Is a GUT Impossible Due to Godel's Incompleteness Theorem?

[Jolb]

Hawking's "End of Physics"

Stephen Hawking wrote a paper a few years ago which discussed a "Godel's Theorem" for Physics. Godel's Incompleteness Theorem is a mathematical result which states that any formal system of mathematics necessarily has true statements which can't be proved using the axioms of the system. Godel's Incompleteness Theorem showed the impossibility of Hilbert's idea that there was an ultimate set of axioms and deductive rules (i.e. a complete formal system of mathematics) with which all true statements could be derived. [Many authors have explained Godel's proof in less advanced language, and I encourage you to look at those if you haven't, since the proof is beautifully mind-f***ish, especially the Epimenides aspect.]

Hawking was discussing that since physics appears to be strongly mathematical, there might be some sort of proof that there will always be more physical truths that can't be derived from any grand unified theory. He seems to think it's possible that a physics result similar to Godel's Incompleteness Theorem might conclusively show that a GUT is impossible.

What do you guys think of this idea? I think it's a very cool idea, actually, and unlike my fellow physics students, I don't dislike the idea that there's no ultimate Grand Unified Physics of God. The universe would be cooler if it were truly beyond any theory or comprehension. Plus there'd always be jobs for physicists!

I think that if such a physical theorem were to be found, it would probably be in the context of information theory. Claude Shannon's work on information has a resonance with statistical physics and quantum theory. Alan Turing's work on the halting problem has relevance to the use of algorithms (i.e., theories) to computationally simulate physical laws. I know this is all vague, but to me, these things seem to ring out as related to the possibility of GUTs. Maybe simulating the GUT in a physical computer (or brain) might be analogous to using a universal turing machine to find out whether a certain algorithm stops.

Any thoughts?

 

[cmb]

I think it's a very cool idea, actually, and unlike my fellow physics students, I don't dislike the idea that there's no ultimate Grand Unified Physics of God. The universe would be cooler if it were truly beyond any theory or comprehension. Plus there'd always be jobs for physicists!

I was debating this point in another thread, in regards whether we could ever get to a point where it is 'no longer science' to push past a certain level of GUT/explanation/understanding.

If we could conceive of some given 'state of a GUT' that takes us, by its own conclusions, to a point where it is 'no longer scientific' to push our comprehension further, then that must be the limit of science, because science does not exist independently of humans - it is a human activity.

*If* there is anything beyond that, then it is not something 'science' will be able to address, so I think you can rest easy that the universe is truly beyond what a theory can offer sentient individuals as an explanation for what they experience and observe.

 

[Dr_Morbius]

That's not what Godel's theorem says.

http://www.science20.com/hammock_physicist/limits_science_god_godel_gravity

 

[Jolb]

That's not what Godel's theorem says.

http://www.science20.com/hammock_physicist/limits_science_god_godel_gravity

Thanks for citing a 20+ page long forum thread. Clearly that's the most authoritative source available.
Please tell me what in my post is inaccurate.

Edit:
Having read your source, Dr. Morbius, I don't think anything I said is inconsistent with Godel's Theorem. [I am referring to the Incompleteness Theorem.] Please let me know if my understanding of Godel's Theorem is at all inaccurate.

 

[Vanadium 50]

Why don't you start by posting a reference to the Hawking paper you are talking about.

 

[Jolb]

Sure:
http://www.hawking.org.uk/godel-and-the-end-of-physics.html

This is a talk, but I'm fairly certain he's written it in some paper also.

 

[Jolb]

I was debating this point in another thread, in regards whether we could ever get to a point where it is 'no longer science' to push past a certain level of GUT/explanation/understanding.

If we could conceive of some given 'state of a GUT' that takes us, by its own conclusions, to a point where it is 'no longer scientific' to push our comprehension further, then that must be the limit of science, because science does not exist independently of humans - it is a human activity.

*If* there is anything beyond that, then it is not something 'science' will be able to address, so I think you can rest easy that the universe is truly beyond what a theory can offer sentient individuals as an explanation for what they experience and observe.

I think String/M theory is pushing the limits of science, according to your definition. Humans have done research into string theory even though the theory itself kind of ensures its own ambiguity (in terms of predictions), and even if we found a less ambiguous version, we'd be far away from being able to test it. (The hopes for the LHC testing some string theory predictions are pretty far-fetched.)

Basically, string theory is the investigation of a possible physical model that we can't yet apply empirical experimentation/observation to. I still think this is science--we are studying a mathematical model that may have relevance to physical reality. This is a productive scientific endeavor. (And what do you say to the people -- not me -- who claim that pure mathematics is a science unto itself?)

I don't think just because we can't test new theories doesn't mean we shouldn't bother studying them. There are counterintuitive results that come out of most of the modern theories--how do we know the mathematics won't lead to a counterintuitive result that is testable?

Plus, we could make useful models that aren't even grounded in reality--at Boltzmann's time, molecular theory was thought of as unphysical, but he made the right predictions! (And now we know that molecular theory is not the classical one he used.) Ptolemy's celestial model worked, and that's why it was a cornerstone of science at the time.

For those reasons, I think the only two possible "ends" to physics would be finding a GUT or proving conclusively that there is no GUT.

 

[bbar]

I don't think you've misrepresented Gödel's incompleteness theorems. He proved that any axiomatizable system is either incomplete, or, if it had an infinite amount of axioms (which would make it complete), would become undecidable.

 

[sophiecentaur]

Although many 'hard boiled' Scientists would strenuously deny it, this has to be a matter of Faith, in the end (not in the religious sense).

It seems to be an article of Faith, with many people that there 'must' be an ultimate answer to everything. This is a very 'optimistic' view of things and is a good way of limiting one's exposure to uncomfortable ideas of fallibility. It's the nearest thing to belief in a god without actually believing in one - 'someone will take care of it in the end'.
Personally, I feel no need to have this comfort blanket. I realize that I will never 'get it all' and that no other individual will, either and I don't feel bad that many other individuals 'get it' a lot better than I do. Even by combining intellects (using aids like 'Deep Thought') there will always be 'cracks' of un-knowledge. No problem.

Demanding a GUT is like asking for a Map which is a map both of itself and of the world. The map on the table has a map of a map of a map. . . . . . . Each successive map needs to contain the previous map plus the table it's on, which is a divergent process.

I think the GUT idea is a misnomer. At best, it will be a GUNT (N='nearly'), which takes us away from absolutes and allows each generation of Physics to get just a bit nearer, in a direction that takes their fancy or suits the current technology. That 'never get there' idea should already be familiar and acceptable to us; we already have c and 0K to deal with - then there's the Big Bang. Just come to terms with it - there isn't a problem.

 

[Ken G]

Hawking was discussing that since physics appears to be strongly mathematical, there might be some sort of proof that there will always be more physical truths that can't be derived from any grand unified theory. He seems to think it's possible that a physics result similar to Godel's Incompleteness Theorem might conclusively show that a GUT is impossible.

I think it's pretty clear that this is true. It seems to me it can be proven directly:
1) Assume we have a GUT that is a mathematical theory from which all physical truths can be derived (let's say that means all possible experiments can be predicted).
2) Godel's theorem says that if the GUT is true, then it will make reference to physical truths that can be set up in the language of the GUT that cannot be proven true from the finite axioms of the GUT (assuming the GUT is both correct and internally consistent).
3) Ergo, physics cannot end, because we can still look for these truths that cannot be proven from the GUT, and which would then need to be axiomatized anew, and the process would continue.

However, I point out that physics lacks the same fascination with logical rigor that mathematical systems require, so given that the physical truths not provable from the GUT might never be encountered or have any practical importance, searching for them would probably not be enough to keep physics going.

Incidentally, I would not be bothered by a well-tested physics theory that can prove something like "no physics that is consistent with this theory can ever explain outcome A of experiment X." That would not be an end of physics for two reasons:
1) physics theories don't prove things because they are generally wrong, and
2) even if we assumed that theory is right (who knows why we'd want to do that), it would just mean that an explanation of outcome A is not part of physics. No one ever said physics was meant to be all things to all people, and most think it's pretty clear that physics will never explain a lot of the things that are important to humans-- which is why we have art in addition to science.

 

[sophiecentaur]

You could say that Physics is not a subset of Maths and Neither is Maths a subset of Physics. There is just a useful overlap which allows to live life more productively.
A Mathematical Proof is not a Physics 'Proof' - but it's a help in the right direction.

 

[jnorman]

"GUNT" - i LOVE that - hilarious...

for just this once, i find myself in agreement with hawking, and i rarely agree with anything he says.

i also concur with the above comment about string/M theory pushing the limits of what can actually be considered "science" - it all sounds too much like they are way out on a branch which has no real connection to the tree...

 

[Jolb]

I think it's pretty clear that this is true. It seems to me it can be proven directly:
1) Assume we have a GUT that is a mathematical theory from which all physical truths can be derived (let's say that means all possible experiments can be predicted).
2) Godel's theorem says that if the GUT is true, then it will make reference to physical truths that can be set up in the language of the GUT that cannot be proven true from the finite axioms of the GUT (assuming the GUT is both correct and internally consistent).
3) Ergo, physics cannot end, because we can still look for these truths that cannot be proven from the GUT, and which would then need to be axiomatized anew, and the process would continue.

I think you'd probably be on the track to a Nobel Prize if this was correct. Here's why I believe your proof doesn't work.

Physical theories aren't the same as mathematical formal systems. Physical theories assume the truth of a certain mathematical formal system, and then tack on a few added "Physical axioms" which apply specifically to the universe. If the GUT, including its mathematical axioms, is true, then if we Godelize the system and cook up a new axiom, then the new axiom is just another mathematical truth. If this new mathematical truth has physical implications within the GUT, it doesn't necessarily imply there's anything wrong with the Physical axioms. It just means a certain mathematical axiom needs to be added within the mathematical axioms, no new physical axioms are necessarily needed.

I.e., it might be like this:

Mathematical axioms
Math Axiom 1
Math Axiom 2
...
Math Axiom M

Physical axioms
Physics Axiom 1
Physics Axiom 2
...
Physics Axiom N
Applying Godel's theorem P times might result in

Mathematical axioms
Math Axiom 1
Math Axiom 2
...
Math Axiom M
Math Axiom M+1
...
Math Axiom M+P

Physical axioms
Physics Axiom 1
Physics Axiom 2
...
Physics Axiom N

These two theories might give completely different predictions. However, notice that the Physical axioms remained the same. The physics of the GUT has remained the same. So to prove Hawking's idea of a Godel's theorem for physics, we would need something that adds another physical axiom. If Physical Axioms 1-N always held, and if using these combined with whatever number of mathematical axioms, we could derive all physical truths, then Physical axioms 1-N are the Grand Unified Theory of Physics. There can still be mathematical godelization going on independently. Even if we chose different numbers of mathematical axioms in different scenarios, if Physical Axioms 1-N were always the correct physical axioms, then Physical Axioms 1-N would be the GUT of physics.

 

[Naty1]

Wikipedia has a discussion here:

http://en.wikipedia.org/wiki/Gödels_incompleteness_theorem

and some links for the adventurous.

 

[Hdeasy]

I would say also that restircted systems like conservation of energy may indeed have exceptions using a Gödel-like argument. Hence groups like Steorn may indeed be creating energy ex nihilo.

 

[sophiecentaur]

Hey all I am grade 10 i found hard understanding many of those topics but its so cool :D

Hello and welcome.
I must say, you jumped in here with both feet and found yourself up to your neck. There are less scary issues discussed on these pages and I'm sure you will have answers for some of the questions you can read on PF.

 

[Constantin]

I'm certain all basic laws of physics will be found. We're just a couple of major steps away. And that's true for any exact science. There are a set of basic rules from which all the complex theorems can be derived.

But physics as a science is very complex and it will get even more complex, so even after no more basic truths will be left to find, there will still be plenty of jobs for physicists. Furthermore, the physics science is so complex even now, that no physicist can be an expert in all its branches. So, even more jobs for physicists as physics is getting more complex.

 

[sophiecentaur]

I'm certain all basic laws of physics will be found. We're just a couple of major steps away.

That's a very 'Optimistic' view. I believe the Victorians had a very similar idea at the end of the 19th Century and then along came QM, Relativity etc. etc.

Perhaps I should ask how big these "major steps" are and how "basic" did you want to go?

Just imagine taking a ruler and drawing a line under the final page in one's Physics notebook.
"Right, that's sorted out, now let's move on to something really difficult. How about todays Sudoku in the Guardian?"

 

[Constantin]

Perhaps I should ask how big these "major steps" are and how "basic" did you want to go?

Those steps might need to be pretty big. As big as Quantum Mechanics or Relativity. At least in my optimistic point of view.

The basic rules have to deal with exactly what the Universe is. What the dimensions are. What the elementary particles are.
Right now it seems the Universe is all about geometry, information and probabilities. So at the basic level the rules should state exactly what the Universe's geometry is, the exact relationship between dimensions and the way the information exists, propagates and interacts.

At the very basic level, the amount of rules has to be small, very small.

Of course we'll still need a large number of complex theories for the more complex phenomena, and here active research may continue for very long, but finding the "basics" can't last for much longer. Just a few decades, at most, in my view.

 

[Zarqon]

I think you'd probably be on the track to a Nobel Prize if this was correct. Here's why I believe your proof doesn't work.
.
.
.

I'm not sure I understand why you need to separate mathematical and physical axioms? Can't you just say that any potential GUT contains a certain set of axioms, both mathematical and physical, and that this total set would then be subject to Gödel's theorem. Then it follows that this set of axioms has to be consistent (because of the principle of explosion) and thus incomplete.

 

[Constantin]

"Grand Unified Theory" (GUT) and "Theory of Everything" (ToE) are two different things.
GUT doesn't have to explain everything, it's just an improvement over the existing Standard Model.

 

[Zarqon]

"Grand Unified Theory" (GUT) and "Theory of Everything" (ToE) are two different things.
GUT doesn't have to explain everything, it's just an improvement over the existing Standard Model.

Ok, but it doesn't change much. Both a GUT and a TOE must be based on consistent sets of axioms, and as such, it appears that both will then necessarily be incomplete.

 

[sophiecentaur]

Ha - so GUT would really be just a MUT (Modest Unified Theory) and not that Grand after all.

 

[Zarqon]

Ha - so GUT would really be just a MUT (Modest Unified Theory) and not that Grand after all.

... although I wouldn't recommend using the word "modest" anywhere near the front page of a grant proposal

 

[sophiecentaur]

J. Swift used it once with a tongue in HIS cheek - as in "A modest proposal".
Would 'Half arsed' go down better?

 

[Jolb]

I'm not sure I understand why you need to separate mathematical and physical axioms? Can't you just say that any potential GUT contains a certain set of axioms, both mathematical and physical, and that this total set would then be subject to Gödel's theorem. Then it follows that this set of axioms has to be consistent (because of the principle of explosion) and thus incomplete.

My apologies for resurrecting an old thread, but it is a thread I really like a lot.

Anyway, I think there is a reason one needs to separate mathematical axioms from physical axioms: the world of math contains a lot of stuff that is impossible in the physical world. E.g., fractals. Fractals do not exist in our universe.

So mathematical axioms apply to the mathematical ("Platonic") universe, whereas physical axioms constrain the mathematical possibilites to only those things which apply to OUR universe, which certainly does not contain all the aspects of the mathematical universe.
[Note that in my previous post I used the word "universe" to refer to the PHYSICAL universe. Here I broaden the term "universe" to include both the physical and "platonic" universes.]

[[Second note: I do believe that there MAY be extant physical truths which cannot be mapped isomorphically to the mathematical "platonic" universe, so the physical universe is not necessarily a subset of the platonic universe. If this were the case, then a ToE/GUT would clearly be impossible, unless we expand our theories to include nonmathematical ones.]]

 

[Reptillian]

To paraphrase Gödel himself "Either the universe is incomprehensible, or the mind is more than a machine."

 

[Jolb]

To paraphrase Gödel himself "Either the universe is incomprehensible, or the mind is more than a machine."

Wow. That's an amazing quote. I never knew that Godel entertained "cognitive science" ideas. Do you have a source for that?

Godel is one of those mathematicians who definitely is underappreciated for his brilliance in physics, philosophy, etc. A true genius. No wonder Einstein and he were such good friends.

 

[Zarqon]

My apologies for resurrecting an old thread, but it is a thread I really like a lot.

Anyway, I think there is a reason one needs to separate mathematical axioms from physical axioms: the world of math contains a lot of stuff that is impossible in the physical world. E.g., fractals. Fractals do not exist in our universe.

Really? The coastline would like a word with you :P

So mathematical axioms apply to the mathematical ("Platonic") universe, whereas physical axioms constrain the mathematical possibilites to only those things which apply to OUR universe, which certainly does not contain all the aspects of the mathematical universe.

I think this doesn't change anything of what I wrote earlier. You may say you put up two sets of axioms (which may share some statements), but I'm simply saying that you can create a total set of axioms, containing all axioms from both sets defined on the combined space of your platonic + physical universe, and that this total set of axioms should still obey Gödel's theorem, i.e. be consistent and hence incomplete.

 

[georgir]

Ok, I am a bit confused.

In physics, we've already had theories which we thought were complete. They turned out to not match reality, so they got changed, but disregarding that, how are they "incomplete" in theoretical sense?

Assume reality were a simple physical model with just point-particles and infinite-speed gravity for example... What unprovable truths could there be in it to spell the "end of physics"?

After all, Goedel's theorem talks about "any" mathematical theory and is not concerned with reality... I can't see its physical equivalent.

 

[Jolb]

Really? The coastline would like a word with you :P

The coastline is not an example of a fractal, and the coastline paradox itself is somewhat nonsensical. I don't want to go on about debunking the coastline paradox, but could the coastline possibly be any longer than a line of n water molecules, where n is the total number of water molecules on Earth (or the solar system, or the galaxy)?

The fact is that quantum physics dictates that once measurements are made on scales smaller than the Planck length, the only structure is randomness. A non-random fractal with infinite self-similarity (like the Mandelbrodt set) is an example of a non-random structure which goes down to infinitely small length scales, and as such cannot exist in a universe constrained by Heisenberg uncertainty on Planck-length scales.

I think this doesn't change anything of what I wrote earlier. You may say you put up two sets of axioms (which may share some statements), but I'm simply saying that you can create a total set of axioms, containing all axioms from both sets defined on the combined space of your platonic + physical universe, and that this total set of axioms should still obey Gödel's theorem, i.e. be consistent and hence incomplete.

I think I'll be repeating myself at this point. Physical axioms are things which constrain the wide world of mathematical possibilities to things that only apply to our universe. If we Godelize your "combined system" and it gives us a new axiom which has no specific reference to the physical universe, then it is a purely mathematical axiom and doesn't change what could be the complete set of physical axioms. The Godelization might only lead to new mathematical axioms and not any new physical axioms, and we could regard the complete set of physical axioms as the be-all-and-end-all theory of physics, despite mathematics being incomplete.

Ok, I am a bit confused.

In physics, we've already had theories which we thought were complete. They turned out to not match reality, so they got changed, but disregarding that, how are they "incomplete" in theoretical sense?

Assume reality were a simple physical model with just point-particles and infinite-speed gravity for example... What unprovable truths could there be in it to spell the "end of physics"?

After all, Goedel's theorem talks about "any" mathematical theory and is not concerned with reality... I can't see its physical equivalent.

You're right. A simple enough physical theory could be complete, at least in the physical sense. There can be mathematical incompleteness without physical incompleteness.

What Hawking explained in the talk I linked earlier is that only now with the advent of M-theory does he think a physical kind of Godel's theorem could exist--it provides the needed complexity that wouldn't be there in your simple model.

 

[Reptillian]

Wow. That's an amazing quote. I never knew that Godel entertained "cognitive science" ideas. Do you have a source for that?

Godel is one of those mathematicians who definitely is underappreciated for his brilliance in physics, philosophy, etc. A true genius. No wonder Einstein and he were such good friends.

The quote from Gödel is: "Either mathematics is too big for the human mind or the human mind is more than a machine"

I've been trying to figure out from whence this quote originates, but haven't had much luck.

I do agree that Gödel was one of the greatest, if not the greatest, mathematican/logician of the 20th century.

 

[sophiecentaur]

The quote from Gödel is: "Either mathematics is too big for the human mind or the human mind is more than a machine"

I've been trying to figure out from whence this quote originates, but haven't had much luck.

I do agree that Gödel was one of the greatest, if not the greatest, mathematican/logician of the 20th century.

I don't find that quote very awe inspiring, as a matter of fact. Why should Mathematics NOT be too big for the human mind? Why should the human mind be more than a machine? The two alternatives are not mutually exclusive and there is too much of a hint of the 'spiritual' about the statement, playing to the vanity of the reader rather than to reason.

I also take issue with the idea of Science consisting of Axioms. It consists of hypotheses which can be verified of falsified. Mathematics happens to be a suitable tool for Science when its axioms happen to produce roughly parallel outcomes to what we observe in Science and that enables 'models' to be built and to allow predictions by extrapolation. There need be no more than that.

 

[Jolb]

I don't find that quote very awe inspiring, as a matter of fact. Why should Mathematics NOT be too big for the human mind? Why should the human mind be more than a machine? The two alternatives are not mutually exclusive and there is too much of a hint of the 'spiritual' about the statement, playing to the vanity of the reader rather than to reason.

The reason mathematics should not be too big for the human mind is because mathematics is a creation of the human mind. Each and every mathematical result is cooked up by a human using some understanding of the mathematics--thus everything in math is designed to be amenable to some kind of understanding. For that reason, I think that there is something 'spiritual' or at least 'transcendental' about saying that mathematics is beyond comprehension.

I also take issue with the idea of Science consisting of Axioms. It consists of hypotheses which can be verified of falsified. Mathematics happens to be a suitable tool for Science when its axioms happen to produce roughly parallel outcomes to what we observe in Science and that enables 'models' to be built and to allow predictions by extrapolation. There need be no more than that.

Well, if science were just about building model after model, why would theoretical physicists be so interested in a "unified theory"? The idea is to reduce the science to a minimal number of hypotheses--and these hypotheses are like the axioms of a formal system.

 

[sophiecentaur]

The reason mathematics should not be too big for the human mind is because mathematics is a creation of the human mind. Each and every mathematical result is cooked up by a human using some understanding of the mathematics--thus everything in math is designed to be amenable to some kind of understanding. For that reason, I think that there is something 'spiritual' or at least 'transcendental' about saying that mathematics is beyond comprehension.


Well, if science were just about building model after model, why would theoretical physicists be so interested in a "unified theory"? The idea is to reduce the science to a minimal number of hypotheses--and these hypotheses are like the axioms of a formal system.

I agree that Maths can be looked upon as an invention but it is easy to invent something that develops more complexity than you can cope with. Just take a simple fractal, for instance. Imo, we attach a 'transandental' nature to what Maths delivers to us, after being invented, because that's how the mind works when things get too hard to deal with.

A unified theory of Science is only an attempt to reduce what we see to a single set of rules - again this is only another example of how our minds attempt to simplify and generalise our view of our world (to cope). But we approach Science from the starting point of what we have observed rather than, as with Maths, by 'what if we set up this set of rules?' - or axioms. I really do think they are two separate approaches.