🚀 go-pugleaf

In article <1992Nov5.060400.14203@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>Ah, now this is starting to sound very interesting and helpful.  What I
>don't understand here is how "living in R^n" is of itself creating
>complexity and obscurity.  I can see that the arbitrariness of f might
>get in the way.  But where does the complexity and obscurity creep in
>if for example we define a manifold to be a smooth retract of an open
>subset of R^n?  (This is essentially taking the existence of tubular
>neighborhoods as definitive of manifolds, and is how Bill Lawvere likes
>to think of them.)

Are you just playing devil's advocate here?  The devil has enough
advocates already and doesn't need your help.  :-)  But anyway, one
point is that many nice ways of constructing new manifolds from old ones
are a pain in the butt to describe if your manifolds all have to be
living in some R^n.  E.g. - constructing manifolds by surgery (cutting
and pasting).  Just 'cause each of the pieces live in some R^n doesn't
mean that when you stick 'em together you get something that can be
embedded in R^n (for the same n).  You can use a larger value of n, but
you should soon get the feeling, "why am I bothering?  What's so great
about living in R^n, anyway?"  

When you tell folks spacetime is curved they always want at first to
visualize it as sitting inside some "hyperspace," an R^n of higher
dimension.  This is a blatant violation of Occam's razor and leads to
silly questions about using hyperspace as a tricky form of rapid
transportation.  (Note: I'm not saying that one should never consider
these questions - just that, upon thought, it turns out that they are
unproductive.)  The fact is, we have no strong evidence that we are
living in R^3, or R^4 -- all we know is that it looks *locally* like R^3
or R^4, so we might as well face up to the intrinsic notion of manifolds.