Note 159 — Introductory material

So much, then, is à priori necessary to metrical Geometry. The restriction of the dimensions to three seems, on the contrary, to be wholly the work of experience[166]. This restriction cannot be logically necessary, for as soon as we have formulated any analytical system, it appears wholly arbitrary. Why, we are driven to ask, cannot we add a fourth coordinate to our x, y, z, or give a geometrical meaning to x4? In this more special form, we are tempted to regard the axiom of dimensions, like the number of inhabitants of a town, as a purely statistical fact, with no greater necessity than such facts have.