SCIENCE AND HYPOTHESIS 1 



BY PROF. JULES HENRI POINCARE, UNIVERSITY OF PARIS 



PART I NUMBER AND MAGNITUDE 



On the Nature of Mathematical Reasoning 



THE very possibility of mathematical science seems an insoluble 

 contradiction. If this science is only deductive in appearance, from 

 whence is derived that perfect rigor which is challenged by none? 

 If, on the contrary, all the propositions which it enunciates may be 

 derived in order by the rules of formal logic, how is it that mathe- 

 matics is not reduced to a gigantic tautology? The syllogism can 

 teach us nothing essentially new, and if everything must spring from 

 the principle of identity, then everything should be capable of being 

 reduced to that principle. Are we then to admit that the enunciations 

 of all the theorems with which so many volumes are filled, are only 

 indirect ways of saying that A is A? 



No doubt we may refer back to axioms which are at the source of all 

 these reasonings. If it is felt that they cannot be reduced to the 

 principle of contradiction, if we decline to see in them any more than 

 experimental facts which have no part or lot in mathematical neces- 

 sity, there is still one resource left to us: we may class them among 

 a priori synthetic views. But this is no solution of the difficulty 

 it is merely giving it a name; and even if the nature of the synthetic 

 views had no longer for us any mystery, the contradiction would not 

 have disappeared; it would have only been shirked. Syllogistic rea- 

 soning remains incapable of adding anything to the data that are given 

 it ; the data are reduced to axioms, and that is all we should find in the 

 conclusions. 



No theorem can be new unless a new axiom intervenes in its dem- 

 onstration ; reasoning can only give us immediately evident truths 

 borrowed from direct intuition; it would only be an intermediary 



i This is a translation of Prof. Poincare"'s celebrated treatise entitled 

 La Science et I'Hypothese. It is presented here in the nature of collateral 

 reading to the lectures on Mathematics and other scientific lectures deliv- 

 ered at the International Congress of Arts and Science. 



