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From: beugnard@sein.enst-bretagne.fr (Antoine Beugnard)
Newsgroups: comp.ai.philosophy
Subject: continuous vs discrete
Message-ID: <381@sein.enst-bretagne.fr>
Date: 9 Apr 91 13:32:21 GMT
Organization: enst-bretagne Brest, FRANCE
Lines: 67


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 ??

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