17-2 .S(;MK I'ROiiLli.M.S ()]•■ I'M \ SUM. CONTINUITY. 



ficies, a moving superficcs produces a l)ody, and a moving " now " pro- 

 duces time. Tiiese things lieing so produced and imagined, though 

 indeed they do not so talsc place in reality, we grasp the aforesaid defini- 

 tion. For if a moving point produces a h'ne, all the parts of the line an- 

 united hy the point. And since in every i)art of tb.e line we can in this 

 way imagine a point, to which apart from all other considerations any 

 other particle is constantly related, hence the line is called continuous. 



Yet this fundamental, and apparently simple, idea of con- 

 tinuity is full of dialectical difficulties for the philosopher. The 

 most ancient of the prohlems of physical continuity is one which 

 still puzzles the logicians. Is the continuum in physics a reality, 

 or just a delusion of the senses? More than 2,400 years ago 

 Zeno of h^lea appears to have i)Ut forward a series of argument.^ 

 that comljated the reality of motion and multiplicity. What 

 Zeno's personal sentiments were W'e cannot now discover, since 

 the only records of his ojHnions are in the words of hostile 

 critics — Plato and .Vristotle. 



But one of his alleged arguments deals a clever blow at the very 

 notion of continuity. It is the well-worn paradox of Achilles 

 and the Tortoise, which every novice in " Logic "' has at some 

 time set liimself to answer. .Vnd the sophism has received a new 

 lease of life in our day on acct)unt of the respectftil agreement 

 with which it has been revived liy Bergson and his followers. 



Let us suppose, Zeno might liave said, that there is such a 

 thing as continuous fjuantity, say a racecourse where Achillea, 

 the cham])ion runner, shall race with the tortoise. Give the latter 

 the least possible advantage of a start in the race. Every time 

 that Achilles reaches the ])oint wliere his slow cotiipetitor is, the 

 latter will have moved on a little further. True enough, the tor- 

 toise will have gained less and less over Achilles ; but since, t'.i 

 hypothcsi couununi all continuous s]x-ice is infinitely divisible, it 

 will take Achilles an infinite time to win the race, i.e.. he will 

 never win it. And all because yoit have sitpposed an absurd 

 thing — that continuous (|uantit}' is a real and not an imaginary 

 thing. It is, in fact, a conflict of intellect and imagination, and 

 one must check the fancy by the straight rules of Logic. 



Perhaps it might be better to overhaul the logical apparatu.> 

 of the argument, in order to make stire that everythintr is quite 

 in accordance with reason. It is ([uite clear that at no possible 

 point postulated in the premises of this jjlausible argitment can 

 Achilles overtake the tortoise, and, on the other hand, there is 

 an infinite number of such points postulated. Outside infinity. 

 where is one to find new i)oint> to save the credit of .\chilles? 



The most coniplete reply to this accumulation of sophisms 

 that I have seen, was given by Mr. C. D. Broad.* two years ago. 

 He begins b\- pointing out that you have not necessarily ex- 

 hausted all the points in a series, because you take an infinite 

 number of them. You might, e.g., take an infinite number of 

 even numbers, and leave the etjually infinite number of odd num- 

 bers. It is true enough that at no point given in the construc- 



* In Mind. April, 1913, 



