"manifold" in mathematics
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I wonder if anybody has any comment to make on the following passage from EB holt? (Remember, I am the guy who tends to ask questions of PEOPLE when he should look them up, so feel free to ignore me here.)
Holt (1914) writes: "If one is walking in the woods, and remarks that "All this is Epping Forest," one may mean that this entire manifold of some square miles is the forest; or else, that every twig and leaf which one sees, in short, every least fragment of the whole is Epping Forest. The former meaning is the true one; the latter meaning is absolutely false. Everyone admits that while a circle is a manifold of points, a single point is not a circle; while a house is a manifold of bricks, boards and nails and any single brick is not a house. "
I am interested in this concept of "manifold" . Can anybody make the metaphor come alive for me? Is it like a shroud?
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
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This is an interesting use of "manifold".
In mathematics, a manifold has a well-defined meaning. The structure of a manifold is encoded by a collection of charts that form an atlas. A chart is a mapping between the manifold and a simple (euclidean) space, see http://en.wikipedia.org/wiki/Manifold If science is a manifold, then each science - physics, chemistry, biology, psychology etc. - is a chart, and all sciences together form the atlas of the manifold. Sunny Y. Auyang makes this point in her book "Foundations of complex-system theories". If understanding is a manifold, then each analogy is a chart, and all analogies together form the atlas of the manifold, see http://is.gd/22u44 Holt uses the term manifold in the sense of an emergent property: a forest is an emergent entity made of trees, plants, soil, etc. The form of plants and trees can be described by fractal patterns. There are clearings and trails in the forest which can be described by patterns as well. Thus one can perhaps say if an emergent entity is a manifold, then each pattern is a chart, and all patterns together form the atlas of the manifold. This pattern could be a simple physical pattern, or a basic agent interaction pattern (http://is.gd/22uZ6) for an agent based system, for instance. -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 |
Re: "manifold" in mathematics
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In reply to this post by Nick Thompson
A manifold is something that can't be a function because it is multi-valued where a function must be single-valued.
A circle, the set of points which satisfy the equation x^2 + y^2 = r^2, is a manifold of points because there are two values of y that satisfy the equation for each value of x, -r < x < r. If we restricted ourselves to y >= 0 (or to y <= 0) then we would get a set of points which is a function of x. -- rec -- On Tue, Aug 4, 2009 at 11:12 AM, Nicholas Thompson <[hidden email]> wrote:
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In reply to this post by Nick Thompson
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 |
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In reply to this post by Nick Thompson
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). 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. > > ============================================================ > 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 |
Re: "manifold" in mathematics
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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. > > > > ============================================================ > > 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 [hidden email] 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 |
Re: "manifold" in mathematics
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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 <[hidden email]> wrote:
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Re: "manifold" in mathematics
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In reply to this post by Russell Standish
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: ============================================================ 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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In reply to this post by Nick Thompson
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,
Clark University ([hidden email])
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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,
Clark University ([hidden email])
============================================================ ============================================================ 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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Of course, it would be unreasonable to expect Holt to conform to the modern
definition of "manifold", as formalized mathematics has changed quite a bit in
the last 100 years. While it is not quite as conceptually elaborate as it could
be, our friend wikipedia has a bit on the history of the term
(http://en.wikipedia.org/wiki/History_of_manifolds_and_varieties). I suspect
Holt uses the term manifold to refer to a description of the intrinsic
properties of complex surfaces. Presumably an entire forest can be thought of
as a complex surface...? And the part of the forest I am responding to could be
another more or less complexly described surface...? etc., etc. Does anyone
know anything more about historic uses of the term?
Eric ============================================================ 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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In reply to this post by Nick Thompson
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,
Clark University ([hidden email])
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In reply to this post by QEF@aol.com
Clay, et al. - I would submit that the term (and especially concept) of manifold significantly predates the mathematical use. This concept and usage would seem to have provided the motivation for the mathematical term itself. It also might provide a little more leeway in expanding the application to reference or accommodate self-similarity in some way, as you suggest. From: English Etymology Dictionary manifold O.E. monigfald (Anglian), manigfeald (W.Saxon), "varied in appearance," from manig "many" + -feald "fold." A common Gmc. compound (cf. O.Fris. manichfald, M.Du. menichvout, Swed. m?ngfalt, Goth. managfal?¢®s), perhaps a loan-translation of L. multiplex (see multiply). And from: Webster's Revised Unabridged Dictionary (1913) Manifold \Man"i*fold\, a. [AS. manigfeald. See Many, and Fold.] 1. Various in kind or quality; many in number; numerous; multiplied; complicated. O Lord, how manifold are thy works! --Ps. civ. 24. I know your manifold transgressions. --Amos v. 12. 2. Exhibited at divers times or in various ways; -- used to qualify nouns in the singular number. ``The manifold wisdom of God.'' --Eph. iii. 10. ``The manifold grace of God.'' --1 Pet. iv. 10. My working definition (sharpened in the process of this discussion, thank you Nick!) is to "be many and varied in nature" with some alternate but related working definitions. "a mechanical multiplexer, as in a gas manifold such as used for intake fuel mixture and exhaust gas in an internal combustion engine." This usage seems to reference the complex topology requiring something of a patching together of various tube-shaped surfaces". and "multi-fold, in the sense perhaps of complex origami pieces or even the simplicity of carbon-paper copying... or multifold writing". As for your suggestion... I submit that for my purposes, geometric artifacts exhibiting self-similarity (often fractals) do in fact suggest a valuable variation on the use of "manifold". One might say that such artifacts have sub-components which are "many and varied in nature", though the definition of "varied" is challenged a bit by the self-similarity. They are clearly varied (being similar, not identical) yet, are not-so-varied (being similar, not different). Fractals (and other self-similar geometries) embedded as surfaces in 3 dimensional spaces do strike me intuitively as being "barely" or "almost" a manifold. They are "barely" in the sense (again) of their subcomponents (regions) being many and vairied but "almost" in the sense of not being as diverse as I would normally want. A *real-world* self-similar system such as your referenced _forest of trees of branches of twigs with leaves having at least a few levels of self-similar structure_ would seem to be very apt and perhaps where some of the earliest referenced usage of the *concept* ("O Lord, how manifold are thy works!") arose. <nostalgic anecdote> My first encounter with the term was on the 390 CI engine of my 1964 Ford Thunderbird in High School. I confronted my learned (aka geek-nerd) friend whose father was an engineer and who bent and welded up his own headers for the ford 289 CI in his 1963 Ford Fairlane coupe with: "what is the difference between a manifold and a header?". His response was somewhat like Doug's or Owen's to many of Nick's questions here, but I persisted until his father overheard the conversation and weighed in himself. As an engineer he was more interested in the practical properties rather than what I was seeking of the "essential" properties and proceeded to explain to me all about pulse-tuning of the exhaust system, how impedence worked in complex, dynamic systems of compressible fluids (exhaust gas in particular) down to how the cooling of the exhaust gasses in the exhaust system effected the pressure/volume and how that should be accounted for in the tuning even. Once we got past the pragmatics, I was able to engage him in my *real* question which was why a welded-up header was not also a manifold (even then I knew how to read a dictionary and had at least been introduced to non-euclidean geometry and topological spaces). What we settled on (so that my friend's mother didn't kill us all for holding up dinner with this long-winded conversation) was quite obvious, as the headers were an array of exhaust pipes connected for convenience by a single flange for bolting up in place of the manifold. We agreed that perhaps the *other end* of the header-system where they all joined to a single exhaust pipe (well 4-1 for each of 8 cylinders and 2 exhaust pipes) was a bit of a "manifold" in the original sense but that since the joints were roughly non-differentiable, not really. They were a "multiplex" of 2D surfaces joined in a complex way, but not (quite) a manifold in the more strict sense. </nostalgic anecdote> I shall now return to my manifold deadlines and projects (all joined together in a singular and continuously differentiable high-dimensional artifact embedded in a higher dimensional space) - Steve PS. If you read this far Gattiker, drop me an e-mail! ============================================================ 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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In reply to this post by Nick Thompson
Hi Nick.
This is an interesting discussion on the concept of Manifolds. But I sense you have a wider interest than just this one word, right? My guess is one of the most important books in this field is Spivak's Calculus on Manifolds. Thus Spivak's interest was in the unification of ideas across several areas of mathematics. He succeeded brilliantly. Where is your interest? History/Philosophy of Mathematics? Fascination with words and language? As a component of CAS? As a core mathematical theme to be mastered, somewhat like the epsilon/ delta concept of the Limit? -- Owen On Aug 4, 2009, at 11:12 AM, Nicholas Thompson wrote: > I wonder if anybody has any comment to make on the following passage > from EB holt? (Remember, I am the guy who tends to ask questions of > PEOPLE when he should look them up, so feel free to ignore me here.) > > Holt (1914) writes: "If one is walking in the woods, and remarks > that "All this is Epping Forest," one may mean that this entire > manifold of some square miles is the forest; or else, that every > twig and leaf which one sees, in short, every least fragment of the > whole is Epping Forest. The former meaning is the true one; the > latter meaning is absolutely false. Everyone admits that while a > circle is a manifold of points, a single point is not a circle; > while a house is a manifold of bricks, boards and nails and any > single brick is not a house. " > > I am interested in this concept of "manifold" . Can anybody make > the metaphor come alive for me? Is it like a shroud? > > Nick ============================================================ 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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Owen brings up an interesting (and important in it's relevance?) question...
> > > Where is your interest? History/Philosophy of Mathematics? > Fascination with words and language? As a component of CAS? As a > core mathematical theme to be mastered, somewhat like the > epsilon/delta concept of the Limit? What might the relevance of manifolds (in the strict mathematical sense) be to CAS? ABMs on arbitrary manifolds anyone? In 1983, I had the good fortune of hanging out at the Cellular Automata conference with the likes of Farmer, Packard, Crutchfield, Wolfram, Langton, Fredkin, Toffoli, Gosper and even John H. Conway (and his wife and two young children who also attended). The agenda and topics covered were manifold (I couldn't resist). At the time, CAs were pretty much the only/best example of what we call ABMs today. Despite the varied and diverse approaches to CA, the one area that was not covered (but was very interesting to me) was how CA's behaviour might be qualitatively different if embedded in higher dimensions, in non-euclidean spaces and more interestingly in more complex topologies. Planes, Tori, and Spheres were the only topologies of interest it seemed and all were considered to be locally euclidean. I was struggling with formally specifying (so that I might implement) something I called a "Network Automata" which today would roughly be a formalization of ABM's in general, and was trying to understand the relationship between the spatial embedding and the topological embedding of these automata. Wolfram was struggling to classify all 1D, 2 state, CA into 4 classes while Farmer (I think) was trying to demonstrate that higher D, numerous State CA could be emulated in 1D, 2 State CA (making Wolfram's work applicable across arbitrary spatial dimension and (discrete) state space). I fortuitously met an anesthesiologist who had postulated that some part of consciousness arose out of the electro-mechanical properties of microtubules in the neurons of the brain (and rest of the nervous system). I was (apparently) the only one at the conference who thought that the embedding space of a CA was interesting (despite some pretty wide interests at that time in that domain). To that end, I struck up a collaboration and proceeded to build a CA embedded in a 13 unit, 6-off helical space (the way "normal" microtubules are arranged) for him. The results were not that interesting in many respects, but we *did* manage to demonstrate that information processing could occur in that "small" of a space. We did not demonstrate anything more interesting than information (soliton or glider-like) propogation along the main axis (and soliton-anti-soliton anhillation). The constriction of one axis (13 units) seemed to prevent any interesting lateral propogation (the head of one propogating structure would encounter it's own tail, or more aptly, the hip encountering the opposite shoulder, do to the helix). Sadly this work is over the aft (redshifted into near invisibility) information horizon and exists only on paper (archaic concept?)... I have a scanned PDF of it somewhere if anyone might be entertained by trying to read fuzzed-out grey words originally mastered on a typewriter (yup, those were the days!) with many hand-drawn figures. The recent ABM art-piece that Ilan and Ben (and Stephen) built for a show at the Complex began to approach the same question, though technically, their ABMs were confined to a complex 2D geometric region in euclidean space. I believe that their original intent was to model the complex surface (topologically a 3 hole torus?) with it's geometric (polygonal cross-sections) properties to constrain and inform the ABMs directly. Surely there has been some work in the area of ABMs and their ilk embedded in more complex manifolds than the euclidean plane? Any references, insights, ideas? - Steve > > -- Owen > > > On Aug 4, 2009, at 11:12 AM, Nicholas Thompson wrote: > >> I wonder if anybody has any comment to make on the following passage >> from EB holt? (Remember, I am the guy who tends to ask questions of >> PEOPLE when he should look them up, so feel free to ignore me here.) >> >> Holt (1914) writes: "If one is walking in the woods, and remarks >> that "All this is Epping Forest," one may mean that this entire >> manifold of some square miles is the forest; or else, that every twig >> and leaf which one sees, in short, every least fragment of the whole >> is Epping Forest. The former meaning is the true one; the latter >> meaning is absolutely false. Everyone admits that while a circle is a >> manifold of points, a single point is not a circle; while a house is >> a manifold of bricks, boards and nails and any single brick is not a >> house. " >> >> I am interested in this concept of "manifold" . Can anybody make the >> metaphor come alive for me? Is it like a shroud? >> >> Nick > > ============================================================ > 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 |
Re: "manifold" in mathematics
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In reply to this post by Nick Thompson
On Wed, Aug 05, 2009 at 10:37:58AM -0600, Nicholas Thompson wrote:
> 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. > My suggestion that the patches need to be continuous immediately rules out arbitrary (embedded) sets from being manifolds. As I said, I find it hard to conceive that a mathematician would call the Cantor set a manifold. It sounds like a tortuous abuse of language. It might even be that the patches need to be diffeomorphic, aside from a set of measure zero. This would allow the surface of a cube to be a manifold, but not say the boundary of the Mandelbrot set. Note that the only manifolds I ever studied were smooth manifolds (ie surface of a cube is not a smooth manifold). It seems Wikipedia only considers smooth manifolds too: http://en.wikipedia.org/wiki/Manifold But then the article http://en.wikipedia.org/wiki/Categories_of_manifolds explictly generalised the concept of smooth manifold (eg piecewise linear, topological, etc). It seems the concept is that for every point on the manifold, there is a neighbourhood N that is homeomorphic to to a Euclidean space R^n. Homeomorphic means the map f:N->R^n is continuous, but also its inverse f^{-1}:R^n->N. Cheers -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [hidden email] 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 |
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In reply to this post by Nick Thompson
To anybody following the Holt thread:
In an offline communication, Eric Charles has pointed out that I misspoke when I wrote:
"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."
In Holt's view, "my" consciousness is the slice of the world implied by my behavior. So mind is more like the manifold itself than it is like the mechanism (the brain?) that generates the manifold.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
============================================================ 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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In reply to this post by Nick Thompson
Lee,
You do seem to be on the list. If everyone apologized for nearly content-free posts, the number of nearly content-free posts would be greatly multiplied. The word manifold is certainly not being used in its mathematical sense when Epping Forest is called a manifold. I think it means a collective noun there, so that "all this" means the collection of all this, not the individual components of all this. Still, I would be interested in what you have to say. --John ________________________________________ From: [hidden email] [[hidden email]] Sent: Wednesday, August 05, 2009 12:49 PM To: [hidden email]; [hidden email] Cc: John Kennison Subject: Re: [FRIAM] "manifold" in mathematics On 5 Aug 2009 at 10:37, Nicholas Thompson wrote: > There is a topologist on the list (at least one) Before attempting a substantive reply to this post, I am going to try posting to the list to see if I *am* indeed on the list. John, do you know for sure that you are on the list? If not, you might try the same... (Apologies to all if, in fact, I am on the list, and everyone gets this nearly content-free post.) ============================================================ 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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