I lO 



NA TURE 



[November 25, 1909 



■solid foundation for the operations of mathematics, of 

 geometry as well as of analysis. 



The movement, however, was not without its dis- 

 advantages. Mathematics gradually became more and 

 more abstract, and the relations of mathematics to the 

 applied sciences tended to fall into the background. On 

 one hand it was manifestly impossible for the physicist 

 and tTie engineer to keep themselves abreast of the de\'elop- 

 ments of pure mathematics; on the other, the rapid 

 extension of physics and engineering made it difficult for 

 the mathematician, even when he had the desire, to under- 

 stand the problems in the investigation of which mathe- 

 matics might have been useful. The mathematics of the 

 secondary schools was not affected to any considerable 

 e.xtent by the critical movement, but it probably became 

 more formal and lost contact with the applied sciences. 



Towards the close of the century complaints were rife 

 especially among the engineering dommunitv, that mathe- 

 matics had lost touch with realitv, and demands were 

 ■made for a radical change in the mathematical training 

 •of the schoolboy. The feelings of dissatisfaction were not 

 confined to any one group, and men who represented the 

 most widely separated interests took a keen and active 

 part in the discussions. Many of the views expressed 

 respecting the methods of mathematics were far from 

 new, but the emphasis with which thev were ur^ed mav 

 perhaps be taken as an indication of the extent to which 

 in the opinion of many competent iudges, the deductive 

 element in mathematics had overshadowed all others 



It may be conceded that, in the claims that have often 

 been advanced for the efficiency of mathematics as an 

 educational instrument, far too much has been made of 

 the deductive aspect of mathematical studies ; but in view 

 ot what has been said about the character of eighteenth- 

 century mathematical methods, the assertion that mathe- 

 matics knows nothing of induction is surely inaccurate. 

 It IS besides, I believe, a complete misunderstanding of 

 the critical school to suppose that induction is barred as 

 a mathematical method. Bv induction I do not here 

 mean simply what is called " mathematical induction " 

 or that method of demonstration which shows that if a 

 theorem is true in one case it is true for the succeeding 

 one : I am using the word in the sense it generally bearl 

 m speaking of scientific method. Induction as a method 

 of discovering new truths or generalising known theorems 

 has always been recognised to be of very great value, and 

 13 in constant use in advanced as well' as in elementary 

 mathematics The critical objection to it was ™ n 

 respect of is use in a systematic development of mathe- 

 matica truth (Euclid's "Elements," for example, embody 

 a systematic development of geometry in which the 

 theorems are linked together by a chain of deduct he 

 rrftics"'"^'- ;^% Weierstrass, one of the greatest of he 

 ciitics, says, ' it is a matter of course that every road must 

 be open to the searcher as long as he seeks; it is onTy a 

 question of the systematic demonstration " ^ 



\J^^r,.,m /'^™5sion of mathematical methods it is 

 Stv of . T '" '^'"'' *^''" "^-^ conviction of the 

 me hnH .f *^°'''"'" L' "°' dependent on any single 



method of proof, even though one may strive to furnish 



IntXf 'r°" "^'''■' ^°;!f°'''"= *° some- prescribed system 

 In mathematics, as in other sciences, conviction comes from 



Weher ZZT"' .^""^ ''"' ""'^^^ =^''"°^f '^^ that Jhe e 

 higher ma hematics enters into the work of the physicist 

 or the engineer the conviction that comes from th^ Jogi'ca 

 iZ?t'"7.,°' ■;! ■^'^^ematical demonstratTon is ^ es 

 mportant than the conviction that is due to insight into 



enceteel fh'' '""l '° "?^ F""^P^'°" °f '^e correspond" 

 ence between the mathematical representation and the data 



^ waves'".?""'- / "''"'? i''^' P"'' "^^thematicians have not 

 always given due weight to the instinct of the trained 

 experimenter, and that, for the physicist, the true source 

 of the conviction of the validity of "existence theorems" 

 hanin LVT""^ in the disciplined imagination rather 

 than in the cogency of the mathematical analysis On 

 rl^u^f e°""'i\''^^.^^^^"*'^' ^^^"--^cy of many of ^e 

 brexolatne'd"'?^^-^' -S'^^T^^f'V^entury mathematicians may 

 DL„h;ni nH '• '"' P'^'f""''' '"'""«= prevented them from 

 pushing a theory or method too far. 



oii'thLjf- '\ ^^ ^"■'T'^'^ '^^' induction is a recognised 

 mathematical method, it is hard to understand how 



NO. 2091, VOL. 82] 



observation and experiment can be dispensed with, because 

 these are essential preliminaries to induction. In the 

 development of mathematical knowledge it is quite certain 

 that the predominance of deductive methods was of com- 

 paratively late growth, and that in the earlier stages 

 observation played the leading part. It is unfortunate that 

 so little of the work of the early Greek geometers has 

 been preserved, but it is undoubtedly the case that geo- 

 metry in its beginnings was essentially surveying or 

 mensuration, and many of Euclid's theorems were known 

 long before they were incorporated in a systematic treatise. 

 There was, in fact, a " natural history " stage in the 

 development of scientific geometry which the perfection of 

 Euclid's deductive treatise has tended to obscure. The 

 stage in which geometry appears as a logically consistent 

 system was preceded by a period in which geometrical 

 theorems were discovered as the result of observation and 

 the consideration of many particular cases ; in this 

 formative period induction based on observation had full 

 scope. 



The evolution of scientific algebra has followed similar 

 lines. The introduction of fractions in arithmetic, for 

 example, and of negative and imaginary numbers in 

 algebra, was due to their convenience in handling prac- 

 tical problems ; the rules for their use were usually estab- 

 lished, so far as proof was considered necessary, by appeal- 

 ing to numerous particular cases. The logical consistency 

 of the scheme of operations was seldom discussed ; so 

 long as a rule led to results which gave a solution of 

 the particular problems under investigation the need for 

 a systematic presentation was not even felt. This stage 

 • — the " natural history " stage — of the development of 

 algebra is well known to us by the works that have been 

 preserved of the early writers on algebra ; it would perhaps 

 be true to say that a great part of elementary algebra has 

 not advanced in actual school teaching beyond this stage. 



The advance of mathematics to the position of a logically 

 consistent system of truth has thus been governed by the 

 same principles as regulate the progress of every science. 

 Induction based on observation and confirmed by tests or 

 verifications was constantly employed in extending the 

 range of the science, and it was only gradually that deduc- 

 tion became the predominant, though never the exclusive, 

 method of mathematical study. 



In the recent discussions on elementary mathematics the 

 guiding principle that has emerged seems to me to be the 

 explicit recognition of the essential part that observation 

 and induction play in the acquisition of mathematical 

 knowledge. With this recognition is associated the idea 

 that in the early training of the pupil it is scientifically 

 unsound and practically hurtful to emphasise the deductive 

 element ; his training should, in its broad outlines, be 

 modelled on the course that the historical development of 

 mathematics has followed. Mathematics has now reached 

 the stage in which it is possible to treat it as a deductive 

 science, but it does not follow that it is either necessary 

 or possible to teach it to beginners entirely as a deductive 

 science. To do so is to mistake the meaning of its history 

 and to deprive it of its place as an exponent of scientific 

 method. Observation, classification, and induction are 

 essential elements of scientific method, and these are well 

 illustrated in the historical development of mathematics. 

 The recent discussions have shown that, in the opinion of 

 many experienced teachers, it is not only possible, but 

 necessary, to make full use of these methods in mathe- 

 matical teaching, and the conviction is widely held that 

 they are of special importance in geometry, the branch of 

 elementary mathematics where deduction lins so long had 

 the leading place. The excellence of the intellectual dis- 

 cipline to be obtained from a study of Euclid is, in my 

 opinion, not to be questioned, but I think there is no 

 doubt that it Is contrary to all scientific order to take 

 Euclid as our guide for an introduction to geometry. It 

 Is necessary for the oupil to acquire a knowledge of the 

 forms of material, objects before he can reasonably be 

 expected to demonstrate the geometrical properties that 

 are implied in the definitions of geometrical bodies. In 

 acquiring this knowledge observation and classification are 

 essential, and deductive reasoning will have little place. 

 The knowledge thus gained may be quite entitled to the 

 name of scientific ; if the course is carefully planned and 



