908 



SCIENCE. 



[N. S. Vol. VIII. No. 208. 



gen, herausgegeben von H. Burkhaedt und 



AY. P. Meyer. Leipzig, Teubner. 1898. Band 



I. Heft 1. Pp. 112. 



Tliis is an undertaliing of extraordinary im- 

 portance and promise. 



Its aim is to give a consecutive presentation 

 of the assured results of the mathematical 

 sciences in their present form, while, by careful 

 and copious references to the literature, giving 

 full indications regarding the historic develop- 

 ment of mathematical methods since the begin- 

 ning of the nineteenth century. The work be- 

 gins with twenty-seven pages on the foundations 

 of arithmetic by Hermann Schubert, of Ham- 

 burg. Schubert's reputation was made by 

 his remarkable book on enumerative geometry. 

 He has since applied the modern ideas in an 

 elementary arithmetic, and is known in Amer- 

 ica as a contributor to the Monist. 



Unfortunately Schubert has made in public 

 some strange slips. In an article, ' On the na- 

 ture of mathematical knowledge,' in the Mon- 

 ist, Vol. 6, p. 295, he says : " Let me recall the 

 controversy which has been waged in this cen- 

 tury regarding the eleventh axiom of Euclid, 

 that only one line can be drawn through a point 

 parallel to another straight line. The discus- 

 sion merely touched the question whether the 

 axiom was capable of demonstration solely by 

 means of the other propositions or whether it 

 was not a special property, apprehensible only 

 by sense-experience, of that space of three di- 

 mensions in which the organic world has been 

 produced, and which, therefore, is of all others 

 alone within the reach of our powers of repre- 

 sentation. The truth of the last supposition 

 affects in no respect the correctness of the 

 axiom, but simply assigns to it, in an epistemo- 

 logical regard, a different status from what it 

 -would have if it were demonstrable, as was one 

 time thought, without the aid of the senses, and 

 solely by the other propositions of mathe- 

 matics." 



If Schubert had written this seventy-five years 

 ago it might have been pardonable. Just at 

 the beginning of this century Gauss was trying 

 to prove this Euclidean parallel-postulate. Even 

 up to 1824 he was in Schubert's state of mind, 

 for he then writes Taurinus : ' ' Ich hahe daher 

 wohl zuweilen im ScJierz den Wunsch geaussert. 



dass die Euclidische Geometrie nicht die Wahre 

 ware." 



But the joke had even then gone out of the 

 matter if Gauss had but known it, for in 1823 

 Bolyai Janos had written to his father : " Prom 

 nothing I have created a wholly new world." 



Of the geometry of this world as given also 

 by Lobachevski, Clifford wrote: "It is quite 

 simple, merely Euclid without the vicious as- 

 sumption." 



But this assumption is only vicious if sup- 

 posed to be ' apprehensible by sense-experience' 

 or ' demonstrable by the aid of the senses.' 



That ' the organic world has been produced ' 

 in Euclidean space can never be demonstrated 

 in any way whatsoever. On the other hand, the 

 mechanics of actual bodies might be- shown by 

 merely approximate methods to be non-Eu- 

 clidean. Therefore, Schubert's contribution on 

 the foundations of arithmetic may fairly be read 

 critically. He begins with counting, and de- 

 fines number as the result of counting. This is 

 in accord with the theory that their laws alone 

 define mathematical operations, and the opera- 

 tions define the various kinds of number as their 

 symbolic outcome. 



There is no word of the primitive number- 

 idea, which is essentially prior to counting and 

 necessary to explain the cause and aim of 

 counting. This primitive number-idea is a crea- 

 tion of the human mind, for it only pertains to 

 certain other creations of the human mind 

 which I call artificial individuals. The world 

 we consciously perceive, is a mental phenom- 

 enon. Yet certain separable or distinct things 

 or primitive individuals, we cannot well help 

 believing to subsist somehow ' in nature ' as 

 well as in conscious perception. Now, by taking 

 together certain of these permanently distinct 

 things or natural individuals, the human mind 

 makes an artificial individual, a conceptual 

 unity. 



Number is primarily a quality of such an 

 artificial individual. 



The operation of counting was made to apply 

 to such an individual to identify it with one of 

 a standard set of such artificial individuals, 

 and so to get the exact shade of its numeric 

 quality. These standard individuals were pri- 

 marily sets of fingers. Then came the written 



