cosmology - 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..