I suspect we still have a ways to go before we exhaust the manifold definitions...
--Doug On Tue, Aug 4, 2009 at 7:40 PM, russell standish <[email protected]>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 ([email protected]) > > http://home.earthlink.net/~nickthompson/naturaldesigns/<http://home.earthlink.net/%7Enickthompson/naturaldesigns/> > > > > > > > > > > > [Original Message] > > > From: Jochen Fromm <[email protected]> > > > To: The Friday Morning Applied Complexity Coffee Group < > [email protected]> > > > 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 > > -- > >
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