Mike, Non-linear does not need to be defined with equations. Non-linear can be a process having continuity. Any process that begins and ends with continuity (i.e. w/o discontinuity) is inherently non-linear because it requires finite periods of that have all derivatives all of the same sign. It 'only' requires is developing a calculus for physical system rates. Search for 'continuity' on my site for some things. It's a new non-linear kind of math.
Phil Henshaw ¸¸¸¸.·´ ¯ `·.¸¸¸¸ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: [EMAIL PROTECTED] explorations: www.synapse9.com <http://www.synapse9.com/> -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Michael Agar Sent: Tuesday, June 19, 2007 10:41 AM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] Seminal Papers in Complexity This thread is sliding around some, but still Id like to add this overlong comment in case its useful. The emails have been good brain food. The problem I keep worrying about in my own work is, I use many core concepts metaphorically because they work at the human organizational scale in powerful and useful ways that I believe respect their scientific origins but at the same time allow the human/social world to see and understand and act differently. But I also want to be clear on those origins, to know and describe when and where and how Im stretching the concepts. The problem I have is, up close the conceptual basis of complexity more often than not turns to mush. Mea culpa much of the time, Im sure, but look what happened to reductionism in this thread. Even Wikipedia has several entries. I dont know how much credence to give them, but here they are: 0. <http://en.wikipedia.org/wiki/Reductionism#Varieties_of_reductionism> 1 Varieties of reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Ontological_reductionism> 1.1 Ontological reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Methodological_reductionism> 1.2 Methodological reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Methodological_individualism> 1.3 Methodological individualism 0. <http://en.wikipedia.org/wiki/Reductionism#Theoretical_reductionism> 1.4 Theoretical reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Scientific_reductionism> 1.5 Scientific reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Set-Theoretic_Reductionism> 1.6 Set-Theoretic Reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Linguistic_reductionism> 1.7 Linguistic reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Greedy_reductionism> 1.8 Greedy reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Eliminativism> 1.9 Eliminativism And now emergence. Ive heard it used in several ways. Way back when, we used it in anthropology as a form of methodological defense against the usual social science model of everything planned in a modular way before the research started. Emergence was shorthand for I cant tell you what Im going to do until I get there and learn whats worth learning and how to learn it. Then its also used more generally as shorthand for surprise, the presence and nature of which depends on perspective and prior knowledge of observer. Then its used for the end result of a deterministic process that has characteristics unlike the elements of that process, like water out of hydrogen and oxygen. Then its used for the need for different concepts and methods for different levels of a phenomenon, like phonology, morphology and syntax in linguistics. Then its used for unexpected evolutionary and historical transitions, like the Cambrian explosion. Probably many other uses if we sampled a lot of texts and conversations. Probably some of the sources cited already in the thread help with the problem. I need to read them. Maybe the field has outgrown the concepts that got it started. If true, thats probably a good sign. So I think Ill work on nonlinearity for awhile. Russell writes: most of my readers understand perfectly well what a linear function is: one that obeys f(a*x+b*y) = a*f(x)+b*f(y). Thats clear, resembles the definition in the Wikipedia entry. But then he writes : If neither * or + are defined for your objects of discussion, you cannot talk about (non-)linearity. That wont do. I have to be able to talk about nonlinear effects of, say, mental health policy on local programs in a qualitative way. I know it makes sense to do so from experience. Problem is to make it clear what the term means in that context. If the math wont do it, something else has to. Ill puzzle over the NECSI definition and the opening pages of Strogatz book for awhile. So maybe nonlinearity wont be so easy either. Theres the famous Einstein quote for inspiration: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Maybe we need a new nonlinear kind of math. Maybe it exists. Enough already. Mike
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