790 



SCIENCE 



[N. S. Vol. XXXVIII. No. i 



of it pretty carefully, just as, in order to 

 know what good art is, it is, in general, 

 essential either to have produced good art 

 or to have attentively examined some 

 specimens of it, or to have done both of 

 these things. Here, then, at the outset our 

 historian would meet a serious difficulty, 

 unless his audience chanced to be one of 

 mathematicians, which is (unfortunately) 

 not likely, inasmuch as the great majority 

 of mathematicians are so exclusively inter- 

 ested in mathematical study or teaching or 

 research as to be but little concerned with 

 the philosophical question of the human 

 worth of their science. It is, therefore, 

 easy to see how our lecturer would have to 

 begin. 



Ladies and gentlemen, we have met, he 

 would say, to open a course of lectures deal- 

 ing with the role of rigorous thinking in 

 the history of civilization. In order that 

 the course may be profitable to you, in order 

 that it may be a course in ideas and not 

 merely or mainly a verbal course, it is 

 essential that you should know what rigor- 

 ous thinking is and what it is not. Even I, 

 your speaker, though a historian, might 

 reasonably be held to the obligation of 

 knowing that. 



It is reasonable, ladies and gentlemen, it 

 is reasonable to assume, he would say, that 

 in the course of your education you neg- 

 lected mathematics, and it is, therefore, 

 probable or indeed quite certain that, not- 

 withstanding your many accomplishments, 

 yoii do not quite know, or rather, perhaps 

 I should say, you are very far from know- 

 ing, what rigorous thinking is or what it is 

 not. Of course, as you know, it is, gener- 

 ally speaking, much easier to tell what a 

 thing is not than to tell what it is, and I 

 might, he would say, I might proceed by 

 way of a preliminary to indicate roughly 

 what rigorous thinking is not. Thus I 

 might explain that rigorous thinking, 



though much of it has been done in the 

 world, and though it has produced a large 

 literature, is nevertheless a relatively rare 

 phenomenon. I might point out that a vast 

 majority of mankind, a vast majority of 

 educated men and women, have not been 

 disciplined to think rigorously even those 

 things that are most available for such 

 thinking. I might point out that, on the 

 other hand, most of the ideas with which 

 men and women have constantly to deal 

 are as yet too nebulous and vague, too little 

 advanced in the course of their evolution, 

 too little refined and defined, to be avail- 

 able for concatenative thinking and rigor- 

 ous discourse. I should have to say, he 

 would add, that, on these accounts, most of 

 the thinking done in the world on a given 

 day, whether done by men in the street or 

 by farmers or factory-hands or merchants 

 or administrators or physicians or lawyers 

 or jurists or statesmen or philosophers or 

 men of letters or students of natural science 

 or even mathematicians (when not strictly 

 employed in their own subject), comes far 

 short of the demands and standards of 

 rigorous thinking. 



I might go on to caution you, our 

 speaker would say, against the current 

 fallacy, recently advanced by eloquent 

 writers to the dignity of a philosophical 

 tenet, of regarding what is called success- 

 ful action as the touchstone of rigorous 

 thinking. For you should know that much 

 of what passes in the world for successful 

 action proceeds from impulse or instinct 

 and not from thinking of any kind; you 

 should know that no action under the con- 

 trol of non-rigorous thinking can be strictly 

 successful except by the favor of chance or 

 through accidental compensation of errors; 

 you should know that most of what passes 

 for successful action, most of what the 

 world applauds and even commemorates as 

 successful action, so far from being really 



