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Jon -<br>
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style="font-size:small;color:rgb(51,51,51)"><span
style="color:rgb(0,0,0);font-size:13.44px"><font
face="monospace"><i>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"? </i></font></span><br>
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style="color:rgb(0,0,0);font-family:Verdana,Geneva,Helvetica,Arial,sans-serif;font-size:13.44px"><br>
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<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">What
about Eddington's measurement or gravitational lensing? These
both</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">appear
situated in a phenomenological domain, and so we seem to have</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">another
candidate domain for talking meaningfully about <i>bent</i>
or <i>curved</i> space.</div>
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I guess I'm not saying that nobody (in this case Eddington and many
others after him) have not experimentally observed light failing to
follow the Euclidean "straight line" we intuitively? expect to
see? I agree that we don't have any (much?) direct experience
otherwise (mirage, underwater with density/temp flux?). I'm still
harping on "bent" as it implies that something was "meant to be
straight is deformed by some process or force"?. <br>
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<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333"> Still,
I suspect I am missing something important in</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">your
emphasis on <i>apprehension.</i> Can you say a bit more about
what you mean?</div>
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I'm not sure if it is important... or obvious... or not. I am using
"apprehension" to be as broad of a word/concept as possible to
describe "taking something in", it could range from observing to
experiencing to measuring (and fitting to an apt model?) to
"intuiting" to <i>grokking</i>? I guess the key is being
convinced that what you are "taking in" is real, even of course, if
you are wrong. This might include Dave's conversations with Joseph
Smith, though that would be on one extreme.<br>
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<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">you
write:</div>
<div class="gmail_default"
style="font-size:small;color:rgb(51,51,51)"><span
style="color:rgb(0,0,0);font-size:13.44px"><font
face="monospace"><i>Which I think is analogous or at least
similar to Guerin's "least action paths"?</i></font></span><br>
</div>
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</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">Yeah,
I suspect so too. MacLane's book intentionally focuses on
developing</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">Lagrangian
and Hamiltonian mechanics. Mechanics, as far as I am
concerned,</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">is
the prototypical home for these ideas.</div>
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At least from an embodied, motile creature's perspective, it seems
like it would be? Not that I can imagine being anything else
really? My orbital mechanics dreams are just the latest variant on
a whole suite of dreams I've had as far as I can remember and they
all involve various forms of locomotion/navigation through space
(flying, levitating, soaring, running, jumping, seven-leaguing,
swimming, swinging from branches, spidermanning, surfing, skiing,
skating, tumbling, etc.).<br>
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<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">You
write:</div>
<div class="gmail_default"
style="font-size:small;color:rgb(51,51,51)"><span
style="color:rgb(0,0,0);font-size:13.44px"><font
face="monospace"><i>feels a bit more to me like an
"algebra of cliche's"?</i></font></span><br>
</div>
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<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">Thanks
for that. Upon further reflection, I completely agree with
you.</div>
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And thank you for "snowclone" between you and Glen, I am feeling
hopelessly out of date on modern terminology. I feel like my
grandfather (born 1890ish) when he came to live with us in the 60's,
and it felt like he had to have explained every other term that I
didn't even recognize as slang! ("hey man", "cool").<br>
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†) At the unintended risk of moving the conversation into the
<i>meta</i>††,</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">I
am including a link here to a page <a
href="https://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks#TheBasicIdeaOfSheaves"
moz-do-not-send="true">motivating the development of sheaves</a>.</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">In
section 2 the author invents a game where he thinks up a space
and the</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">player
can query the author about how other spaces map into it.<br>
<br>
†† <i>Meta</i> in that sheaves themselves offer a more
flexible paradigm for</div>
<div class="gmail_default"
style="font-family:verdana,sans-serif;font-size:small;color:#333333">reasoning
about generalized spaces than we get from manifold theory.</div>
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<br>
<p>I'm gonna have to give this a lot more attention than the quick
read I tried just now!</p>
<p>I'm barely versed enough in CT to *read* it with a lexicon in my
hand.</p>
<p>- Steve<br>
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