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The search for a neat Theory of Everything (ToE) which unifies the entire set of fundamental forces of the universe (as well as the rules which govern dark energy, dark matter and anti-matter realms) has been the subject of a long-standing all-out effort of physicists since the early 20th century. Some complicated theories such as Quantum Field Theory and M-Theory have been developed along these lines. However, the ultimate theory of everything still seems far out of reach and highly controversial.(A related debate: 1, 2).

The difficulty of the hopeless situation brought some physicists, such as Hawking, on the verge of total disappointment. Their general idea was that maybe such an ultimate theory is not only out of reach (with respect to our current knowledge of the universe) but also fundamentally non-existent. In his Gödel and the End of the Universe lecture Hawking stated:

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. 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.

A glimpse of Hawking's lecture makes it clear that the argument which he uses for refuting the possibility of achieving the Theory of Everything in his lecture, is loosely (inspired by and) based on Gödel's incompleteness theorems in mathematics. Not to mention that Hawking is not the only person who brought up such an argument against ToE using Gödel's theorems. For a fairly complete list see here and here. Also, some arguments of the same nature (using large cardinal axioms) could be found in this related MathOverflow post.

Hawking's view also shares some points with Lucas-Penrose's argument against AI using Gödel's incompleteness theorems, indicating that human mind is not a Turing machine (computer) and so the Computational Theory of Mind's hope for constructing an ultimate machine that has the same cognitive abilities as humans will fail eventually.

There have been a lot of criticism against Lucas-Penrose's argument as well as the presumptions of Computational Theory of Mind. Here, I would like to ask about the possible critical reviews on Hawking's relatively new idea.

Question: Is there any critical review of Hawking's argument against the Theory of Everything in his "Gödel and the End of the Universe" lecture, illustrating whether it is a valid conclusion of Gödel's incompleteness theorem in theoretical physics or just yet another philosophical abuse of mathematical theorems out of the context?

Articles and lectures by researchers of various background including mathematicians, physicists, and philosophers are welcome.

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Morteza Azad
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    this argument was not specific for Hawking, it has been brought up and discussed by several scholars, see the links in https://en.wikipedia.org/wiki/Theory_of_everything#Gödel's_incompleteness_theorem – Carlo Beenakker Mar 14 '18 at 09:22
  • @CarloBeenakker I see! Thanks for the clarification, Carlo! Let me edit the post in order to make this point clear. – Morteza Azad Mar 14 '18 at 09:29
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    I won't claim to understand the physics, but I find it a bit odd to try to apply Gödel's incompleteness theorem to show that all of physics can't derive from a finite (and presumably small) number of principles, when Gödel's theorem illustrates, precisely, that a finite small number of principles can still give rise to a very complicated system! (Obviously what we can't do is find every consequence of these principles, but I don't think anyone believes that anyway.) – Gro-Tsen Mar 14 '18 at 11:00
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    The above argument for refuting the possibility of achieving the Theory of Everything in physics is loosely based on Godel's incompleteness theorems in mathematics. --- To me, Hawking's quote does not say this or even suggest this. It seems to me that he was using Godel's theorems and mathematics as an ANALOGY for a certain situation that he believes may be the case in physics (namely, that M-Theory is sufficiently malleable to keep physicists working indefinitely, or something like this). – Dave L Renfro Mar 14 '18 at 11:21
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    (MOMENTS LATER) Having now read the entire essay, I now see that in other parts of the passage Hawking does make a more direct link to Godel's theorems. – Dave L Renfro Mar 14 '18 at 11:27
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    Professor Stephen Hawking passed away today. – Joonas Ilmavirta Mar 14 '18 at 11:36
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    @DaveLRenfro Yes, exactly. For example, he mentions: "What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted." – Morteza Azad Mar 14 '18 at 11:53
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    @JoonasIlmavirta Unfortunately yes! Early this morning I woke up just to receive the sad news. The post is actually a logical tribute to his memory. May he rest in peace. :-( – Morteza Azad Mar 14 '18 at 11:58
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    I am not physicist. I'd would like to point out anyway that maybe gravity does not exist at the Plank scale just because gravity is an emergent property of spacetime, not a fundamental force. That would imply that quantum mechanics and general relativity cannot be unified because there would be nothing to unify. Some physicists reading my comments will be certainly able to explain better than I can this new (emergent !) point of view in theoretical physics. – Philippe Gaucher Mar 14 '18 at 14:30
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    The thing about Gödel's theorems is that they rely on infinity, which generally is not a problem in mathematics. Physics, however, is another matter, most physical theories do not assume or require physical infinities. And adding an assumption of physical infinity is really, really high on the Occam's Razor scale (though few physicists seem to realize this). – RBarryYoung Mar 14 '18 at 14:38
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    Perhaps Thomas Breuer, Quantenmechanik: ein Fall fur Gödel?, Spektrum Akademischer Verlag 1997, MR1453883 (99m:00006), should be added to the list. "Gödel's incompleteness theorem, or more precisely, problems of self-reference, place limits on the experimental investigability of the universe, which makes impossible the answering of important questions, even if we had a TOE." – Gerry Myerson Mar 14 '18 at 22:47
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    @GerryMyerson (+1) Nice related quote! I added it to the original post. Thanks for mentioning, Gerry! – Morteza Azad Mar 15 '18 at 05:08
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    I have no intention of dismissing the question outright, but let's make no mistake: there's no mathematical argument to analyse here, because Hawking (deliberately!) didn't make one. He was a brilliant and eminent physicist among other things, but I don't think he was, or ever tried to be, a logician. He presumably couldn't have stated Goedel's theorems, and would have been fully aware of this at the time of writing. This suggests to me, even in spite of his phrasing, that he never intended to invoke Goedel's theorems as mathematical statements, just as epistemological heuristics. – Billy Mar 15 '18 at 09:18
  • related: https://physics.stackexchange.com/questions/14939/does-g%c3%b6del-preclude-a-workable-toe –  Feb 28 '20 at 01:33

4 Answers4

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  • Alon Amit: "There are some things that break my heart more thoroughly than reading nonsensical conclusions from Gödel's Theorems to the limitations of physics published by eminent scientists, but they are few."
  • Johannes Koelman: "Stating that Gödel (or Turing, or gravity) implies the logical impossibility of a TOE, is the same as stating that because of the incompleteness theorem an axiomatic logic can not be constructed. This is simply wrong."
  • Stanley Jaki: a more favorable review by the scientist who first argued, in 1966, that because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete.
Carlo Beenakker
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    Hawking, as quoted above, uses Gödel's theorem as a metaphor, not as a lemma; I don't see how the criticism applies. In a way, Gödel broke the dam for the idea that some questions may not have a final answer, and that you will always have something left to explore. – darij grinberg Mar 14 '18 at 17:08
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    @darij Hawking wrote, in that article, “Thus a physical theory is self-referencing, like in Gödel’s Theorem. One might therefore expect it to be either inconsistent or incomplete.” Even at the level of “expecting” things, this “therefore” is just plain false. There’s no reason to expect this. – Alon Amit Mar 21 '18 at 06:54
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The answers so far (by Carlo Beenakker and p6majo) do not directly address Hawking's main argument, which is presented near the end of his lecture.

Hawking first presents a heuristic, motivational argument based on Gödel's theorem.

In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed and can contain an arbitrary amount of information without affecting the universes they describe. But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete.

This mention of Gödel's theorem is metaphorical, so strictly speaking one cannot accuse Hawking of misapplying Gödel's theorem. However, the analogy does seem a little strained, since earlier in the lecture Hawking emphasizes that "incompleteness" in the sense of unprovable propositions of arithmetic is not the kind of "incompleteness" he is really interested in—he is instead concerned about whether physical theories can, in principle, predict the future to arbitrary accuracy. But let us give Hawking a pass here since his main point isn't really about Gödel's theorem per se. Rather, his main point is that physicists, and hence presumably physical theories, necessarily reside in the physical universe that they are trying to describe. Hawking is suggesting that the "obvious" fact that physicists, and hence presumably physical theories, reside in the universe may place certain constraints on the predictive power of physical theories.

The crucial part of Hawking's argument comes in the next paragraph.

The black hole limit on the concentration of information is fundamental, but it has not been properly incorporated into any of the formulations of M theory that we have so far. They all assume that one can define the wave function at each point of space. But that would be an infinite density of information which is not allowed. On the other hand, if one can't define the wave function point wise, one can't predict the future to arbitrary accuracy, even in the reduced determinism of quantum theory.

Not knowing much about M theory, I can't be certain that I totally understand what Hawking is saying here, but I think that he is saying that current physical theories model wave functions as being (locally, at least) arbitrary complex-valued functions of an open subset of ${\mathbb R}^n$ (or perhaps ${\mathbb Q}^n$)—or even if they are not completely arbitrary functions, at least the space of possible wave functions in a finite region is infinite-dimensional. Hence, if a wave function is a determinate member of this function space, then it encodes an infinite number of bits of information inside a finite region, which contradicts the black hole limit on the concentration of information.

I do not consider myself competent to evaluate whether this is a cogent criticism of M theory, but let us give Hawking the benefit of the doubt and assume that it is cogent. Even then, I see no indication here of a "fundamental" barrier of the type that Hawking metaphorically alludes to in the previous paragraph. First, no appeal is being made to our biological limitations ("if our brains were simple enough for us to understand, then we would be too stupid to understand them"). Second, Hawking also makes it clear that practical limits on our computational power to solve large systems of equations is not his concern either. And third, Hawking never exhibits any worry that there will always be some conceivable bigger and better experiment in some unexplored regime that might falsify our purported theory of everything.

The only thing resembling a fundamental barrier is the possibility that the information required to specify the future with arbitrary accuracy may exceed the information capacity of the universe itself. However, even with a very generous reading of Hawking's argument, I believe that he is conflating the information required to state the axioms of a physical theory with the information needed to instantiate a particular universe. For example, let us hypothesize that the universe is more or less Newtonian but somehow there is "not enough room" in the universe to "write down" the positions and velocities of everything in the universe. Then this would seem to be a situation where the information required to specify the future of the universe exceeds the information capacity of the universe itself, yet the physical principles in the theory of everything can be stated very compactly and can easily fit inside the universe. Since Hawking doesn't care about practically computing the future, but only about grasping the principles underlying the universe, there is no need to "write down" all the positions and velocities; we just need to write down a small number of laws of physics.

In short, even if we accept Hawking's specific argument from quantum gravity at face value, he has given us no reason, Gödelian or otherwise, to think that there is any fundamental obstacle to formulating a theory of everything.

Timothy Chow
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  • (+1) Thanks for your comprehensive answer, Timothy! I personally do agree with your final conclusion about Hawking's argument, that he actually doesn't provide enough evidence supporting his claim on the possible non-existence of the Theory of Everything in physics! Also, this particular use of Goedel's incompleteness theorems out of the context actually seems to add no strength to his already obscure justification. – Morteza Azad Mar 24 '18 at 18:25
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Based on some of the comments I have seen, some commentators are finding Hawking's arguments vague or unconvincing. These are legitimate criticisms. But I think he has been partially, if not completely vindicated, by two recent advances that were not fully developed at the time the paper was published.

  1. The undecidability of the spectral gap problem.

Hawking states: if there are mathematical results that can not be proved, there are physical problems that can not be predicted. Since Hawking wrote this article, Cubitt, Perez-Garcia and Wolf showed that the spectral gap is undecidable by proving that the spectral gap decision problem is equivalent to the halting problem. And the halting problem is the theoretical computational equivalent of Godel's first theorem. This result does indeed suggest that some problems regarding the in quantum physics cannot be predicted, and many of the most challenging and long-standing open problems in theoretical physics concern the spectral gap,

  1. David Wolpert's inference engine proof

Hawking states: "But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Godel’s theorem."

The Wolpert "inference engine" proof is , similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, and, indeed, even the the Spectral Gap problem. It relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.

Philippe M. Binder, a physicist at the University of Hawaii at Hilo, suggests that Wolpert's theory implies researchers seeking unified laws cannot hope for anything better than a “theory of ALMOST everything.”

Jeffrey Harrison
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    I mean, if you're willing to count physical manifestations of undecidability you could just observe that we can build computers. But the fact that we can build computers using Newtonian mechanics doesn't mean that we are fundamentally incapable of figuring out Newtonian mechanics! – Solveit Feb 28 '20 at 01:11
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    Very interesting answer. – Nik Weaver Feb 28 '20 at 04:10
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A physical theory usually follow from fundamental principles, eg. gauge symmetries (similar to axioms) and the requirements of consistency, eg. the absence of anomalies. It is supposed to give answers to all questions (similar to theorems) within the scope of the theory. This point of view might be understood as a door opener for Gödel's incompleteness theorem.

However, answers of the physical theory are not only subject to mathematical reasoning (ie. proofing theorems) but they primarily have to be consistent with experiments.

In my opinion, this is clearly bypassing Gödel's incompleteness theorem. A theory of everything would not be required to be completely prooven mathematically. It would be sufficient, if it was confirmed by every conceivable experiment within the scope of the theory (in the case of toe: every experiment).

p6majo
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