2o 4 POPULAR SCIENCE MONTHLY 



had temporarily served as support and which was afterward useless 

 was rejected; there remained only the construction itself, irreproach- 

 able in the eyes of the logician. And yet if the primitive image had 

 totally disappeared from our recollection, how could we divine by what 

 caprice all these inequalities were erected in this fashion one upon 

 another ? 



Perhaps you think I use too many comparisons; yet pardon still 

 another. You have doubtless seen those delicate assemblages of sili- 

 cious needles which form the skeleton of certain sponges. When the 

 organic matter has disappeared, there remains only a frail and elegant 

 lace-work. True, nothing is there except silica, but what is interest- 

 ing is the form this silica has taken, and we could not understand it 

 if we did not know the living sponge which has given it precisely this 

 form. Thus it is that the old intuitive notions of our fathers, even 

 when we have abandoned them, still imprint their form upon the logical 

 constructions we have put in their place. 



This view of the aggregate is necessary for the inventor ; it is equally 

 necessary for whoever wishes really to comprehend the inventor. Can 

 logic give it to us ? No ; the name mathematicians give it would suffice 

 to prove this. In mathematics logic is called analysis and analysis 

 means division, dissection. It can have, therefore, no tool other than 

 the scalpel and the microscope. 



Thus logic and intuition have each their necessary role. Each is 

 indispensable. Logic, which alone can give certainty, is the instrument 

 of demonstration; intuition is the instrument of invention. 



VI. 



But at the moment of formulating this conclusion I am seized with 

 scruples. At the outset I distinguished two kinds of mathematical 

 minds, the one sort logicians and analysts, the others intuitionalists 

 and geometers. Well, the analysts also have been inventors. The 

 names I have just cited make my insistence on this unnecessary. 



Here is a contradiction, at least apparently, which needs explana- 

 tion. And first, do you think these logicians have always proceeded 

 from the general to the particular, as the rules of formal logic would 

 seem to require of them? Not thus could they have extended the 

 boundaries of science; scientific conquest is to be made only by gen- 

 eralization. 



In one of the chapters of ' Science and Hypothesis/ I have had 

 occasion to study the nature of mathematical reasoning, and I have 

 shown how this reasoning, without ceasing to be absolutely rigorous, 

 could lift us from the particular to the general by a procedure I have 

 called mathematical induction. It is by this procedure that the an- 

 alysts have made science progress, and if we examine the detail itself 



