S0M1£ PROBLEMS OF Pin-.SKAL CONTINUITN'. I73 



tion of Zeno's argument can Achilles and the tortoise meet. But 

 the possible points of meeting are by no means exhausted by 

 thinking of the points where the tortoise stops before it is over- 

 taken. Outside the large number of such points there may be 

 one or more where the tortoise is actually overtaken. 



The whole argument is vitiated by the implicit refusal to 

 consider such a point possible. 



This can best be illustrated fsays Mr. Broadj liy considering a series 

 of numbers instead of one of points, and the real relation of "greater 

 than" instead of that of " lieyond.'' Consider the series whose general term 



is 2 — — — — - where 11 can l)c any integral value including o. It is clear 



That its first term is i. It is further clear that it has an infinite (i.e., 

 indefinite) number of terms. Finally, 2 is greater than every term of the 

 series. Hence if we had an analogous proposition to that assumed by the 

 supporters of the Achilles, we should have to say: "2 is infinitely greater 

 than I, for it is infinitely greater than every term of an infinite series 

 whose first term is i." Tlie obvious absurdity of this shows the absurdity 

 of the implicit premises without which the Achilles cannot draw its con- 

 clusions. 



There is, therefore, no sound reason to hold that our 

 imagination deceives itself or contradicts the higher judgment of 

 reason in holding that the continuum is both a reality and in- 

 finitely divisible. 



But I am inclined to be grateful to Zeno for the worry that 

 he has caused all the philosophers by ineans of this ingenious 

 argument. He has taught us to sound some of the hidden 

 depths of simple concepts. Or, to put it in the words of Mr. 

 Wm. James* Zeno. 



Gives a dramatic character to llie difficulty inherent in understanding 

 intellectually any phenomenon whatsoever of continuous change. 



The difficulty applies not only to continuous change, but to every 

 species of continuous quantity. 



But when we Ijcgin to probe into the nature of physical 

 continuity, we are faced with many of the special difficulties of 

 the infinite. If the continuum were a reality, we are told by the 

 old Greek sophist, you would require an infinite time to traverse 

 it. For it can be divided into an infinity of points ; and no 

 matter how small the period of time required to pass one of 

 these i:)oints, the time required to pass the whole series would be 

 infinite. Which is absurd : and so also is the notion of con- 

 tinuity into which the eye and the finger decov us. 



The solution of this purely logical knot is nearly as old as 

 the original difficulty. Aristotle, i who ])reserved the contuidruni, 

 has also furnished us with the simplest solution of it: "Now it 

 is not possible to touch things infinite in regard to number in 

 an infinite time, but it is possible to touch things infinite in regard 

 to divisibility; for time itself is also infinite in this sense. So 

 that, in fact, we go through an infinite (space") in an infinite 



* Article on "The Philosophy of Bergson." in Hibbcrf .Journal. April, 

 jgcq. 



t" Physics," 6 [2]. 



