THE HISTORY OF MATHEMATICS 387 



rules of reasoning which accompanied them and became embodied into 

 a system of logic It was, however, in proceeding backward to dis- 

 cover the foundations that the whole aspect of the subject changed. 

 While many of the hypotheses were suggested by observation and were 

 known by numerous tests to be applicable to the discussion of natural 

 phenomena, it became evident that the actual hypotheses used were in- 

 dependent of the phenomena. The laws which are at the basis of 

 geometrical reasoning are not necessarily natural laws : they can equally 

 well be regarded as mere productions of the brain in the same sense 

 as we might imagine a race of intelligent beings on another planet 

 free from some of our limitations or restricted by limitations from 

 which we are free. It is seen to be the same with the rules of symbolic 

 reasoning which have gradually grown up. A geometry without 

 Euclid's axiom of parallels has been constructed perfectly consistent 

 in all its parts. This is built up of a set of axioms which constitute its 

 foundation together with a code of reasoning by which we develop the 

 consequences of the axioms. The same is true of geometries which 

 involve space in more than three dimensions. It is somewhat easier to 

 imagine symbolic developments which have their foundations different 

 from those of our school and college algebras because there is no 

 obvious connection between these rules and the phenomena of nature. 



It may be asked what limitation is there in the development of 

 mathematical theories if any set of axioms may be laid down. 

 Theoretically there is none, except that if we retain our code of reason- 

 ing about them such axioms must not be inconsistent with one another. 

 A certain sense of the fitness of things restrains mathematicians from 

 a wild overturning of the law and order Which have been established 

 in the development of mathematics, just as it restrains democracies 

 from trying experiments in government which overturn too much the 

 existing order of affairs. Changes proceed in mathematics just as in 

 politics by evolution rather than by revolution. The slowly built up 

 structure of the past is not to be lightly overturned for the sake of 

 novelty. 



The developments traced above apply mainly to the subject of 

 mathematics apart from its applications to the solution of problems 

 presented by nature. Applied mathematics is a method of reasoning 

 through symbols by which we can discover the consequences of the 

 assumed laws of nature. The symbolism which we adopt and the 

 rules we lay down with which to reason are immaterial, provided they 

 are convenient for the objects we have in view. One feature must not 

 be forgotten. We can never deduce the existence of phenomena through 

 mathematical processes which were not implicitly contained in the laws 

 of nature expressed at the outset in symbols. One cannot take out of 

 the mathematical mill any product which was not present in the raw 



