Let me add another inquiry to this - how do we reconcile this notion of manifold with the idea of self-similarity? If Epping Forest is a manifold, but the leaves and twigs are not, yet the leaves and twigs have some self-similarity, is Holt truly thinking in terms of the mathematical definition of manifold, as Roger gave us, or is the metaphor missing something (or am I)?
- Claiborne Booker - -----Original Message----- From: Nicholas Thompson <nickthomp...@earthlink.net> To: friam@redfish.com Sent: Wed, Aug 5, 2009 12:39 am Subject: Re: [FRIAM] "manifold" in mathematics Is an organism a manifold? ? Do the parts have to be heterogeneous?? Dictionary definition would seem to suggest so.? Thus a regiment would not be a manifold (except insofar as it contains soldiers of different ranks).? ? n ? Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University (nthomp...@clarku.edu) http://home.earthlink.net/~nickthompson/naturaldesigns/ ? ? ? ? ----- Original Message ----- From: Robert Cordingley To: The Friday Morning Applied Complexity Coffee Group Sent: 8/4/2009 8:03:00 PM Subject: Re: [FRIAM] "manifold" in mathematics So to return to the forest question... Sherwood Forest is I presume another manifold.? I know it is now discontiguous, separated by urban development and such (perhaps Epping Forest is too).? Is it still a manifold?? I could ask the same question about the British Isles: lots of little places, some bigger ones, surrounded by water. Also while the twig is in the forest it is part of the forest until someone removes it.? Does it's history keep it part of the manifold?? Or can I declare it as such and it is so? Robert C. russell standish wrote: On Tue, Aug 04, 2009 at 03:51:38PM -0600, Nicholas Thompson wrote: This is why I like to ask questions of PEOPLE: because when you get conflicting answers, you have somewhere to go to try and resolve the conflict. So I have three different definitions of a manifold: 1. A patchwork made of many patches 2. The structure of a manifold is encoded by a collection of charts that form an atlas. 3. a "function" that violates the usual function rule that there can be only y value for each x value. (or do I have that backwards). I can map 1 or 2 on to one another, but not three. i think 3. is the most like meaning that Holt has in mind because I think he thinks of consciousness as analogous to a mathematical formula that generates outputs (responses) from inputs(environments). 1 & 2 were different ways of saying the same thing - one does need a definition of patch or chart, though. I think (although I could be mistaken), each chart (or patch) must be a diffeomorphism (aka smooth map), although it may be sufficient for them to be continuous. The reason I say that, is that I don't believe one could consider the Cantor set to be a manifold. Most of my experience of manifolds have been smooth manifolds (every point is surrounded by neighbourhood with a diffeomorphic chart/patch), with the occasional nod to piecewise smooth manifolds (has corners). The surface of a sphere is a smooth manifold. The surface of a cube is not, but it is piecewise smooth. No 3 above was just a way of saying that graphs of suitably smooth functions are manifolds, but not all manifolds are graphs of functions. Thanks, everybody. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University (nthomp...@clarku.edu) http://home.earthlink.net/~nickthompson/naturaldesigns/ [Original Message] From: Jochen Fromm <jfr...@t-online.de> To: The Friday Morning Applied Complexity Coffee Group <friam@redfish.com> Date: 8/4/2009 6:31:57 PM Subject: Re: [FRIAM] "manifold" in mathematics A manifold can be described as a complex patchwork made of many patches. If we try to describe self-consciousness as a manifold then we get - the patch of a strange loop associated with insight in confusion (according to Douglas Hofstadter) - the patch of an imaginary "center of narrative gravity" (according to Daniel Dennett) - the patch of the theater of consciousness which represents the audience itself (according to Bernard J. Baars) have I missed an important patch ? -J. ============================================================ 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 ============================================================ 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 ============================================================ 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
============================================================ 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
