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SCIENCE 



[N. S. Vol. XLV. No. 1170 



long before the construction of such lines 

 was undertaken, so mathematics has been 

 preparing thought roads for sciences long 

 before their development was seriously 

 begun. Hence it does not appear inappro- 

 priate for a body of scientists to pause now 

 and then for a few moments to reflect on 

 the methods and ideals which have actu- 

 ated the mathematical investigator. Such 

 reflections may be inspired by a sense of 

 respect for all that contributes to scientiflc 

 progress, but they should also prove help- 

 ful in the formation of most comprehensive 

 notions in regard to the great problems 

 which confront us as a united band of 

 workers to secure light, to dispel more of 

 the superstitions and to present far-reach- 

 ing thoughts in the simplest manner. 



Among the questions which scientists as 

 a body might be inclined to ask the mathe- 

 matician is the following : What is the atti- 

 tude of mind which has contributed most 

 powerfully to mathematical progress? 

 Such a profound question would naturally 

 be answered somewhat diiferently by differ- 

 ent men, and your speaker to-night is not 

 so completely ignorant of his own limita- 

 tions as to suppose that he can furnish a 

 final answer to this question. He hopes, 

 however, that he may not be entirely un- 

 successful in making some illuminating re- 

 marks on it, and in interesting you by 

 directing your attention to common 

 thoughts which underlie the varied efforts 

 by which we as a body aim to enrich the 

 world. 



"With this reservation I would be inclined 

 to say that modesty is the attitude of mind 

 which has contributed most powerfully to 

 mathematical progress. The great "Ele- 

 ments" of Euclid, for instance, are can- 

 didly based on assumptions or axioms and 

 do not claim to prove everything db initio. 

 Moreover, this great work does not concern 

 itself directly with such fundamental ques- 



tions as truth, reality, life, death, etc., but 

 it confines itself to deductions relating to 

 matters which might appear as trivialities 

 when compared with many other problems 

 which then confronted and now confront 

 the human race. 



To understand the modesty of Euclid 

 and his geometric predecessors it is neces- 

 sary to bear in mind the fact that the "Ele- 

 ments" of Euclid were written at a time 

 when other sciences made little or no de- 

 mand for such results as these "Elements'' 

 embodied. Even such a closely related 

 subject as surveying could then make little 

 direct use of these results in view of the 

 theoretic form in which they were pre- 

 sented. The work which is said to have 

 passed through more editions than any 

 other book except the Bible, which con- 

 siders diametrically opposite questions, 

 must have appeared to many of Euclid's 

 contemporaries as dealing with compara- 

 tively trivial matters in a modest way, since 

 it made no attempt to trace its fundamental 

 principles to their sources, but explicitly 

 started with axioms or postulates. 



As another evidence of modesty in 

 mathematics I may mention the special 

 symbols for unknowns and the use of equa- 

 tions in these unknowns. The scientific 

 method embodied in equations involving at 

 least one unknown implies the careful study 

 of relations involving something with re- 

 spect to which we have openly acknowl- 

 edged our ignorance. Like the axioms or 

 postulates of geometry, it makes no preten- 

 tion of complete knowledge, but is satisfied 

 with an humble attitude of mind. The basis 

 of mathematical development is thus seen 

 to be characterized by a modesty which has 

 led the investigator to do what he can do 

 thoroughly rather than to try to do that 

 which would naturally interest him more 

 but which lies beyond his power. 



Even in its most primitive form, count- 



