Re: "manifold" in mathematics

Posted by Robert Cordingley on
URL: http://friam.383.s1.nabble.com/manifold-in-mathematics-tp3385914p3388638.html

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 ([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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