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From: beugnard@zeus.enst-bretagne.fr (Antoine Beugnard)
Newsgroups: comp.ai.philosophy
Subject: Re: Continuous vs discrete
Summary: In continuous models limits are NEVER reached, therefore
	 Achilles never reaches the turtle !! why is it false ?
Message-ID: <383@zeus.enst-bretagne.fr>
Date: 12 Apr 91 08:49:06 GMT
References: <382@batz.enst-bretagne.fr> <1991Apr10.182036.29916@beaver.cs.washington.edu>
Organization: enst-bretagne Brest, FRANCE
Lines: 54



In article <382@batz.enst-bretagne.fr> beugnard@batz.enst-bretagne.fr (Antoine Beugnard) writes:
>>
>>Is the model wrong, or are the hypotheses of continuity false?
>>Why the classical model works (matches our experience) while Zeno's one don't?
>>
>>Our interpretation is that the world is by essence discontinuous.
>>

mikew@cutthroat.cs.washington.edu (Mike Williamson) replies in <1991Apr10.182036.29916@beaver.cs.washington.edu> 

>The problem with Zeno's paradox is that it makes makes use of the
>concept of infinity, while ignoring the concept of the infintessimal.

>A quick precis of Zeno's paradox is:  "Assume space is continuous.
>Then there are an infinite number of points between any two points A
>and B.  To travel from A to B, you must first travel to each of these
>infinite number of points.  This could not be done in finite time.
>Therefore, if you are able to travel from A to B, space must not be
>continuous."

>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...

Our experience shows it is wrong, but why?

In a discreet world and with a reasonning and modelling similar to Zeno's one. The calculus terminates *claiming* the end of the model validity...

In a continuous world you will have to decide: "Well, Achilles reaches the
turtle when, say, T(n) - A(n) < 10^(-100) , and then I have to change my model".

It is a physicist behaviour, pragmatic, and, well, "discreetizing" reasonning, no?

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


Ps: We do not call into question mathematics and its powerfull use. Even 
continuous mathematics!!. It works...but it may just be an abstraction of the 
reality, a usefull tool that has not to be related to the essence of world...

The question is not only philosophical, people thinking discreet machines cannot
become "intelligent" assumes the world is continuous ... which is not so obvious ...