biep@klipper.UUCP (05/20/87)
This is a text I sent to those who asked me about a remark I made about Zeno's paradoxes and their intention. It has already been posted to sci.math and sci.physics, but I am reposting it here in hope of stirring up some discussion. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= About the (really recommended) reference: Gerald J. Whitrow "The natural philosophy of time", Thomas Nelson and Sons Ltd., London In my (arrgh) *abridged* *translation* (all I could get), it is Chapter III, sections 4 and 5. Whitrow is I think the greatest philosopher on the subject of time, and he has the nice property of always first giving a fairly complete overview of what other philosophers have written on a subject, before he gives his own ideas. He also tries to give merit to all viewpoints. The paradoxes: (in my own wordings, and as far as I understand them. The great Genii themselves do not always agree on what Zeno has meant) A There is no smallest time interval: (so let's suppose there is one) 1) Suppose one shoots an arrow. "During" every smallest time interval it has to move a bit, i.e. at every new point in time it has to have a different position. If we put numbered dots on the arrow, so that the amount of movement is just the distance between two dots, we get the following image: t=n 1 2 3 4 5 6 ... t=n+1 1 2 3 4 5 6 ... Now suppose another arrow is passing in the other direction, and at t=n they are just one above the other. We get: A:t=n 1 2 3 4 5 6 ... A:t=n+1 2 3 4 5 6 7 ... B:t=n 1 2 3 4 5 6 ... B:t=n+1 1 2 3 4 5 ... Saying: A:t=n+1 2 3 4 5 6 7 ... A/B:t=n 1 2 3 4 5 6 ... B:t=n+1 1 2 3 4 5 ... Which shows that the situation A: 2 3 4 5 6 B: 1 2 3 4 5 is missing. Hence, the time interval chosen is not the shortest. This one seems solvable. The next one is very vague, in fact it doesn't even seem to handle about time: 2) If I drop one grain of wheat, I hear nothing. If I drop a whole bunch of it, I do hear something. Aristotle just threw this onbe away with the comment "the human ear is not perfect. One grain of wheat does make a sound, but we cannot hear it". Many others since have said Aristotle didn't understand the paradox. Some claim it was meant allegorically: if at each point of time there is no movement (no sound), how can there be if I just take a lot of time points together. This explanation does seem to make sense (at least it now handles about time), but also sounds rather solvable. B: Time is not continuous: (so let's claim it is) 1) Achilles can never overtake (is that the word?) the tortue, since in order to do so he has to do an infinite number of things first. (i.e getting at time t(i) where the tortue was at time t(i-1)) Zeno happily agrees that *if* Achilles could do it, he would be able to do it in a finite time (e.g. if V(A)=10V(T), and S(T,0)=S(A,0)+1 then S(T,1/9) = S(A, 1/9)), but claims ce cannot do it. Of course just pointing at infinitesimals in math doesn't make this problem go away (in fact, mathematicians have had a hard time with this, till they decided it was a physical problem, and they would just *postulate* it was OK. A pity that, as far as I know, the physicists have never seriously taken it up again, also mainly because many of them thought (and still think) the mathematicians have solved it already) 2) The dichotomy problem: much like the above: a movement can never start because in order to move from A to B you have first to move from A to (B-A)/2. The endless recursion in this definition makes this problem more understandable to computer scientists that A&T, perhaps. The main difference between the two is, that while A can get as close to T as he wants to, i.e. an approximation of arbitrary precision can be reached, in the dichotomy problem one doesn't even get started, let alone approach a solution. Also, while the A&T problem talks about a limit lim n , which is countable n -> oo the dichotomy problem talks about n lim 2 , which is uncountable n -> oo Biep biep@cs.vu.nl (via mcvax) -- Biep. (biep@cs.vu.nl via mcvax) Never confound "power", "command" with "right", especially not when it concerns our own body!