Posted by
Russell Standish on
URL: http://friam.383.s1.nabble.com/manifold-in-mathematics-tp3385914p3388546.html
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 (
[hidden email])
>
http://home.earthlink.net/~nickthompson/naturaldesigns/>
>
>
>
> > [Original Message]
> > From: Jochen Fromm <
[hidden email]>
> > To: The Friday Morning Applied Complexity Coffee Group <
[hidden email]>
> > 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.
> >
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> > FRIAM Applied Complexity Group listserv
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>
>
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