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From: kohout@drinkme.cs.umd.edu (Robert Kohout)
Newsgroups: comp.ai.philosophy
Subject: Re: Continuous vs discrete
Message-ID: <32913@mimsy.umd.edu>
Date: 13 Apr 91 19:13:07 GMT
Article-I.D.: mimsy.32913
References: <382@batz.enst-bretagne.fr> <1991Apr10.182036.29916@beaver.cs.washington.edu> <383@zeus.enst-bretagne.fr>
Sender: news@mimsy.umd.edu
Reply-To: kohout@drinkme.cs.umd.edu (Robert Kohout)
Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742
Lines: 45

In article <383@zeus.enst-bretagne.fr> beugnard@zeus.enst-bretagne.fr (Antoine Beugnard) writes:
>
>
>In article <382@batz.enst-bretagne.fr> beugnard@batz.enst-bretagne.fr (Antoine Beugnard) writes:
>>>
>
>>What this ignores, of course, is the fact that some infinite series
>>have a finite sum.  Zeno's "paradox" may have baffled an ancient
>>Greek, but it shouldn't fool anyone who knows calculus.  The proper
>>conclusion to draw in the Achilles and Turtle scenario is:  "Achilles
>>never reaches the Turtle, until he does."
>
>We are aware of that. The problem is that **a limit is NEVER reached**.
>Zeno's model never becomes wrong...and Achilles never reaches the Turtle because
>the limit cannot be reached. The model is actually troncated both in time and
>space, that is T(n) and time(n) have limits that are never reached...
>

What are you talking about? What do you mean **a limit is NEVER reached**.
Is that supposed to be some property of limits? When we say, "the limit
of f(x), as x approaches infinity = Z", we don't mean "if only we could
ever get there". for example

0.1 + 0.01 + 0.001 + 0.0001 + 0.00001 .... = 1/9 .

The sum EQUALS 1/9. '=' does NOT mean "would be 1/9 if we could ever add
all of these things up." IF the hare travels 10 times as fast as the tortoise,
each summand (is that the word) of Zeno's paradox describes a distance 
1/10 as large as the previous distance, and/or a time 1/10 the size of
the previous increment. This sum has a limit, AND THEREFORE THE LIMIT IS
REACHED!!! (See, I can emphasize too)

The ancient Greeks may have had problems adding up infinitely many things,
but just because one can break a quantity into infinitely many parts
in no way implies that the quantity is infinite. For example, the
number 1 can be written 0.9999....., which is just 0.9 + 0.09 + 0.009 +
0.0009+.... ad infinitum. 1 is not for this reason infinite.

I fear I must be missing your point. Emphasizing "a limit is NEVER reached"
makes it no less opaque. Knowing Zeno's paradox, you must realize that the
limit is reached every time you overtake a slower moving object. Since
this seems to be the crux of your argument, please clarify what you 
mean by asserting that "a limit is NEVER reached".

Bob Kohout