Maech 3, 1899.] 



SCIENCE. 



325 



separate detailed study of the Algebra of Logic 

 aud of the Calculus of Extension ; the second 

 volume will contain a separate detailed study 

 of Quaternions and Matrices and a detailed 

 comparison of the symbolic structures of the 

 several algebras. The main idea of the work 

 is not unification of the several methods, nor 

 generalization of ordinary algebra so as to in- 

 clude them, but rather the comparative study 

 of their several structures. But, it may be 

 asked, if the branches of universal algebra are 

 essentially distinct from ordinary algebra and 

 from one another, what bond is there to con- 

 nect them into one whole? A connecting bond 

 is found in the generalized conception of space; 

 the properties and operations involved in that 

 conception are found capable of forming a uni- 

 form method of interpretation of the various 

 algebras. 



The work is well and clearly written and, 

 when completed, will form an admirable presen- 

 tation of the subject from the formal view of 

 mathematical analysis. One escellent feature 

 is conservatism in the vise of symbols ; by this 

 means the author makes his pages easier read- 

 ing to those who have already studied some of 

 the special branches. 



Another excellent feature of the volume con- 

 sists in the Historical Notes appended to some 

 of the chapters. In these Mr. Whitehead gives 

 a brief history of the development of the special 

 branch, so far as known to him, without mak- 

 ing an exhaustive research. The importance 

 of the Historical Notes probably calls for a 

 more exhaustive research, as the work covers a 

 great and growing province of mathematics 

 and will, when completed, be considered one of 

 the best authorities on its subject in the Eng- 

 lish language. 



The feature which is most open to discussion 

 is the view which the author takes of the fun- 

 damental nature of mathematics ; and it is 

 most important, for it determines the whole 

 plan of the work. In the preface the author 

 thus states his view, in very plain terms : 

 " Mathematics is the development of all types 

 of formal, necessary, deductive reasoning. The 

 reasoning is formal in the sense that the meaning 

 of propositions forms no part of the investiga- 

 tion. The sole concern of mathematics is the 



inference of proposition from proposition. The 

 justification of the rules of inference in any 

 branch of mathematics is not properly part of 

 mathematics ; it is the business of experience or 

 philosophy. The business of mathematics is 

 simply to follow the rules. In this sense all 

 mathematical reasoning is necessary, namely, 

 it has followed the rules. Mathematical 

 reasoning is deductive in the sense that it 

 is based upon definitions which, as far as 

 the validity of the reasoning is concerned 

 (apart from any existential import), need only 

 the test of self-consistency. Thus no external 

 verification of definitions is required by mathe- 

 matics as long as it is considered merely as 

 mathematics. Mathematical definitions either 

 possess an existential import or are conven- 

 tional. A mathematical definition with an ex- 

 istential import is the result of an act of pure 

 abstraction. Such definitions are the starting 

 points of applied mathematical sciences ; and, 

 in so far as they are given this existential im- 

 port, they require for verification more than 

 the mere test of self-consistency. Hence a 

 branch of applied mathematics, in so far as it is 

 applied, is not merely deductive, unless in some 

 sense the definitions are held to be guaranteed 

 a priori as being true in addition to being self- 

 consistent. A conventional mathematical defi- 

 nition has no existential import. It sets before 

 the mind, by an act of imagination, a set of 

 things with fully-defined self-consistent types 

 of relation. In order that a mathematical sci- 

 ence of any importance may be founded upon 

 conventional definitions, the entities created by 

 them must have properties which bear some 

 aflSnity to the properties of existing things. 

 Thus the distinction between a mathematical 

 definition with an existential import and a con- 

 ventional definition is not always very obvious 

 from the form in which they are stated. In 

 such a case the definitions and resulting prop- 

 ositions can be construed either as referring 

 to a world of ideas created by convention or 

 as referring exactly or approximately to the 

 world of existing things." 



In reply, it may be asked : Is geometry a 

 part of pure mathematics ? Its definitions have 

 a very existential import ; its terms are not con- 

 ventions, but denote true ideas; its propositions 



