the propositions within the system, it is obvions that he does not have to think of the meaning of his symbols at all; it does not make the slightest difference to his calculations what they signify, he is concerned with them only in so far as they satisfy the axioms of the system, or, in his mathematical language, that certain equations hold between them. This is absolutely everything he has to know, and nothing else can possibly enter into the system of theoretical physics, as it appears in any scientific paper or test book.
This state of affairs has first been clearly recognized with respect to Geometry, if by this word we mean the science of Space, expressing certain truths about points, plains, straight lines etc. in physical space, is not a branch of pure mathematics, but forms part of physics.
This was already seen by Newton, who declared it to be "the most general part of mechanics". The first representation of geometry as a coherent system is due to Euclid who already gave to it the classical form of a set of axioms from which all other geometrical propositions are derived. The derivation of a proposition from the axioms is called the proof of the proposition. Closer inspection of Euclid's proofs soon reveals the fact that they are by no means purely logical derivations, but consist of a mixture of logical deductions and appeals to drawings or the observation of the behaviour of rules and compasses. Drawings, rules, and compasses are physical objects, and an appeal to their observation is really an appeal to experience. Philosophers who did not wish geometrical truths to be based on brute facts of experience have denied this and have maintained that drawings etc. are not really the source of geometrical knowledge, but only artificial representations of some original "pure intuition" that precedes all experience and is independant of it. This doctrine (most vigorously advanced by Kant) encounters insuperable difficulties, but this is not the place to criticize it, in any case, it just tried to secure and indicate a situation which to Euclid, if he had been fully aware of it, would have seemed very deplorable and needing correction: namely, that the proofs of his propositions were not of a purely logical nature. The mathematicians (who have always been the most ardent and most scrupulous logicians in the world) were very much troubled and dissatisfied and set to work in order to purge all geometrical proofs from everything that was not purely
