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
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