發信人: abracadabra.bbs@aidebbs.edu.tw (abracadabra philosopher), 看板: philosophy
標  題: 時間與運動的詭論
發信站: DCI HiNet (Mon Aug 31 13:52:53 1998)
轉信站: fhl-bbs!news.seed.net.tw!feeder.seed.net.tw!news.ntu!news.mcu!news.cs.


古希臘哲學家 Zeno 的 Paradoxes:

1.運動員永遠不能抵達賽跑的終點
2.飛行中的箭, 事實上是不動的

這兩個詭論, 對於後世的數學、科學、哲學的發展, 都產生了重大的
思想衝擊。他的詭論, 一直要到近兩千年之後, 由于人類對「時間」
和「運動」這兩個概念的本質, 能夠作出更精緻的詮釋下, 才得以破
解。

中國古代墨學支流的別墨名家, 也曾經提出非常類似的詭論:

1.一尺之棰, 日取其半, 萬世不竭
2.飛鳥之影, 未嘗動也
3.今日適越而昔來

可惜由於文獻的湮滅和學術的失傳, 所以中國哲學無法發展出足以匹
敵西方哲學的精緻思維。


quote:
----------------------
What is the solution to Zeno's paradoxes? 

     In about 445 B.C., Zeno offered several arguments that led 
to conclusions contradicting what we all know from our physical 
experience.  The paradoxes had a dramatic impact upon the later 
development of mathematics, science, and philosophy. Zeno argued 
that a runner will never reach the goal line because he first 
must have time to reach the halfway point to the goal, but after 
arriving there he will need time to get to the 3/4 point, then 
the 7/8 point, and so forth.  If the distance to the goal is 1, 
then the runner must cover a distance of 1/2 + 1/4 + 1/8 + .... 
Zeno believed this sum was infinite and concluded that the run-
ner will never have the infinite time it takes to reach the in-
finitely distant goal.  Because at any time there is always 
more time needed, motion can never be completed.

A similar paradox by Zeno shows that motion can never be initi-
ated either. Consider a runner's first step. Any step may be 
divided into a first half and a second half.  Before taking 
a full step, the runner must have time to take a 1/2 step, but 
before that a 1/4 step, and so forth. The runner will need an 
infinite amount of time just to take a first step, and so will 
never get going. Zeno's arrow paradox takes a different approach 
to challenging the coherence of the concepts of time and motion.
Consider one instant of an arrow's flight. For the entire instant 
the arrow occupies a region of space equal to its total length, 
so at that instant the arrow isn't moving.  If at every instant 
the arrow isn't moving, then the arrow can't move.  Yet another 
argument of Zeno's attacks the notion of plurality. Consider a 
duration of one second. It can be divided into two non-
overlapping parts. They, in turn, can be divided, and so on. 
At the end of this infinite division we reach the elements. If
these elements have zero duration, then adding an infinity of 
zeros yields a zero sum, and the total duration is zero seconds. 
Alternatively, if that infinite division produced elements having 
a finite duration, then adding an infinite number of these 
together will produce an infinite duration. So, a second lasts 
either for no time at all or else for an infinite amount of time.
 
     Zeno's paradoxical arguments have the form logicians call 
"reductio ad absurdum." They are valid, given his assumptions 
about space, time, motion and mathematics, and they revealed the 
underlying incoherence in ancient Greek thought, an incoherence 
that wouldn't be resolved until two millennia later. The way out 
of Zeno's paradoxes requires revising the concepts of duration, 
distance, instantaneous speed, and sum of a series. The relevant 
revisions were made by Leibniz, Newton, Cauchy, Weierstrass, 
Dedekind, Cantor, and Lebesque over two centuries. The notion of 
infinite sums of numbers had to be revised so that an infinite 
series of numbers that decrease sufficiently rapidly can have a 
finite sum. Although 1/2 + 1/3 + 1/4 +... is infinite, the more 
rapidly decreasing series 1/2 + 1/4 + 1/8 +... is 1. The other 
key idea was to appreciate that durations and distances must be 
topologically like an interval of the linear continuum, a dense 
ordering of uncountably many points.  Although individual points 
of the continuum have zero measure ('total length'), the modern 
notion of measure on the linear continuum does not allow the 
measure of a segment (continuous region) to be the sum of the 
measures of its individual points, as Zeno had assumed in his 
argument against plurality. The new notions restore the coherence 
of mathematics and science with our experience of space and time, 
and they are behind today's declaration that Zeno's arguments are 
based on naive and false assumptions. 
---------------------------
unquote.

來源: http://www.utm.edu/research/iep/t/time.htm#ZENO



abracadabra