X-Message-Number: 22117
Date: Wed, 2 Jul 2003 09:50:36 -0700
Subject: No physical infinities.
From: Peter Merel <>

Michael Price writes,

> I must commend Peter Merel on his entertaining dialogue between
> Achilles and the Tortoise about their fabled race.  That's what I get
> for mentioning Aristotle!  Still, a bit of light relief with a 
> classical
> parable is always welcome.

I meant to include the middle you'd accused me of excluding, and so 
sketch the flaw in limit theory. Limits are specifications of actions 
that cannot be performed. They paper over the fact that ambiguity lies 
within distinctions - refer to ZF again.

> Unfortunately it relates to the subdivision
> of finite quantities and is, therefore, more about the existence of
> infinitesimals rather than the sort of physical infinities I'm 
> interested
> in.

There are no infinitessimals in the tortoise's construction. Only 
intervals and their finite subintervals distinguished by means of the 
agreeable interaction of T&A. Just as with their racecourse, if we 
start with a universal interval, we can never subdivide it an infinite 
number of times, never obtain infinitessimals, so never get into strife 
asking questions about the interactions of combinations of same.

Then instead of picking some arbitrary process and defining it as the 
standard unit - the meter in France - we would have to take as our 
standard some process that enumerates subdivision of the extent of the 
observable universe. Something based on an interferometer, probably.

> Ah, but I must stick with a particular model, since the existence
> of physical infinities is, by definition, an empirical claim, not a
> logical claim.  My claim relates to the number of (real) photons
> emitted when an electric charge is shaken.

You've agreed we can obtain no apparatus sensitive enough to register 
this infinity. Surely that relegates the claim to the realm of theory.

You might construct an experiment that, if it proceeded unexpectedly, 
would require refinement of any accepted physical theory. But no 
experiment can prove theory. Changing the method of theoretic 
construction might lead to revolutionary empirical results, as we saw 
in the start of the 20th century with QM & GR, but never empirical 
proof. Only disproof.

> It difficult to imagine that any reformulation of "the basics" will 
> eliminate the photon.
> The photon is the package of energy that light (or electromagnetic
> radiation generally) comes in.  Photomultiplier tubes count photons
> in the same way that Geiger counters count alpha or beta particles.
> Any theory that tried to do away with photons would have to do
> away with the reality of everything else, it seems to me -- banishing
> reality to banish infinities seems a rather desperate course of action,
> to put it mildly.

Light was particles. Then waves. Then quanta. Then non-local quanta. 
With every step in our understanding the photon has acquired a new 
model. I might call my leather settee a photon if it pleased me, but 
that's not the point: the process you call a photon may be freely 
reformulated in a manner that is not commensurable with any popular 
theory today, with implications we could never expect from the popular 
theory.

If we began, for example, by considering a photon's entire process from 
birth to death, and then subdividing that, we might find our 
descriptions of our experiments change significantly. I say "a photon" 
where there might only be behaviors of a unified field - if that - and 
us making up arbitrary names for their combinations.

But let's pursue your popular theory further. If there are an infinity 
of photons resulting from some event, then may we say there an infinite 
number of ripples on the ocean? How about in a pond? A raindrop? A 
single water molecule? A hydrogen atom? An electron? A quark? Are there 
an infinite number of ripples inside the infimum of your popular 
physical theory? Are there an infinite number of ripples inside every 
ripple?

You see the tortoise's point - there must be a physical question we 
can't trouble ourselves to answer. An ambiguity. Including that 
ambiguity in the framing of the elements of theory seems a relatively 
untried method of physical representation.

Peter Merel.

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