🚀 go-pugleaf

berrpm@aur.alcatel.com (Patrick M. Berry) wrote:

>In article <333fb739.18406442@news.mindspring.com>, equus@mindspring.com writes:
>> On Wed, 26 Mar 1997 18:35:54 -0800, Mark Simpson
>> <marksimpson@sprintmail.com> wrote:
>> 
>K,
>> >Here's another one for all to ponder.
>> >In order to travel from point A to point B, one must first travel 1/2
>> >the distance.  In order to travel half way, you must first travel 1/4 of
>> >the way, etc., etc., etc. In mathematical terms, the denominator
>> >approaches infinity and we know that you can't travel an infinite
>> >distance in a finite amount of time, therefore all motion is impossible.
>> 
>> 1/ infinity isn't infinity.
>
>No, it's zero -- implying that you would travel zero distance in less than
>an infinite amount of time.  At least that's what I think he was trying to
>say.
>
>The fallacy in all this, of course, is that 
>the universe isn't infinitely 
>divisible.  For example, there's
> no way to travel half the diameter of an
>electron.
>

Being (gently) picky, you're missing words like  "theoretically" above
a couple of times.  Be that as it may, whilst the above perhaps makes
disproving the original argument easier, it's not necessary.  The true
fallacy is much older, and more basic.

 The original post above is just a slightly mangled version of Xeno's
paradox, which has been around for a couple of thousand years, and
rests on the premise that the sum of an infinite series of positive
finite terms is infinite.  I remember it something like as follows:

("It takes you a finite time to go half way. Then a finite time to go
half the remaining distance.  Then a finite time to go half the new
remaining distance.  And so on for ever - no matter how steps you
take, there's still a non-zero distance left to cover.

Now, every step takes a finite time.  And the sum of an infinite
number of finite intervals must, logically, be infinite. Therefore it
takes you an infinite time to cross *any* distance. 

Therefore motion is impossible") . 


The key premise is  invalid - and was demonstrably so, even in Xeno's
day.   For the purposes of blowing the logic of an argument, it's only
necessary to find *one* counter-example to any premise on which it's
built .  In this case, the way the distance to be travelled is
continuously subdivided gives an infinite series which clearly has a
finite sum - therefore the premise is false, therefore the argument
fails. 


(As for the relevance of theoretical quantum concepts like the
Heisenberg limit - I can't resist pointing out that the maths to
formally sum the series at the heart of the above fallacy was actually
invented by precisely the same man who put *pre*-quantum mechanics on
a formal footing, Sir Isaac Newton.  
Imbed black-market emoticon of your choice here.)

- Ian Noble
(Been there, done that, got the headache...)