<div dir="auto">Steve,<div dir="auto"><br></div><div dir="auto">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.<br></div><div dir="auto"><br></div><div dir="auto">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??"</div><div dir="auto"><br></div><div dir="auto">Frank</div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto">Frank</div><div dir="auto"><br></div><div dir="auto">Frank<br><div data-smartmail="gmail_signature" dir="auto">---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM</div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, May 29, 2020, 2:47 PM Steve Smith <<a href="mailto:sasmyth@swcp.com">sasmyth@swcp.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p>Jon -</p>
<p>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.<br>
</p>
<blockquote type="cite">
<div dir="ltr">
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">Frank,
Steve,</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br>
</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">My
favored approach is to say that <i>space is like a manifold</i>.</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">For
me, space is a <i>thing</i> and a manifold is an <i>object</i>.
The former</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">I
can experience free from my models of it, I can continue to</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">learn facts(?)
about space not derived by deduction alone</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">(consider
Nick's posts on inductive and abductive reasoning).</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">I
concede here that we talk about an objectified space, but</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">I
am not intending to. I am using the term space as a place-</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">holder
for the thing I am physically moving about in. OTOH</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">manifolds
are fully <i>objectified</i>, they exist by virtue of their</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">formality.
Any meaningful question <i>about a manifold</i> itself</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">is
derived deductively from its construction. Neither in their</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">own
right are metaphors, the metaphor is created when we</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">treat
space <i>as if it were</i> a manifold. Just my two cents.</div>
</div>
</blockquote>
<p>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. <br>
</p>
<p>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?<br>
</p>
<blockquote type="cite">
<div dir="ltr">
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">At
the beginning of MacLane's <i>Geometrical Mechanics,</i> (a
book</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">I
have held many times, but never found an inexpensive copy</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">to
buy) MacLane opens his lecture's with '<i>The slogan is:
Kinetic</i></div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><i>energy
is a Riemann metric on configuration space</i>'. What a
baller.</div>
</div>
</blockquote>
<p>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... <br>
</p>
<p>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? <br>
</p>
<blockquote type="cite">
<div dir="ltr">
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">Glen,</div>
<div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br>
</div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><span style="font-family:verdana,sans-serif">I love that you
mention the </span><font face="monospace"><placeholder></font><font face="verdana, sans-serif">, ultimately reducing</font></div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><font face="verdana, sans-serif">the argument to a <i>snowclone</i>.
Because the title of the </font><span style="font-family:verdana,sans-serif">thread</span></div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><span style="font-family:verdana,sans-serif">actually </span><span style="font-family:verdana,sans-serif">implicates a
discussion of metaphor, and </span><span style="font-family:verdana,sans-serif">because I may</span></div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><span style="font-family:verdana,sans-serif">have </span><span style="font-family:verdana,sans-serif">missed your point
about </span><i style="font-family:verdana,sans-serif">xyz,</i><span style="font-family:verdana,sans-serif"> please allow me this
question.</span></div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><span style="font-family:verdana,sans-serif">Do you feel </span><font face="verdana, sans-serif">that </font><i style="font-family:verdana,sans-serif">snowclones</i><span style="font-family:verdana,sans-serif"> are necessarily </span><span style="font-family:verdana,sans-serif">templates for making</span></div>
<div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><span style="font-family:verdana,sans-serif">metaphors, </span><span style="font-family:verdana,sans-serif">or do you feel that </span><span style="font-family:verdana,sans-serif">a snowclone is
somehow different?</span></div>
</div>
</blockquote>
<p><i>Snowclone</i> (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?</p>
<p>- Steve<br>
</p>
<br>
</div>
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