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Newsgroups: comp.ai.philosophy
Subject: Re: continuous vs discrete
Message-ID: <1991Apr11.230145.16643@fivegl.co.nz>
From: carl@fivegl.co.nz (Carl Reynolds)
Date: 11 Apr 91 23:01:45 GMT
Reply-To: carl@fivegl.co.nz (Carl Reynolds)
References: <381@sein.enst-bretagne.fr>
Organization: 5GL International Ltd, Auckland, New Zealand
Lines: 112

(Long article)
In article <381@sein.enst-bretagne.fr> beugnard@sein.enst-bretagne.fr (Antoine Beugnard) writes:
>
>Is Zenon paradox a proof of the world discontinuity??
>
>We give a personal interpretation of the Zenon paradox leading to a strange
>conclusion about our universe. But we may certainly have missed something, could
>you please help us?
>
>The paradox...
>Achilles and a turtle are about to run a race. Obviously as Achilles can run
>faster than the turtle, he decides to let her start N meters ahead. They start 
>running at the same time, but Achilles never reaches the turtle...
>
>Zenon explanation follows with modern mathematical notation:
>
>	T(n) denotes the successive locations of Turtle.
>	A(n) denotes the successive locations of Achilles.
>
>When Achilles reaches the previous Turtle location, she has gone forward.
>d(n) denotes the distance she has covered while Achilles was reaching her 
>previous location. So,
>
>	A(0) = 0 and A(n) = T(n-1) defines A(n).
>	T(0) = N and T(n) = T(n-1) + d(n) defines T(n).
>
>According to Zenon, d(n) is positive then Achilles never reaches the Turtle.
>
>We may calculate d(n) assuming constant velocities for Achilles (Va) and for the
>Turtle (Vt). We introduce B = (Vt/Va) with B < 1. Then,
>
>	d(n) = N.B**(n+1).
>
>d(n) is strictly positive,...QED
>
>Assuming all the hypotheses we made are true, Achilles never reaches the Turtle.
>
>But in every day life, we can notice that Achilles overtakes her.
>
>We do not call into question the mathematical modelling. So where is the 
>problem? Most people would say our model is wrong since it does not describe the
>real world.
>We do ask why it is wrong!!
>
>Let us  sumarize:
>- given a set of hypothesis H we deduce a conclusion C wich is not true.
>- two interpretations are allowed:
>	- one of the hypotheses is wrong,
>	- our deduction is erroneous.
>- Assuming our deduction is valid, we are lead to call in question one hypothesis.
>
>The hypotheses are:
>	- The constant velocities...axiom (why not?)
>	- Vt < Va ... 
>	- The world is continuous.
>
>The last one may seem weird, but let us explain it. Assuming the world is 
>continuous (this hypothesis is too strong, We could just use the fact that 
>between two points (time or space ?) there is always another one (see Rational 
>numbers)), we can find a unit in which B is a real number. Then d(n) is a real 
>number and d(n) is an infinite sequence of real numbers, and Achilles never 
>reaches the Turtle, which does not match the real world!
>
>Therefore the world is discontinuous !! No ??
>

Unfortunately this is not a proof when one considers the time element.
As Achilles approaches the turtle (assuming that Achilles moves at a constant
velocity),

   let X(n) be the time taken between successive locations of Achilles
   i.e. X(n) = Time taken between A(n) and A(n+1)

Now X(n) will gradually decrease as Achilles moves closer and closer to the
turtle. If the Zenon paradox held, then the point in time when Achilles
actually reached the turtle would never, could never, occur! However in our
real world, time SEEMS continuous, and we certainly never have this problem.

And, of course, any decreasing (as happens in our world) sequence of X(n)
will eventually converge to a value.
This is the theory of limits, where an infinite sequence converges to a finite
sum. And remember, the real number line is continuous, not discrete.

For those unacquainted with this 

Imagine Achilles is twice as fast as the turtle.
Suppose Achilles starts off being a distance away from the turtle that he can
travel in one second.
Then by the time Achilles reaches the turtle (1 second) the turtle is only
half a second away. Then a quarter. Then an eighth. At what point will he
reach the turtle? After 2 seconds because

 (and I can't use mathematical notation, so in words...)
 the limit, as i approaches infinity, of the sum from 1 to i of 1/(2*i)
 EQUALS 1
 (Note emphasis on EQUALS. Not "approximately equals". 0.(9 recurring) = 1)

 (plus the first second equals 2)

Therefore after 2 seconds Achilles reaches the turtle, even in a continuous
universe.
Now personally, I like to think that the world is discrete. Maybe.

>	Antoine Beugnard and Didier Guy
>	ENST de Bretagne, LIBr, Brest, France
>	beugnard@enstb.enst-bretagne.fr
>	guy@enstb.enst-bretagne.fr

Carl Reynolds
5GL International Ltd, Auckland, New Zealand
carl@fivegl.co.nz
Generic Question: Why? | Sarcastic Retort: Why not?