That space is three dimensional is one of those propositions drilled into us as schoolchildren, and which most of us accept on faith.

Yet, since it is a statement of the real world, it can only be validated by experiment and not by a mathematical proof. Trying to contort one's fingers or sticks in various mutually perpendicular orientations isn't satisfactory since we could be limited by lack of imagination of the dexterity of our hands.

The existence of physical phenomena satisfying the inverse square laws is one of those observations which constrains the geometry of the world.