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

[Brian Hart]

On Sat, Oct 11, 2008 at 6:59 PM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> Absent a generally accepted Theory of Everything it is premature to
> speculate on whether such a theory would meet all the conditions of
> Gödel's first incompleteness theorem.  But assuming it does, those
> finding such a theory useful will be no more handicapped than number
> theorists and computer scientists, some of whose most useful theories
> already do meet the condition of the theorem yet whose productivity
> seems none the worse for it.

Dyson would agree with you.  He viewed Goedel's incompleteness results
optimistically in that they demonstrate the inexhaustability of human
intuition as it finds usage within science and mathematics and the
unending stream of scientific inquiry that future generations of
scientists would be priveleged to experience.  Weyl, on the other hand
(and for this very reason), saw this as a "constant drain on his
[scientific] enthusiasm", for he saw that the enterprise of science
was open-ended and humanity's quest for knowledge would forever be
incomplete.

>
> None of Gödel's theorems would appear to have adversely impacted the
> productivity of either applied mathematicians or physicists, and I'm not
> aware of anyone else's theorems that have done so either.  Gödel's first
> incompleteness theorem can only be considered "alarming" as you put it
> by those who believe that the continued progress of pure and applied
> mathematics will ultimately depend on the eventual success of Hilbert's
> program of wir müssen wissen, wir werden wissen.  The proper reaction to
> Gödel's theorem by such believers should be the acknowledgment that
> their belief is a lost cause: for those theories meeting the conditions
> of the theorem, we will never know because we cannot know.  People
> should be allowed to nurse their lost causes in private but not so as to
> be a burden to others.

There are, of course, a multitude of limitations to human knowledge in
practice and in theory and we must be wary of them and accept them
with a stiff upper lip as you say and as Wittgenstein advised:
"Whatever we cannot speak about must be passed over in silence."

>
> Vaughan Pratt
>
> Brian Hart wrote:
>> Why doesn't Godel's 1st Incompleteness Theorem imply the
>> incompleteness of any theory of physics T, assuming that T is
>> consistent and uses arithmetic?  Shouldn't the constructors of the
>> Theory of Everything be alarmed?
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