1. Sep 7, 2004

### Tom McCurdy

Today we were talking about how the greek philosphers came up with many different theories about space. We got talking about how could a person ever get to the other side of the room if he kept walking half the distance to the other side. I said you would run into a problem because you would get to plank's length and you couldn't go any smaller. The teacher than said I should look into Zenos paradox and I found a site on it and it comes in 4 different parts
I will post all four parts sepratly along with the sites given solution
the source for the first one is
http://mathforum.org/isaac/problems/zeno1.html
the site has a lot of good puzzles

The answer is in white highlight if you would like to see it
note it is from a differnt site but it should be adapted easily ennough

A couple of common responses are not adequate. One might -- as Simplicius ((a) On Aristotle's Physics, 1012.22) tells us Diogenes the Cynic did by silently standing and walking -- point out that it is a matter of the most common experience that things in fact do move, and that we know very well that Atalanta would have no trouble reaching her bus stop. But this would not impress Zeno, who as a paid up Parmenidean held that many things are not as they appear: it may appear that Diogenes is walking or that Atalanta is running, but appearances can be deceptive and surely we have a logical proof that they are in fact not moving at all. And if one doesn't accept that Zeno has given a proof that motion is illusory -- as we hopefully do not -- one then owes an account of what is wrong with his argument: he has given reasons why motion is impossible, and so an adequate response must show why those reasons are not sufficient. And it won't do simply to point out that there are some ways of cutting up Atalanta's run -- into just two halves, say -- in which there is no problem. For if you accept all of the steps in Zeno's argument then you must accept his conclusion (assuming that he has reasoned in a logically deductive way): it's not enough to show an unproblematic division, you must also show why the given division is unproblematic.

Another response -- given by Aristotle himself -- is to point out that as we divide the distances run we should also divide the total time taken: there is 1/2 the time for the final 1/2, a 1/4 of the time for the previous 1/4, an 1/8 of the time for the 1/8 of the run and so on. Thus each fractional distance has just the right fraction of the finite total time for Atalanta to complete it, and thus the distance can be completed in a finite time. Aristotle felt that this reply should satisfy Zeno, however he also realized (Physics, 263a15) that this could not be the end of the matter (and surely Zeno would have made the same point if presented with Aristotle's response). For now we are saying that the time Atalanta takes to reach the bus stop is composed of an infinite number of finite pieces -- …, 1/8, 1/4, and 1/2 (of the total time) -- and isn't that an infinite time?

Of course, one could again claim that some infinite sums in fact have finite totals, and in particular that the sum of these pieces is 1 × the total time, which is of course finite (and again a complete solution would demand a rigorous account of infinite summation, like Cauchy's). However, Aristotle did not make such a move. What he said is worth noting because it had a considerable influence on later thinking about Zeno. In his response Aristotle drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts. Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are two distinct things: and that the later is only ‘potentially’ derivable from the former. Next, Aristotle takes the common-sense view that time is like a geometric line, and considers the time it takes to complete the run. We can again distinguish the two cases: on the one hand there is the continuous run from start to finish, and on the other there is the run divided into Zeno's infinity of half-runs. The former is ‘potentially infinite’ in the sense that it could be divided into latter ‘actual infinity’. Here's the crucial step: Aristotle thinks that since these times are geometrically distinct they must be physically distinct. But how could that be? He claims that the runner must do something at the end of each half-run to make it distinct from the next: she must stop. (Why stop rather than cough or something? Because if the time is discontinuous then so is the motion.) And so Aristotle's full answer to the paradox is that Zeno's question -- whether the infinite series of runs is possible or not -- is ambiguous. One the one hand, the answer is ‘yes’ if one means the potentially infinite series that form the continuous run. On the other the answer is ‘no’ if one means the actual infinity of pieces that form the discontinuous run.

It is hard -- from our modern perspective perhaps -- to see how this answer could be completely satisfactory. In the first place it assumes that a clear distinction can be drawn between potential and actual infinities, something that was never fully achieved. Second, suppose that Zeno's problem turns on the claim that infinite sums of finite quantities are invariably infinite. Then Aristotle's distinction will only help if he can explain why potentially infinite sums are in fact finite (and couldn't I potentially add 1 + 1 + 1 + …, which does not have a finite total); or if he can give a reason why potentially infinite sums just don't exist. Or perhaps Aristotle did not see infinite sums as the problem, but rather whether completing an infinity of finite actions is metaphysically and conceptually and physically possible, an idea discussed at length in recent years: see ‘Supertasks’ below. In this case we need an account of actions that makes precise the sense in which the continuous run is indeed a single action (using rest to individuate motions seems problematic, for humans are probably never completely still, and yet we perform distinct motions -- breathing, eating, skipping and so on). Finally, the distinction between potential and actual infinities has played no role in mathematics since Cantor tamed the transfinite numbers -- certainly the potential infinite has played no role in the modern mathematical solutions discussed here.

One last point: Zeno's argument seeks most obviously to establish the impossibility of motion, but he also intended it (and the following arguments) as further refutations of plurality -- certainly, Plato interprets Zeno's intentions in this way. How might the argument seek to establish this conclusion? Presumably Zeno has in mind the view that spatial (and perhaps temporal) distances have a plurality of parts; parts which are infinitely divisible into two. Given that assumption, supposedly finite distances (or times) can be decomposed into an infinity of finite parts with no first (or alternatively, last) one. And how can such distances be finite after all? And if the pluralist also believes in motion, how can such a distance be traversed? It seems it cannot be.