From Osher Doctorow Ph.D.
I have been posting to the Math Forum (access it by those keywords) in the subforums Geometry-Research and Math-History-List and also to ResearchMathematics@yahoogroups.com and Theory-Edge@yahoogroups.com and somewhat to CHORA@liverpool.ac.uk (a philosophy forum in England) on what I will here call the Edge of Connectedness. Mathematical and Theoretical Physics (physics respectively done by Mathematicians and theorists with mathematical knowledge inside physics) appears to be presently extremely dominated by the one-step-at-a-time one-path-at-a-time people whom I call the Curvilinear School because their paths look like straight lines or bent/curved lines. Opposed to that school are a few theorists like me who adopt an Expansion- Contraction approach in which simultaneous motion in different directions is the key thing. In my postings at the above sites, I am finding that just as the Edge of Chaos plays a critical role in Fractals and Chaos theory, so the Edge of Connectedness plays a critical role in Expansion-Contraction theory. What is the Edge of Connectedness? The Expansion-Contraction school roughly maintains that the Universe itself is mostly connected, in opposition to the Curvilinear school which has concluded (to the extent that it concludes anything) that the Universe is in its foundations mostly DISCONNECTED. The Edge of Connectedness is where Connected things come close to getting Disconnected but not quite. It is exemplified by a Hole in a Connected object. Unless the Hole is so big that it tears the object in two, it ordinarily does not disconnect the object entirely. Some connected objects can be "shot through" with holes and still stay connected. We say that they are porous for example. For those members of FractInt who want to explore this, take a look at my postings above. They tie together things that have not been tied together before in the Mainstream research literature and put them in a new light. After mastering the rather simple language, you might want to take a look at the rather forbidding-sounding names Pseudodifferential Operators and Parametrix and Parametrices, and Singular Support. These keywords lead to very complicated mathematical and physics literature, but look at the abstracts and the summaries and even some of the rough English which occasionally emerges because at that difficult level even quantitative people find a certain need to gather their intuitions verbally (not so much in Curvilinear mathematics, however). Osher Doctorow