8o MYSTICISM AND LOGIC 



used to be at least as full of doubt as any other part of 

 philosophy, order and certainty have replaced the con- 

 fusion and hesitation which formerly reigned. Philo- 

 sophers, of course, have not yet discovered this fact, and 

 continue to write on such subjects in the old way. But 

 mathematicians, at least in Italy, have now the power of 

 treating the principles of mathematics in an exact and 

 masterly manner, by means of which the certainty of 

 mathematics extends also to mathematical philosophy. 

 Hence many of the topics which used to be placed among 

 the great mysteries for example, the natures of infinity, 

 of continuity, of space, time and motion are now no 

 longer in any degree open to doubt or discussion. Those 

 who wish to know the nature of these things need only 

 read the works of such men as Peano or Georg Cantor ; 

 they will there find exact and indubitable expositions of 

 all these quondam mysteries. 



In this capricious world, nothing is more capricious 

 than posthumous fame. One of the most notable examples 

 of posterity's lack of judgment is the Eleatic Zeno. This 

 man, who may be regarded as the founder of the philo- 

 sophy of infinity, appears in Plato's Parmenides in the 

 privileged position of instructor to Socrates. He invented 

 four arguments, all immeasurably subtle and profound, 

 to prove that motion is impossible, that Achilles can 

 never overtake the tortoise, and that an arrow in flight 

 is really at rest. After being refuted by Aristotle, and 

 by every subsequent philosopher from that day to our 

 own, these arguments were reinstated, and made the 

 basis of a mathematical renaissance, by a German pro- 

 fessor, who probably never dreamed of any connection 

 between himself and Zeno. Weierstrass, 1 by strictly 



1 Professor of Mathematics in the University of Berlin. He died in 

 1897. 



