This piece isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments on 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 they sometimes simply suggest - 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 Everything 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 of 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.
For example, 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:
"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. (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 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 cannot be proved, there are physical problems that cannot 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 to 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) positivist 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 constructivist/anti-realist/positivist (take your pick!) pragmatism. 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 part of the same mathematical system. Hence the requirement for metamathematics (or a metalanguage/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 they 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 the 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.”
Proof and the 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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