Definition of the number of dimensions of a vector space

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I understand that the definition of the number of dimensions of a vector space, but somehow that doesn't really help me with physical dimensions. How in practice do we know that our space is 3-dimensional?

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James R
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How in practice do we know that our space is 3-dimensional?

We seem to be able to specify the location of any particular object in our universe with 3 numbers.

Our space might have more than three dimensions, if speculative theories such as string theory turn out to be correct. However, any "extra" dimensions would be "curled up" in such a way that we don't notice them in our daily lives.

I was looking for that answer...because I'm a bit skeptical about how. How do we prove that we can unambiguously specify any (and every) point in space with just 3 coordinates?

asdf60 said:
I was looking for that answer...because I'm a bit skeptical about how. How do we prove that we can unambiguously specify any (and every) point in space with just 3 coordinates?
I actually think that's a really good question. It's probably because we generally assume space is homogeneous and isotropic, and since we've always been able to describe the positions of everything with three numbers we assume it works everywhere in the universe. There is no reason to believe there are more than three macroscopic spacial dimensions, so there's no reason to have a physical model that uses any number of macroscopic spacial dimensions than three. The assumption that space is homogeneous and isotropic dates back to Galileo, and so far it's proven to be a valuable postulate.

We, human, create mathematics and physics to describe the behaviour of the universe.
perhaps we can create an other way to describe the universe with extra demension.
(M-theory describes the universe with 11 demension???)

this is what I THINK.

reilly
We use three dimensions because it seems to work for everyday life. Check out a topographical map -- they always hit the nail on the head, well almost always. And, we really can't draw a 4 or higher dimensional object. Nature makes our perceptions intelligible in three (or less) dimensions; why? Who knows. (The extra dimensions of string theory are just that, theoretical concepts. )

Regards,
Reilly Atkinson

This isn't something I know much about, so perhaps there is someone here that does know about it. I read recently that we can prove we are in three dimensional space because we can tie knots. Apparantely knots are only possible in three dimensions, but like I said, I don't really know any topology or knot theory.

HallsofIvy
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reilly said:
We use three dimensions because it seems to work for everyday life. Check out a topographical map -- they always hit the nail on the head, well almost always. And, we really can't draw a 4 or higher dimensional object. Nature makes our perceptions intelligible in three (or less) dimensions; why? Who knows. (The extra dimensions of string theory are just that, theoretical concepts. )

Regards,
Reilly Atkinson
"well, almost always"!! I have a recently published map that has an entire mountain on the wrong side of a highway!

robphy
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I recall a paper by Ehrenfest that provided some classical physics arguments (stability of orbits, Huygens Principle) that suggests that space is three dimensional. I presume that
"P. Ehrenfest, Proc. Amsterdam Acad. 20, 200 (1917).
P. Ehrenfest, Ann. Physik 61, 440 (1920)."
are the references, taken from the references of Max Tegmark's paper "On the dimensionality of spacetime" http://arxiv.org/abs/gr-qc/9702052 , which I haven't read.

For more references, you might try scholar-googling
"dimensionality of spacetime"
"dimensionality of space"

pervect
Staff Emeritus
asdf60 said:
I understand that the definition of the number of dimensions of a vector space, but somehow that doesn't really help me with physical dimensions. How in practice do we know that our space is 3-dimensional?
You might want to take a look at this thread

especially the last post by Mathwonk. Come to think of it, it's short, so I'll just quote it.

esources: "dimension theory" by hurewicz and wallman, "why space has 3 dimensions" by poincare.

poincare's essay is for the general public on the notion of dimension. he says basically that he calls a finite set zero dimensional for starters. then a set is 1 dimensional if it can be separated by removing a zero dimensional set. e.g. as matt grime pointed out, R^1 is disconnected by the removal of any one point, hence is one dimensional.

R^2 is not disconnected by removing one point, but is disconnected by removing a copy of R^1 hence R^2 is two dimensional. etc etc..
There are other approaches - my personal favorite approach is the "Lebesque covering dimension". This allows one to derive the notion of dimension from the notion of "neiborhood". See the previous thread for more details.

So ultimately our notion of distance is what defines the dimensionality of space, because our notion of distance is what defines the "neighborhood" of a point, and we can determine the dimension of a space given only it's characterization as a topological space (the notion of "neihborhood").

Note that if we include time in our notion of "distance", we get a 4-d space-time, rather than a 3-d space.

There aren't any obvious candidates to extend the notion of dimensionality beyond 4. It is possible that there could be more dimensions that are "rolled up", so that they are so small they do not affect distances very much on a macroscopic scale.