Inquiry Driven Systems • Discussion 5

Re: Laws of FormLeon Conrad

LC:
As someone who has worked on, teaches, and uses the CoI [Calculus of Indications] to make classical syllogistic logic much easier to practice and more visually intuitive than any of the visualisations we have to date, I would be very interested in finding out more about your work in applying GSB’s work to logical tables, particularly if it does a similar thing.

Ahoy Leon!  Welcome aboard, a-synchronicity being what it is, it may be September before I get both my hands back on this deck myself as I’ve got a bunch of long-procrastinated home and garden and auto and health-related matters to deal with.

If you’re an old time web surfer like we all used to be way back when I could leave you with a link or two to follow up on your own recognizance — I know, I know, these days it’s more like you can link a horse to whatev but you can’t make em click it.

I will try to write something more coherent later today but failing that here’s a link to an omnibus Survey page for my blog, where you can find what’s been occupying my trains of
thought for the past half-century.  The last-numbered links under each topic include and update all the earlier entries.

Best regards,

Jon

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Differential Logic • 11

Transforms Expanded over Ordinary and Differential Variables

As promised in Episode 10, in the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how the differential operators \mathrm{E} and \mathrm{D} act on that set.  There being some advantage to singling out the enlargement or shift operator \mathrm{E} in its own right, we’ll begin by computing \mathrm{E}f for each of the functions f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.

Enlargement Map Expanded over Ordinary Variables

We first encountered the shift operator \mathrm{E} in Episode 4 when we imagined being in a state described by the proposition pq and contemplated the value of that proposition in various other states, as determined by the differential propositions \mathrm{d}p and \mathrm{d}q.  Those thoughts led us from the boolean function of two variables f_{8}(p, q) = pq to the boolean function of four variables \mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q) = \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)}, as shown in the entry for f_{8} in the first three columns of Table A3.  (Let’s catch a breath here and discuss what the rest of the Table shows next time.)

\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded over Ordinary Variables}~ \{ p, q \}

Ef Expanded over Ordinary Variables {p, q}

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Differential Logic • Discussion 14

Re: Differential Logic • Discussion • (12) (13)
Re: FB | Peirce SocietyΧριστο Φόρος

Another bit of work I did toward a Psych M.A. was applying my Theme One program to a real-live dataset on family dynamics.  A collection of notes on that project is linked below.

Resources

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Differential Logic • Discussion 13

Re: Differential Logic • Discussion 12
Re: FB | Peirce SocietyΧριστο Φόρος

Χριστο Φόρος asked whether the difference between qualitative and quantitative information was really all that much of a problem, especially in view of mixed datasets.  As I have encountered it in practice the rub is not so much between different types of data as between the two cultures of quantitative and qualitative research paradigms.

As it happens, my mix of backgrounds often found me employed consulting on statistics at the interface between quantitative and qualitative researchers.  On the qual side back in the 80s and 90s we were just beginning to develop software for ethographic methods, massaging linguistic, narrative, and verbal protocols toward categorical variables and non‑parametric statistics.  I worked a lot on concepts and software bridging the gap between qual and quant paradigms.

The program I spent the 80s developing and eventually submitted toward a Master’s in Psych integrated a Learning module (Slate) and a Reasoning module (Chalk).  The first viewed its input stream as a two-level formal language (“words” and “phrases”) and sought to induce a grammar for the language its environment was speaking to it.  The second was given propositional expressions describing universes of discourse and had to find all the conjunctions of basic qualitative features (boolean variables) satisfying those descriptions.  There’s a report on this work in the following paper.

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Differential Logic • Discussion 12

Re: Category Theory • John Baez (1) (2)

JB:
One thing I’m interested in is functorially relating purely qualitative models to quantitative ones, or mixed quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it.  That’s a situation we often find ourselves in:  having a mixture of quantitative and qualitative information about what’s going on in a complicated system.
When I say “functorially”, I mean for starters:  there should be a functor from “quantitative models” of system dynamics to “qualitative models”.

Dear John,

This is something I’ve been working on.  In a turn of phrase I once concocted, it’s like passing from the qualitative theory of differential equations to the differential theory of qualitative equations.  See the Chategory topic on Differential Logic.

Regards,
Jon

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Inquiry Driven Systems • Discussion 4

Re: Category TheoryEduardo Ochs

EO:
Do you have links on how to teach Logical Graphs to children (and to people like me!) and how to use them as a basis for learning Propositional Calculus and quantifiers?

Dear Eduardo,

There’s a lot of stuff I’ve put on the web over the last twenty years devoted to CSP and GSB and my own versions of Logical Graphs — I’m still working at organizing all that in a step-by-step tutorial fashion.  I’ll be doing more of that over time on a number of local streams and topics, e.g.

You might try sampling my Inquiry blog for the daily rushes and discussions or my OEIS user page and OEIS workspace to see if anything engages your interest.

Cheers,
Jon

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Inquiry Driven Systems • Discussion 3

Re: Category TheoryHenry Story

HS:
Could one re-invent the whole curriculum from age 5 onwards
built on new [category theoretic] concepts?

If I were starting from scratch, and I’m always starting from scratch, I would ease my way up to the pons asinorum of logic and math using the types of logical graphs laid down by Peirce and Spencer Brown.  That is because I think it’s crucial to firm up propositional logic before taking on quantifiers and to grasp classical logic before intuitionistic.

The climb from “zeroth order logic” to first order logic is a lot more interesting and richer in adventure once you have a truly efficient calculus for propositional logic at the ready.  An approach to categories, combinators, etc. can then be made via the propositions as types analogy.  For the kiddies, Smullyan’s Mockingbird would be the primer of choice.

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Inquiry Driven Systems • Discussion 2

Re: Category TheoryHenry Story

Way back in the Summer of Love I met a girl who had just graduated in Chemistry and was thinking about grad school in Education, the hot new field of Instructional Media, we got to talking and dreamed up a vision of using media, at first just shapes in motion, to teach people math from scratch.  Long time passing, we got married, she did a dissertation — The Effect of the Hausdorff–Besicovitch Dimension of Figure Boundary Complexity on Hemispheric Functioning — studying the effects of fractal figure complexity on cognitive processing, Mandelbrot gave her permission to use a series of his figures and ranked them by fractal dimension for her, and I pursued an array of parallel lives in math, statistics, computing, philosophy, and psych.

Here is one of our later collaborations aimed toward integrating inquiry learning and information technology into education.

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Differential Logic • Discussion 11

Re: Differential Logic • Discussion 9

Let’s look more closely at the “functor” from \mathbb{R} to \mathbb{B} and the connection it makes between real and boolean hierarchies of types.  There’s a detailed discussion of this analogy in the article and section linked below.

Assorted types of mathematical objects which turn up in practice often enough to earn themselves common names, along with their common isomorphisms, are shown in the following Table.

\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}

Analogy Between Real and Boolean Types

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Inquiry Driven Systems • Discussion 1

Re: Topos LabMathFoldr Project
Re: Category TheoryValeria de Paiva

Dear Brendan and Valeria,

I’ve been a participant∫observer in web-ontology knowledge projects for a couple of decades and they always give far more attention to knowledge as a product than due reflection on the dynamics of inquiry required to develop our provisional knowledge.  Many such projects have come and gone with the winds of fashion and it’s my guess the lack of balance between process and product orientation is one of the reasons why.

So I’ve been working on that … here’s a few links to the model of knowledge development sketched in my work on Inquiry Driven Systems.

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