March 13, 1903.] 



SCIENCE. 



40;3 



coucerued— while it is equally clear that 

 these symbols are in the abstract geometry 

 superfluous. 



In his dissertation on euclidean geom- 

 etry, Mr. Veblen, following the example 

 of Paseh and Peano, takes as undefined 

 sjTubols 'point' and 'between,' or 'point' 

 and 'segment.' In terms of these two 

 symbcls alone he expresses a set of inde- 

 pendent fundamental postulates of euclid- 

 ean geometry, in the first place develop- 

 ing the projective geometry, and then as 

 to congruence relating himself to the 

 point of view of Klein in his 'Erlangen 

 Programm, ' whereby the group of move- 

 ments of euclidean geometry enters as 

 a certain subgroup of the group of col- 

 liueations of projective geometry. Here 

 arises an interesting question as to the 

 sense in whitli the undefined symbol 'con- 

 gnience' is superfluous in the euclidean 

 geometry based upon the symbols 'point,' 

 'between.' One sees at once that a defini- 

 tion of 'eongnienee' involves parametric 

 points in its expression, while on the other 

 hand a definition of the sy.stem of all 

 'planes,' that is, of the general concept 

 'plane,' involves no such parametric ele- 

 ments. But, again, just as there exist dis- 

 tinct definitions of 'congruence,' owing to 

 a variation of the parametric points, so 

 there exist distinct definitions of the gen- 

 eral concept 'plane,' as was indicated a 

 moment ago. One has the feeling that the 

 state of affairs must be as follows : In any 

 interpretation of, say, Hilbert's symbols, 

 wherein the postulates of Hilbert are valid, 

 every valid statement which does not in- 

 volve the symbol 'plane' in direct connec- 

 tion with the general logical symbol (=) 

 of symbolic definition, remains valid when 

 we modify it in accordance with either of 

 the definitions of 'plane' previously re- 

 ferred to. On the other hand, this state 

 of affairs does not hold for the symbol 

 'congruence.' The proof of the former 



statement would seem to involve funda- 

 mental logical niceties. 



The compatibility and the independence 

 of the postulates of a system of postulates 

 of a special deductive science have been 

 up to this time always made to depend 

 upon the self-consistency of some other de- 

 ductive science ; for instance, geometry de- 

 pends tlras upon analysis, or analysis upon 

 geometry. The fundamental and still un- 

 solved problem in this direction is that of 

 the direct proof of the compatibility of the 

 postulates of arithmetic, or of the real 

 number system of analysis. ( To the society 

 this morning Dr. Huntington exhibited two 

 sets of independent postulates for this real 

 number system. ) This is the second of the 

 twenty-three problems listed by Hilbert in 

 his address before the Paris Mathematical 

 Congress of 1900. 



The Italian writers on abstract mathe- 

 matics for the most part make use of 

 Peano's symbolismT One may be tempted 

 to feel that this symbolism is not an essen- 

 tial part of their work. It is only right 

 to state, however, that the symbolism is 

 not difficult to learn, and that there is 

 testimony to the effect that the symbolism 

 is actually of great value to the investigator 

 in removing from attention the concrete 

 connotations of the ordinaiy terms of gen- 

 eral and mathematical language. But of 

 course the essential difficulties are not to 

 be obviated by the use of any symbolism, 

 however delicate. 



Indeed the question arises whether the 

 abstract mathematicians in making precise 

 the metes and bounds of logic and the spe- 

 cial deductive sciences are not losing sight 

 of the evolutionary character of all life- 

 processes, whether in the individual or in 

 the race. Certainly the logicians do not 

 consider their science as something now 

 fixed. All science, logic and mathematics 

 included, is a function of the epoch— all 

 science, in its ideals as well as in its achieve- 



