40 UNIVERSITY OF MISSOURI BULLETIN 



mention later. Just now I should insist that this is the chief 

 description of mathematics, and that those more superficial 

 signs by which we judge whether a piece of work is mathe- 

 matical or not, for example, equations, and algeBraic signs, and 

 other marks, have nothing to do with the matter. I shall point 

 out later that these are rather only the shorthand which has 

 been forced upon mathematics in order to express the im- 

 mense volume of existing mathematical knowledge. This 

 shorthand is no more mathematics than is the usual shorthand 

 to be regarded as physics or literature or law, or any other 

 subject in which a stenographer may take notes. The con- 

 fusion of the shorthand of algebra and the language which 

 we have built up in our enormous terminology, for mathe- 

 matics itself is a very popular and very misleading error. 

 Mathematics itself consists in processes which can be carried 

 out always without any such shorthand, and without this mass 

 of technical words, except that perhaps no human being woula 

 have the time for it. It is distinct from formal logic in that 

 it draws its hypotheses from experimental science and renders 

 again to experimental science its conclusions, and it is qualified 

 also by the demand that its results must be useful. I should 

 emphasize the real distinction made above : that formal logic 

 is a study of the laws of thought, while mathematics, taking 

 these laws, applies them to assumptions made in science and 

 to generalizations of the theories thus formed. 



Alliance to Science 



Referring to what I have just said, let me insist that my 

 essential restriction on the field of mathematics was its alli- 

 ance to science, that its assumptions arose in science, and that 

 its justification lay in science. I should always mention also 

 the elaboration of theories thus formed to complete them and 

 to render them satisfying to the mind. Perhaps no stronger 

 bond could possibly exist between two great branches of know- 

 ledge. Indeed mathematics typifies absolutely one phase of 



