Impact of Gödel's incompleteness theorems on a TOE

[Lievo]

Where do you get this from? My original post motivated the plausibility of no end to new phenomena and laws, and then proposed that there is no evidence for a finite number. Was there something unclear in my post??

No, but I came to say that Godel's incompletness implicates an infinite number of law if you also assume your TOE is consistent. And then you state this supports your view, while it's contradicting it. But I guess it was my post which was not clear.

Let's explain with Conway's game of life: sure you can describe the evolution with a finite set of laws. But it's not completely true, in the sense that there is something interesting to say that is not in the basic rules. As CGL can behave as an universal Turing machine, that means that you can encode something which says ''Conway's game of life is consistent''. The evolution of this will never halt: this is true, but you can't prove it with the basic rules of CGL. So there something interesting to say about the evolution of CGL which is not specified in [nor derivable from /added] the basic rules. Let's add it!

Now you have a CGL plus the law that CGL is consistent: CGL+CON(CGL). Fine. But then it's not [provably /added] complete. Meaning you can arrange the CGL+CON(CGL) to encode something that says ''Conway's game of live plus the assumption that it is consistent is consistent''. This is true, but you can't prove it with the basic rules of CGL. So there something interesting to say about the evolution of CGL+CON(CGL) which is not specified in the basic rules. Let's add it!

Now we have CGL+CON(CGL)+CON(CGL+CON(CGl))...

At first sight this seems quite artificial and of no uses. I disagree. This is what guarantee you that some state never halt, meaning that while runing CGL you will always find some new configuration you never saw before.

Your point was that, when we go back closer and closer to the big bang, we can expect to find new law again and again. This is quite the same with Conway's game of life: for [STRIKE]certain well chosen[/STRIKE] [any interesting] initial state, you'll always find new configuration again and again.

 

[jmoorhea]

We could be a long long way off a complete and self consistent TOE even with the major advances in all these physical theories like String Theory, so I'm not sure we are in a position to even ask the question "How does Godels Theorems impact on a TOE" never mind answer it yet.
Maybe you might be interested in Max Tegmark's theory (detailed for the lay reader on his personal website) which basically postulates that the universe is nothing other than mathematics at the most fundamental level (Maybe someone could correct me on his exact theory). One could say that this is a TOE. I was very interested in this theory but I did question in the back of my mind how Godels Theorems impacted on Tegmarks Theory. He does actually go into this subject in his personal website although I can't remember what he said about Godel's Incompleteness Theorems.

http://space.mit.edu/home/tegmark/crazy.html

 

[PAllen]

No, but I came to say that Godel's incompletness implicates an infinite number of law if you also assume your TOE is consistent. And then you state this supports your view, while it's contradicting it. But I guess it was my post which was not clear.

Let's explain with Conway's game of life: sure you can describe the evolution with a finite set of laws. But it's not completely true, in the sense that there is something interesting to say that is not in the basic rules. As CGL can behave as an universal Turing machine, that means that you can encode something which says ''Conway's game of life is consistent''. The evolution of this will never halt: this is true, but you can't prove it with the basic rules of CGL. So there something interesting to say about the evolution of CGL which is not specified in [nor derivable from /added] the basic rules. Let's add it!

Now you have a CGL plus the law that CGL is consistent: CGL+CON(CGL). Fine. But then it's not [provably /added] complete. Meaning you can arrange the CGL+CON(CGL) to encode something that says ''Conway's game of live plus the assumption that it is consistent is consistent''. This is true, but you can't prove it with the basic rules of CGL. So there something interesting to say about the evolution of CGL+CON(CGL) which is not specified in the basic rules. Let's add it!

Now we have CGL+CON(CGL)+CON(CGL+CON(CGl))...

At first sight this seems quite artificial and of no uses. I disagree. This is what guarantee you that some state never halt, meaning that while runing CGL you will always find some new configuration you never saw before.

Your point was that, when we go back closer and closer to the big bang, we can expect to find new law again and again. This is quite the same with Conway's game of life: for [STRIKE]certain well chosen[/STRIKE] [any interesting] initial state, you'll always find new configuration again and again.

I orginally intended to ignore this because I was busy, but now have some time and am completely confused. To try to be totally clear, let me define:

A) Godel's theorem applies to the universe and thus every finite system of laws is incomplete.

B) Godel's theorem may not apply to the universe, but (suggestively) there might be other principles or simply the fact that any finite system of laws is an incomplete theory or the univers.

C) Evidence of pattern of discovery of laws through history.

D) Evidence that there the universe can be described by a finite number of laws.

My original post may be described as:

A or B are likely, or at least plausible, C supports this and D does not exist (so far as I know).

Lievo states: if A, then the number of laws is definitely infinite and this contradicts your statement.

Under what laws of logic is this possibly true? It seems to me that it supports a subcase of my proposal, thus supporting my original post, as I originally responded.

 

[Lievo]

I think the confusion comes from

Godel's theorem (...) every finite system of laws is incomplete.

Godel's theorems are not limited to finite systems of laws, but extend to every recursively enumerable theories. It means: if you can mechanically construct the axioms (as I did in the previous post), then the system can be infinite and still subjects to Godel's theorems.

Better?

 

[PAllen]

I think the confusion comes from

Godel's theorems are not limited to finite systems of laws, but extend to every recursively enumerable theories. It means: if you can mechanically construct the axioms (as I did in the previous post), then the system can be infinite and still subjects to Godel's theorems.

Better?

Ok, but then this isn't contradiction of my post, but an observation of a minor wording inaccuracy completely secondary to the thrust of the post or the thread.

 

[Lievo]

Ok, but then this isn't contradiction of my post

As I understood it you were saying: A: there is a likely an infinite number of laws describing the universe; B: Godel's theorems apply to certain finite set of laws; A+B: Godel's theorems do not apply to the universe.

If that's your view there is the flaw I mentionned. If that's not, I did not understand your view. Can you explain again?

an observation of a minor wording inaccuracy completely secondary to the thrust of the post or the thread.

Sorry, but are you sure you want an honest discussion about the validity of your view? In this thread I already spent several posts arguing with some persons that turns out to be more interested in pretending than in understanding. I'm a bit fed up of this kind of discussion, so if you want me to stay and discuss I would like you to lower your tone. Please.

 

[PAllen]

As I understood it you were saying: A: there is a likely an infinite number of laws describing the universe; B: Godel's theorems apply to certain finite set of laws; A+B: Godel's theorems do not apply to the universe.

If that's your view there is the flaw I mentionned. If that's not, I did not understand your view. Can you explain again?

No, that's not at all what I meant. There was a lot of discussion in this thread about whether you could say Godel's theorem actually applied or could apply to the universe. There appeared to be an implication that if the universe didn't formally meet the criteria for Godel's theorem to apply, then one should assume a finite number of laws would suffice. I wanted to say: maybe it (Godel's theorem) does apply, maybe it doesn't, but even if it doesn't, it could be taken to suggest we seriously consider that no finite set of laws describes the universe. Then I tried to briefly support the idea of an infinite number of laws, and question why the converse was believed with no real evidence.

Sorry, but are you sure you want an honest discussion about the validity of your view? In this thread I already spent several posts arguing with some persons that turns out to be more interested in pretending than in understanding. I'm a bit fed up of this kind of discussion, so if you want me to stay and discuss I would like you to lower your tone. Please.

Sorry about any tone, but I truly could not understand how I was being so misunderstood despite my best attempts to be clear. I could not begin to comprehend the claimed contradiction with with what I said. Even in my clarification post, I explicitly said my intent was to say A or B (meaning Godel's theory may apply).

 

[Lievo]

about any tone

Thank you

There appeared to be an implication that if the universe didn't formally meet the criteria for Godel's theorem to apply, then one should assume a finite number of laws would suffice.

Ok I think that's the part I did not understand. I fully agree this idea is strange.

I wanted to say: maybe it (Godel's theorem) does apply, maybe it doesn't, but even if it doesn't, it could be taken to suggest we seriously consider that no finite set of laws describes the universe.

Here in bold the point I was contradicting: assuming Godel's theorem (and consistency) necessarly provides an infinite number of law, so one can't take GT doesn't apply to suggest that no finite set of law describe the universe. Do you see my point here?

I tried to briefly support the idea of an infinite number of laws, and question why the converse was believed with no real evidence.

Here I fully agree. What I tried to explain was that if you think that the universe is consistent, it can't really escape Godel's theorem, and it can't escape having an infinite number of laws. So now I see and understand why you were saying this supports your view. You were right, sorry I did not understand it first time.

 

[Chalnoth]

No, that's not at all what I meant. There was a lot of discussion in this thread about whether you could say Godel's theorem actually applied or could apply to the universe. There appeared to be an implication that if the universe didn't formally meet the criteria for Godel's theorem to apply, then one should assume a finite number of laws would suffice.

Well, if the only criterion for Godel's theorem to apply is that the theory be recursively-enumerable, then because we are described by the theory of everything (whatever that theory may be), if we ever discover our theory that will be a demonstration of its recursive enumeration.

 

[PAllen]

Thank you


Here in bold the point I was contradicting: assuming Godel's theorem (and consistency) necessarly provides an infinite number of law, so one can't take GT doesn't apply to suggest that no finite set of law describe the universe. Do you see my point here?

Maybe I see a bit of your point, but I'm not sure. Let me try to expand my point here. Prior to Godel, it was considered 'obvious' by almost all mathematicians that a finite set of consistent axioms formed a complete system; there may be an infinite number derivable statements, but any statement about the domain covered by the axioms was expected to be provably true or false. Godel changed this. I am simply making an analogy (in case Godel doesn't formally apply to the universe): physicists have often 'assumed' some finite set of laws will ultimately explain the universe at a fundamental level (I have assumed this, tacitly, until encountering this thread). Independent of whether Godel applies to the universe, physicists should beware of tacitly assuming that which is convenient: that a finite system of laws can constitute a complete foundation for the universe. I'm making an analogy, not a formal argument here.

 

[friend]

And what physical entity corresponds to an "axiom"?

 

[Lievo]

Independent of whether Godel applies to the universe, physicists should beware of tacitly assuming that which is convenient: that a finite system of laws can constitute a complete foundation for the universe. I'm making an analogy, not a formal argument here.

Yes I see this point. In a sense, what I'm saying is that one can translate this analogy into a formal argument.

And what physical entity corresponds to an "axiom"?

There is no need that the physical entities correspond to one axiom. For exemple in Conway's life there exists 'physical entities' such as "[URL and http://en.wikipedia.org/wiki/Gun_(cellular_automaton)" , but this does not corresponds to any axiom (although it is a logical consequence of it).

 

[Chalnoth]

And what physical entity corresponds to an "axiom"?

There would be no correspondence between physical entities and axioms. Rather, if the theory can be expressed in an axiomatic way (which I strongly suspect is possible), physical entities are a consequence of the combination of axioms that make up the theory.

 

[friend]

There is no need that the physical entities correspond to one axiom. For exemple in Conway's life there exists 'physical entities' such as "[URL and http://en.wikipedia.org/wiki/Gun_(cellular_automaton)" , but this does not corresponds to any axiom (although it is a logical consequence of it).

I take it that the axioms consist of the rules in which to move these physical items in this game (or to go from one configuration to the next). But these rules are abstract, made up by human intelligence and are not physically represented or proven necessary. Is reality equivalent to our model of it? Is the wavefunction physically real?

physical entities are a consequence of the combination of axioms that make up the theory.

So you're saying that physics is derived (is a consequence of) logic (a system of abstract axioms)?

 

[Lievo]

I take it that the axioms consist of the rules in which (...) to go from one configuration to the next (...) Is reality equivalent to our model of it? Is the wavefunction physically real?

Here you're asking for if the TOE would be the correct interpretation. To me this is outside of science, the same way it's outside science to decide whether many-worlds is better than Copenhagen interpretation. So saying that the wavefunction is the real thing, or saying that it's only an accurate way for computing experimental predictions... up to you.

 

[Chalnoth]

Here you're asking for if the TOE would be the correct interpretation. To me this is outside of science, the same way it's outside science to decide whether many-worlds is better than Copenhagen interpretation.

Well, if you wanted an analogy that is "outside of science", you picked a really poor one. The Copenhagen interpretation is blatantly false because it makes no statements about when a wave function does or does not collapse. Detailed observations of how wavefunction collapse occurs agree with the many worlds interpretation.

 

[Chalnoth]

Is reality equivalent to our model of it? Is the wavefunction physically real?

If our model of reality were completely correct, then yes, reality would be equivalent to our model of it.

However, we know this isn't the case. All of our physical models have places where they provide nonsensical results, as well as places where they disagree with one another. So this means that our current models are mere approximations of reality, not equivalent to reality. But I think the eventual goal of theoretical physics is to find a model that is equivalent to reality.

 

[Lievo]

The Copenhagen interpretation is blatantly false

That's what many-worldist say.

 

[friend]

If our model of reality were completely correct, then yes, reality would be equivalent to our model of it.

Physical theories consist of ideas in our heads that we translate into scribbles on paper. No way can it be equivalent to reality itself. I guess the best we can do is to completely understand the universe in term of these scribbles we put on our paper. Perhaps this concern about Godel's Incompleteness Theorem is just quibbling about our scribbling ;-)

 

[Chalnoth]

That's what many-worldist say.

And attacking the conclusion instead of discussing the argument is what people who are wrong say...

 

[Chalnoth]

Physical theories consist of ideas in our heads that we translate into scribbles on paper. No way can it be equivalent to reality itself. I guess the best we can do is to completely understand the universe in term of these scribbles we put on our paper. Perhaps this concern about Godel's Incompleteness Theorem is just quibbling about our scribbling ;-)

If the physical theory is completely correct, then it is absolutely equivalent. The difficulty is that we don't know if we'll ever be able to find a completely correct physical theory. None of our current ones are.

Bear in mind that mathematical structures that look different on the surface can be the exact same thing deep down. For instance, it was mentioned before that the natural numbers (1, 2, 3, 4, ...) are the same as all integers (... -3, -2, -1, 0, 1, 2, 3, ...). This may seem strange to you. It is certainly the case that the two look very different. However, I can reorder the set of all integers like so: (0, 1, -1, 2, -2, 3, -3, ...). Reordered in this way, I can associate each integer with a natural number, and the two different numbering systems become just different ways of talking about the exact same system.

So if we ever did find the theory of everything, it would be merely one way of writing down said theory. There could be a great number of different, but completely equivalent ways of writing down the same theory. Given that the theory of everything is correct, they would also be equally equivalent to reality.

For example, if we consider string theory, there are five different types of string theory: HO, HE, I, IIA, IIB. But we are now discovering that the different string theories are all actually describing the exact same underlying theory, they're just different ways to look at it.

 

[Fra]

Is reality equivalent to our model of it? Is the wavefunction physically real?

If our model of reality were completely correct, then yes, reality would be equivalent to our model of it.

However, we know this isn't the case. All of our physical models have places where they provide nonsensical results, as well as places where they disagree with one another. So this means that our current models are mere approximations of reality, not equivalent to reality. But I think the eventual goal of theoretical physics is to find a model that is equivalent to reality.

I object to this reasoning.

IMO, there is no way to determine, measure of infer that our knowledge of reailty is "correct" and equivalent to the "actual reality". Also, our actions are expected to depend only on our knowledge of reality, not on reality itself. Only the feedback of our action depends on the unknown, and that's how we get informed.

IMHO, the only reasonable conclusion is that it's the very notion of reality or "actual reality" that is just obscure, redundant and confusing. I also don't think that the goal of theoretical physics is to reveal what reality really is. I think it's about trying to predict nature, by producing rational expectations based on what we know.

That the actual future is different from the expectation does not mean the expectation was irrational. On the contrary, the UNexpected changes is as I see it the essence of the collapse in the first place.

I think that when people try to expect the unexpected, they are IMHO missing an essential point in inductive learning, and inference. IF there was no unexpected events, everything would essentially be trivial as there are no interactions = no new information.

/Fredrik

 

[Lievo]

That's what many-worldist say

And attacking the conclusion instead of discussing the argument is what people who are wrong say...

Read my sentence again. This is wrong and attacking your conclusion, really?

If you really want an honest discussion on this (why do I have a doubt? ), please consider the following statement:

The many-world interpretation is blatantly false because it makes no statements about when a wave function does or does not branched.

Think about it: this is exactly the same problem. So if you think Copenhagen interpretation is blatantly false for this reason, you have little choice but to make the same statement for MWI.

... and, one last thing, please do not assume what is my favorite interpretation before I expressed an opinion myself. :grumpy:

 

[Chalnoth]

I object to this reasoning.

IMO, there is no way to determine, measure of infer that our knowledge of reailty is "correct" and equivalent to the "actual reality".

Well, if there were no possible limitations on the behavior of reality, I would agree to this. However, if we constrain reality to be self-consistent, then that places very significant constraints on possible theories of everything.

Now, if it just so happened that we managed to build a full list of possible "theories of everything", and demonstrate that it was a full listing, and furthermore demonstrate experimentally that only one of these possible theories fits the reality we observe, then we would have found the theory of everything.

There are a lot of if's here, of course. But it isn't in principle impossible to find the theory of everything.

That said, I should mention that any theory of everything that we do find won't tell us everything about our world. Even if we could narrow it down to just one possible theory of everything, we would still run into the classical problem of inference when it comes to, for instance, determining the behavior of stars.

 

[Chalnoth]

The many-world interpretation is blatantly false because it makes no statements about when a wave function does or does not branched.

Think about it: this is exactly the same problem. So if you think Copenhagen interpretation is blatantly false for this reason, you have little choice but to make the same statement for MWI.

This is incorrect. Decoherence is the mechanism of "branching" in MWI. You may also wish to read up on einselection.

 

[Hurkyl]

The many-world interpretation is blatantly false because it makes no statements about when a wave function does or does not branched.

Actually, MWI says exactly how the wave function evolves -- via the Schrödinger equation, always (or the appropriate analog, depending on what variant of QM you're interpreting).

The Copenhagen interpretation is blatantly false because it makes no statements about when a wave function does or does not collapse.

What you meant to say is that the CI has an incomplete description of how the wave-function evolves over time.

 

[Lievo]

This is incorrect. Decoherence is the mechanism of "branching" in MWI. You may also wish to read up on einselection.

Good to know you have solved this problem. You may wish to read up onhttp://www-physics.lbl.gov/~stapp/bp.PDF" .

 

[Lievo]

EDIT -sorry misreading

 

[Chalnoth]

What you meant to say is that the CI has an incomplete description of how the wave-function evolves over time.

I don't know if that's what I meant to say, but that is another accurate way of saying the same thing. I was just pointing out that the point of incompleteness is at wavefunction collapse (which is ill-defined in the CI).

 

[Fra]

Well, if there were no possible limitations on the behavior of reality, I would agree to this. However, if we constrain reality to be self-consistent, then that places very significant constraints on possible theories of everything.

This is using the logic of expectations again. How do we infer these limitations of reality? ie. some theory space that nature MUST be constrained do? Again, there can be nothing but an expected constraints as well.

Also, what exactly does self-consistency mean when applies to nature? That's not a simple question I think.

Usually self-consistency mean that consistent parts must not make inconsisting or conflicting implications.

But there is in fact not a large difference between inconsistency based "conflitcs" and interactions due to systems not beeing equilibrated. It's quite possible, that the inconsistencties you refer to (typically that different observers EXPECT/predict different things) are in fact a key feature, and the origon of interactions. (This is what I personally think is the case).

So, inconsistencies are not necessarly fatal, they merely imply an interaction that will served to restore the consistency. At full consistency between two systems, I see them as beeing in equiblrium.

So your assumption of equiblirium, for me, is the same as to assume that the universe is at equilibrium. This is likely a good approximation for some interactions, but not for all.

But it isn't in principle impossible to find the theory of everything.

Yes, it's not impossible that that we are in an equiblirium. But to me it's a possibility, not something that is obvious. I also personally think that this is probably not at all likely in the full sense. I also think that the assumption of these things makes it harder to understand how things work. I see them as realist-illusions that are well POSSIBLE, but that generally inhibit progress by causing misfocus.

In my personal view, it's EXPECTATIONS that need to be self-consistent. Different systems(observer) encoding different expectations need NOT be consistent, except when at equibrium. Then equilibrium corresponds at best to a known deterministic transformation rules between the deviations (just like transformations are responsbile for every single symmetry in our standard models)

I just claim, that this is apparently yet not understood well, and that it's exactly hte meaning of expectations, information updates etc that we need to understand better. Copenhagen interpretation is certainly not the answer, but I think dissing the notion of information updates and subjective information as false is rushing into very doubtful conclusions.

/Fredrik

 

[Fra]

I don't know if that's what I meant to say, but that is another accurate way of saying the same thing. I was just pointing out that the point of incompleteness is at wavefunction collapse (which is ill-defined in the CI).

What is the problem with this?

The schrödinger equation essentialy is the computation of our expectations. It is the EXPECTED self-evolution, given the past.

But if new information arrives, of course the expectations needs to be updated (ie to put in the new prior).

And of course an information update is not predictable, that would indeed be a contradiction.

/Fredrik

 

[Chalnoth]

This is using the logic of expectations again. How do we infer these limitations of reality? ie. some theory space that nature MUST be constrained do? Again, there can be nothing but an expected constraints as well.

Also, what exactly does self-consistency mean when applies to nature? That's not a simple question I think.

Self-consistency merely means that every sufficiently-specified statement is definitively either true or false. In other words, every ambiguous statement can be made true or false by specifying it more precisely. We may not always be aware of whether a statement is true or false. But under this assumption one or the other must be the case.

And I, for one, am perfectly fine with making this assumption because it is required for reality to make sense. If we allow inconsistencies in physical theory, even if we are careful to limit those inconsistencies so that they don't make the entire theory meaningless, those areas where the inconsistencies arise are still nonsensical. A theory of everything must describe everything, but an inconsistency in the theory of everything means that there are some things it cannot describe.

Usually self-consistency mean that consistent parts must not make inconsisting or conflicting implications.

But there is in fact not a large difference between inconsistency based "conflitcs" and interactions due to systems not beeing equilibrated. It's quite possible, that the inconsistencties you refer to (typically that different observers EXPECT/predict different things) are in fact a key feature, and the origon of interactions. (This is what I personally think is the case).

This makes no sense to me. What you have said here is that interactions arise to recover consistency in the theory, which is the same thing as saying that the nature of physical law is constrained by consistency.

But no, finding a theory of everything would not be a discovery that we are in equilibrium. The two are completely and utterly different things. I can make neither heads nor tails of what you mean by equilibrium in your post, but it clearly has nothing whatsoever to do with the thermodynamic meaning.

 

[Chalnoth]

What is the problem with this?

The schrödinger equation essentialy is the computation of our expectations. It is the EXPECTED self-evolution, given the past.

But if new information arrives, of course the expectations needs to be updated (ie to put in the new prior).

And of course an information update is not predictable, that would indeed be a contradiction.

/Fredrik

I don't see how this has anything to do with my post. I was merely commenting on the incompleteness of the Copenhagen interpretation of quantum mechanics, that the Copenhagen interpretation makes no statements one way or another about how or when wave function collapse occurs. It merely states that when wave function collapse does occur, you can calculate the probability of various outcomes based upon the wave function before collapse. But the Copenhagen interpretation simply doesn't tell you when the system has or has not collapsed.

 

[Fra]

On my way to bed, some more short comments...

Self-consistency merely means that every sufficiently-specified statement is definitively either true or false.

In a given formal system, yes. I suggested previously that in nature, there is no unique such, rather each subsystem is a different formal system.

This makes no sense to me. What you have said here is that interactions arise to recover consistency in the theory, which is the same thing as saying that the nature of physical law is constrained by consistency.

Yes almost! but with the critical distinction that physical law themselves are evolving. They are not eternally true, neither objective. Objectivity could also be emergent.

This means that interactions between two subsystems, can be thought of as two interacting theories; implictly encoded in the matter.

Or at least this is a possibility. Nature does not NEED laws or "actual reality" notions to make sense. It could even be that seeing that laws evolved from common simple codes, may help alot.

/Fredrik

 

[Chalnoth]

Let me put it this way: if it is possible to describe reality as a set of distinct but interrelated physical systems, then it is also possible to describe reality as one physical system. If, in one description of reality, some physical law changes with time, then in another description the physical laws remain unchanged while the apparent change is explained by the dynamics of the unchanging theory.

Basically, if there is a way that reality behaves, then there is a way to accurately describe that behavior. Because of this, it must be possible to narrow it all down to one single self-consistent structure (though that structure may be extremely complex).