How can we be so sure the universe has 3 dimensions? I'm just a layman here so please bear with me if I don't get all the words or theories by correct name or whatever. I hope it will be clear enough what my question is. Here goes:
When I read articles or watch documentaries about Einsteins theories, M-theory, string-theory and so forth it always seems as it is common knowledge that we as objects in space stretches out in 3 dimensions: length, width and height. I get that this is very convenient on Earth when building book shelves and perhaps when calculating for space travel and such, that we plaster some sort of abstract grid onto space so we then can use geometry for calculating distances and forces and what not. What I DON'T get is how this is applicable when thinking about space in cosmology, to me it seems like the universe would need to stretch out ONTO this grid instead of BEING the grid for this way of thinking to apply.
Discussion 1:
If the universe would only be made up of two infinite small particles, it would be a 1 dimensional universe because we only have the distance between the two. If we were to add a particle it would be a 2 dimensional universe since each of the particles now have to relate to each of the other two. If we add another one we have a 3 dimensional universe, yes? If we add another particle it would be a 4 dimensional universe, and for each particle added a new relation to each of the other particles is needed, which means that we have as many dimensions as the number of particles minus 1. I guess this argument gets more complicated when adding waves and width of particles etc. But please correct me where I'm wrong.
Discussion 2:
Purely hypothetical now, if all the particles and everything that make up the universe would align on any one axis, they still have the POSSIBILITY to move away from that axis but just don't, doesn't that mean that the universe would be 2 dimensional? If not, just because of the possibility, wouldn't that mean that the above argument in discussion 1 is sound?
This has been bugging me for a while that every where I look it seems like it is something we take for granted (just because 3 dimensions is handy on a blue print), and I never ever have seen or heard an explanation on why we think the universe has 3 dimensions. Then again, could be just me..
 A: Going by the title of your question rather than the relational arguments, that Ron has addressed, we can't be sure that the universe is three dimensional. All we can do is determine the dimension experimentally, and of course this doesn't prove anything because the extra dimensions may be too small to measure.
This is pretty much covered by the question recipriversexclusion cited. That question concentrated mostly on electromagnetism, so as a complement to it I'd point out that gravity has been observed to obey the inverse square law down to about a micron. See http://arxiv.org/abs/1106.3848 for details. So if extra dimensions exist (e.g. as suggested by string theory) they must be smaller than this.
A: If one measures the distances $d_k(x)$ of a point $x$ in physical space from any $k\ge 3$ given points in space, and thinks of the results as points $d(x)$ in a k-dimensional space, one finds that upon varying $x$ the points only fill a 3-dimensional submanifold (to current experimental accuracy) of this $k$-dimensional space.
This determines the intrinsic dimension of physical space to be 3. (More dimensions may be interesting theoretically but haven't been observed.)
A: You are describing the idea of a relational space-time, so that one point particle would have 0 dimensions (because it cannot have position), two point particles would have only 1 dimension (because only their relation matters), three point particles would have 2 dimensions, and in general, N point particles would have N-1 dimensions, which describe their relative distances.
But we live in a space where the relative distances are constrained. If you have 3 points, in general they live in 2 dimensions. But if the points are A,B,C, and the distance between A and C is equal to the sum of the distances between A and B and B and C, then you can see that A,B,C are really lying on a line.
The idea of a relational spacetime than makes it seem weird that we live in 3 dimensions. This would mean that the distances between different points obey the analogous relations, which convert the N(N-1)/2 different pairwise distances between N points to order 3N parameters, which are what is required to specify the positions of particles.
This reduction seems unnatural, and you might expect that particles can "really" have arbitrary distance relations to one another, and it might just be a coincidence that the particles are lying on a three-dimensional sheet.
This point of view is difficult to incorporate naively into physics. The reason is that we don't build space-time up in a relational way in physics today. The closest thing to this is loop quantum gravity, and this program is far from complete. The space-time in traditional physics is not completely relational, in that it is a background on which everything moves.
The relational idea is interesting however.
The evidence for 3 space dimensions is then simply the statement that for every 5 particles, there is an exact algebraic relation between the 10 pairwise distances. This relation should hold exactly when the points are close together, so that space is locally flat. It is not an unjustified assumption--- it is the same assumption as saying you can only wriggle your hands in three perpendicular directions.
