🚀 go-pugleaf

Toon Moene writes:

Well, ok, I know (and appreciate) that one. I, however, still don't  
understand why it is a question *whether* the manifold describing *our*
space-time structure wouldn't be embedded in some R^n. I read the  
aforementioned theorem as saying: Either you prove that your manifold  
isn't seperable, or it *is* embedded in some R^n. Whether there is a  
constructive proof of it doesn't interest me. Besides, it bothers me that  
one needs this to be able to prove or disprove that a manifold can 'bump
into itself'. This somehow contradicts your last paragraph.

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The spacetime manifold might as well be assumed to be separable.  That's
not the point.

We're getting mired in semantics here.  The mathematical theorem says
any separable manifold CAN BE embedded in some R^n.  It doesn't say it
IS embedded in some R^n.  It's also easy to prove that any separable
manifold CAN BE embedded in an infinite-dimensional Hilbert space.
That's different than saying it IS so embedded.  If you can't see the
distinction, I hope someone else can explain it to you.  I'm afraid I
won't have the patience.

More to the point, we're doing physics here, not math.  Just because our
spacetime CAN BE mathematically embedded in some R^n does not mean we
HAVE TO think of it as being so embedded, and physically it makes not a
whit of difference whether we think of it as being so embedded or not,
unless these two viewpoints give us different physical predictions.  In
bringing up the issue of whether our spacetime could "bump into itself,"
I was trying to imagine a sort of theory that might take advantage of
the idea of our spacetime being embedded in a higher-dimensional space.