130 PART II. SOME IMPORTANT METHODOLOGICAL TERMS. 



striving to frame incontrovertible definitions, and when the 

 anterior inductive enquiry is compressed into a definition. 

 Should these epitomised statements, however, be excessively 

 difficult to reach, then definitions as rigid as possible should 

 be the goal of scientific endeavour. 



54. (D) DEFINITENESS IN SCIENTIFIC WORK GE- 

 NERALLY. This leads us naturally to recognise the need for 

 definiteness generally, leaving no more of the activities in a 

 fog or undetermined than is inevitable. Such an attitude of 

 mind will prick any number of iridescent bubbles which un- 

 warrantably arrest our untrained attention; it will secure that 

 we do not pass to the second stage before the first stage is 

 completed; it will circumvent mountains of errors; and it will 

 ensure a rapid and safe advance. 



In a word, a true methodology demands definiteness (a) in 

 terms, (b) in general statements, and (c) in work generally, 

 requiring mathematical treatment wherever practicable. Pro- 

 cedure may be said to be exact when it is mathematical, and 

 when definiteness is aimed at in terms, in statements, and in 

 work generally. 



55. (E) MATHEMATICAL AND NON-MATHEMATICAL 

 PROCEDURE. To conclude. In view of the opinion which 

 widely obtains that mathematics is separated, as it were, by a 

 gulf from the inductive sciences through its abstractness, its 

 irresistible demonstrations, and its mode of procedure, it is 

 interesting to find that, of recent years, three mathematicians 

 have sought to show that no such breach exists. The last of 

 the three utterances is that of Prof. E. W. Hobson, F.R.S., and 

 is contained in his Presidential Address to the Mathematical 

 and Physical Science Section of the British Association in 1910. 

 We shall let him speak: 



"In the first place, it is a fact that frequently, and at various times, 

 differences of opinion have existed among mathematicians, giving rise to 

 controversies as to the validity of whole lines of reasoning aad affecting 

 the results of such reasoning; a considerable amount of difference of 

 opinion of this character exists among mathematicians at the present time. 

 In the second place, the accepted standard of rigour, that is, the standard 

 of what is deemed necessary to constitute a valid demonstration, has 

 undergone change in the course of time. Much of the reasoning which 

 was formerly regarded as satisfactory and irrefutable is now regarded as 

 insufficient to establish the results which it was employed to demonstrate. 

 It has even been shown that results which were once supposed to have 

 been fully established by demonstrations are, in point of fact, affected 

 with error." (British Association's Report of 1910, p. 514.) "That oldest 

 text-book of science in the world, Euclid's Elements of Geometry, has 

 been popularly held for centuries to be the very model of deductive 

 logical demonstration. Criticism has, however, largely invalidated this 

 view." (Ibid., p. 516.) "The actual evolution of mathematical theories 

 proceeds by a process of induction strictly analogous to the method of 

 induction employed in building up the physical sciences : observation, 

 comparison, classification, trial, and generalisation are essential in both 

 cases." (Ibid., p. 520.) 



