[FOM] Godel's First Incompleteness Theorem as it possibly relatesto Physics

Bill Taylor W.Taylor at math.canterbury.ac.nz
Wed Oct 15 00:05:53 EDT 2008

Hendrik correctly notes:

-> Only intractable in the sense that there isn't a neat formula 
-> that solves it -- there's just a differential equation. 

Exactly so, no "closed-form" solution.

-> Nothing prevents the numerical analyst from computing arbitrarily close 
-> approximations except time and effort.  Isn't that more or less the 
-> definition of computable reals?

Again, precisely so.

And it's intriguing to note the remarkable interplay here, between
theory and computation.  As is known, the behaviour of 3 or more
Newtonian bodies can become chaotic, which was what the original
poster on this issue presumably had in mind.  This would be a problem
for computations, as, though in principle it remains as Hendrik has said,
in practise the accumulation of (round-off and other) errors would make
the computation method useless in practice.  I dare say there are many
people here who have tried such simulations, working only with Newton's
basic equations, and found that (even for 2 bodies and their ellipses)
the answers rapidly become useless.

HOWEVER; there are several "emergent properties" that are not directly
contained in Newton's equations, but well known to all schoolboy
physicists and beyond.  The most obvious are conservation of energy
and conservation of angular momnetum (in three components).  If these
are used as "moderating influences" (I forget the proper technical term)
on the simulations, the simulations are continually brought under control,
(rather analogously to the quantium Zeno effect!), and the computations
are made good for very long times into the future.  This is standard
astro-modelling procedure OC, but easily overlooked by the beginner.

It strikes me that this moderating effect due to emergent properties,
is one of the most wonderful features of this whole simulation enterpise;
and is, as I say, a great example of the interplay between theory and
computing practice.

Sorry to be so long-winded.

-- Long Winded William

P.S.  In answer to the thread title, I have noted often before that
      Godel's theorems can have NO POSSIBLE influence on physics,
      due to this being a category mistake.  However I know this
      observation will fall on deaf ears, as it has done before.

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