that which persists in thumping us on the head until we get what it’s trying to tell us.  Are there formal realities, forms that drive us in that way?

Discussions like these tend to begin by supposing we can form a distinction between external and internal.  That is a formal hypothesis, not yet born out as a formal reality.  Are there formal realities that drive us to recognize them, to pick them out of a crowd of formal possibilities?

Posted in Aesthetics, Category Theory, Coherentism, Communication, Connotation, Form, Formal Languages, Foundations of Mathematics, Higher Order Propositions, Illusion, Information, Information Theory, Inquiry, Inquiry Into Inquiry, Interpretation, Interpretive Frameworks, Intuition, Language, Logic, Mathematics, Objective Frameworks, Objectivism, Peirce, Philosophy, Philosophy of Mathematics, Pragmata, Pragmatics, Pragmatism, Recursion, Reflection, Semantics, Semiotics, Set Theory, Sign Relations, Syntax, Translation, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Signs Of Signs • 2

Re: Michael HarrisLanguage About Language

I compared mathematics to a “consensual hallucination,” like virtual reality, and I continue to believe that the aim is to get (consensually) to the point where that hallucination is a second nature.

I think that’s called coherentism, normally contrasted with or complementary to objectivism.  It’s the philosophy of a gang of co-conspirators who think, “We’ll get off scot-free so long as we all keep our stories straight.”

Posted in Aesthetics, Category Theory, Coherentism, Communication, Connotation, Form, Formal Languages, Foundations of Mathematics, Higher Order Propositions, Illusion, Information, Information Theory, Inquiry, Inquiry Into Inquiry, Interpretation, Interpretive Frameworks, Intuition, Language, Logic, Mathematics, Objective Frameworks, Objectivism, Peirce, Philosophy, Philosophy of Mathematics, Pragmata, Pragmatics, Pragmatism, Recursion, Reflection, Semantics, Semiotics, Set Theory, Sign Relations, Syntax, Translation, Triadic Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Signs Of Signs • 1

Re: Michael HarrisLanguage About Language

There is a language and a corresponding literature that approaches logic and mathematics as related species of communication and information gathering, namely, the pragmatic-semiotic tradition passed on to us through the lifelong efforts of C.S. Peirce.  It is by no means a dead language, but it continues to fly beneath the radar of many trackers in logic and math today.  Still, the resource remains for those who are ready intuit to dip.

Posted in Inquiry, Peirce, Logic, Information, Mathematics, Semiotics, Foundations of Mathematics, Philosophy, Pragmatism, Pragmatics, Type Theory, Higher Order Propositions, Reflection, Interpretation, Form, Formal Languages, Category Theory, Aesthetics, Recursion, Communication, Inquiry Into Inquiry, Syntax, Information Theory, Intuition, Triadic Relations, Sign Relations, Semantics, Philosophy of Mathematics, Pragmata, Illusion, Set Theory, Objective Frameworks, Interpretive Frameworks, Language, Connotation, Translation, Coherentism, Objectivism | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Animated Logical Graphs : 9

Re: Ken ReganThe Shapes of Computations

The insight that it takes to find a succinct axiom set for a theoretical domain falls under the heading of abductive or retroductive reasoning, a knack as yet refractory to computational attack, but once we’ve lucked on a select-enough set of axioms we can develop theorems that afford a more navigable course through the subject.

For example, back on the range of propositional calculus, it takes but a few pivotal theorems and the lever of mathematical induction to derive the Case Analysis-Synthesis Theorem (CAST), which provides a bridge between proof-theoretic methods that demand a modicum of insight and model-theoretic methods that can be run routinely.

Posted in Amphecks, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Diagrammatic Reasoning, Graph Theory, Inquiry Driven Education, 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 | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Animated Logical Graphs : 8

Re: Ken ReganThe Shapes of Computations

The most striking example of a “Primitive Insight Proof” (PIP❢) known to me is the Dawes–Utting proof of the Double Negation Theorem from the CSP–GSB axioms for propositional logic. There is a graphically illustrated discussion at the following location:

I cannot hazard a guess what order of insight it took to find that proof — for me it would have involved a whole lot of random search through the space of possible proofs, and that’s even if I got the notion to look for one in the first place.

There is of course a much deeper order of insight into the mathematical form of logical reasoning that it took C.S. Peirce to arrive at his maximally elegant 4-axiom set.

Posted in Amphecks, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Diagrammatic Reasoning, Graph Theory, Inquiry Driven Education, 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 | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Animated Logical Graphs : 7

Re: Ken ReganThe Shapes of Computations

There are several issues of computation shape and proof style that raise their heads already at the logical ground level of boolean functions and propositional calculus.  From what I’ve seen, there are three dimensions of variation that appear most prominent at this stage:

  • Insight Proofs vs. Routine Proofs
  • Model-Theoretic Methods vs. Proof-Theoretic Methods
  • Equational (Information-Preserving) Proofs vs.
    Implicational (Information-Reducing) Proofs

More later, after I dig up some basic examples …

Posted in Amphecks, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Diagrammatic Reasoning, Graph Theory, Inquiry Driven Education, 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 | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Survey of Theme One Program • 1

This is a Survey of previous blog and wiki posts on the Theme One Program that I worked on all through the 1980s. The aim of the project was to develop fundamental algorithms and data structures to support an integrated learning and reasoning interface, looking toward the design of an Automated Research Tool (ART) that could do double duty as a medium for Inquiry Driven Education (IDE). I wrote up a running pilot version of the program well enough to get a Master’s degree out of it, but I’m still “getting around to” writing up the complete documentation.

Wiki Hub

Documentation in Progress

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.
Posted in Algorithms, Animata, Artificial Intelligence, Automated Research Tools, Boolean Algebra, Boolean Functions, Cactus Graphs, Cognitive Science, Computation, Computational Complexity, Computer Science, Computing, Constraint Satisfaction Problems, Cybernetics, Data Structures, Diagrammatic Reasoning, Diagrams, Differential Analytic Turing Automata, Education, Educational Systems Design, Educational Technology, Equational Inference, Functional Logic, Graph Theory, Indicator Functions, Inquiry, Inquiry Driven Education, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Knowledge, Learning, Learning Theory, Logic, Logical Graphs, Machine Learning, Mathematics, Mental Models, Minimal Negation Operators, Painted Cacti, Peirce, Programming, Programming Languages, Propositional Calculus, Propositional Equation Reasoning Systems, Propositions, Research Technology, Semeiosis, Semiosis, Semiotics, Sign Relations, Surveys, Teaching, Theorem Proving, Triadic Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment