芝诺残篇

禅海一粟

1.
芝诺的关于运动的辩论，给那些试图解决这些问题的人，带来了极大的麻烦，他的辩论共有四条。



2.
二分法。第一条，他断言运动不存在，因为运动的物体，当它到达终点前，必须先到达全程的一半，故此还必须到达一半的一半等等，已致于无穷，所以运动是不可能的。。由此，芝诺的辩论造成了一个错误的假设，即断言物体在有限时间内从此运动到彼，或者在有限时间内来对应无限的事物，是不可能的。因为有两个方面，对于时间和长度，还有任何连续的东西，其被称为无限可分的：它们被这样称呼，在于连续物之可分割的无限性和可取端点的无限性。所以一样东西在有限的时间内不可能来对应数量上无穷的事物，它只可能由无限可分性来和无穷的事物相对应：因为就这一点，时间本身就是无限可分的，所以我们发现，时间跨度了一个无限的时间，而与其对应的是在数量上并非有限，而是无限的时间片段。




3.
阿基里斯和乌龟。第二条，是所谓的阿基里斯，在一个赛跑比赛中，跑得最快的永远追不上最慢的，因为追逐者必须先到达被追逐者的起点，而当他到了被追逐者的起点时，被追逐者又跑了一段距离了。所以被追逐者总是领先的。这个辩论原则上和上面的二分法是一样的，只是我们在这里处理的空间，不必将其一分为二而已。



4.
飞箭。第三条，即飞箭是静止的。任何东西在任何瞬间总是占据与自身相等的处所，所以是静止的。这一结果由时间是由片段组成的假设得到的：如果这一假设不成了，其结果也将不成立。



5.
运动的行队。第四条，是关于两个行队的，每个行队都包括相同的数目和大小，在一个赛场中，以相同的速度面对面地相向运动，相同时间后，其中一行占据的是终点和中点之间的空间，另一行占据的是起点和中点之间的空间。也即行行之间的距离是行与参照静止物之间距离的两倍。由此，他断言说，一个给定时间的一半和其是相等的。



6.
如果事物是叠加而成的，那么事物既是大的又是小的；大到于尺寸无穷，小到根本没有尺寸。如果事物没有尺寸，那么事物是不存在的。因为，如果没有尺寸的事物叠加，事物得不到增加，结果还是没有尺寸。反过来，如果将没有尺寸的事物从某一事物上拿去，其原理和前者是一样的，事物的尺寸没有丝毫减少。但是，如果事物是有一定尺寸的，其一部分必然有在其前的部分，以此类推，前者还有前者，以致于无穷，那么这些尺寸叠加起来，也会使事物的尺寸无穷。所以如果事物是可叠加的，它的尺寸既大又小，大到无穷，小到没有。



7.
如果事物是可叠加的，事物必须是和其本身一样多的，那么事物就是有限的。而另一方面，事物又是无限的:因为事物的两端必有中间点，而中间点和两端又有中间点，以此类推，事物有无限中间点，将事物分为无限段，故此无限段相加而得到的事物是无限的。


Zeno
1.
Zeno’sarguments about motion, which cause such trouble to those who try to solve theproblems that they present, are four in number.(Aristotle, Physics 239b 9)


2.
Thestadium. The first asserts the non-existence of motion on the ground that thatwhich is in locomotion must arrive at the half-way stage before it arrives atthe goal. (Aristotle, Physics 239b 11)


3.
HenceZeno’s argument makes a false assumption in asserting that it is impossible fora thing to pass over or severally come in contact with infinite things in afinite time. For there are two senses in which length and time and generallyanything continuous are called infinite: they are called so either in respectof divisibility or in respect of their extremities. So while a thing in afinite time cannot come in contact with things quantitatively infinite, it cancome in contact with things infinite in respect of divisibility: for in thissense the time itself is also infinite; and so we find that the time occupiedby the passage over the infinite is not a finite but an infinite time, andcontact with the infinites is made by means of moments not finite but infinitein number. (Aristotle, Physics 233a 21)


4.
Achillesand the tortoise. The second is the so-called Achilles, and in amounts to this,that in a race the quickest runner can never overtake the slowest, since thepursuer must first always hold a lead. This argument is the same in principleas that which depends on bisection, though it differs from it in that thespaces with which we successively have to deal are not divided by halves. (Aristotle,Physics 239b 14)


5.
Theflying arrow. The third is… to the effect that the flying arrow is at rest,which result follows from the assumption that time is composed of moments: ifthis assumption is not granted, the conclusion will not follow. (Aristotle,Physics 239b 30)


6.
Themoving rows. The fourth argument is that concerning the two rows of bodies,each row being composed of an equal number of bodies of equal size, passingeach other on a race-course as they proceed with equal velocity in oppositedirections, the one row originally occupying the space between the goal and themiddle point of the course and the other that between the middle point and thestarting-post. This, he thinks , involves the conclusion that half a given timeis equal to double that time. The fallacy of the reasoning lies in theassumption that a body occupies an equal time in passing with equal velocity abody that is in motion and a body of equal size that is at rest; which isfalse. (Aristotle, Physics 239b 33)


7.
Ifthere is plurality, things will be both great and small; so great as to beinfinite in size, so small as to have no size at all. If what is had no size,it would not even be. For it were added to something else that is, it wouldmake it no larger; for being no size at all, it could not, on being added,cause any increase in size. And so what was added would clearly be nothing. Againif, when it is taken away, the other thing is no smaller, just as when it isadded it is not increased, obviously what was added or taken away was nothing. Butif it is, each thing must have a certain size and bulk, and one part of it mustbe a certain distance from another; and the same argument holds about the partin front of it-it too will have some size and there wil be something in frontof it. And it is the same thing to say this once and to go on saying itindefinitely; for no such part of it will be the last, nor will one part everbe unrelated to another. So, if there is a plurality, things must be both smalland treat; so small as to have no size at all, so great as to be infinite.(Fr.1 and 2)


8.
Ifthere is a plurality, things must be just as many as they are, no more and noless. And if they are just as many as they are ,they must be limited. If thereis a plurality, the things that are infinite; for there will always be otherthings between the things that are, and yet others between those others. And sothe things that are infinite.