Clairborne,
Here is what I think Holt is up to. He is using a model of mathematical induction for his understanding of mind. Mathematical induction is actually a form of logical DEduction in which the combination of a principle with a single case is used to generate a second case, and then a third, etc., ad infinitum. (It is all strangely reminiscent of Rosen's Life Itself which tries to understand life in terms of recursive sets.) In Holt's system, I think, a mind is analogous to the principle in a mathematical induction and the cases are "the world". So each mind is a kind of logical engine that generates a slice of the world in much the same way that a tune is an engine that generates a pattern of touches on a piano keyboard. (I am sorry; I didnt do that very well, but I had to try!) Now, one might be tempted to simply say that a mind is a function where the argument is facts about the world and the ! output is behavior. But if calling it a function would limit the values that y can take with respect to any given x (or vice versa, I can NEVER remember), then Holt might be induced to call a mind a manifold (rather than a function) to free himself of that constraint.
I dont think he speaks to the question of whether the leaves and twigs are manifolds, only to the question of whether they are the forest. (They could, after all, be manifolds WITHIN larger manifolds.) He seems to be arguing with a very strange proposition, that he attributes to idealists, that the forest IS each and every one of its parts. It sounds like an argument only a philosopher could love, but he takes it very seriously and he is still banging on about it a hundred pages later.
There is a topologist on the list (at least one) who, I am hoping, will offer at least one more definition of manifold. I say hoping, because at present, I dont understand why "set" or "metaset" is not a perfectly good definition of the non-roger definitions of manifold so far offered.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
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----- Original Message -----
Sent: 8/5/2009 5:57:03 AM
Subject: Re: [FRIAM] "manifold" in mathematics
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 <[hidden email]>
To: [hidden email]
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,
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----- Original Message -----
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 (nthompson@...)
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[Original Message]
From: Jochen Fromm <jfromm@...>
To: The Friday Morning Applied Complexity Coffee Group <friam@...>
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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