374 
ZENO OF ELEA 
one and all sophisms. After two thousand years of continual 
refutation, these sophisms were reinstated, and made the 
foundation of a mathematical renaissance, by a German 
professor who probably never dreamed of any connexion 
between himself and Zeno. Weierstrass, by strictly banishing 
all infinitesimals, has at last shown that we live in an 
unchanging world, and that the arrow, at every moment of its 
flight, is truly at rest. The only point where Zeno probably 
erred was in inferring (if he did infer) that, because there 
is no change, the world must be in the same state at one time 
as at another. This consequence by no means follows, and in 
this point the German professor is more constructive than the 
ingenious Greek. Weierstrass, being able to embody his 
opinions in mathematics, where familiarity with truth elimi 
nates the vulgar prejudices of common sense, has been able to 
give to his propositions 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.’ 1 
Thus, while in the past the arguments of Zeno have been 
treated with more or less disrespect as mere sophisms, we have 
now come to the other extreme. It appears to be implied that 
Zeno anticipated Weierstrass. This, I think, a calmer judge 
ment must pronounce to be incredible. If the arguments of 
Zeno are found to be ‘ immeasurably subtle and profound ’ 
because they contain ideas which Weierstrass used to create 
a great mathematical theory, it does not follow that for Zeno 
they meant at all the same thing as for Weierstrass. On the 
contrary, it is probable that Zeno happened upon these ideas 
without realizing any of the significance which Weierstrass 
was destined to give them; nor shall we give Zeno any less 
credit on this account. 
It is time to come to the arguments themselves. It is the 
four arguments on the subject of motion which are most 
important from the point of view of the mathematician; but 
they have points of contact with the arguments which Zeno 
used to prove the non-existence of Many, in refutation of 
those who attacked Parmenides’s doctrine of the One. Accord 
ing to Simplicius, he showed that, if Many exist, they must 
1 Bertrand Russell, The Principles of Mathematics, vol. i, 1903, pp. 
347, 348.