Impact of Gödel's incompleteness theorems on a TOE

[Lievo]

No, it supports it.

So, if there is a finite number of law, this supports your view. And if there is an infinite number of laws, this also supports your view. Well, good for your view.

 

[S.Daedalus]

Turning the question around, I asked myself: Have I ever seen even the slightest actual evidence that the universe can be described by a finite set of laws (let alone laws motivated by encompassing just known phenomenology, e.g. 4 forces) ? My answer is that I am not aware of *any* evidence *at all* that this should be expected.

Well, the existence of a finite set of laws describing the evolution of systems that manifestly are subject to incompleteness constitutes such evidence, as in the Game of Life example for instance. In fact, if the universe can in principle be simulated on a computer, since we can finitely describe computers, we can finitely describe the universe, as well. This may of course be intractable, or even in principle impossible -- as there's no guarantee that the universe can be simulated on a computer --, but the possibility exists, and it's arguably the conservative one: only if we are certain that no such description can be found (a state I can't see how to arrive in) should we abandon the attempt.

 

[PAllen]

So, if there is a finite number of law, this supports your view. And if there is an infinite number of laws, this also supports your view. Well, good for your view.

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??

 

[PAllen]

Well, the existence of a finite set of laws describing the evolution of systems that manifestly are subject to incompleteness constitutes such evidence, as in the Game of Life example for instance. In fact, if the universe can in principle be simulated on a computer, since we can finitely describe computers, we can finitely describe the universe, as well. This may of course be intractable, or even in principle impossible -- as there's no guarantee that the universe can be simulated on a computer --, but the possibility exists, and it's arguably the conservative one: only if we are certain that no such description can be found (a state I can't see how to arrive in) should we abandon the attempt.

Ok, that is a substantive response. However, all it says is that it is possible there are a finite number of laws, which I don't doubt (actually, I never thought about these questions until this thread appeared). I still don't see it as evidence in favor of a finite number of laws. I also don't see that assuming finite until proven otherwise is the conservative position. It feels more like the wishful thinking position.

I also don't see that admitting the plausibility or likelihood of no finite number has much effect on the practice of physics, any more than Godel's theorem had much effect on the practice of number theory. The only significant case I know of where a meaningful hypothesis turned out to be independent of other axioms is the continuum hypothesis. People speculate about Goldbach's or P<>NP, but these are just that - speculations. Similarly, I would expect the phenomena outside the scope of some finite set of laws to be ever more exotic as physics advances, with exceedingly small contribution to the universe.

[EDIT] Actually, I can see one positive effect: less wrangling about a search for a TOE, and more focus on effective theories for known phenomena that also make some new predictions. The attitude that a 'theory of everything so far' is worthwhile and all you can really know, I think is good for physics.

 

[S.Daedalus]

Ok, that is a substantive response. However, all it says is that it is possible there are a finite number of laws, which I don't doubt (actually, I never thought about these questions until this thread appeared). I still don't see it as evidence in favor of a finite number of laws. I also don't see that assuming finite until proven otherwise is the conservative position. It feels more like the wishful thinking position.

It's conservative in so far that we know scores of examples of systems that can be simulated on a computer (up to arbitrary finite precision given enough computing power), and none that can't. So assuming that there exist such systems is unwarranted.

 

[PAllen]

It's conservative in so far that we know scores of examples of systems that can be simulated on a computer (up to arbitrary finite precision given enough computing power), and none that can't. So assuming that there exist such systems is unwarranted.

Assuming no end to laws, it would still be true that all phenomena within the scope of of some finite set could be simulated. Thus, at all times, it would be true that everything we currently understand could be simulated. Unless I misunderstand your point, I don't get its significance.

 

[S.Daedalus]

Assuming no end to laws, it would still be true that all phenomena within the scope of of some finite set could be simulated. Thus, at all times, it would be true that everything we currently understand could be simulated. Unless I misunderstand your point, I don't get its significance.

There's no meaning to claiming that nature 'actually' is described by an inexhaustible set of laws if we can capture it to arbitrary precision with a finite one; no experiment would be able to tell the difference.

 

[PAllen]

There's no meaning to claiming that nature 'actually' is described by an inexhaustible set of laws if we can capture it to arbitrary precision with a finite one; no experiment would be able to tell the difference.

An example of how this could play out is that in the first moments of the universe, and final moments of collapse, there are a plethora of fundamentally new laws that become significant, that otherwise are not. Given the limited information content of cosmic microwave background and other residual signals, and absence of information from inside event horizons, we could find ourselves unable to simulate such things with any precision until such conditions could be reproduced and studied; and we might never get a complete description / simulation.

Just one example of how my statement might not be without meaning.

 

[S.Daedalus]

An example of how this could play out is that in the first moments of the universe, and final moments of collapse, there are a plethora of fundamentally new laws that become significant, that otherwise are not. Given the limited information content of cosmic microwave background and other residual signals, and absence of information from inside event horizons, we could find ourselves unable to simulate such things with any precision until such conditions could be reproduced and studied; and we might never get a complete description / simulation.

Just one example of how my statement might not be without meaning.

But then, we have the case that experiment disagrees with expectation, and hence, a violation of the condition that we should be able to capture nature to arbitrary precision; in such a case, of course one would have to add new or revise old laws. But still, this provides no justification for the hypothesis that the actual laws of nature are inexhaustible: the revised laws ought to be taken as fundamental up to experimental falsification.

At every point in this chain, one is in the situation, as stipulated, that the known laws agree with all known experimental data (a situation obviously far removed from the present one). Of course it's always possible that new experimental data might upset this state of affairs, but at any given point, hypothesizing the existence of new laws without experimental necessity is a violation of parsimony.

 

[Fra]

but at any given point, hypothesizing the existence of new laws without experimental necessity is a violation of parsimony.

I fully agree with this viewpoint.

I would have said that an expectation lacking evidence pointing in it's direction, is simply irrational. There is no rationale for maintaining such a expectation in anything but a as a fluctuation because encoding expectations occupy resources. It seems highly unlikel that irrational systems would be observed in nature.

Also the question is not what WILL happen in the future, because no one ones and no one can compute it, period. What we do have, is expectations of the future, and it is what influences our actions. So all we need to decided, is what actions to take, based on our present knowledge. It happens all the time that we are wrong, but then we will revise our information states in a given theory, as well as the theory itself, when new evidence points to an inconsistency of the current theory.

So I think it's not unreasonable to think that any systems, instantly acts AS IF, there are a finite set of laws, simply because it's the self-imposed constraint of that system. But the action will get more complex on a finite scales is it will involve the systems revisions of prior conceptions. I think even this can partly be inductively computed by a much more complex observer that can monitor the system and it's environment as a subsystem.

/Fredrik

 

[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.

 

[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).

 

[D H]

This thread appears to be verging into metaphysics as opposed to physics. Some problems as of late:

1. As far as I know, the Copenhagen and many-worlds interpretations will always yield the same results. Arguing that one is right and one is wrong is taking this thread off-track. Besides, a TOE, if one is ever developed, will almost certainly say that both are wrong.

2. There is a continued misunderstanding / misrepresentation of what a TOE would entail. A TOE will describe the particle zoo and all the ways they can interact. Period. As far as physicists are concerned, the production rules of Conway's Game of Life are a "theory of everything" for that game. The Peano axioms, including induction, similarly are the "theory of everything" for the natural numbers. If the physical TOE is incomplete in the sense of Gödel's incompleteness theorems. So what? Physicists wouldn't care. Their TOE would still be everything that physicists mean by a TOE. You are dealing in metaphysics, not physics.

 

[Deveno]

my, my, y'all seem to be a bunch of very smart peeps. by comparison, i feel rather stupid. but be that as it may, i do have thoughts about this particular subject (odd isn't it? as i am neither a physicist nor a mathematician).

at the risk of showing just how stupid i really am, i feel it necessary to point out something is happening. unless we're sharing some vivid mass hallucination (a possibility, i suppose, but a faint one), there really is a universe out there, doing its thing. and it appears that we understand "it" better than we used to.

a long time ago, when i was in high-school, we were told that F = ma. now, without being pedantic about this, just the existence of that equation means we need the notion of a multiplicative structure to even make heads or tails out of it. in fact, if one regards "a" as a vector-valued function of time (not so unusual, or so i hear), them boom! you're already into the world of 4-dimensional real vector spaces. i hear hilbert spaces are popular with quantum physicists. even if these are crude models of reality, they ARE models of reality. we expect something (knowledge of some sort) from them.

my point is, that mathematics, and the consistency of mathematical theories, has a direct impact on how we communicate those theories. the "TOE's", even if they are just symmetry groups (or n-branes, or whatever) to explain particle interactions, aren't abstract curiosities, but intended to communicate real information about the world as we think it actually may be. as long as we use mathematical theories as languages to describe physical systems, then mathematical theorems (in those theories) imply some actual knowledge about the real world. in just such a way, an undecideable statement, in a mathematical theory we take to be an accurate translation of the way the universe works, filters down to some kind of existential statement about reality.

in other words, if math truly is the appropriate language for describing science, then godel's theorem strongly suggests there are real facts about the universe we can never know. perhaps these facts aren't interesting, that's a subjective call. i find it a bit disturbing to contemplate that we would desire a model of the world that was accurately predictive of all desired information over long periods of time.

one hopes, but i must profess this is more a tenet of faith with me, that certain problems remain intractible, that subatomic interactions (or perhaps super-galactic ones) are complicated enough, that so many possibilities remain, that we never know our future. i hope that even if mr. godel's theorem isn't the relevant one, some other constraint stops us from fully understanding "it all".

several people have expressed the opinion, that such philosophical concerns are not any physicist's primary concern. perhaps not. and yet, i find it intriguing that the theory of relativity, to pick a random example, was born out just such kind of "fruitless" speculation...what kind of structure fits if things are actually like this, instead of that?

no one denies nowadays, the usefulness of high-speed computers in research, and yet i find it surprising that so many people consider very basic questions about the limits of computability to be irrelevant. the limits of our mathematical theories ought to be of some concern, as well, unless we wish to take the accumulated knowledge of the last 500 years, flush it down, and start over.

complete theories do exist, and it is possible that some axiomatic treatment of physics within such a theory is possible, but i doubt it. no one has come up with a logical system that can do what the real numbers do, but without all the fuss. and I'm fairly certain that the real numbers are categorical, if you have a system with their properties, you might as well call it the real numbers as well, and it automatically inherits a natural number object as a subclass, and it will be (mathematically) incomplete.

today's abstract mathematical curiosity may well be tomorrow's pressing concern. (some)physicists seem to (over the years) acquired the bad habit of quietly co-opting the utility of abstraction, while claiming to do the opposite.

i mean no disrespect to any of the posters here. if nothing else, you've all given me several hours of enjoyable reading, and much food for thought.

 

[friend]

2. ... If the physical TOE is incomplete in the sense of Gödel's incompleteness theorems. So what? Physicists wouldn't care. Their TOE would still be everything that physicists mean by a TOE. You are dealing in metaphysics, not physics.

I think that's the point of contention. If a physical theory IS incomplete, then by definition it does NOT describe ALL possible physical events, right?

 

[Chalnoth]

I think that's the point of contention. If a physical theory IS incomplete, then by definition it does NOT describe ALL possible physical events, right?

Nope. All possible physical events would still be consequences of a true TOE. It's just that we would be doomed to never know all of the consequences of the theory, in that whatever list of proven-true or proven-false statements we manage to come up with, it is guaranteed that there are still more true or false statements out there that we have yet to prove.

 

[Chalnoth]

1. As far as I know, the Copenhagen and many-worlds interpretations will always yield the same results. Arguing that one is right and one is wrong is taking this thread off-track. Besides, a TOE, if one is ever developed, will almost certainly say that both are wrong.

This is false. The Copenhagen interpretation makes no statement about the nature of wave function collapse. So whenever you are dealing with an experimental situation near the boundary of collapse, the Copenhagen interpretation provides no results at all, while the many worlds interpretation makes a very clear prediction for the result (one which has so far held up against experiment).

2. There is a continued misunderstanding / misrepresentation of what a TOE would entail. A TOE will describe the particle zoo and all the ways they can interact. Period. As far as physicists are concerned, the production rules of Conway's Game of Life are a "theory of everything" for that game. The Peano axioms, including induction, similarly are the "theory of everything" for the natural numbers. If the physical TOE is incomplete in the sense of Gödel's incompleteness theorems. So what? Physicists wouldn't care. Their TOE would still be everything that physicists mean by a TOE. You are dealing in metaphysics, not physics.

This I agree with. Except for the specious "metaphysics not physics" claim.