Steve, FWIW the more basic mathematical concepts that are used to define manifolds are topological space and thus open set, continuity, functions, and R^n (for an n-dimensional manifold). They are usually, but not necessarily, assumed to be Hausdorff and paracompact. Hausdorff means that distinct points are in non-intersecting open sets. For details see Baez, previously cited.
I usually forget the metaphor and think of the abstract definition. Maybe that's why I have trouble with the relationship to applications. Once Hywel and I were reading the definition and I was digging the abstractness. He said, "I see where they're going with this". I asked, "Where?" He said something like, "An electron in an energy state..." When he finished I asked, "What??" Frank Frank Frank --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Fri, May 29, 2020, 2:47 PM Steve Smith <[email protected]> wrote: > Jon - > > This is a nicely crisp and dense description which I found myself > responding to several times (inline) and having to start over, as multiple > readings (and partial responses) did help me unpack it somewhat better I > hope. If this response makes it through my internal editor, it is probably > still sloppy or incomplete. > > Frank, Steve, > > My favored approach is to say that *space is like a manifold*. > For me, space is a *thing* and a manifold is an *object*. The former > I can experience free from my models of it, I can continue to > learn facts(?) about space not derived by deduction alone > (consider Nick's posts on inductive and abductive reasoning). > I concede here that we talk about an objectified space, but > I am not intending to. I am using the term space as a place- > holder for the thing I am physically moving about in. OTOH > manifolds are fully *objectified*, they exist by virtue of their > formality. Any meaningful question *about a manifold* itself > is derived deductively from its construction. Neither in their > own right are metaphors, the metaphor is created when we > treat space *as if it were* a manifold. Just my two cents. > > Can we agree that the term "manifold" is a signifier for a mathematical > object which we have chosen to use as a formalism for describing something > we have (presumably) a more intuitive sense of? The space we "move around > in" (propriocept?) and "apprehend through action-at-a-distance" (see, hear, > grasp, feel-the-heat-from)? The mathematical construct we call a > "manifold" is built up from simpler mathematical concepts of "dimension" > and "point" and "set" "curve" and "surface" (and n-d analogs). I *think* > the distinction between intrinsic and extrinsic curvature might be the > formalism related to what I am trying to gesture at when I talk about > "apprehending" the curvature of a space directly, and why both "bent" and > "curved" space are a little dubious to me. > > I suppose your terminology of "the metaphor is created when we treat space > *as if it were* a manifold* can work for me, though I might instead say > that the source domain of the metaphorical description of "bent" or > "curved" space IS the formal mathematical construction of "a manifold"? > To say "bent" (IMO) requires an additional layer of something like a > homogenous substance with plastic (but not elastic?) deformability? > Colloquially "bent" is a fair standin for "curved" but I think only > intrinsic curvature is really meaningful in this context? > > At the beginning of MacLane's *Geometrical Mechanics,* (a book > I have held many times, but never found an inexpensive copy > to buy) MacLane opens his lecture's with '*The slogan is: Kinetic* > *energy is a Riemann metric on configuration space*'. What a baller. > > Which I think is analogous or at least similar to Guerin's "least action > paths"? And what I *think* I (imagine that I) experience in my orbital > mechanics dreams (albeit without any direct obvious intuitive grounding, > just one extrapolated from experiences like aerobatics, acrobatics, > high-diving, swimming under-water... > > This all reduces to what qualifies for a direct apprehension, a deep > grounded intuition, a (legitimate) gut-feeling? I'm beginning to suspect > that I might be the only one who has or at least needs that kind of > grounding for formalisms? > > Glen, > > I love that you mention the <placeholder>, ultimately reducing > the argument to a *snowclone*. Because the title of the thread > actually implicates a discussion of metaphor, and because I may > have missed your point about *xyz,* please allow me this question. > Do you feel that *snowclones* are necessarily templates for making > metaphors, or do you feel that a snowclone is somehow different? > > *Snowclone* (new word to me) feels a bit more to me like an "algebra of > cliche's"? Which is another hazard of "loose" metaphors... they are > prone to becoming canalized as/into cliche's? > > - Steve > > -- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . > ... ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/ >
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