November 25, 1909 



NA TURE 



carried out, it will be quite possible to obtain a system 

 that is not a mere aggregate of isolated details, but a 

 coherent structure. The importance of a practical course 

 is now generally recognised in its bearing on deductive 

 geometry ; its value, however, in relation to the apprecia- 

 tion of scientific method is equally great. 



The early stages of algebra are usually found to be 

 very difficult, and are too often of little scientific value ; 

 the' subject is more abstract than geometry, and the 

 temptation to let the teaching degenerate into a mere 

 mechanical application of rules is very great. I cannot 

 but think, however, that the spirit of De Morgan's chapter 

 on " The Study of Algebra " in his book " On the Study 

 and Difficulties of Mathematics," written so long ago as 

 1831, is in full accord with scientific method, and is 

 worthy of being more completely realised in practice than 

 it has yet been. I cannot refrain from quoting a few 

 sentences that indicate his view of the way in which a 

 reasonable conviction may be obtained. After pointing 

 out the value of maXhcmalical induction, he says : — 

 " The beginner is obliged to content himself with a less 

 rigorous species of proof though equally conclusive as 

 far as moral certainty is concerned. Unable to grasp the 

 generalisations with which the more advanced student is 

 familiar, he must satisfy himself of the truth of general 

 theorems by observing a number of particular simple 

 instances which he is able to comprehend. For example, 

 we would ask anyone who has gone over this ground 

 whether he derived more certainty as to the truth of the 

 binomial theorem from the general demonstration (if 

 indeed he was suffered to see it so early in his career), or 

 from observation of its truth in the particular cases of 

 the development of (a+fc)', (a + 6)', &c., substantiated by 

 ordinary multiplication. We believe firmly that to the 

 mass of young students general demonstrations afford no 

 conviction whatever ; and that the same may be said of 

 every species of mathematical reasoning when it is entirely 

 new." 



There can, I think, be no doubt that it is now generally 

 recognised that it is in accordance with true scientific 

 method to keep the purely deductive element in the back- 

 ground so far as the early training in mathematics is 

 concerned, and that by so doing the general methods 

 characteristic of scientific procedure are more fully illus- 

 trated. This recognition, however, does not imply that 

 the characteristically deductive side of a mathematical 

 training is to be neglected : it means rather that deduc- 

 tion, which is surely a scientific method, will be used 

 with a fuller comprehension of its place and even of its 

 necessity. The time and the manner of the passage to 

 deduction are not to be easily decided ; much depends on 

 the pupil, and it is one of the hardest tasks of the teacher 

 to determine the appropriate correlation of metliods. 

 Induction is essential as an instrument of research, but 

 deduction is also essential to the systematic development 

 of mathematical science, and no training in mathematics 

 can be considered satisfactory that does not show the 

 complete process by which mathematical knowledge 

 advances from the stage of observation to that of a science 

 in which deduction plays the principal part in the coordina- 

 tion of its contents. 



In this conception of elementary mathematics we have 

 the leading characteristics of scientific method, and have 

 them, as I think, in great simplicity. It is on this ground 

 that the study of mathematics seems to me to be a valu- 

 able, if not indeed an essential, factor of modern educa- 

 tion. Science has effected a great revolution in the 

 material conditions of life, but it has also produced a pro- 

 found change in the mental attitude of all thinking men. 

 Our civilisation is not intelligible unless account is taken 

 of the influences, material and intellectual, that are due 

 to the progress of science. The right study of mathe- 

 matics, even in its humblest forms, offers an easily 

 accessible road to the appreciation of the fundamental 

 characteristics of scientific method. 



It is of interest to note further that the more recent 

 mfthods of treating elementary mathematics, which are 

 inductive rather than deductive in their character, lead 

 in a natural manner to an appreciation of some of the 

 cardinal ideas and methods of pure mathematics. Thus 

 the notion of a continuously varying function, the con- 

 KO. 2091, VOL. 82] 



ception of a limit and the method of successive approxima- 

 tion, cannot fail to be impressed upon a pupil who has 

 been adequately disciplined in graph tracing. 



The complexity of the problems confronting modern 

 scientific research, with the vast accumulation of detail 

 so characteristic of it, demands a careful training in the 

 discrimination of the essential from the accidental, in the 

 search for the underlying principles that coordinate or 

 explain the details, and in the selection of the most general 

 points of view from which to survey the field that has 

 been worked. In this training, quite apart from the direct 

 utility of the more advanced mathematical processes, much 

 assistance is to be obtained from a mathematical course : 

 the processes of thought involved in any serious study of 

 mechanical or physical phenomena have much in common 

 with those developed in the study of mathematics. It is 

 the special task of the teacher to determine the extent to 

 which the rigorous methods of pure mathematics are to 

 be carried. Rigour is relative, not absolute, and will 

 always be conditioned by circumstances of subject and 

 person, and even by the prevailing fashions of the day. 

 Restrictions corresponding to the nature of the subject and 

 to the intellectual development of the student have always 

 been recognised as essential. Many assumptions are either 

 tacitly or explicitly made, fundamental theorems the 

 demonstration of which offers special difficulty are frankly 

 taken for granted until the necessity or the expediency of 

 their demonstration arises and the logical completeness of 

 a course is therefore impaired; but progress is all but 

 impossible on anv other lines, and much may be gained 

 from demonstrations that are in parts confessedly incom- 

 plete. The real danger to the student lies in a demonstra- 

 tion that has the appearance of being complete and yet 

 conceals serious assumptions. It is a great advantage 

 that in mathematics general theorems can often be tested 

 by particular cases that are easily handled, and practice of 

 this kind will often produce that working conviction \yhich 

 is so essential for fruitful applications. One is reminded 

 in such cases of the saying attributed to D'Alembert, " Go 

 forward and faith will come to you." 



Up to this point I have been considering the methods 

 of mathematics almost solely in relation to the function of 

 mathematics as a factor of general education or as the 

 auxiliarv of the applied sciences in their more elementary 

 stages. ' The considerations that I have thus hastily 

 sketched seem to me to involve the conclusion that this 

 phase of mathematics is to be justified neither by its use- 

 fulness alone nor by its disciplinary power alone, but by 

 the degree to which the training combines these elements. 

 In a properlv balanced mathematical course the character- 

 istic features' of scientific method will receive due recogni- 

 tion, and the mental horizon of the learner will be 

 gradually enlarged ; but the choice of material and of 

 method 'will prepare him for the application of mathe- 

 matical processes in various fields, and the study as a 

 whole will powerfully react on his mental development. 



It must not be forgotten, however, that the claims of 

 mathematics are not exhausted by such developments as I 

 have indicated. I have deliberately avoided all reference 

 to what is called pure mathematics, and have confined 

 myself to those aspects of mathematical study that are of 

 genera! interest. It is difficult for anyone who is not a 

 professed student of mathematics to realise the position of 

 the subject in its modern developments. The great critics 

 of the nineteenth centurv were not less successful in ex- 

 tending the boundaries of mathematical science than in 

 securing by a just title the territory acquired, and to-day 

 the range of subjects that fall properly within the domain 

 of mathematics has an extent that the contemporaries of 

 Newton and Leibnitz never dreamed of. As the result of 

 their labours mathematics ranks as a science worthy of 

 cultivation for the intrinsic value of the conceptions which 

 it embodies, for the appeal it makes to the constructive 

 imagination, for the light it casts on the processes of 

 thought, and for the inherent beauty of form_ that 

 characterises manv of the theories comprised within its 

 domain ; but any 'attempt at reviewing, within the limits 

 of time allotted to me, the present state of the science 

 would certainlv fail to give any adequate conception of the 

 nature of its contents. To the mathematical student, how- 

 ever, the assurance can be given that he need not fear 



