574 VIII. ON THE STRUCTURE OF MATHEMATICS
with the complexity of human problems and language. For when we have understood the simplest notions, which happen to be mathematical, then only shall we be able better to understand our human problems, which are in comparison so difficult and so confused.
Any reader who has a distaste for mathematics will benefit most if he overcomes his semantic phobia and struggles through these pages, even several times. As a result of so. doing he will find it simple although not always easy. It is always semantically useful to overcome one's phobias; it liberates one from unjustified fears, feelings of inferiority, . The main point of this whole discussion is to evoke the semantic components of a living Smith, when he habitually uses the method which will be explained herewith. This method is so simple and so fundamental that in the form given by a-system and further simplified according to the gifts of the teacher, it will some day be introduced into elementary schools without technicalities, as a preventive semantic method against 'insanity', un-sanity and other nervous and semantic difficulties, and as a foundation for a training in sanity and adjustment.
Section B. On the Differential Calculus.
1. GENERAL CONSIDERATIONS
As we have already seen, the structural notion of a function is strictly connected with that of the variable. The variable on one level does not 'vary'; it is a selection by Smith of a definite value from a given set. As these processes are going on inside of the skin of Smith he might experience on a different level a feeling of 'change'. The method of dealing with such problems is given by the mathematical differential and integral calculus.
The beginnings of methods dealing with 'change' are to be found even among the ancients. Galileo, Roberval, Napier, Barrow, and others were interested in 'fluxional' methods, before Newton and Leibnitz.1 The epoch-making discoveries of the last two mathematicians consisted not only in perfecting the knowledge they had and in inventing new methods, but also and this is perhaps the most important they formulated a general theory of these structural methods and invented a new notation suitable for their purpose. The definite abandonment of the old tentative methods of integration in favour of methods in which integration is regarded as the inverse of differentiation was especially the work of Newton. Leibnitz' main work was in the field of precise formulation of simple rules for differentiation in special cases and the introduction of a very useful notation.
It is not an exaggeration to say that the calculus is one of the most inspiring, creative, structural methods in mathematics. There is little doubt that the analysis of the foundations of mathematics, and their revision, was suggested by a study of the methods of the calculus. It is structurally and semantically the 'logic' of sanity and, as such, can be given ultimately without technicalities by the present A -system and semantic training, with the aid of the Structural Differential.