CONCEPTIONS AND METHODS OF MATHEMATICS 461 



sary conclusions to be drawn? The answer clearly implied is, from 

 any premises sufficiently precise to make it possible to draw neces- 

 sary conclusions from them. In geometry, for instance, we have a 

 large number of intuitions and fixed beliefs concerning the nature 

 of space: it is homogeneous and isotropic, infinite in extent in every 

 direction, etc.; but none of these ideas, however clearly defined 

 they may at first sight seem to be, gives any hold for exact reasoning. 

 This was clearly perceived by Euclid, who therefore proceeded to 

 lay down a list of axioms and postulates, that is, specific facts which 

 he assumes to be true, and from which it was his object to deduce all 

 geometric propositions. That his success here was not complete 

 is now well known, for he frequently assumes unconsciously further 

 data which he derives from intuition; but his attempt was a monu- 

 mental one. 



III. The Abstract Nature of Mathematics 



Now a further self-evident point, but one to which attention seems 

 to have been drawn only during the last few years, is this: since we 

 are to make no use of intuition, but only of a certain number of 

 explicitly stated premises, it is not necessary that we should have 

 any idea what the nature of the objects and relations involved in 

 these premises is. 1 I will try to make this clear by a simple example. 

 In plane geometry we have to consider, among other things, points and 

 straight lines. A point may have a peculiar relation to a straight 

 line which we express by the words, the point lies on the line. Now 

 one of the fundamental facts of plane geometry is that two points 

 determine a line, that is, if two points are given, there exists one and 

 only one line on which both points lie. All the facts that I have just 

 stated correspond to clear intuitions. Let us, however, eliminate our 

 intuition of what is meant by a point, a line, a point lying on a line. 

 A slight change of language will make it easy for us to do this. In- 

 stead of points and lines, let us speak of two different kinds of objects, 

 say A-objects and ^-objects; and instead of saying that a point 

 lies on a line we will simply say that an ^-object bears a certain 

 relation R to a 5-object. Then the fact that two points determine 

 a line will be expressed by saying: If any two A -objects are given, 

 there exists one and only one 5-object to which they both bear the 

 relation R. This statement, while it does not force on us any specific 

 intuitions, will serve as a basis for mathematical reasoning 2 just as 

 well as the more familiar statement where the terms points and lines 



1 This was essentially Kempe's point of view in the papers to be referred to 



Eresently. In the geometric example which follows it was clearly brought out 

 y H. Wiener: Jahresbericht d. deutschen Mathematiker-Vereinigung, vol. I (1891), 

 p. 45. 



2 In conjunction, of course, with further postulates with which we need not 

 here concern ourselves. 



