402 



SCIENCE. 



[N. S. Vol. XVII. No. 428. 



are devoted to the codification in Peano's 

 symbolic language of the principal mathe- 

 matical theories, and to researches on ab- 

 stract mathematics. General interest in 

 abstract mathematics was aroused by Hil- 

 bert's Gauss-Weber Festschrift of 1899: 

 'Ueber die Grundlagen der Geometric,' a 

 memoir rich in results and suggestive in 

 methods; I refer to the reviews by Som- 

 mer,* Poineare,f Halsted,J Hedriek§ and 

 Veblen.|| 



We have as a basal science logic, and as 

 depending upon it the special deductive 

 sciences which involve undefined symbols 

 and whose propositions are not all capable 

 of proof. The symbols denote either 

 classes of elements or relations amongst 

 elements. In any such science one may 

 choose in various ways the system of un- 

 defined symbols and the system of undem- 

 onstrated or primitive propositions, or pos- 

 tulates. Every proposition follows from 

 the postulates by a finite number of logical 

 steps. A careful statement of the funda- 

 mental generalities is given by Padoa in a 

 paper ^ before the Paris Congress of Phi- 

 losophy, 1900. 



Having in mind a definite system of un- 

 defined symbols and a definite system of 

 postulates, we have first of all the notion 

 of the compatibility of these postulates; 

 that is, that it is impossible to prove by a 

 finite number of logical steps the simul- 

 taneous validity of a statement and its con- 

 tradictory statement ; in the next place, the 

 question of the independence of the postu- 



* Bull. Amer. Math. Soc. (2), vol. 6 (1900), p. 

 287. 



^ Bull. Sciences MatMm., vol. 26 (1902), p. 249. 



% The Open Court, September, 1902. 



^Bull. Amer. Math. Soc. (2), vol. 9 (1902), p. 

 158. 



\\ The Monist, January, 1903. 



T[ ' Essai d'une thfiorie alggbrique des nombres 

 entiers, prgcgdfi d'une Introduction logique a une 

 theorie deductive quelconque,' Bihliotheque du 

 Congres International de Philosophie, vol. 3, p. 309. 



lates or the irreducibility of the system 

 of postulates; that is, that no postulate 

 is provable from the remaining postulates. 

 Padoa introduces the notion of the irre- 

 ducibility of the system of undefined sym- 

 bols. A system of undefined symbols is 

 said to be reducible if for one of the sym- 

 bols, X, it is possible to establish, as a log- 

 ical consequence of the assumption of the 

 validity of the postulates, a nominal or sym- 

 bolic definition of the form X = A, where 

 in the expression A there enter only the 

 undefined symbols distinct from X. For 

 the purpose of practical application, it 

 seems to be desirable to modify the defini- 

 tion so as to call the system of undefined 

 symbols reducible if there is a nominal 

 definition X = A of one of them X in 

 terms of the others such that in any inter- 

 pretation of the science the postulates re- 

 tain their validity when instead of the 

 initial interpretation of the symbol X there 

 is placed the interpretation A of that sym- 

 bol. If the system of symbols is reducible 

 in the sense of the original definition it is 

 in the sense of the new definition, but not 

 necessarily conversely, as appears for in- 

 stance from the following example, occur- 

 ring in the foundations of geometry. 



Hilbert uses the following undefined 

 symbols: 'point,' 'line,' 'plane,' 'incidence' 

 of point and line, 'incidence' of point and 

 plane, 'between,' and 'congruent.' Now 

 it is possible to give for the symbol 'plane' 

 a symbolic definition in terms of the other, 

 undefined symbols— for instance, a plane 

 is a certain class of points (as Peano showed 

 in 1892), or again, a plane is a certain 

 class of lines; while the notion 'incidence' 

 of point and plane receives convenient, 

 definition. It is apparent from the fact 

 that these definitions may be given in these 

 two ways that Hilbert 's system of unde- 

 fined symbols is not in Padoa 's sense irre- 

 ducible, at least, in so far as the symbols 

 'plane,' 'incidence' of point and plane are 



