Proof of only 3 macroscopic spacial dimensions is shown by the existance of knots, imagine creating one in higher dimensions, it cant be done, in fact the easiest way to gain a feel for higher dimensions (which Ill note dimensions technically do not exist in the strictest sense) is to imagine tying your shoelaces in them. Knots can only exist in 3 dimensions.
Just imagine tying your shoelace, only in 3 dimensions are the degrees of freedom the correct amount. Returning to the shoelace scenario imagine the portion where you have to cross the laces, this is not possible in 2D and in 4+ spacial Dimensions it is not possible to create a knot. Picture it, you have enough degrees of freedom where there is no unique action which can be taken that results in a knot. That is, given that many degrees of freedom, no series of motions can be taken where the creation of an entangled loop occurs, it is more or less impossible.
Note that I mean 3 spacial dimension, indeed there are 4 total, including time, nor am arguing agaisnt the possibility of N dimensional space or objects, simply given an interesting proof of the fact that we live in a 3D world.
There are signs in nature as well, in the inner ear the semi circular canal used in determing orientation and balance, are possesed of 3 interconnected specifically oriented tubular organs (or whatever) .
The eye is controled by 3 muscles,for up <-> down , left <-> right movement and a further one which deforms the lens along the z axis, the eye's shape from shifts from an ellipsoid flattened in the vertical axis to sphere to one flattened in the horizontal.
Interesting fact - Carrots > Baby Corn > Green Pea things