Differential Logic, Dynamic Systems, Tangent Functors

[Jon Awbrey]

Cf: Differential Logic, Dynamic Systems, Tangent Functors • 1
https://inquiryintoinquiry.com/2018/09/05/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-1/

All,

Seeing as how quasi-neural models and the recurring issues of
logical-symbolic vs. quantitative-connectionist paradigms have
come round again, as they do every dozen or twenty years or so,
I thought I might refer again to this work I started initially
in that context, seeking logical-qualitative-symbolic analogues
of systems proposed by McClelland, Rumelhart, and the PGP Group,
and especially Stephen Grossberg's cooperative-competitive models.

Note. I posted on this topic back in the Fall of 2018 but
the lion's share of links got broken when the InterSciWiki
went off the live web. I am fixing those as I go along.

***

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, that of 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 of the ways this problem arises has to do with describing change
in logical, qualitative, or 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 discovery/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,
to discover/invent 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.

Reading continues here:

Overview
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

Part 1
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Part 2
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Part 3
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3

Part 4
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4

Part 5
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
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[Jon Awbrey]

Cf: Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9
http://inquiryintoinquiry.com/2020/07/19/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-9/

All,

I have a post with a lot of math formatting, but
unicode hasn't always been getting through lately,
so I'll just post the blog link for now plus some
test characters, and I'll post a transcript later
if everything seems to working okay at the moment.

α β γ δ ε

Later,

Jon

[Jon Awbrey]

Cf: Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9
http://inquiryintoinquiry.com/2020/07/19/differential-logic-dynamic-systems-tangent-functors-%e2%80%a2-discussion-9/

Re: FB | Systems Sciences
https://www.facebook.com/groups/2391509563/permalink/10158218873839564/
Re: Kenneth Lloyd
https://www.facebook.com/groups/2391509563/permalink/10158218873839564/?comment_id=10158219473119564

Dear Kenneth, All ...

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 R, the various dimensions of
real-value spaces, R^n, R^m, and so on, along with the various types
of mappings between those spaces. There are mappings from the real
line R into an n-dimensional space R^n — we think of those as curves,
paths, or trajectories. There are mappings from the plane R^2 to
values in R — we picture those as potential surfaces over the plane.
More generally, there are mappings from an n-dimensional space R^n to
values in R — we think of those as scalar fields over R^n — say, the
temperature at each point of an n-dimensional volume. There are
mappings from R^n to R^n and mappings from R^n to 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 γ through an n-dimensional space
would be typed as a function γ : R → R^n, where the functional domain R
would ordinarily be regarded as a time dimension. A mapping α from the
plane to a real value would be typed as a function α : R^2 → R, where
we might be thinking of α(x, y) as the altitude of a topographic map
above each point (x, y) of the plane. A scalar field β defined on an
n-dimensional space would be typed as a function β : R^n → R, where
β(x_1, …, x_n) is something like the pressure, the temperature, or the
value of some other dependent variable at each point (x_1, …, 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 : R^n → R^m.

To be continued …

Regards,

Jon