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Edward Green:
>> >Unless I am mistaken, Euclidean space has no preferred origin.
Me:
>> You are not mistaken, but many people are, and persistently make
>> the error of taste and judgment of identifying the vectorspace
>> R^n (or R(n) as Lefty writes it) with the affine space E_n, which
>> they then call Euclidean space. Of course Euclid's lines, planes,
>> and space only become affine spaces (of dimensions 1, 2, and 3,
>> respectively) upon interpretation--but they can only become
>> vectorspaces upon *mis*interpretation.
>
Lefty:
>So Euclidean space has no origin ?
Correct (to people of good taste and good judgment, like me).
>Just a well ordering
I do not think that "well ordering" means what you appear to think
it means.
>and a metric ?
No, not "Just" that--of course. It also has, among other things,
a prefered set of subsets called "straight lines" (which, as it
turns out, can be described using only the metric--by which, by
the way, *I* mean the "metric" as in "metric space"; coming from
sci.physics as you appear to do, you may want to use "metric" to
mean "Riemannian metric" but I don't, though of course again *here*
they end up being equivalent, though they are obviously not the same
because they are different sorts of things entirely).
>Sure
>about that ?
Certainly I am--always with my proviso about taste and judgment.
>According to
>http://en.wikipedia.org/wiki/Euclidean_space
>Real coordinate space together with dot product and the associated norm and
>metric is called Euclidean space often denoted by En. (Many authors refer to
>Rn itself as Euclidean space, with the Euclidean structure being
>understood). The Euclidean structure on En gives it the structure of an
>inner product space (in fact a Hilbert space), a normed vector space, and a
>metric space.
The author(s) wikipedia article betray)s( poor taste and poor judgment.
As I said, many people do. In this particular, I do not.
>The contrast I'm trying to draw is the difference between En, (or Rn), and
>the physical universe. The distinction between En and Rn is perhaps somewhat
>less signifigant in this effort - but certainly worth noting.
The contrast I'm trying to draw--because I'm a mathematician and I
am not a physicist--is between a vectorspace, which *has* an origin
as part of its defining structure, and an affine space, which *doesn't*.
Of course you, the wikipedia, Uncle Tom Cobleigh and all, are welcome
to use language as you wish. But in this case you are using it so as
to confound things, and you would do well to do otherwise.
As a physicist, you may have come across phrases like "homogeneous
space" (as a single construct: *not* just the application of the
adjective "homogeneous" to the noun "space), "principal bundle",
and so on. In those terms (as understood by mathematicians: I have
no idea how physicists understand them) one can, and should, think
of Euclidean space E_n (which has no preferred origin) as a
principal homogeneous space (i.e., a principal bundle over a
single point!) for the (Lie group with underlying) vectorspace
R^n (which does, of course, have a "preferred origin"). With
an arbitrary choice of origin, any principal homogeneous space
M of a group G automatically becomes a group itself (in natural
isomorphism with G), but it isn't a group until the origin is
chosen, and that choice can be completely arbitrary within M.
I don't expect people who have grown used to the bad usage to
correct themselves, but I'd obviously prefer that they did.
The good usage keeps "Euclidean" a bit closer to what Euclid
presumably thought he was talking about (surely you don't
think that *he* imagined his lines, planes, or space to have
non-arbitrary origins? much less that he thought they came
equipped with particular coordinate systems!), and makes a
distinction that is deeply meaningful. The bad usage encourages
sloppy thinking, stupidity, and pointless arguments.
Lee Rudolph
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