[FRIAM] Manifold Enthusiasts

[Steven A Smith]

Nick -

I do know that reading my missives *can be* excruciatingly painful but I
do trust those without such masochistic tendencies to use their <delete>
or <next> buttons freely.

Frank -

Sorry I can't commiserate better with your physical pain... but in an
ironic reversal of roles, my pain is entirely abstract (existential
angst) while yours sounds to be entirely embodied!

- Steve

On 3/9/19 4:23 PM, Nick Thompson wrote:
>
> Sorry, everybody,
>
>  
>
> I am experiencing phantom pain in Steve’s body.
>
>  
>
> Gotta read these threads  more carefully.
>
>  
>
> Nick  
>
>  
>
> Nicholas S. Thompson
>
> Emeritus Professor of Psychology and Biology
>
> Clark University
>
> http://home.earthlink.net/~nickthompson/naturaldesigns/
>
>  
>
> *From:*Friam [mailto:friam-bounces at redfish.com] *On Behalf Of *Steven
> A Smith
> *Sent:* Saturday, March 09, 2019 4:17 PM
> *To:* The Friday Morning Applied Complexity Coffee Group
> <friam at redfish.com>
> *Subject:* Re: [FRIAM] Manifold Enthusiasts
>
>  
>
> Nick -
>
>     All I can say is, for a man in excruciating pain, you sure write
>     good.  Your response was just what I needed. 
>
> Something got crossed in the e-mails.   *I'*m not in excruciating
> pain... that would be (only/mainly/specifically) Frank, I think.  But
> thanks for the thought!
>
> Any excruciating pain I might be in would be more like existential
> angst or something... but even that I have dulled with a Saturday
> afternoon Spring sunshine, an a cocktail of loud rock music, cynicism,
> anecdotal nostalgia, and over-intellectualism.  Oh and the paint fumes
> (latex only) I've been huffing while doing some touch-up/finish work
> in my sunroom on a sunny day is also a good dulling agent.
>
>     Now, when I think of a manifold, my leetle former-english-major
>     brain thinks shroud, and the major thing about a shroud is that it
>     /covers/ something.  Now I suspect that this is an example of
>     irrelevant surplus meaning to a mathematician, right?  A
>     mathematician doesn’t give a fig for the corpse, only for the
>     properties of the shroud.  But is there a mathematics of the
>     relation between the shroud and the corpse?  And what is THAT
>     called. 
>
> Hmm... I don't know if I can answer this fully/properly but as usual,
> I'll give it a go:
>
> I think the Baez paper Carl linked to has some help for this in that. 
> I just tripped over an elaboration of a topological boundary/graph
> duality which might have been in that paper.    But to be as direct as
> I can for you, I think the two properties of /shroud/ that *are*
> relevant is *continuity* with a surplus but not always irrelevant
> meaning of *smooth*.  In another (sub?)thread about /Convex Hulls/, we
> encounter inferring a continuous surface *from* a finite point-set.  
> A physical analogy for algorithmically building that /Convex Hull/
> from a point set would be to create a physical model of the points and
> then drape or pull or shrink a continuous surface (shroud) over it.   
> Manifolds needn't be smooth (differentiable) at every point, but the
> ones we usually think of generally are.  
>
>      So, imagine the coast of Maine with all its bays, rivers and
>     fjords.  Imagine now a map of infinite resolution of that
>     coastline, etched in ink.  I assume that this is a manifold of sorts.
>
> In the abstract, I think that coastline (projected onto a plane) IS a
> 1d fractal surface (line).  To become a manifold, it needs to be
> *closed* which would imply continuing on around the entire mainland of
> the western hemisphere (unless we artificially use the non-ocean
> political boundaries of Maine to "close" it).
>
>      Now gradually back off the resolution of the map until you get
>     the kind of coastline map you would get if you stopped at the
>     Maine Turnpike booth on your way into the state and picked a
>     tourist brochure.  Now that also is a manifold of sorts, right? 
>     In my example, both are representations of the coastline, but I
>     take it that in the mathematical conception the potential
>     representational function of a “manifold” is not of interest?
>
> I think the "smoothing" caused by rendering the coastline in ink the
> width of the nib on your pen (or the 300dpi printer you are using?)
> yields a continuous (1d) surface (line) which is also smooth
> (differentiable at all points)... if you *close* it (say, take the
> coastline of an island or the entire continental western hemisphere
> (ignoring the penetration of the panama canal and excluding all of the
> other canals between bodies of water, etc. then you DO have a 1D (and
> smooth!) manifold.
>
> If you zoom out and take the surface of the earth (crust, bodies of
> liquid water, etc), then you have another manifold which is
> topologically a "sphere" until you include any and all natural
> bridges, arches, caves with multiple openings.  If you "shrink wrap"
> it  (cuz I know you want to) it becomes smooth down to the dimension
> of say "a neutrino".   To a neutrino, however, the earth is just a
> dense "vapor" that it can pass right through with very little chance
> of intersection... though a "neutrino proof" shroud (made of
> neutrino-onium?) would not allow it I suppose.
>
> This may be one of the many places Frank (and Plato) and I (and
> Aristotle) might diverge...   while I enjoy thinking about manifolds
> in the abstract,  I don't think they have any "reality" beyond being a
> useful archetype/abstraction for the myriad physically instantiated
> objects I can interact with.  Of course, the earth is too large for me
> to apprehend directly except maybe by standing way back and seeing how
> it reflects the sunlight or maybe dropping into such a deep and
> perceptive meditative state that I can experience directly the
> gravitational pull on every one of the molecules in my body by every
> molecule in the earth (though that is probably not only absurd, but
> also physically out of scale... meaning that
> body-as-collection-of-atoms might not represent my own body and that
> of the earth and I think the Schroedinger equation for the system
> circumscribing my body and the earth is a tad too complex to begin to
> solve any other way than just "exisiting" as I do at this location at
> this time on this earth.)
>
> If you haven't fallen far enough down a (fractal dimensioned?) rabbit
> hole then I offer you:
>
>     https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on
>
> Which to my reading does not answer the question, but kicks the
> (imperfectly formed, partially corroded, etc.) can on down the  (not
> quite perfectly straight/smooth) road, but DOES provide some more
> arcane verbage to decorate any attempt to explain it more deeply?
>
> - Steve
>
> PS.  To Frank or anyone else here with a more acutely mathematical
> mind/practice, I may have fumbled some details here...  feel free to
> correct them if it helps.
>
>  
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> archives back to 2003: http://friam.471366.n2.nabble.com/
> FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://redfish.com/pipermail/friam_redfish.com/attachments/20190309/122c7647/attachment.html>