This
piece, of course, isn't about deflating Kurt Gödel's metamathematics
or even deflating his own comments about physics. It's about
deflating other people's applications of Gödel's theorems to
physics.

Indeed
Gödel
himself wasn't
too keen on applying his findings to physics – especially to
quantum physics. According
to John
D. Barrow:

“Godel
was not minded to draw any strong conclusions for physics from his
incompleteness theorems. He made no connections with the Uncertainty
Principle of quantum mechanics....”

More
broadly, Gödel's theorems may not have the massive
and important applications to physics which some philosophers and
scientists believe they do have.

**For and Against Gödelised Physics**

Some
scientists are unhappy with the claim that Kurt Gödel's theorems can
be applied to physics. Others are very happy with it. More explicitly,
many people in the field claim that Gödel incompleteness means –
or sometimes simply

*suggests*- that any Theory of Everything must fail.
For
example, way back in 1966 the Hungarian Catholic priest and
physicist, Stanley Jaki, argued
that any Theory of Every is bound to be a

*consistent*mathematical theory. Therefore it must also be incomplete.
On
the other side of the argument, in 1997 the German computer
scientist, Jürgen Schmidhuber, argued
against this defeatist - or simply modest/humble – position.
Strongly put, Schmidhuber says that Gödel incompleteness is
irrelevant for computable physics.

Thus,
despite such pros and cons, it's still the case that many physicists argue that Gödel incompleteness doesn't mean that a
Theory of Everything can't be constructed. This is because they also
believe that all that's needed for such a theory is a statement of the rules which underpin all physical theories.
Critics of this position, on the other hand, say that this simply
bypasses the problem of our

*understanding*all these physical systems. Clearly, that lack of understanding is partly a result of the application of Gödel incompleteness to those systems.**The Gödel-Physics Analogy**

Despite
all the above, the relation between Gödel incompleteness and physics
often seems analogical; rather than (strictly speaking)

*logical*.
The
incompleteness of physical theories taken individually (or even as
groups) has nothing directly - or logically - to do with
Gödel incompleteness (which is applied to mathematical systems). The
latter is about essential or inherent incompleteness; the former
isn't. Or, to put that differently, science isn't about

*insolubility:*it's about*incompletablity*. (Though it can be said that incompleteness implies - or even entails - insolubility.)
This
analogical nature is seen at its most explicit when it comes to
scientists and what may be called their

*scientific humility or modesty*.
Stephen Hawking, in his 'Gödel
and the End of Physics', said:

“I'm
now glad that our search for understanding will never come to an end,
and that we will always have the challenge of new discovery. Without
it, we would stagnate. Gödel’s theorem ensured there would always
be a job for mathematicians. I think M theory will do the same for
physicists. I'm sure Dirac would have approved.”

This position is backed up by the words of Freeman Dyson. He
wrote:

“Gödel
proved that the world of pure mathematics in inexhaustible... I hope
that an analogous situation exists in the physical world.... it means
that the world of physics is also inexhaustible....”

Stephen
Hawking originally believed in the possibility of a/the Theory of
Everything. However, he came to realise that Gödel's theorems will
be very relevant to this theory. In 2002 he
said (to an audience):

"Some
people will be very disappointed if there is not an ultimate theory,
that can be formulated as a finite number of principles. I used to
belong to that camp, but I have changed my mind."

However,
Hawking does seem to be ambivalent on this issue. Specifically when it comes
to the analogical nature of Gödel incompleteness and incompleteness in physics.

Stephen
Hawking himself uses the word “analogy”; at least within one
specific context. That context is “a formulation of M theory that
takes account of the black hole information limit”. He then, rather
tangentially or loosely, says that

“our
experience with supergravity and string theory, and the analogy of
Gödel's theorem, suggest that even this formulation will be
incomplete”.

Here
Hawking isn't talking about

*Gödel incompleteness*. He's simply talking about*incompleteness*– the incompleteness of a “formulation” of a theory (i.e., M theory). More specifically, it's about incomplete information or incomplete knowledge. Gödel incompleteness certainly isn't about incomplete information or incomplete knowledge.
There's
another statement from Hawking that's also really about analogies. (Then again, with
his use of the word “reminiscent”, Hawking - more or less - says
that himself.) Hawking says:

“Maybe
it is not possible to formulate the theory of the universe in a
finite number of statements. This is very reminiscent of Gödel’s
theorem. This says that any finite system of axioms is not sufficient
to prove every result in mathematics.”

The
question is how

*reminiscent*is*reminiscent*? Is it vague or strong? Is is substantive or simply analogical? Indeed, on the surface, it's hard to know how to connect the statement that “any finite system of axioms is not sufficient to prove every result in mathematics” to physics generally. Apart form the fact that, yes, physics utilises mathematics and can't even survive without it.**Mathematical Models**

Hawking
himself states a strong relation between Gödel and physics. It comes
care-of what he calls the “positivist philosophy of science”.
According to such a philosophy of science, “a physical theory is a
mathematical model”. That, for one, is a very tight link
between physical theory and maths. Hawking says that

“if
there are mathematical results that can not be proved, there are
physical problems that can not be predicted”.

Despite
mentioning that tight link, it's a jump from the “mathematical
results that cannot be proved” bit to “there are physical
problems that can not be predicted” conclusion. The argument must
be this:

i)
If physical models are mathematical,

and
the mathematics used in such models contains elements which can't be
proved,

ii)
then the predictions which use those models can't be proved either.

That
means that mathematical incompleteness (if only in the form of a model in
physics) is transferred to the incompleteness of our predictions.

Is
“proof” an apposite
word when it comes to physical predictions?

Hawking
stresses one reason why physics can be tightly connected with
mathematics in a way which moves beyond the essential usefulness and
descriptive power of maths. He cites the “standard positivist
approach” again.

In
that approach, “physical theories live rent free in a Platonic
heaven of ideal mathematical models”. Thus one (logical) positivist
(i.e., Rudolf Carnap) argued
that one's theory (or “framework”) determines which objects one
“posits”.
Similarly, in Hawking's words, “a [mathematical] model can be
arbitrarily detailed and can contain an arbitrary amount of
information without affecting the universes they describe”. This,
on the surface, sounds like Hawking is describing an extreme case of

*constructivism*in physics. Or, since Carnap has just be mentioned, is this simply an example of (logical) positivistic pragmatism or instrumentalism?
The
least that can be said about this stance is that the mathematical
model must – at least in a strong sense - come first: then
everything else will follow (e..g., which
objects
exist, etc.). At the most radical, we can say that all we really have
are mathematical models in physics. Or, as with ontic structural
realists, we can say that all we have is

*mathematical structures*. We don't have objects or “things”.
Hawking
doesn't appear to like this extreme
constructivism/anti-realism/positivist
pragmatism (take your pick!). Firstly he says that the mathematical
modelers
“are not [people] who view the universe from the outside”. He
also states the interesting (yet strangely obvious) point that “we
and our models are both part of the universe we are describing”.
Thus, just like the axioms and theorems of a system, even if there
are many cross-connections (or acts of self-reference) between them,
they're all still all part of the same mathematical system. Hence the
requirement for metamathematics (or a metalanguage or metatheory in
other disciplines).

Finally,
all this stuff from Hawking is tied to Gödel himself.

Hawking
says that mathematical modelers
(as well as their models and “physical theory”) are “self
referencing, like in Gödel's theorem”. Then he makes the obvious
conclusion: “One might therefore expect it to be either
inconsistent or incomplete.”

Isn't
all this is a little like a dog being unable to catch its own tail?

**Self-Reference and Paradox**

Self-reference
and dogs have just been mentioned. Here the problem gets even worse.

Gödel’s
metamathematics is primarily about self-reference (or
meta-reference). As Hawking puts it:

“Gödel’s
theorem is proved using statements that refer to themselves. Such
statements can lead to paradoxes. An example is, this statement is
false. If the statement is true, it is false. And if the statement is
false, it is true.”

Now
how can self-referential statements or even paradoxes have anything
to do with the world or physical theory? Indeed do the
realities/theories of quantum mechanics even impact on this question?
(Note Gödel's own position on QM as enunciated in the introduction.)
Are there paradoxes in quantum mechanics? Are there cases of
self-reference? Yes, there are highly counter-intuitive things (or
happenings) in QM; though are there actual paradoxes? I suppose that

*one thing being in two places at the same time*may be seen as being paradoxical. (Isn't that only because we insist on seeing subatomic particles, etc. as J.L. Austin's “medium-sized dry goods” - indeed as*particles*?) Some theorists, such as David Bohm, thought that QM's paradoxes will be ironed out in time. So too did Einstein.
The
ironic thing is that - according to Hawking - Gödel himself tried to
iron out paradoxes from his mathematical theories (or systems).
Hawking continues:

“
Gödel went
to great lengths to avoid such paradoxes by carefully distinguishing
between mathematics, like 2+2 =4, and meta mathematics, or statements
about mathematics, such as mathematics is cool, or mathematics is
consistent.”

Here
again the problem is

*self-reference*. The solution was - and still is - to distinguish mathematics from metamathematics. Alfred Tarski, in the 1930s, did the same with metalanguages and object languages in semantics. Indeed, even before Gödel and Tarski, Bertrand Russell had attempted to do the same within set theory when he distinguished sets from classes (as well the members of sets and classes) in his “theory of types” (a theory established between 1902 and 1913).**Proof and the/a Theory of Everything**

Wouldn't
a/the Theory of Everything
be a summing up (as it were) of all physical laws? Thus wouldn't it be partly - and evidently - empirical in nature? Surely that would
mean that mathematics couldn't have the last - or the only – word on
this.

It
can also be argued that a/the Theory of Everything wouldn't demand
that every physical truth could be proven in the mathematical/logical sense;
even if every physical truth incorporates mathematics.

This
is also a case of whether or not proof is as important in physics as
it is in mathematics. Indeed, on certain arguments, there can be no
(strict) proofs about physical theories.

For
example, some have said that the/a Theory of Everything will need to
expressed as a proof. Nonetheless, that proof will still be partly observational (or partly empirical – i.e, not
fully logical). However, even if only partly observational and
largely mathematical, how can it still guarantee a proof? How can there be
any kind of proof when a theory includes observations or experimental
evidence?

Again,
the Theory of Everything would be a final theory which would explain
and connect all known physical phenomena. This – to repeat - would
be partly empirical in nature. It would also be used to predict the
results of future experiments. These predictions would be partly
empirical or observational; not (to use a term from semantics)

*proof-theoretic*.
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