REVIEWS 



MATHEMATICS 



Synopsis of Linear Associative Algebra. A Report on its Natural Development 

 and Results reached up to the Present Time. By James Byrnie Shaw, 

 Professor of Mathematics in the James Millikan University. [Pp. 145.] 

 (Washington, D.C. : Published by the Carnegie Institution of Washington, 

 1907.) 

 Although this "Publication No. 78" of the Carnegie Institution was published 

 many years ago, it is interesting and important to review and read it at the 

 present time, both for its own sake and because it throws light on a recent book 

 by Prof. Shaw on the philosophy of mathematics, which is reviewed elsewhere in 

 the present number of this Journal. The Synopsis aims to set forth the present 

 state of the theory of linear associative algebra, " not in a comparative study of 

 different known algebras, nor in the exhaustive study of any particular algebra, 

 but in tracing the general laws of the whole subject" (p. 5). In view of the 

 historical and critical work referred to on p. 5, no historical review is given in this 

 memoir. However, a bibliography to some extent takes the place of a historical 

 survey, and the bibliography (pp. 133-145) to this work is long and fairly 

 complete. It would have been preferable to refer to some of the numerous papers 

 by Grassmann as well as to the two separately-published books of this author ; 

 there are other relevant papers, and at least one book, by Peano besides those 

 mentioned ; it would surely have been better to refer more in detail to the papers 

 of Hamilton and Sylvester than merely by the words " numerous papers " ; De 

 Morgan's work ought to have been noticed, and surely Weierstrass did not publish 

 a paper in English ! 



Prof. Shaw's memoir is divided into three parts : " General Theory," " Par- 

 ticular Algebras," and "Applications." In the first part "is given the develop, 

 ment of the subject from fundamental principles, no use being made of other 

 mathematical disciplines, such as bilinear forms, matrices, continuous groups, and 

 the like" (p. 5), and the other two lines of development of linear associative 

 algebra are then described (cf. p. 6). In the last part there is " a sketch of the 

 theory of general algebra, placing linear associative algebra in its genetic relations 

 to general linear algebra" (p. 7). Here it is that we meet those views which 

 explain the attitude of the author to work on the logical foundations of mathematics. 

 "The foundations of mathematics," we read on p. 75, "consist of two classes of 

 things, the elements out of which are built the structures of mathematics, and the 

 processes by which they are built. The primary question for the logician is : What 

 are the primordial elements of mathematics? ... To the mathematician these 

 elements do not convey much information as to the processes of mathematics. 

 The life of mathematics is the derivation of one thing from others, the transition 

 from data to things that follow according to given processes of transition." Of 

 course, the logical relations between the entities of mathematics are included 

 among what Prof. Shaw calls the " elements "—though Prof. Shaw hardly seems 



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