logical, i.e. from all appeals to the meaning of the words occurring in the propositions, no matter whether this meaning was provided by experience or by Kant's mysterious "pure intuition". A proof is purely logical if it is valid by virtue of its form only, independently of the meaning of its terms (As the simplest example you can take the old modus barbara: if all M's are P. and if all S's are M's, then all S are P, whatever the meaning of the terms M, S, P, may be).
Now, what has become of geometry after the purification of all non-logical elements? Since all its deductions or proofs can now be carried out by some one who is not at all acquainted with the meaning of the symbols, the whole system can be considered as such, with regard only to its interior coherence and without regard to its signification. It will then no longer be a physical science — (for in a physical science all symbols must stand for physical things, or events, they must mean something) — it has become "pure" geometry, something that is of interest to the pure mathematician only, who enjoys transforming expressions into one another without caring what they express; it does not tell us anything about space any more, even if the word "space" should occur in it continually; it has lost all contact with reality; it is a frame, that frames nothing; it is mere structure without content. If no interest is taken in the application of the structure, the particular set of axioms of the systems becomes unimportant, and the mathematician can amuse himself by introducing arbitrary changes of it. This led to the invention of "non-Euclidean" geometries, which would, at first, be regarded as empty creations of the human mind, until physical applications happened to be found for some of them, for instance in connection with the theory of Relativity.
It was this pure geometry, obviously, that Bertrand Russell was thinking about when he gave his famous definition of Mathematics as the Science in which we don't know what we are talking about nor whether what we say is true. As a matter of fact, if the meaning of our symbols is disregarded, we are evidently not speaking of anything particular, and before a meaning is given to them, the question whether we are speaking truly or falsely cannot be asked. I do not think Mr. Russell would stick to his definition now; he would hardly be able to make it fit arithmetic as he himself conceives it, and it gives the wrong impression as if mathematics were really a science
