5i8 



NATURE 



[June 30, 1910 



The optical appearances connected with sunrise and 

 sunset are somewhat briefl}' treated, reference being 

 especially made to Kiessling's monograph " llntor- 

 suchungen iiber Dammerungserscheinungen " for a 

 more complete description of the phenomena. Some 

 account of the observed effects due to the Krakatoa 

 eruption and other similar disturbances is included. 



Dr. Exner has followed Prof. Pernter in the careful 

 reproduction of the best recorded and historically in- 

 teresting observations of the phenomena. He has 

 himself emphasised the impossibility of reproducing 

 the charm of Pernter's work, dependent as it was on 

 the latter's extensive knowledge of the literature of 

 the subject as well as on his critical judgment. This 

 volume will, however, be welcomed both as a fitting 

 completion of the task undertaken by Prof. Pernter 

 and as a valuable survey of the progress which has 

 been made in the interpretation of the phenomena with 

 which it deals. 



THE PHILOSOPHY OF MATHEMATICS. 

 Methodologisches und Philosophisches zur Elementar- 

 Mathemaiik. By Dr. G. Mannoury. Pp. viii + 279. 

 (Haarlem : P. Visser Azn., 1909.) Price Ss. lod. 



THIS work is the outcome of lectures delivered by 

 the author at the University of Amsterdam, and 

 retains in different ways the marks of its origin. Its 

 frequent digressions, its general discursiveness, and its 

 rather sketchy character make it difificult to describe ; 

 and many of the topics are so controversial that where 

 one reader will agree with the author, another, equally 

 competent, will entirely dissent. Still, it is an honest 

 and interesting attempt to deal, from the philosophical 

 side, with the fundamental difiiculties of mathematics, 

 and as such deserves attention. 



The first part contains five chapters dealing respec- 

 tively with unity and plurality, finite and infinite 

 numbers, the elementary laws of arithmetic applied 

 to whole numbers, the extension of the idea ol 

 number, and, finally, the definition of irrationals. 

 The second part is devoted to geometry, and its four 

 chapters discuss respectively mathematical logic ; 

 elementary constructions, postulates, and theory of 

 measurement ; non-Euclidean geometries ; and the 

 notion of space from the standpoint of physiology and 

 psychology. 



A few examples must suffice to illustrate the merits 

 and the defects of the author's procedure. Take the 

 question of defining a unit. After pointing' out, 

 rightly enough, that there is no such thing as an 

 objective unit directly perceived, he gives as a formal 

 definition (p. 31): — "Units are sensation-complexes 

 (Empfindungs-Komplexe), and a plurality (Vielheit) 

 consists of mutually related units." Now, if there is 

 one thing that recent mathematics has done, it is 

 surely to clarify and make precise the notion of a unit 

 apart from all elements of sensation. Verbal defini- 

 tion of a unit is a small matter, of course ; the thing 

 to be desired is the complete notion. As a matter of 

 fact, everybody does acquire the notion more or less 

 exactly, long before thinking about defining it ; and 

 as to the definition, a kindergarten teacher will suc- 

 NO. 2122, VOL. 83] 



ceed where a philosopher will fail. "These are toys; 

 each toy is a unit among the toys"; "You are my 

 class ; each of you is a unit of the class " ; such 

 examples will convey the meaning of the term "unit" 

 better than any formal definition. At the same 

 time, if we must have a metaphysical example of a 

 unit, the ego seems to be the best, for it cannot be 

 denied, or affirmed to be a plurality, without an in- 

 trinsic contradiction in terms. If Jones makes a 

 statement or forms an opinion, however erroneous, it 

 is his, and this "he" is an irreducible entity which 

 has a preeminent claim to be called a unit. It may 

 be remarked that Dr. Mannoury expressly objects to 

 this line of argument, apparently on the ground that 

 the idea of the ego is a derivative one ; this may be 

 admitted in a sense, as a fact in the development of 

 an individual consciousness, but it does not make the 

 ego derivative, any more thi^Ot'the deciphering of 

 hieroglyphics in recent times affects the date at which 

 they were carved. Is not this one of those cases 

 w^here psjchology is appealed to where it is really 

 irrelevant, the question being one concerning meta- 

 physical data? We must have something a priori and 

 undefined in any science ; the question is, how few and 

 how fundamental (or elementary) may we assume 

 these data to be? 



A more striking example of the same sort, of thing 

 is to be found on p. 263, where the author speaks of 

 "the four-dimensional space-time-notion which is to 

 be regarded as an image of the whole group of sensa- 

 tions." It is almost impossible to give any sense to 

 this phraseology, consistent with either popular or 

 mathematical usage. If it merely means that in 

 abstract kinematics in three-dimensional space there 

 are foiir independent variables {x, y, z, i), it is a very 

 unsatisfactory way of stating a simple fact ; and it is 

 very doubtful whether kinematics is, properly, an 

 image of sensations, any more than our sensation of 

 the colour of homogeneous light is imaged by its 

 wave-length. 



In treating of the elementary laws of arithmetic, the 

 author, in the text, mainly follows those who appeal 

 to the principle of analogy or "permanence"; he does 

 not give a detailed discussion of the elementary opera- 

 tions. The definition of irrationals is Dedekind's, 

 which is wrongly attributed to Dirichlet ; there is a 

 brief account of Peano's system of shorthand, and a 

 section on mathematical induction, with quotations 

 from Poincare, Couturat, and others. Dr. Mannoury 

 is evidently dissatisfied with Poincar^'s arguments, 

 but here, as in other cases, he does not bring forward 

 any very definite statements of his own. 



In the geometrical section there are several features 

 of interest, and this is the most readable part of the 

 book. A fair account is given of the different types 

 of three-dimensional geometry, of Hilbert's non- 

 Desarguian system, and of metrical geometry based 

 on a movable standard assumed to be rigid. But 

 there is no discussion of a system of definitions, and 

 the only element treated in any detail is the straight 

 line. With regard to the different types of geometry, 

 the author adopts the sensible attitude that it is now, 

 and always will be, impossible to fix on one as the 



