MATHEMATICS AND METAPHYSICIANS 8i 



banishing from mathematics the use of infinitesimals, 

 has at last shown that we live in an unchanging world, 

 and that the arrow in its flight is truly at rest. 

 Zeno's only error lay in inferring (if he did infer) 

 that, because there is no such thing as a state of 

 change, therefore the world is in the same state j 

 at any one time as at any other. This is a conse- 

 quence which by no means follows ; and in this respect, 

 the German mathematician is more constructive than 

 the ingenious Greek. Weierstrass has been able, by 

 embodying his views in mathematics, where famiharity 

 with truth eliminates the vulgar prejudices of common 

 sense, to invest Zeno's paradoxes with the respectable 

 air of platitudes ; and if the result is less delightful to the 

 lover of reason than Zeno's bold defiance, it is at any 

 rate more calculated to appease the mass of academic 

 mankind. 



Zeno was concerned, as a matter of fact, with three 

 problems, each presented by motion, but each more 

 abstract than motion, and capable of a purely arith- 

 metical treatment. These are the problems of the 

 infinitesimal, the infinit e, and continuity . To state 

 clearly the difficulties mvolved, was to accomplish perhaps 

 the hardest part of the philosopher's task. This was done 

 by Zeno. From him to our own day, the finest intellects 

 of each generation in turn attacked the problems, but 

 achieved, broadly speaking, nothing. In our own time, 

 however, three men Weierstrass, Dedekind, and Cantor 

 have not merely advanced the three problems, but have 

 completely solved them. The solutions, for those ac- 

 quainted with mathematics, are so clear as to leave no 

 longer the slightest doubt or difficulty. This achieve- 

 ment is probably the greatest of which our age has to 

 boast ; and I know of no age (except perhaps the golden 



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