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

Antonino Drago drago at unina.it
Mon Oct 13 19:14:03 EDT 2008

From: "Brian Hart" Friday, October 10, 2008 4:43 AM

> 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?

First, a physical theory, contrarely to what hoped Hilbert, cannot be
axiomatised. At least, it is impossible to give the connection axioms, for
relating experimental statements to mathematical statements.

Second. Nagel discusses in which way a physical law can be formalised
through quantifiers; but when one gives an idealised meaning to the law,
i.e. ethernal with respect to the time t, and omnicomprehenisve with respect
to the ranges of all its variables; of course this idealisation is out of
the any experimental verification. The fact that the theoretical physicst
make use of the laws in this idealised way does not mean that they are
making physics; rather they are making mathematics in order to reach through
calculations any kind of result fitting with experiemental data.

Third. In theoretical physics quantifiers play an important role only when
they are included in some principle: " *Any* body either in rest or in
uniform motion perseveres in its state
unless a force changes its motion" (Newton 1687). Of course, this statement
is incomplete in the sense that Newton did not predicated this statement
about *all* bodies if not in an idealised way, i.e. in a non experimental
way. Is physics this? Again, it is a conjecture in order to obtain results
fitting experimental data. But notice that theoretical physics is capable to
disregard this kind of principles which include quantifiers; the same
inertia principle was stated without quantifiers by Lazare Carnot (1803) as
follows: "Once a body is at rest it does not change its state; when in
motion it is not capable to change its speed and its direction". Hence
physics is capable to dismiss at all the quantifiers; quantifiers come occur
in a way of speaking that make use of more elaborated statements. In the
more adequate statements to the experimental reality, no quantifier does
exist.  In such a case there is not reason to apply Goedel's theorem.

Fourth.  The pieces of mathematical theory a physicst uses - for instance
what is interesting for him of the real number theory - are incomplete
without no more  problem for him than for the practice of a mathematician.

> Shouldn't the constructors of the
> Theory of Everything be alarmed?

Please, take care of physicists' inappropriate way of speaking. In this case
mean no more than to state a bit of as much as possible things; not to state
all about everything.

Rather, let us remark that in theoretical physics some *universal* theorems
occur. For instance, in the simple case of thermodynamics, Carnot's theorem
holds true for *every* heat engine. Let us remark that this theorem is not
proved by means of mathematics, but by a logical argument only . More
precisely it is an ad absurdum argument. Even more interesting is that also
in mathematics the same occurs: Lobachevsky's theory (Geoemetrische
Untersuchungen... 1840, Engl. Transl. In Bonola Non-Euclidean Geometry)
concludes (proposition 22: "It follows that in *all* rectilinear trangles
the sum of the three anngles... etc.". "The second assumption can likewise
be admitted without leading to *any* contradiction in the results..."
In which sense Goedel theorem apply to these two theories, constituted by ad
absurdum theorems?

Best regards at all
Antonino Drago

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