Inquiry Into Inquiry

Re: Differential Propositional Calculus • Discussions • (1) • (2) • (3)
Re: Laws of Form • Lyle Anderson (1) (2)

Dear Lyle,

I’ve been meaning to get back to your comments linked above — the connections you observed to finite difference calculus, ordinary and partial differential equations, and differential geometry are very apt — but preparing the ground for a smooth transition to differential logic takes time, plus I needed to deal with the details of the logical graph duality I’ve been wanting to give their due for decades.

The flat out fastest key to the highway of differential logic is still the Casual Introduction I wrote for Part One of Differential Propositional Calculus.  It affords direct access to the basic intuitions and motivations of the subject but stops short of the syntactic mechanics needed to really take off.  The jump from that point to the more aggressive approaches of Differential Logic and Differential Logic and Dynamic Systems has long been a challenge.  I went looking for materials to bridge the gap and was pleased to find a few old writings I had almost forgotten but wrote when I myself was passing through a similar transition.  Perhaps one of those will help the intrepid reader hit the ground running in this field.

I’ll take up one of those pieces next time.

Resources

• Survey of Differential Logic
• Differential Logic and Dynamic Systems • Overview

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

Re: FB | Differential Logic • Rajib Hossain Pavel

RHP:

How can Differential Logic find Optimal Condition in a Game setting for an Individual Player (Individual Choice) and Overall System (Social Welfare)?

Dear Rajib,

Differential logic is a general framework for analyzing aspects of change and difference in systems amenable to qualitative description, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  To apply its concepts and methods to a concrete case one has to define the state space and the objective function one desires to optimize.  This may indeed be the hardest part of the problem.  It helps to break ground if one can think up a simple example from the class of systems one has in mind.

References

• Awbrey, S.M., and Awbrey, J.L. (May 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.

Resources

• Survey of Differential Logic
• Differential Logic and Dynamic Systems • Overview

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

Re: Michael Harris • Does Mathematics “Progress”? • Comment

In several places I can’t find right now I described formalization as an arrow.  A related idea occurs in a paper by Susan Awbrey and myself where we discussed “a dimension of increasing formalization in our mental models of the world” as an obstacle to integrating knowledge across various styles of inquiry.  An excerpt follows.

References

• Awbrey, S.M., and Awbrey, J.L. (May 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.

Resources

• Survey of Differential Logic
• Differential Logic and Dynamic Systems • Overview

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.

Robert Musil • The Man Without Qualities

Re: Michael Harris • Does Mathematics “Progress”?

Just off-hand, by way of getting grounded and oriented, there are a few questions I’d have to ask first.

• Is mathematics a science, or a form of scientific inquiry?
• If so, what is the place of mathematics within the sciences?
• Does science progress?
• (I mean progress in the sense of progress toward a goal, not necessarily progressive jazz or progressions in music generally.)
• If so, what is the goal of science?
• If science has a goal, does mathematics serve it?
• If so, how?

By science or scientific inquiry I meant to focus on species of goal-directed activity with the specific goal of “knowledge” and to ask whether mathematical arts and crafts and rites fall within that ballpark.  We come back to the antic Socratic question of whether we put another quarter in the machine for the sake of an external gain or simply to continue the play for its own sake.

MH:

In this post I was less concerned with philosophy than with philology — which, by the way, is another example of a term that was once seen as hopelessly antiquated and musty but that has been revived recently.

But when you introduce “knowledge” you have to grapple with “truth” and “objective reality”.  Philosophy has been so unsuccessful at pinning those down that some would prefer to give up on them altogether.  We can choose to call what mathematics generates “knowledge” but that just leads to the (philological) question of what this has to do with what other practices call “knowledge”.

Not that I don’t love wisdom and words, however often their stars may cross, but I invoked Musil rather for the way he syzygied the way of the engineer, the way of the mathematician, and the recursive point where their ways diverge.  It’s in this frame I think of the word entelechy, which I got from readings in Aristotle, Goethe, and Peirce and promptly gave a personal gloss as end in itself, partly on the influence of Conway’s game theory.

And that’s where I remember all those two-bit pieces I gave up to pinball machines in the early 70s and the critical point in my own trajectory when I realized I would never beat those machines, not that way, not ever, and I turned to the more collaboratory ends of teaching machines how to learn and reason.

♩ On A Related Note ♪ The Music Of The Primes ♫

Now there’s a progression of progressions I could enjoy, musically speaking, ad infinitum, and yet this pilgrim would consider it progress, mathematically speaking, if he could understand why the sequentiae should be sequenced as they are.  Would that understanding add to my enjoyment?  On jugera …

Resources

• Survey of Differential Logic
• Differential Logic and Dynamic Systems • Overview

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form