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?
Easy answer:How in practice do we know that our space is 3-dimensional?
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.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?
"well, almost always"!! I have a recently published map that has an entire mountain on the wrong side of a highway!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. )
You might want to take a look at this threadasdf60 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?
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.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..