SECTION 16. DEFINITE, EXACT, AND MATHEMATICAL PROCEDURE. 123 



however colourable a deduction may appear to be, it should 

 be scrupulously verified above all, needless to say, in matters 

 of life and death. The time should have passed when men, 

 in particular scholars and propagandists, are satisfied with a 

 collection, small or large, of affirmative instances or pretentious 

 deductions. We should insist that he who claims to be a 

 specialist, should, as an elementary duty, rigorously verify his 

 observations, generalisations, and deductions. 



SECTION XVI. DEFINITE, EXACT, AND MATHEMATICAL PRO- 

 CEDURE. 



50. (A) THE CASE FOR MATHEMATICAL PROCE- 

 DURE. If it be queried why highly developed sciences should 

 tend to assume mathematical form, and why complete absence 

 of mathematical apparatus argues crudity in any sphere of 

 knowledge, the answer is near at hand. The sciences ruled by 

 mathematics appear to be the only exact sciences, and accord- 

 ingly every science, since it strives to be exact, must needs 

 strive to be mathematical. 1 Out in the world of practice, ideas 

 are protean in character: a new meaning develops out of an 

 old one because a new need has arisen, and this meaning 

 gradually passes, for the same reason, from one shape to 

 another, a single word representing the multifarious meanings. 

 Analyse the terms morality, duty, virtue, ought,- for instance, 

 and observe how they alter in signification from age to age, 

 from country to country, and, to some extent, from individual to 

 individual, and even from occasion to occasion. These terms, 

 which are among our current ethical coin, symbolise countless 

 attitudes and actions, and this is approximately true of words 

 in general. Now mathematical procedure rescues us almost 

 totally from this vertiginous chaos. It measures phenomena, 

 and reduces data to a form which is as inflexible and universal 

 as the every-day terminology is accommodating and individual. 

 Once we have reached the mathematical level, it is no longer 

 necessary to define by indefinable terms, or explain by offering 

 equivocal illustrations. We are able then to make statements 

 which it is practically impossible to misconstrue, and which 

 therefore convey precisely the same signification to all persons 

 alike. Hence, perhaps, no science can be said to be fully estab- 

 lished so long as it is entirely or even fractionally non-mathe- 

 matical. 



1 "Inquiries into nature have the best result, when they begin with physics 

 and end in mathematics." (Bacon, Novum Organum, bk. 2, 8.) 



- "The word represented by 'cause' has sixty-four meanings in Plato and 

 forty-eight in Aristotle. These were men who liked to know as near as might 

 be what they meant; but how many meanings it has had in the writings 

 of the myriads of people who have not tried to know what they meant by 

 it will, I hope, never be counted." (W. K. Clifford, Lectures and Essays, 

 ed. 1918, p. 35.) 



