Give one valid counter-example to "everything is finite": a piece of evidence that the infinite is actually...

[ Editor: from threads in sci.philosophy.tech and sci.logic, "Infinity
as a proper mathematical concept", "What does it mean that 'actual
infinities' exist?" and "Actual infinities DO exist!".  No reply to the
challenge is in the collected material. ]

Leif Sterner writes (sci.logic):
> I would say that it's pretty clear that infinities actually do exist.
And later on:
> ........ . An interesting question is if actual infinities could arise in the
> physical world. Recently I made some postings to the sci.physics newsgroup
> on this subject. ( ... stuff deleted ... ), ... if anyone is interested I can
> send the articles by @mail.

Very strange kind of reasoning !  There ARE actual infinities, but you
quest if they could arise in everyday (physical) REALity.  You seem to
assume that there exists some other "reality" out there.  Let's call it
the Heaven of Mathematics, or even better: Plato's good old World of
Ideas.

Gene Ward Smith writes:
> Of course, outside of pure mathematics the trouble with dismissing infinity
> is even greater. As was pointed out, the universe might very well be infinite.
> Quantum field theory has to deal with infinite numbers ... and "renormalize".
And again to Eli Chiprout:
> I notice you did not deal with the point that actual, physical infinities
> seem to be apart of the physical universe. What about it ?
The answer of Eli could have been mine:
> They do ? That's a new one to me ! What part of the physical universe have
> you discovered that I don't know about ?

Serious now.
Of course, there is NO physical evidence that infinities actually do
exist.  Actually, there IS physical evidence that actual infinities do
NOT exist.  I have done my best to collect some substantial, physical,
arguments.  Most of these are certainly NOT my own. (Remember !)

1. If the universe was infinite in space and time, then it wouldn't
   become dark at night.  The whole sky would be as bright as the sun
   itself.  There would be an infinite number of stars, and all their
   light would have reached us, since there would be an infinite time
   for it to travel.

2. What do you think of the ever increasing entropy in an infinite
   universe?

3. Infinite vector spaces ARE used in Quantum Mechanics.  However, they
   all have a complete base, AS IF they were only vectorspaces with a
   very BIG dimension.  In fact, these Hilbert spaces cannot be
   distinguished from a big, but finite vectorspace, as every physicist
   knows from practice.

Therefore, from a physical point of view, the Universe seems to be
FINITE.  Big Bang, Creation or Whatsoever ...  If it is finite in space,
it must also be finite in time, and vice versa: the theory of
relativity leaves no other choice to us.

So far about the infinitely BIG things.  What about the infinitely
SMALL?

4. Fluid flow is described by a set of coupled partial differential
   equations.  However, it is very clear that this must be considered
   as a mere illusion.  The "differential" volumes are NOT REALLY
   infinitely small.  Far from that !  They should contain A LOT OF
   molecules, for the approximation to be valid.  There is a beautiful
   text about this (due to Perrin) in Benoit Mandelbrot's "The Fractal
   Geometry of Nature", first few pages (6,7).

5. Space and time seem to be the only things that remain infinitely
   divisible.  But there are also strong arguments against that !
   Quoting without permission from Landau and Lifshitz "Quantum
   Electrodynamics", Introduction, page 2+3:
   > In the rest frame of the electron, the least possible error in the measure
   > ment of its coordinates is:
   >                                 dq = h/mc.
   >
   Here q=position, h=Planck's constant, m=electron mass, c=velocity of
   light.  If you try to measure the position of an electron, you must
   send for example a photon to it.  The electron is disturbed by this
   photon, which is known as the Compton effect.  As a consequence of
   this, there is a definite bound on the accuracy with which you can
   locate the electron.  This is expressed by the above formula.  Any
   attempt to locate the electron *within* this interval is "severely
   punished" by:
   > ..... the (in general) inevitable production of electron-positron pairs in
   > the process of measuring the coordinates of an electron. This formation of
   > new particles in a way which cannot be detected by the process itself
   > renders meaningless the measurement of the electron coordinates.

6. There is another (excellent) book, written by Leon Brillouin.  It is
   called "Relativity Reexamined", Academic Press (1970).  Quoting
   without permission:
   > Einstein's clocks were supposed to emit extremely short signals and to
   > measure accurately time intervals between signals emitted and received.
   > In a word, *an Einstein clock was a radar system*, and its requirements
   > were thus very different from those of a frequency standard. ...
   The modern atomic clocks are referred to by the above "standard".  It
   can be shown by a simple thought-experiment that, if one tries to
   measure an electron by such an Einstein radar pulse, this pulse
   always must be much WIDER than the following:

                                   dt = h/mc^2        (t=time)

   It thus is definitely impossible to measure (electron) time more
   accurately than this amount, due to the same Compton effect as in
   (5).

7. There are some pathological theories in physics where infinities DO
   arise, like Quantum Electro Dynamics (QED).  In these theories,
   notions like "point" charges, with infinitely small spatial
   dimensions, are used explicitly.  But what happens here is NOT a
   proof that "Actual infinities DO exist".  Instead it should be
   looked upon as a

   SEVERE WARNING:
|| Absolute mathematical concepts like "point", "continuity" and "Identity" :-)
|| (yes!) are not void of danger if they become an integral part of a difficult
|| physical theory. Then the foundations of mathematics suddenly change into
|| a piece of real world physics. Once our mathematical axioms are absorbed by
|| a physical theory, they become, in fact, assumptions about Nature.

I don't think most mathematicians did design their basics for such a
purpose.  Therefore, in my opinion, physics should UNDERLY mathematics,
in some sense.

QED suffers from its infamous divergencies ever since it came into
existence.  Folks like Richard P. Feynman have devised some dirty
"renormalization" tricks, in order to cure the very worst symptoms of
this "infinity" disease.  At least, QED nowadays gives SOME of the
desired answers.  But, as Feynman himself points out in the famous
"Feynman's Lectures on Physics" (part II), the difficulties already
showed up in classical electromagnetic theory.  The self-energy of
point charges is infinite.  Electrons cannot move, if "actual
infinities DO exist" !

Physical theories CAN suffer from an inadequate foundation of their
mathematics, I think.  Somebody should have the guts to postulate the
following

         Basic axiom: EVERYTHING IS FINITE.
         ==================================

Challenge to everyone:
|| Give me ONE valid counter-example: a piece of evidence that the Infinite is
|| actually useful in physics, in a theoretical or practical sense (independent
|| of wishful mathematical thinking, of course).

-- Han de Bruijn, TU Computing Centre, Delft, The Netherlands