PRESENT PROBLEMS OF GEOMETRY 563 



example, the physical, the physiological, the psychological, the meta- 

 physical, but the interest of mathematicians has been confined to the 

 purely logical aspect. The main results in this direction are due to 

 Peano and his co-workers; but the whole field was first brought 

 prominently to the attention of the mathematical world by the 

 appearance, five years ago, of Hilbert's elegant Festschrift. 



The central problem is to lay down a system of primitive (unde- 

 fined) concepts or symbols and primitive (unproved) propositions 

 or postulates, from which the whole body of geometry (that is, the 

 geometry considered) shall follow by purely deductive processes. 

 No appeal to intuition is then necessary. " We might put the axioms 

 into a reasoning apparatus like the logical machine of Stanley Jevons, 

 and see all geometry come out of it" (Poincare). Such a system of 

 concepts and postulates may be obtained in a great (indeed end- 

 less) variety of ways: the main question, at present, concerns the 

 comparison of various systems, and the possibility of imposing lim- 

 itations so as to obtain a unique and perhaps simplest basis. 



The first requirement of a system is that it shall be consistent. 

 The postulates must be compatible with one another. No one has yet 

 deduced contradictory results from the axioms of Euclid, but what 

 is our guarantee that this will not happen in the future? The only 

 method of answering this question which has suggested itself is the 

 exhibition of some object (whose existence is admitted) which fulfills 

 the conditions imposed by the postulates. Hilbert succeeded in con- 

 structing such an ideal object out of numbers; but remarks that the 

 difficulty is merely transferred to the field of arithmetic. The most 

 far-reaching result is the definition of number in terms of logical 

 classes as given by Pieri and Russell; but no general agreement is 

 yet to be expected in these discussions. Will the ultimate conclu- 

 sion be the impossibility of a direct proof of compatibility? 



More accessible is the question concerning the independence of 

 postulates (and the analogous question of the irreducibility of con- 

 cepts). Most of the work of the last few years has been concentrated 

 on this point. In Hilbert's original system the various groups of 

 axioms (relating respectively to combination, order, parallels, con- 

 gruence, and continuity) are shown to be independent, but the dis- 

 cussion is not carried out completely for the individual axioms. In 

 Dr. Veblen's recently published system of twelve postulates, each 

 is proved independent of the remaining eleven. 1 This marks an ad- 

 vance, but, of course, it does not terminate the problem. In what 

 respect does a group of propositions differ from what is termed a 

 single proposition? Is it possible to define the notion of an absolutely 

 simple postulate? The statement that any two points determine a 

 straight line involves an infinity of statements, and its fulfillment for 

 1 Trans. Amer. Math. Soc., vol. v (1904). 



