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<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>
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cite="mid:CAH5Jek2q-O8G3c=vi1mqQ_AZKf2ZCQiXZfzuC8UKnFpN18-Wtg@mail.gmail.com">
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<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>
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<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>
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<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>
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