Spencer Heath Archive

Item 403

Random taping by Spencer MacCallum during conversation with Heath in New York City.

June 23, 1958

     What about non-Euclidean geometries? Euclid took a mathematical chess board (he didn’t have mathematics, but in effect he did. I’ll say!)

Euclidean geometry is like a chess board on which a game is played by a prescribed set of rules called axioms, or premises. The Euclidean premises are arbitrary, not drawn from experience, though they have a certain rational validity. The only reason Euclid did not have many kinds of (non-Euclidean) geometries is because he did not use many different sets of premises. When other geometricians choose other premises and build upon them rational structures, as Euclid did on his, they call these new rational structures non-Euclidean.  A common characteristic of geometries, including that of Euclid, is that they are not built upon premises taken from the objective world of experience by the human sensory apparatus (with or without the aid of instruments—instruments between the sense receptors and the objective world being merely extensions of that world).

A hearing aid or a pair of glasses is not an extension of the sensory apparatus but an extension of the objective world you are looking at.