Dkcembeb 10, 1904.] 



SCIENCE. 



boo 



SCIENTIFIC BOOKS. 



POINCARE's ' SCIENCE AND HYPOTHESIS.' 



La Science et I'Hypothese. Par H. Poincare, 



Membre de I'lnstitut. Paris, 1903. Pp. 



284. 

 Wissenschaft und Hypothese. H. Poincare. 



Autorisierte deutsche Ausgabe, mit erlauter- 



enden Anmerkungen, von F. und L. Linde- 



JiANN. Leipzig, 1904. Pp. xvi + 342 ; the 



notes, pp. 245-333. 



A work from the pen of one of the dis- 

 tinguished savants who have so recently been 

 the guests of the x\inerican scientific public 

 is doubly interesting at the present time. 

 Among the several domains of pure and ap- 

 plied mathematics which M. Poincare has 

 enricLid by his researches, not the least im- 

 portant is that of the fundamental concepts 

 and logical development of various branches 

 of science. Like its predecessors, the work 

 under consideration here is remarkable for 

 the clear, incisive and succinct fashion in 

 which it deals with the difficult and elusive 

 problems lying at the foundation of mathe- 

 matical knowledge. 



The work is divided into four parts, pre- 

 ceded by a short introduction, viz.: First 

 Part : ' Number and Magnitude,' p-^,. 9^8. 

 Second Part: 'Space,' pp. 49-109. Third 

 Part: 'Force,' pp. 110-166. Fourth Part: 

 'Nature,' pp. 167-281. 



The first chapter is entitled, ' On the 

 Nature of the Eeasoning of Mathematics.' 

 At the very outset, even the existence of 

 the science of mathematics seems to present 

 an irreconcilable contradiction. If mathe- 

 matics is deductive, drawing all its conclu- 

 sions strictly from their antecedent premises, 

 how can it be more than a huge tautology? 

 How are all the ponderous tomes of mathe- 

 matical theory aught else than devious ways 

 of saying A is A? If, on the other hand, the 

 conclusions of mathematics say more than 

 their antecedent premises, how is the unques- 

 tioned perfect rigor of mathematics main- 

 tained ? 



M. Poincare finds the answer to these ques- 

 tions in the so-called 'mathematical induc- 

 tion ' which proceeds from the particular to 

 the more general, but at the same time does 



so by steps of the highest degree of certitude. 

 In this process he sees the creative force of 

 mathematics, which leads to real proofs and 

 not mere sterile verifications. The illustra- 

 tions used to make the thought clear are 

 taken from the beginnings of arithmetic, 

 where mathematical thought has remained 

 least elaborated and uncomplicated by the 

 difficult questions related to the notion of 

 space. In successive instances it is shown 

 how more general results are obtained from 

 "fundamental definitions and from previous 

 results by means of mathematical induction. 

 In each case the advance is made by virtu© 

 of that " power of the mind which knows that 

 it can conceive of the indefinite repetition of 

 the same act as soon as this act is at all pos- 

 sible. The mind has a direct intuition of 

 this power and experience gives only the op-- 

 portunity toi use it and to become conscious 

 of it" (pp. 23-4). 1 



The conviction that the method of mathe- 

 matical induction is valid our author regards 

 as truly an a priori synthetic judgment; the 

 mind can not tolerate nor conceive its contra- 

 dictory and could not even draw any theoretic 

 consequences from the assumption of the con- 

 tradictory. No arithmetic could be built up, 

 rejecting the axiom of mathematical induc- 

 tion, as the non-Euclidean geometries have 

 been built up, rejecting the postulate of 

 Euclid. 



The second chapter terminates the first part 

 and is entitled, ' Mathematical Magnitude and 

 Experience.' It deals with irrational num- 

 bers and the creation of the mathematical 

 continuum, concluding that 'this notion has 

 been created by the mind, but that experience 

 furnished the occasion' (p. 35). " The mind 

 has the power of creating symbols, and by 

 this means it has constructed the mathemat- 

 ical continuum which is merely a particular 

 system of symbols. This power is limited 

 only by the necessity of avoiding contradic- 

 tion, but the mind makes use of it only when 

 experience furnishes the warrant" (p. 40). 



The second part, devoted to ' Space,' con- 

 sists of chapters on ' The non-Euclidean 

 Geometries,' ' Space and Geometry ' and ' Ex- 

 perience and Geometry.' 



