Note 130 — THE MONIST. (continued 3)

Mr. Peirce correctly says that the axioms of geometry are now exploded. This, however, does not overthrow the reliability of formal mathematics; on the contrary, it places it on a safer basis than that of unprovable assumptions, which must be taken for granted.

We look upon the whole system of geometry as a product of mental operations. We perform some operations and note what their products are. We do something and mind the consequences of what we do. The problem of modern geometry is to invent a method by which we can construct in the simplest manner possible a straight line and a plane. Euclid still presupposes the existence of the plane and assumes it to be such that parallel lines do not meet. When we are able to construct the plane of Euclidean geometry, we can dispense with the axiom of parallels, for, in that case, the plane will possess the qualities it has by construction. We can very well execute other constructions in which parallel lines possess other qualities, and we shall on the basis of such an altered plan of operation be able to produce entirely different systems of geometry.

We must distinguish between the space of our mathematicians and real space. Experience teaches us that real space has three dimensions which means that from a given point every other point is determinable by three magnitudes. We might doubt (although I think there is little occasion to do so) whether the real space of our experience is truly three-dimensional, but we cannot doubt that the truths developed in the one-dimensional system of numbers, in the two-dimensional system of plane geometry, in the three-dimensional system of solid geometry, and also in n-dimensional systems each in their respective domain are perfectly reliable, for they are unequivocally determined, they are eindeutig bestimmt. There is no application of the theory of probabilities in a field where the products are not due to chance but result with certainty.