636 SCIENCE AND HYPOTHESIS 



ever, a simple particular case. It is something quite different. We 

 may not therefore even say in the really analytical and deductive part 

 of mathematical reasoning that we proceed from the general to the 

 particular in the ordinary sense of the words. The two sides of the 

 equality (2) are merely more complicated combinations than the two 

 sides of the equality (1), and analysis only serves to separate the ele- 

 ments which, enter into these combinations and to study their relations. 



Mathematicians therefore proceed " by construction," they " con- 

 struct " more complicated combinations. When they analyze these 

 combinations, these aggregates, so to speak, into their primitive ele- 

 ments, they see the relations of the elements and deduce the relations 

 of the aggregates themselves. The process is purely analytical, but it 

 is not a passing from the general to the particular, for the aggregates 

 obviously cannot be regarded as more particular than their elements. 



Great importance has been rightly attached to this process of " con- 

 struction," and some claim to see in it the necessary and sufficient 

 condition of the progress of the exact sciences. Necessary, no doubt, 

 but not sufficient! For a construction to be useful and not mere 

 waste of mental effort, for it to serve as a stepping-stone to higher 

 things, it must first of all possess a kind of unity enabling us to see 

 something more than the juxtaposition of its elements. Or more ac- 

 curately, there must be some advantage in considering the construc- 

 tion rather than the elements themselves. What can this advantage 

 be? Why reason on a polygon, for instance, which is always decom- 

 posable into triangles, and not on elementary triangles? It is be- 

 cause there are properties of polygons of any number of sides, and 

 they can be immediately applied to any particular kind of polygon. 

 In most cases it is only after long efforts that those properties can be 

 discovered, by directly studying the relations of elementary triangles. 

 If the quadrilateral is anything more than the juxtaposition of two 

 triangles, it is because it is of the polygon type. 



A construction only becomes interesting when it can be placed side 

 by side with other analogous constructions for forming species of the 

 same genus. To do this we must necessarily go back from the particu- 

 lar to the general, ascending one or more steps. The analytical 

 process " by construction " does not compel us to descend, but it 

 leaves us at the same level. We can only ascend by mathematical 

 induction, for from it alone can we learn something new. Without 

 the aid of this induction, which in certain respects differs from, but 

 is as fruitful as, physical induction, construction would be powerless 

 to create science. 



Let me observe, in conclusion, that this induction is only possible if 

 the same operation can be repeated indefinitely. That is why the 

 theory of chess can never become a science, for the different moves of 

 the same piece are limited and do not resemble each other. 



