MATHEMATICS AND METAPHYSICIANS 89 



if everything has been taken, there is nothing left to add. 

 Cantor has a proof that there is no greatest number, and 

 if this proof were valid, the contradictions of infinity 

 would reappear in a sublimated form. But in this one 

 point, the master has been guilty of a very subtle fallacy, 

 which I hope to explain in some future work. 1 



We can now understand why Zeno believed that Achilles 

 cannot overtake the tortoise and why as a matter of fact 

 he can overtake it. We shall see that all the people who 

 disagreed with Zeno had no right to do so, because they 

 all accepted premises from which his conclusion followed. 

 The argument is this : Let Achilles and the tortoise start 

 along a road at the same time, the tortoise (as is only 

 fair) being allowed a handicap. Let Achilles go twice as 

 fast as the tortoise, or ten times or a hundred tunes as 

 fast. Then he will never reach the tortoise. For at every 

 moment the tortoise is somewhere and Achilles is some- 

 where ; and neither is ever twice in the same place while 

 the race is going on. Thus the tortoise goes to just as 

 many places as Achilles does, because each is in one place 

 at one moment, and in another at any other moment. 

 But if Achilles were to catch up with the tortoise, the 

 places where the tortoise would have been would be only 

 part of the places where Achilles would have been. Here, 

 we must suppose, Zeno appealed to the maxim that the 

 whole has more terms that the part. 2 Thus if Achilles were 



1 Cantor was not guilty of a fallacy on this point. His proof 

 that there is no greatest number is valid. The solution of the puzzle 

 is complicated and depends upon the theory of types, which is explained 

 in Principia Mathematica, Vol. I (Camb. Univ. Press, 1910). [Note 

 added in 1917.] 



2 This must not be regarded as a historically correct account of 

 what Zeno actually had in mind. It is a new argument for his con- 

 clusion, not the argument which influenced him. On this point, see 

 e.g. C. D. Broad, "Note on Achilles and the Tortoise," Mind, N.S., 

 Vol. XXII, pp. 318-19. Much valuable work on the interpretation of 

 Zeno has been done since this article was written. [Note added in 191 7.] 



