[FRIAM] Metaphor [POSSIBLE DISTRACTON FROM]: privacy games

[Frank Wimberly]

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 <sasmyth at swcp.com> 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
>
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