X-Message-Number: 22106
Date: Mon, 30 Jun 2003 10:08:25 -0700
Subject: No Physical Infinities.
From: Peter Merel <>

Michael Price writes,

> No, we don't know that limits don't correspond to reality.   It is an
> assumption that Aristotle made, but I see no reason to go along
> with it.  We have the edge on Aristotle -- we know about the
> photoelectric effect and that energy comes in quanta, which is the
> source of the infinities in question.  You have not demonstrated
> Aristotle's claim, just asserted it, whereas mainstream physicists,
> make a counter claim -- and have the detailed theory and
> calculations to back their claim up.

My view is hardly mainstream, but it is not Aristotelean. I'm hoping 
our conversation will provide useful pointers for me to follow to 
refine or correct it.

Now plainly we're off to Zeno's great race. Achilles, fleetest of 
Greeks, and the tortoise with the head start. To pass the tortoise 
Achilles must first cover half the distance. Then half the remaining 
distance. And so on, never quite doing it. Limit theory put Achilles 
past the post, and he hasn't looked back yet.

But now he does look back, and sees things from a different 
perspective. Throughout the course of the race both contestants 
occupied the same race track. This is a unit interval describing two 
ontologies, time and space. Of course there are other ontologies 
implied, but we'll keep things simple for the sake of Achilles, among 
whose many virtues wisdom did not number.

Now Achilles takes his compass and subdivides his interval. He's free 
to divide it any way he likes, but naively he chops it into thirds.

Throughout the first third he was trailing the tortoise. Throughout the 
last third he was in the lead. In the middle third, however, there is 
an ambiguity. Sometimes he was trailing, and sometimes he was leading.

Achilles wants to eliminate the ambiguity, so he redefines this third 
as his unit interval and subdivides it again. Now in the new second and 
third subdivisions he is clearly the winner. But the new first third 
contains the ambiguity.

Ever persistent Achilles continues subdividing, his tongue assuming an 
undignified aspect at the corner of his mouth. Then the tortoise, 
sweating and puffing, cast his long shadow across the diagram in the 
dust.

"You'll have to give it up," he said.

"Why so?" said Achilles, "I'm only attempting to enumerate the point at 
which I passed you."

"Indeed, and congratulations on doing so," said the tortoise, "but will 
you let me use this art another way?"

"I am magnanimous in victory." declared the stalwart. "Be my guest."

"Look then," said the tortoise, sketching, "Instead of the race, make 
the unit interval your whole life, Achilles. In the first third you 
have not raced me and you are not interested in determining any point 
in it. In the last third you have tired of your process of enumeration 
and have gone on to greater things."

"So mote it be!" thundered that mightiest man. "E'en now I have 
received word from Agamemnon by Odysseus that -"

"To the point, friend," interrupted the tortoise, mildly. "Somewhere in 
this middle third of your life you and I race, and you attempt to 
calculate the point at which you passed me."

"Verily and forsooth. But where in that third? Wilt subdivide again?"

"I might, but I don't need to. You can easily see that this third is 
finite, and that a process of subdivision cannot be consciously 
conducted in any less than a finite subinterval of it. Therefore you 
won't have sufficient time in your life to continue the subdivision to 
a point, and your diagram will always include a subdivision that 
describes an ambiguity."

"Hah!" declared Achilles, "we become even, aged reptile - I have bested 
you by foot, but you me by wit! So to horse! For I have word that fair 
Helen, most lovely of -"

Here Achilles is interrupted by your faithless storyteller, for he 
continued on for some time after the tortoise fell asleep.

>> To be more positive, without an empirical necessity we should
>> assume nothing about the physical universe.
>
> Except that infinities don't exist, eh?

I have never seen a unicorn. One might exist. So it is unreasonable for 
me to assume one does not. But if reasonableness enters into it, then 
we must ask ourselves whether it is more reasonable to think that 
infinitely many infinities spring up all over the infinite universe 
from infinitely many instants to infinitely many instants all 
infinitely correlated - whoops, back up -  an infinite number of 
infinite universes each with an infinite number of elements in various 
degrees of infinite cross-universal cross-time correlation, ad nauseam 
- that all that is more likely than that our model is poorly framed.

But I give ground happily. I agree that we should not assume infinities 
don't exist. We should look to utility and economy of expression; if 
infinities can be done away with, and the resulting models maintain 
predictive power, then the infinities were artifacts of poor modelling. 
And if our models disgorge infinities at the drop of a hat, and if the 
calculation of even simple problems becomes too complex for us to 
complete in our lifetimes, we're well served to identify "infinity" 
with ambiguity, and think there is reason to reformulate our number 
systems.

> Please excuse me for not discussing your example about faeries.
> I prefer to stick to photons, for which we have an accepted
> scientific theory.

I used "faeries" because I don't want us getting stuck on the 
particulars of the popular model. If we're reformulating the basics, we 
might wind up with a physical model that has no counterpart to a photon.

Peter Merel.

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