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