A friend mentioned Robert Rosen's 1996 essay "On the Limitations of Scientific Knowledge". It's in a book titled Limits of Scientific Knowledge edited by John Casti of SFI, with nine other essays presented at a workshop of that name held at the Swedish Academy of Sciences in 1995. Rosen's contribution outlined his perspective in a much simpler and clearer way than in his more extensive treatments. I think my 1995 theorem answers his complaint simply. Rosen 1995 [www.synapse9.com/ref/Rosen_On_Limitations_of_Sci.pdf] pfh 1995 [http://www.synapse9.com/drtheo.pdf]
What I now gather to be Rosen's central complaint is that in choosing to study the behavior of only convergent mathematical series (determinate math), science has arbitrarily discarded the study of divergent processes. In his observation, both emergence in complex systems and the behaviors of life are subjects of divergent processes. My theorem approaches the same complaint from another side. What I prove has to do with what is necessary for things to begin or end, in that things that begin or end don't have a past or future to start from or lead to. That's a fairly clear alternate definition of emergence, and beginning and ending is certainly the main subject of life. The theorem shows that for things to begin or end and avoid having infinite energy flows at some point they need to follow divergent processes. The aspect of divergence that the theorem specifically identifies is the property of having all derivative rates of change of the same sign for a finite period. The hard constraint that makes it necessary is that energy flow has to be continuous by the conservation laws. The result that things begin with divergence leaves open a very large and useful question. If emergence in nature has to begin with a divergent processes, what in some cases changes it to a convergent process later. That it is necessary, and simple observation makes evident it is common, that the beginnings of isolated system events (emergence or life) is with divergent processes (in the form of compound growth of one or another kind) then in those cases you can then look for what sometimes changes them into convergent processes. Observation suggests the switch to convergence occurs when divergent processes run into their environments, a useful scientific result. Perhaps science can't make use of divergent mathematics because equations don't represent environments for themselves to run into... and observing divergence in nature is then a clue to the need to enlarge the system domain in question, to look outside. If you take Rosen's complaint as being as I say, that limiting science to the study of convergent series prevents science from studying emergence or life, then does my proof answer his complaint 'yes' by showing a useful result from scientifically considering them? Best, Phil Henshaw ¸¸¸¸.·´ ¯ `·.¸¸¸¸ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 212-795-4844 680 Ft.Washington Ave NY NY 10040 "it's not finding what people say interesting, but finding the interest in what they say" ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
