630 SCIENCE AND HYPOTHESIS 



parasite. Should we not therefore have reason for asking if the syllo- 

 gistic apparatus serves only to disguise what we have borrowed ? 



The contradiction will strike us the more if we open any book on 

 mathematics; on every page the author announces his intention of 

 generalizing some proposition already known. Does the mathematical 

 method proceed from the particular to the general, and, if so, how 

 can it be called deductive? 



Finally, if the science of number were merely analytical, or could 

 be analytically derived from a few synthetic intuitions, it seems that a 

 sufficiently powerful mind could with a single glance perceive all its 

 truths ; nay, one might even hope that some day a language would be 

 invented simple enough for these truths to be made evident to any 

 person of ordinary intelligence. 



Even if these consequences are challenged, it must be granted that 

 mathematical reasoning has of itself a kind of creative virtue, and is 

 therefore to be distinguished from the syllogism. The difference must 

 be profound. We shall not, for instance, find the key to the mystery 

 in the frequent use of the rule by which the same uniform operation 

 applied to two equal numbers will give identical results. All these 

 modes of reasoning, whether or not reducible to the syllogism, pro- 

 perly so called, retain the analytical character, and ipso facto, lose 

 their power. 



The argument is an old one. Let us see how Leibnitz tried to show 

 that two and two make four. I assume the number one to be defined, 

 and also the operation oH-1 i.e., the adding of unity to a given num- 

 ber x. These definitions, whatever they may be, do not enter into the 

 subsequent reasoning. I next define the numbers 2, 3, 4 by the 

 equalities : 



(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4, and in the same way 

 I define the operation re + 2 by the relation; (4) x + 2 =(x + 1)+ 1. 



Given this, we have : 



2+2=(2+l)+l; (def. 4). 

 (2+1) +1=3+1 (def. 2). 



3+1=4 (def. 3). 



whence 2+2=4 Q.E.D. 



It cannot be denied that this reasoning is purely analytical. But 

 if we ask a mathematician, he will reply : " This is not a demonstra- 

 tion properly so called; it is a verification." We have confined our- 

 selves to bringing together one or other of two purely conventional 

 definitions, and we have verified their identity ; nothing new has been 

 learned. Verification differs from proof precisely because it is analyti- 

 cal, and because it leads to nothing. It leads to nothing because the 

 conclusion is nothing but the premisses translated into another lan- 

 guage. A real proof, on the other hand, is fruitful, because the con- 

 clusion is in a sense more general than the premisses. The equality 



