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 -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpco...@hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ 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
