MATHEMATICS AND TlIK WORLD 251
the names must be related by some relation-word, which, however, may be explicit or implied by the context, the situation, by established habits of speech,. The division of words into the above two classes may seem arbitrary, or to introduce an unnecessary complication through its simplicity ; yet, if we take modern knowledge into account, we cannot follow the grammatical divisions of a primitive-made language, and such a division as I have suggested above seems structurally correct in 1933.
Traditionally, mathematics was divided into two branches: one was called arithmetic, dealing with numbers; the other was called geometry, and dealt with such entities as 'line', 'surface', 'volume',. Once Descartes, lying in bed ill, watched the branches of a tree swaying under the influence of a breeze. It occurred to him that the varying distances of the branches from the horizontal and vertical window frames could be expressed by numbers representing measurements of the distances. An epoch-making step was taken: geometrical relations were expressed by numerical relations; it meant the beginning of analytical geometry and the unification and arithmetization of mathematics.
Further investigation by the pioneers Frege, Peano, Whitehead, Russell, Keyser, and others has revealed that 'number' can be expressed in 'logical terms' - a quite important discovery, provided we have a valid 'logic' and structurally correct non-el terms.
Traditionally, too, since Aristotle, and, in the opinion of the majority, even today, mathematics is considered as uniquely connected with quantity and measurement. Such a view is only partial, because there are many most important and fundamental branches of mathematics which have nothing to do with quantity or measurement - as, for instance, the theory of groups, analysis situs, projective geometry, the theory of numbers, the algebra of 'logic',.
Sometimes mathematics is spoken about as the science of relations, but obviously such a definition is too broad. If the only content of knowledge is structural, then relations, obvious, or to be discovered, are the foundation of all knowledge and of all language, as stated in the division of words given above. Such a definition as suggested would make mathematics co-extensive with all language, and this, obviously, is not the case.
Before offering a semantic definition of mathematics, I introduce a synoptical table taken from Professor Shaw's The Philosophy of Mathematics, which he calls only suggestive and 'doubtless incomplete in many ways'. I use this table because it gives a modern list of the most important mathematical terms and disciplines necessary for the purpose of this work, indicating, also, in a way, their evolution and structural interrelations.