962 



SCIENCE 



[N. S. Vol. XXX. No. 783 



It is to be noted that in usage various 

 kinds of numbers are denoted by the same 

 symbol. This is due to the fact that cus- 

 tom antedates criticism. Thus 2 stands for 

 a cardinal (a class), for a positive integer 

 (a relation), for a rational number or frac- 

 tion (a relation), for an ordinal (a rela- 

 tion), and for a rational real (a class) — 

 neither the classes nor the relations being 

 of the same kind. 



Passing now to the notion of the (linear) 

 continuum, it is to be defined in ordinal 

 terms and without the logically vicious 

 assumption often tacitly made that the 

 continuum to be defined is already im- 

 mersed in a continuum. The following 

 procedure is due to G. Cantor. Let t} 

 denote the order-type of series like that of 

 the rationals taken in so-called natural 

 order. Any series of this type has the 

 following properties, all of them ordinal: 

 (1) it is denumerable; (2) it has neither 

 beginning nor end; (3) it is compact. A 

 series of terms in a series of type jy is said 

 to be fundamental if it is a progression, 

 that is, if it is like the series 1, 2, 3, • • • ; 

 and it is described as ascending or descend- 

 ing according as its terms follow one an- 

 other in the same sense (or direction) as 

 do those of the series i? or in the reverse 

 sense. A term of a series is a limit if it 

 immediately follows (or precedes) a class 

 of terms of the series and does not imme- 

 diately follow (or precede) any one assign- 

 able term of it. It follows that a funda- 

 mental series s of a series -q has a limit if 

 in 7] there is a term that is first after or 

 first before all the terms of s according as 

 s is ascending or descending. A series is 

 said to be perfect if ( 1 ) all its fundamental 

 series have limits and (2) all its terms are 

 limits of fundamental series. It can be 

 proved that a series whose terms are terms 

 of a perfect series and which, besides being 

 denumerable, are so distributed that there 



is one between every two terms of the per- 

 fect series, is a series of type 77. We can 

 now define : a series 6 is continuous if it is 

 perfect and contains a denumerable class 

 of terms such that there is one of them 

 between every two terms of 6. The defini- 

 tion is based upon the pi'operties found to 

 characterize the series of real numbers from 

 inclusive to 1 inclusive. 



The significance of what has been said is 

 by no means confined to analysis. Tet I 

 wish, in closing, to refer explicitly to geom- 

 etry. As a branch of mathematics, geom- 

 etry does not claim to be an accurate or 

 true description of actual or perceptual 

 space, whatever that may be. As for the 

 notion and the name of space, it does not 

 seem to be a modern discovery that they 

 are not essential to geometry, for, as Peano 

 has pointed out, neither the one nor the 

 other is to be found in the works either of 

 Euclid or of Archimedes. What, then, is 

 geometry? And how related to the thesis 

 of modern logistic? The answer must be 

 in terms of form and suhject-matter. As 

 to form, geometry is, as Pieri has said and 

 by his great memoirs has done as much as 

 any one to show, a purely "hypothetico- 

 deductive" science. It is triie indeed that 

 in each of the postulate-systems— whether 

 those of Pieri or of Pasch or of Peano or 

 of Hilbert or of Veblen or of others— that 

 have recently been offered as basis for de- 

 scriptive or projective or metric geometry 

 or for any sub-division of those grand 

 divisions, there occurs at least one postulate 

 in categoric form, as, for example, "there 

 exists at least one point"— thus seeming to 

 assert or to imply that the geometry in 

 question, whatever variety it may be, 

 transcends the hypothetic character and 

 has in fact validity of an extra-theoretic 

 or external kind. Nevertheless, the seem- 

 ing is appearance only. What the geomet- 

 rician really asserts, and he asserts nothing 



