Fermat's Last Theorem solvable in a higher dimension?

x^n+y^n=z^n

n can be 1 and 2. Proved to not exist for n>2.

But why. I start to think of dimensions. We live in a 3d world but can only see 2d truelly and then our brains works out depth and distance to objects, but we dont truelly see 3d objects like we do 2d objects. A 2d object you will see it fully and get all information from it. Look at a ball and you only see one side and not whats on the inside.. So we do not see in 3d but instead we see 2d pictures of 3d objects.

So we can truelly see a square (x^2) and all information it has to give, but we can not truelly see a cube (x^3).

Is this why we dont "see" more solutions to the theorem? If we were 4d creatures would our mathematic system be different so we could solve the theorem when n=3?