MATHEMATICS AND LOGIC. 147 



II. 



What strikes us first of all in the new mathematics 

 is its purely formal character. " Imagine," says Hilbert, 

 " three kinds of things, which we will call points, 

 straight lines, and planes ; let us agree that a straight 

 line shall be determined by two points, and that, in- 

 stead of saying that this straight line is determined by 

 these two points, we may say that it passes through 

 these two points, or that these two points are situated 

 on the straight line." What these things are, not only 

 do we not know, but we must not seek to know. It is 

 unnecessary, and any one who had never seen either a 

 point or a straight line or a plane could do geometry 

 just as well as we can. In order that the -words pass 

 through or the words be situated on should not call up 

 any image in our minds, the former is merely regarded 

 as the synonym of be determined, and the latter of 

 determine. 



Thus it will be readily understood that, in order to 

 demonstrate a theorem, it is not necessary or even 

 useful to know what it means. We might replace 

 geometry by the reasoning piano imagined by Stanley 

 Jevons ; or, if we prefer, we might imagine a machine 

 where we should put in axioms at one end and take 

 out theorems at the other, like that legendary machine 

 in Chicago where pigs go in alive and come out trans- 

 formed into hams and sausages. It is no more neces- 

 sary for the mathematician than it is for these machines 

 to know what he is doing. 



I do not blame Hilbert for this formal character of 

 his geometry. He was bound to tend in this direction, 

 given the problem he set himself. He wished to reduce 



