Impact of Gödel's incompleteness theorems on a TOE

[Chalnoth]

I think the problem is that you do not take encoding of the theory as seriously as I do. Your explanation required more complexity thatn the original observer has control of. So is what your answer, or new theory, lives not on the original observer domain. Therefor it does not address the question.

In what sense? Nobody tries to consider a set of initial conditions in the MWI that includes the full wavefunction. But as long as the interactions between our world and the rest of the wavefunction are negligible, which they have to be to conform with observation, it won't effect the results anyway.

but whenever you compute and expectaion and encode a theory, a single observer is used. Question posed by this observer, can not be answers by a different observer. But yes, the different observer can "explain" why the first observer asks this question and how it perceives that answer.

I'm really not understanding your objection. This is precisely why the appearance of collapse forces us to only consider the probability distribution of results, as decoherence ensures that no single observer has access to the entire wave function.

The expectations observer B has, on observers A interacting with system X, is obviously different than observers A intrinsic expectations.

What? That's silly. The MWI reduces to the CI in the limit of complex observers. It can't predict different expectations for different observers, because CI doesn't.

 

[PAllen]

See here:
http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.2718v1.pdf

Great, just what I'm looking for. This will be my xmas reading.

 

[George Jones]

I don't know if anyone in this tread mentioned

http://arxiv.org/abs/physics/0612253.

 

[Fra]

Chalnoth, the dicussion has become confusing. Before we go on I'd like to just restate that this detour started in post #232 where I mainly objected to your suggestion that we look for a model/theory that unravels the true nature of reality, in an observer independent way. I personally think that may be a doubtful guide, because be construction, models are always inherently observer depedent. So I think the mental image that we can ever get an exernal objective picture of some reality is wrong. And using this goal as a constraint my be misguiding.

This was my main point.

My main point isn't wasn't to debate CI be MWI, because my personal view on this that the problem is not just about interpretations, it's deeper. I think we need a new reconstruction of measurement theory. So pure interpretations of current formalism is a moot discussion for me.

But I defend some traits of CI, as I think the points of informatiom updates, and the existence of both decidable and undecidable changes, and the logic of forming an intrinsic expectations as a basis for action is essential - and will remain key points even in a reworked quantum theory.

The obvious points where CI is bad, is because QM itself is bad. No other interpretation cures it either. So I'm not discussing interpretations, I'm discussing which view is the "best" in order to improve things. Here I think MWI is trying to find an external view of the observer, in a way that explains it away - in a way that is in violation with I consider to be the principles of intrinsic inference.

I see QM as an "extrinsic information theory", where extrinsic refers either to a "classical observer" or an "infiniteley complex QM observer". This is why it makes sense only for subsystems.

What I see is a reformulation, where the theory is intrinsically formualted. Ie. a theory where all elements of the theory in principle are inferrable from a real, realistic finite observer. Some overall predictions of such a programme would be that the noton of theory is an evolving one (there IS no eternal objective realist theory) and that the interaction of physical systems is even invariant with respect to such "true theory". The systems actions are implied by the effective theories.

There are a lot of open question in this, and I don't have the answers. But I feel quite confident about the direction.

/Fredrik

 

[Fra]

In what sense? Nobody tries to consider a set of initial conditions in the MWI that includes the full wavefunction. But as long as the interactions between our world and the rest of the wavefunction are negligible, which they have to be to conform with observation, it won't effect the results anyway.I'm really not understanding your objection. This is precisely why the appearance of collapse forces us to only consider the probability distribution of results, as decoherence ensures that no single observer has access to the entire wave function.

The perspective I have is that there is one natural decomposition. The observer itself, which is defined by what hte observer knowns, AND the remainder of the universe. But the remainder of the universe does NOT mean the entire universe as we konw it in the realist sense. It means the remainder of the encodable part of the observable universe. Which means that the remainder of the universe for a proton, is probably very small! How small I can't not say at this point, but probably the expected action of a proton system at any instant of time is invariant with respect to anything happening outside the laboratory frame. so there is indeed a builting cutoff here, the cutoff is due to that it's impossible for a proton to encode information about the entire universe.

So what you admit is not posible, and seem to solve be common sense and what's "negelctable" etc, I think should be taken seriously any be accounted for in OUR human theory.

Ie. humans "theory" of say particle physics, are an external one, relative to atomic world, this is WHY the current framework did work so well, but there are missing pieces and I think the next revolution may require that we try to understand what "scaling" the theory down to subatomic observers actually does? Most certainly we will see that the interactions scale out in a way that automatically gives us unification.

But the reverse perspective is what I think is more fruitful; to start with a basal low complexity observer, and try to understand how the inference system grows as we add complexity, and see how the unified original interactions split off into the known ones.

In order to do this, we can not STICK to the external perspective (ie. classical obserer, or infinitely compelx observers, or just infinite horizons scattering matrix descriptions of clean in/out) we need to get into the domain where the setup times are so long that expectations based in uncertain theories need to be used. This is a more chaotic domain, and the expectations are interrupted before the output is collected.

What? That's silly. The MWI reduces to the CI in the limit of complex observers. It can't predict different expectations for different observers, because CI doesn't.

CI and standard QM is not my measuring stick here. I think the problem is QM, and I my only point was that the notion of collapse, as beeing and "information update" is an essential ingredient in any theory of inference. There is no way to explain this away. Also, I simply fail to understand what the problem is with this?

An information update is not a problem, it just means that the expectation is updated.

The problem I have is that the action forms are not the result of inference in the current models, they are pulled from quatizing classical models. This is itself vere non-intrinsic. I think the information update; and actions based on expectations are key blocks to construct full expectations of actions from pure interaction historys.

Edit: Merry Xmas to everyone! :)

/Fredrik

 

[Fra]

To clarify what I mean, as this is a key point for me.

Which means that the remainder of the universe for a proton, is probably very small! How small I can't not say at this point, but probably the expected action of a proton system at any instant of time is invariant with respect to anything happening outside the laboratory frame.

I do not mean this in the obvious approximate sense. Because this is obvious to everyone.

I mean that I think that the complexity of a proton, (one problem is how to relate complexity to energy and mass, but certainly I imply here that high confined mass ~ high cmoplexity) is what defines the PHYSICAL cutoff. This number would have to enter somewhere in the expectation computation.

The common method of cutoffs are purely ad hoc, or arbitrary. I think there is a physical motivation for this cutoff that we can understand once we take seriously, the encoding of the theory and information in matter.

So I think this cutoff is exact, it's not just a FAPP type cutoff.

/Fredrik

 

[Careful]

I don't know if anyone in this tread mentioned

http://arxiv.org/abs/physics/0612253.

I think the conclusion from Godel's theorem is pretty simple and not at all what most people imagine it to be. I don't think Hilbert's program is destroyed but only that classical logic fails. For a physicist this should come as no surprise because we already know this to be the case from quantum mechanics. So, Godel's theorem is relevant for a TOE in the sense that the latter has to be defined in a non-classical logic.

Careful

 

[friend]

I don't know if anyone in this tread mentioned

http://arxiv.org/abs/physics/0612253.

This article reads in part as follows:

"The symbols are 0, 'zero', S, 'successor of', +, X, and =. Hence, the number two is the successor of the successor of zero, written as the term SS0, and two and plus two equals four is expressed as SS0 + SS0 = SSSS0."

These are the symbols used in the proof of Godel's Incompleteness Theorem (GIT). My question is does GIT work when a continuum is involved? At first glance it would seem not. Because then any numbers (other than zero) is constructed with an infinite number of "successor" steps of an infinitesimal difference. Thus every number is expressed with an infinite number of S's so that you can not tell one number apart from another.

 

[Careful]

These are the symbols used in the proof of Godel's Incompleteness Theorem (GIT). My question is does GIT work when a continuum is involved? At first glance it would seem not. Because then any two numbers have an infinite number of "successor" steps, where each step is an infinitesimal difference. Thus every number is expressed with an infinite number of S's so that you can not tell one number apart from another.

Godel's theorem is merely a mathematical masturbation of the liar's paradox which is captured by a statement of self reference like ''this statement is false'' (and likewise ''this statement is true'' leads to problems). The mathematical generalization to Turing machines and so on is just that but the deeper underlying message is that you can construct sentences for which it is impossible to determine whether they are false or true. This does not depend upon the kind of technicalities which you are suggesting.

Careful

 

[nomadreid]

To get back to the poster's original question: another point, this time from the viewpoint of the practice of pure mathematics.

Gödel's incompleteness theorems (two of them, note) show that a consistent system cannot prove itself complete or consistent. But one would not get very far if one always moaned that arithmetic has this limitation, so in the practice of mathematics one proves that one's theory is equi-consistent with Peano Arithmetic (PA), and since PA has served us well, one simply continues on one's way with the assumption that PA, and hence anything equi-consistent with it, is consistent. Problems with an theory are expected to come not from PA, which hasn't yielded any contradictions so far, but from the extensions of PA, so this is where mathematicians concentrate their efforts. Also note that, whereas PA cannot prove itself consistent, it can be proven consistent by another theory, call it algebra. True, one then has the problem with algebra not being able to prove itself, but then this can be proven consistent with another system, and so forth. Whereas one never can prove it absolutely, the higher up one goes, the more confidence one has that, if a contradiction is there somewhere, it is pretty remote.

Also, note that Hawking's statement was referring to incompleteness (First Incompleteness Theorem) rather than provable consistency (Second Incompleteness Theorem). That a system is self-consistent is of course concern for a theory, but any inconsistency that is not easily spotted in the formalism will usually pop out eventually in experiment. Nature is often much better at spotting inconsistencies than we are. As far as completeness: so the system is not complete? So what? More's the fun. After all, that was the original statement of the EPR thought experiment: that reality is not complete.

Finally, Gödel's theorems, although they are about PA, have been extended with systems in which a continuum exists. For example, ZFC, which can produce a statement about the existence of the power set of the natural numbers, and so forth.

 

[PAllen]

Godel's theorem is merely a mathematical masturbation of the liar's paradox which is captured by a statement of self reference like ''this statement is false'' (and likewise ''this statement is true'' leads to problems). The mathematical generalization to Turing machines and so on is just that but the deeper underlying message is that you can construct sentences for which it is impossible to determine whether they are false or true. This does not depend upon the kind of technicalities which you are suggesting.

Careful

Just an observation that many mathematician's disagree with this minimizing of Godel. Further, there are a fair number of mathematicians who think two important unsolved problems are likely examples of Godel incompleteteness: the Golbach conjecture, and P=?NP.

 

[Lievo]

Godel's theorem is merely a mathematical masturbation

It will soon be 2011, so let me wish you that your career will include at least one thing as important as any of the Godel's theorems. In the mean time, happy new year!

 

[nomadreid]

...there are a fair number of mathematicians who think two important unsolved problems are likely examples of Godel incompleteteness: the Golbach conjecture, and P=?NP.

The Goldbach conjecture remains a tough egg, but many mathematicians are waiting to see if the latest attempt at proving P<>NP pans out by Vinay Diolalikar from the HP Research labs in at Palo Alto labs : http://www.scribd.com/doc/35539144/pnp12pt. There are always attempts, but this one looks promising. (Unless I have missed something in the ongoing peer review.)

And, as a couple of recent posts emphasized, Gödel's theorems are extremely important. But there are some who, hearing of its importance, try to apply it where it should not be. A misinterpretation that many people fall into is the Lucas-Penrose fallacy. (Penrose has sold a lot of books based on his misinterpretation, even though it was torn apart by the logician Solomon Fefferman.) As well, some people get ontological uncertainty (as in the Heisenberg Uncertainty Principle) and epistemological uncertainty (as in Gödel's Incompleteness Theorems) mixed up.

 

[Careful]

Just an observation that many mathematician's disagree with this minimizing of Godel. Further, there are a fair number of mathematicians who think two important unsolved problems are likely examples of Godel incompleteteness: the Golbach conjecture, and P=?NP.

I don't minimize Godel. If I recall my history right then Godel was mainly interested in showing the inadequacy of logic and only wanted to show that something as mundaine as the liar's paradox has severe consequences for the foundations of mathematics. Godel also was firmly convinced that human beings were not computers but that things like creativity, intuition and so on were grounded in a higher kind of ''logic''. It is Turing who took the opposite side of the debate and who mainly stressed the computational aspect of Godel's work and he firmly believed that humans were simply sophisticated computers (which we clearly are not).

So, it might be that I take Godel more seriously than you do; I actually look beyond classical logic and search for a more general kind of proof method. Mostly what people do, is to take the relativist attitude and regard the axiomatic approach as fundamentally incomplete but prove consistency of one system relative to a bigger one. I think this is the wrong approach for a TOE since it is clearly so that logic is not only relational but also relies upon self reference.

Careful

 

[friend]

I don't think that the facts of reality are isomorphic to mathematical axioms. Which particle is "number one" and which particle is "number two", etc? You can always renumber them differently without affecting their existence. Which axiom applies to one event but not another? And if you can't map numbers or axioms to particles or phenomena, then Gödel's Incompleteness Theorem can not be applied, right?

 

[Careful]

As well, some people get ontological uncertainty (as in the Heisenberg Uncertainty Principle) and epistemological uncertainty (as in Gödel's Incompleteness Theorems) mixed up.

Well, I would not say that some people mix it up. I hate using expensive words but ontology basically means how things are, what their ''reality'' is and epistemology is what we can know about them. Godel's theorem indeed says that we cannot know some things to be true even if they are true; but what I am saying is that this gap between ontology and epistemology is an artificial human construct due to a too limited definition of what knowledge is supposed to be. As a physicist, it is clear that these limitations in knowledge are induced by the way our perceptions impose a natural macroscopic logic upon us; that is, we have too limited access to ontology in order to have a complete epistemology. You could also turn it the other way and say that we use the wrong ontology, that for example the concept of a set with a definite number of elements somehow does not ''exist'', that an absolute empty set does not ''exist''. It is not because we name something in a particular way that it really ''exists'' in a deeper sense; actually, that is what quantum physics teaches us. Anyway, this is getting philosophical...

Careful

 

[Careful]

I don't think that the facts of reality are isomorphic to mathematical axioms. Which particle is "number one" and which particle is "number two", etc? You can always renumber them differently without affecting their existence. Which axiom applies to one event but not another? And if you can't map numbers or axioms to particles or phenomena, then Gödel's Incompleteness Theorem can not be applied, right?

I have mixed feelings about this; one day I get up and tell to myself that a fundamental theory of everything, if it exists, is one which resists rigorous definition within a fixed mathematical context and that we need at least a new kind of ''mathematics'' to proceed. That is, a ''mathematics'' of genuine creation, a theory of understanding, but I severly doubt whether such thing exists and if it will ever be in reach of human activity. On other days, I am more optimistic but certainly these considerations do not apply to something as modest as a theory of quantum gravity.

Careful

 

[Hurkyl]

Godel also was firmly convinced that human beings were not computers but that things like creativity, intuition and so on were grounded in a higher kind of ''logic''. It is Turing who took the opposite side of the debate and who mainly stressed the computational aspect of Godel's work and he firmly believed that humans were simply sophisticated computers (which we clearly are not).

Some people would disagree with your parenthetical, but that's a digression...


No matter how creative a human is, or how strong his intuition, a human will never create a proof that a Turing machine is incapable of discovering.

No matter how creative a human is, or how strong his intuition, a human will never create a list of postulates from which he can prove interesting things that a Turing machine is incapable of discovering.

So, it might be that I take Godel more seriously than you do; I actually look beyond classical logic and search for a more general kind of proof method.

Whatever "proof method" you consider, if you can validate a purported proof, then a Turing machine is capable of coming up with it.



The computational aspect is a rather obvious thing -- if there is a theorem that a human can discover a proof for through his intuition and cleverness, then a Turing machine is also capable of finding it by doing nothing more intelligent than brute force exhausting through all possible combinations of symbols, and checking each one to see if it's a proof of the theorem or not.


But TBH, aside from the sort of silly vague ideas that people like to philosophize about that are only loosely related to Gödel's incompleteness theorems (if at all), I've mostly seen them applied as impossibility proofs in real mathematics and computer science. The first time I was really introduced to the subject was in a theory of computation class, in a proof that there does not exist an algorithm to enumerate the true sentences in any model of integer arithmetic. Of course, the whole notion was old hat to me at the time, since we had already spent time on simpler situations, such as the halting problem.

 

[Careful]

No matter how creative a human is, or how strong his intuition, a human will never create a proof that a Turing machine is incapable of discovering.

No matter how creative a human is, or how strong his intuition, a human will never create a list of postulates from which he can prove interesting things that a Turing machine is incapable of discovering.

That's not the point, I have just commented on that on my personal page. I do not feel any compelling need to ''disprove'' strong AI (whatever that means in a generalized logic), nor do I need to ''prove'' my position. All I am saying is that my strategy is more plausible than yours, a machine will never discover anything unless the seeds for this discovery are already ingrained in it's algorithm. So you will have to systematically add new elements to your algorithm for the latter to be reasonable capable of doing what you already know exists. The point is that I conjecture that machines made by men will never ever create something which gets even close to what human imagination can achieve; that is sufficiently good for me not to adhere to your position.

Whatever "proof method" you consider, if you can validate a purported proof, then a Turing machine is capable of coming up with it.

I don't know, can you prove this ? Penrose gives plausible arguments why this could be doubted. Moreover, and this is the point, the machine will never ever cook up the proof method by itself.

The computational aspect is a rather obvious thing -- if there is a theorem that a human can discover a proof for through his intuition and cleverness, then a Turing machine is also capable of finding it by doing nothing more intelligent than brute force exhausting through all possible combinations of symbols, and checking each one to see if it's a proof of the theorem or not.

Again, this may be false, there is no proof of that. See my previous comment.

But TBH, aside from the sort of silly vague ideas that people like to philosophize about that are only loosely related to Gödel's incompleteness theorems (if at all), I've mostly seen them applied as impossibility proofs in real mathematics and computer science. The first time I was really introduced to the subject was in a theory of computation class, in a proof that there does not exist an algorithm to enumerate the true sentences in any model of integer arithmetic.

This is not how Godel thought about it, surely you do not want to imply that he was silly. BTW: I also learned the Turing version first and read the real history about it only many years later.

 

[Hurkyl]

This is not how Godel thought about it, surely you do not want to imply that he was silly. BTW: I also learned the Turing version first and read the real history about it only many years later.

Actually, I do -- as I recall my history, Gödel had some rather... odd ideas.

But, in any case, history is as history does. Just because Gödel, Einstein, or anyone else is a prominent historical figure in their field does not mean their opinions are right, and that one should dismiss decades of progress simply because the subsequent work (appears to) disagree with the historical figure's point of view.

Moreover, and this is the point, the machine will never ever cook up the proof method by itself.

You sure?

The Turing machine can certainly enumerate all machines -- in particular, it will eventually cook up any machine that implements said proof method.

And a human who is considering proof methods has to have a way to decide which ones are good. If there is any algorithm for making the decision on whether or not a particular proof method is viable, then the aforementioned Turing machine will not only find it, but say "hey, this is a good one!"


I think you underestimate just how much force is available to brute force when there aren't practical constraints.

 

[Careful]

Actually, I do -- as I recall my history, Gödel had some rather... odd ideas.

But, in any case, history is as history does. Just because Gödel, Einstein, or anyone else is a prominent historical figure in their field does not mean their opinions are right, and that one should dismiss decades of progress simply because the subsequent work (appears to) disagree with the historical figure's point of view.

I don't dismiss any subsequent work; as far as I know there has not been any shocking new insight into these matters since then. Actually, my approach to science is twofold: I immerse myself into contemporary results but I also study the original thoughts of the the genius. Usually, the genius was not so far of the mark. Let me give you an example where modern physics has gone silly: if you ask most people about general relativity, many PhD's or post-docs will write you down Einstein-Hilbert action, or even also the higher derivative terms and in the best case, they know Mansouri, Hilbert-Palatini or Holst action. But this is not how Einstein thought about GR: for him, one of the several equivalence principles and general covariance were central and the field equations were merely a simple example of his ideas. Nevertheless, most people will insist that a theory of QG needs to recuperate the Einstein equations in the low energy limit; completely dumb !

You sure?

I am pretty sure, can your invent a computer who has the intellectual power to discover relativity theory, or to invent clifford analysis?

And a human who is considering proof methods has to have a way to decide which ones are good. If there is any algorithm for making the decision on whether or not a particular proof method is viable, then the aforementioned Turing machine will not only find it, but say "hey, this is a good one!"

But I think there is no such algorithm, actually it is a conjecture that every problem that can be solved algorithmically, can be solved by a turing machine. Usually, a really deep proof employs new concepts, new theorems and the existence of an algorithm would require the machine to be genuinely creative. For example, when I would ask the machine to compute the integral on ln(x)/x^3 between 1 and infinity and the machine would only know the Riemann definition of an integral, do you think it could invent partial integration ? It would have to invent derivatives for that and discover that integration and differentiation are the inverse of one and another. Moreover it would have to find out that the differential of ln(x) equals 1/x. You may say: yes, but I can write this down in a symbolic language using only a finite number of symbols. True, but that doesn't guarantuee the machine is going to find it; most likely, this uneducated machine will just apply the definition of the Riemann sum straight away and study all possible partitions of the interval 1 to infinity of length N. That will occupy him an infinite amount of time. I can of course not prove this, but given the current state of computers, it seems most likely.

Moreover, I offered you the possibility of alternative logic, such as modal or dynamical logic. Could you even think of a machine figuring out new methods of reasoning? By definition, the ''thinking'' of a machine is limited by the ground rules of the game, I think it is reasonable to say that a human has the capacity to genuinely invent new types of ''thinking''. It seems utterly implausible that all our creativity and knowledge is encoded in the initial state of the universe.

I think you underestimate just how much force is available to brute force when there aren't practical constraints.

Oh, I know the strength of brute force and when I was younger, I always used it myself in the beginning; I think it is a natural thing and ingrained in our psychology. When you get past your 30-ties, you start to think in a more clever way and you learn to rely more on your intuition.

 

[nomadreid]

Mostly what people do, is to take the relativist attitude and regard the axiomatic approach as fundamentally incomplete but prove consistency of one system relative to a bigger one. Careful

First, although proving consistency of subsystems is one approach, most approaches to consistency prove consistency of one system relative to a smaller one. That is, one proves things like " 'Peano Arithmetic (PA) + 'a measurable cardinal exists' is equiconsistent with PA", and, since one already works in PA with the implicit caveat that it cannot prove itself consistent but it is a steady workhorse, one goes ahead and works with the extended system with the same implicit caveat.

Nor does one regard the axiomatic approach as incomplete, but rather the axiomatic approach must be better understood for what it is, one that is infinitely extensible in many directions, so that one must first ponder which axioms one will select for a given purpose. As far as futile discussions as to whether human intuition goes further than axiomatic systems, or the Church-Turing thesis, given the impossibility of proving or disproving this (see next paragraph), most practical discussions focus on various versions of effective computability rather than this thesis.

"if you can validate a purported proof, then a Turing machine is capable of coming up with it."
I don't know, can you prove this ? Penrose gives plausible arguments why this could be doubted.Careful

Penrose's arguments are not plausible, once one gets into his exposition. He made serious technical errors in his proof, which were pointed out in a classic paper by Professor Solomon Fefferman of Stanford. Since then, Penrose's arguments have been known among logicians as the "Penrose-Lucas fallacy", since Penrose's arguments were essentially the same misinterpretation of the First Incompleteness Theorem that John Lucas had made some years earlier.

 

[Careful]

Penrose's arguments are not plausible, once one gets into his exposition. He made serious technical errors in his proof, which were pointed out in a classic paper by Professor Solomon Fefferman of Stanford. Since then, Penrose's arguments have been known among logicians as the "Penrose-Lucas fallacy", since Penrose's arguments were essentially the same misinterpretation of the First Incompleteness Theorem that John Lucas had made some years earlier.

I was not talking about his ''proof'' of non-computability ! I was thinking about some toy model of the universe he made or a chess game he invented in which he clearly demonstrated that solving these problems requires a higher kind of thought even the most powerful machines are not up to at this moment. Note that he writes that these examples do not constitute a proof against strong AI. Again, I have no conclusive position against strong AI apart from the original Godel objection that classical logic is incomplete; in that respect I blelieve that Penrose tried to climb the wrong mountain.

Careful

 

[Careful]

Nor does one regard the axiomatic approach as incomplete, but rather the axiomatic approach must be better understood for what it is, one that is infinitely extensible in many directions, so that one must first ponder which axioms one will select for a given purpose.

This is a matter of wording and taste, I wouldn't say you mean something different than incomplete here.

As far as futile discussions as to whether human intuition goes further than axiomatic systems, or the Church-Turing thesis, given the impossibility of proving or disproving this (see next paragraph), most practical discussions focus on various versions of effective computability rather than this thesis.

Remember, the title of this thread was the impact of Godel on a TOE. What you say is that Godel makes a TOE impossible; while what I claim is that a TOE is only possible if one gets an understanding of things like human creativity and intuition. So, it is far from futile I would say.

Careful

 

[nomadreid]

he clearly demonstrated that solving these problems requires a higher kind of thought even the most powerful machines are not up to at this moment. Careful

The key expression here is "at this moment." Since present computers cannot reach the performance of a lame cockroach, this is not a relevant argument about what computers could, in principle and in the future, achieve.

"Nor does one regard the axiomatic approach as incomplete, …"
I wouldn't say you mean something different than incomplete here.

Since "incomplete" is ambiguous, having several different meanings (e.g., it would be different as applied to the axiomatic system from its application to a particular axiom system), I should have asked you for your definition of the word, as you were the first to apply it to the axiomatic system. I invite you to provide one, and also to ask whether your definition will not apply equally well to humans.

What you say is that Godel makes a TOE impossible;

No, I never said this. In fact, I suspect that the Incompleteness Theorems will have no real impact on the development of a TOE. That is, the First Incompleteness Theorem just provides a general method of producing undecidable sentences from any extension of Peano Arithmetic (PA) (or, as shown later, from even something as weak as Robinson's Q), but this type of sentence is not the type of sentence that a TOE will try to decide. The undecidability of more relevant sentences, such as the Axiom of Choice or the Axiom of Determinacy from ZF, or the Continuum Hypothesis from ZFC, merely give the physicist to choose whichever is convenient. The Second Incompleteness Theorem shows that the consistency of any extension of PA is another example of such an undecidable statement, but since the mathematics that is used has been shown to be equiconsistent with PA, and since PA is taken as dependable for the sake of physics, this is also not an issue. In other words, although the Gödel Theorems had far-reaching impacts in the foundations of mathematics, and even have been responsible for new fields of mathematics, they have not changed the way that Hamiltonians, tensors, spinors, groups, etc. are calculated; it is in these terms in which a TOE will likely be formulated. The eternal doubt that there can be a better theory does not owe its validity to Gödel; that was implicit already in the work of Lobachevsky and Bolyai (one did not even need the strength of PA for this).

what I claim is that a TOE is only possible if one gets an understanding of things like human creativity and intuition.

Although an understanding of human creativity and intuition could help develop creative artificial intelligence, it may not be necessary, as the complexity of future computers will make it likely that much will evolve without anyone knowing exactly what it was. However, if we are talking about humans, a TOE could be developed by physicists who have no inkling about psychology or neurobiology. Newton seemed to do quite well without them. Being creative does not imply that you know how you are creative.

 

[friend]

I don't think that the facts of reality are isomorphic to mathematical axioms. Which particle is "number one" and which particle is "number two", etc? You can always renumber them differently without affecting their existence. Which axiom applies to one event but not another? And if you can't map numbers or axioms to particles or phenomena, then Gödel's Incompleteness Theorem can not be applied, right?

No, seriously, wouldn't you have to be able to map the axioms of Godel's Incompleteness Theorem in a unique, one-to-one fashion to the axioms or elements of the new system in order to prove the incompleteness of the new system? I mean as soon as you lose unique mapping and can reassign the axioms (still in a one-to-one fashion), then how could you say some axiom or element in the new system is not provable; it could be reassigned as one of the first axioms of the old system.

 

[Careful]

The key expression here is "at this moment." Since present computers cannot reach the performance of a lame cockroach, this is not a relevant argument about what computers could, in principle and in the future, achieve.

Yes, but the point is that computers never will improve by themselves, they are not living creatures. The best they could do is reproduce, imitate and so on, but they will never ever be creative.

Since "incomplete" is ambiguous, having several different meanings (e.g., it would be different as applied to the axiomatic system from its application to a particular axiom system), I should have asked you for your definition of the word, as you were the first to apply it to the axiomatic system. I invite you to provide one, and also to ask whether your definition will not apply equally well to humans.

But my point is that some things cannot be defined, never ever ! Whitehead and Russell have written a beautiful treatise about the meaning of equality, the latter is a referential concept and therefore an absolute definition can never be given. Look at languages, actually nothing is defined in a language and still we can communicate to one and another. Therefore, there exists something which goes beyond what one can grasp in a symbolic language which is always relational. If the world were reduced to mere symbols, we wouldn't get anywhere. For example, try to tell to a computer what the quantifier forall means ! I bet a computer who would not be told how to look for proofs and was ingrained with capacity verifying formal logical laws and be given the notion of continuity would never ever produce a proof that something as simple as the function $x -- > x$ is continuous.

Newton seemed to do quite well without them. Being creative does not imply that you know how you are creative.

I think we are talking about different things here; I guess you mean by a TOE a theory which unifies all known laws of nature. What I mean by a TOE is the metaphysical theory which literally accounts for everything including human creativity. There is no point in arguing for anything else, if you mean by a TOE a hands-on theory of quantum gravity, then indeed Godel will not be very important. But again, such theory will not be complete again and fail on other aspects... That's why I implied from what you said that you meant that Godel's theorem implies that our work will never ever be complete.

Careful

 

[nomadreid]

No, seriously, wouldn't you have to be able to map the axioms of Godel's Incompleteness Theorem in a unique, one-to-one fashion to the axioms or elements of the new system in order to prove the incompleteness of the new system?

No. All your system has to do is to have at least a countably infinite number of possible names with a linear order with least element on them, be able to have some manner of assigning unique codes, and a couple of other similar requirements.

 

[nomadreid]

Yes, but the point is that computers never will improve by themselves, they are not living creatures. The best they could do is reproduce, imitate and so on, but they will never ever be creative.

There are already computers that, according to some criteria, are self-improving, and to some extent creative. However, you may wish to label it simulation, although the question then presents itself as to how human creativity differs in principle. In any case, there is no evidence that an organic base is a prerequisite for the mental processes that make humans creative. But given the state of computers at the moment, whether computers can achieve human creativity is undecidable; an assertion one way or the other belongs to belief, not to physics. This is a physics forum.

But my point is that some things cannot be defined,.

If so, then they are concepts which do not belong to mathematics and hence not to physics.

Russell and Whitehead... equality, the latter is a referential concept and therefore an absolute definition can never be given.,.

You are apparently thinking of the primitive terms in an axiom system. (By the way, in the language of ZFC, set membership has replaced equality as the undefined term; equality is then defined in terms of set membership.) However, since Principia Mathematica, the field of Model Theory has given a more precise formulation of the relationships between syntax and semantics, so that primitive terms are now simply a more solid link between mathematics and physics. The whole concept of referential concepts has been made precise, and do not constitute a reason to think of the corresponding concepts as belonging outside of the formalized framework for physics. Secondly, I am not sure what you mean by an "absolute definition". By its nature, a definition, just as an axiom, is relative. Remember in Alice in Wonderland:
"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means
just what I choose it to mean – neither more nor less."

Look at languages, actually nothing is defined in a language and still we can communicate to one and another.

I always wondered what I had my dictionaries for. But even with that, we don't communicate well enough in natural language for the purpose of physics; hence the language of physics is mathematics, where most things are defined, and undefined terms have a specific role.

Therefore, there exists something which goes beyond what one can grasp in a symbolic language which is always relational. .

Most of mathematics and physics deals with relations which are formalized in symbolic language. True, there is a point where physics stops and metaphysics begins, but this is a physics forum, not a metaphysics forum.

If the world were reduced to mere symbols, we wouldn't get anywhere..

It is precisely because of our ability to use symbols that our species has been able to achieve what it has.

For example, try to tell to a computer what the quantifier forall means ! ..

Check out a book on Model Theory.

I bet a computer who would not be told how to look for proofs and was ingrained with capacity verifying formal logical laws and be given the notion of continuity would never ever produce a proof that something as simple as the function $x -- > x$ is continuous...

See my comments in the first paragraph above.

I think we are talking about different things here; I guess you mean by a TOE a theory which unifies all known laws of nature...

More or less, yes. This is the Physics Forum, under the Rubric "Beyond the Standard Model", in which "TOE" refers to the hoped-for theory of physics which will be a type of GUT. I believe that is what most of the physicists reading this understand by the term TOE in this context.

What I mean by a TOE is the metaphysical theory which literally accounts for everything including human creativity...

This is a PHYSICS forum. Not neurobiology, computer science, psychology, or metaphysics. A TOE is supposed to be the base for further applications, although it is probable that an eventual understanding of human creativity will only use the physics already known today, so that, Roger Penrose notwithstanding, the presence or absence of a TOE will probably not be a deciding factor in the understanding of human creativity.

if you mean by a TOE a hands-on theory of quantum gravity, then indeed Godel will not be very important. ...

OK, if we have stopped talking at cross-purposes, we have agreement on that point.

But again, such theory will not be complete again and fail on other aspects...

I am still waiting for your definition of "complete". But yes, a physical TOE as presently envisioned will not mean the end of physics. No reasonable physicist expects it to, any more than Maxwell's equations meant the end of the study of electromagnetism. As far as it failing in "other aspects", it is hard to know what it will fail at, if anything, before it has been formulated and tested. But there is no theoretical reason that a TOE will necessarily fail in the task that has been defined for it. True, it will not solve your metaphysical problems, but it isn't supposed to even try, so this will not, at least in physics, be seen as a failure.

 

[Careful]

i don't think there is much point in continuing the discussion; you seem to be unaware that your position is equally a (very unplausible) belief and moreover you seem to indulge yourself in the comfort that your view belongs to physics and mathematics while mine doesn't. I sharply disagree with that in the case of physics, in the case of mathematics I could be more forgiving. Physics is not appied mathematics. We agree that mathematics is relational; that why I tried to tell you cannot tell to a computer what the word forall means, something which he will need if he wants to prove that the function x --> x is continuous. Therefore what I tried to tell you, and what Penrose tries to convey is that these undefinable qualities associated to meaning and understanding are necessary to do mathematics. Since we are a part of nature, a TOE should be able to discribe that as well, and it basically never ever will. I agree that symbolic language has been the main driver of human progress and knowledge but again the quality which manipulates this symbolic language cannot be defined in terms of it. Moreover, I am not trying to even say that this issue is the end mathematics and certainly not of physics as I understand it! On the contrary, I think the most basic laws of nature will be defined in terms of very general principles like general covariance and so on which by themselves cannot be defined accurately. It is a particular projection of them, by adding more relational context than necessary which will allow for study in terms of the language of mathematics. This is precisely what Einstein stressed throughout his whole life, if we can learn something of the old man, then it is this!

As a final comment, I would say that physicists and mathematicians should become more open for interdisciplinary study regarding the other sciences. They are also sciences and have meaningful aspects to communicate to us, the reductionist view will always fail and as a physicist/mathematician I have certainly not the pretense that my activities would somehow be better than the one of a biologist.

Careful

 

[nomadreid]

i don't think there is much point in continuing the discussion;

Aw, and we had just gotten to agree on the original question of the post, that whether a TOE would be influenced by the incompleteness theorems depended on how you defined "TOE".

But you're right, since the other issues that came up until we got to this point were side issues about which we have put down our respective arguments, we can either let other readers expand upon them or let this post finally come to an end.

Cheers

 

[Careful]

Aw, and we had just gotten to agree on the original question of the post, that whether a TOE would be influenced by the incompleteness theorems depended on how you defined "TOE".

But you're right, since the other issues that came up until we got to this point were side issues about which we have put down our respective arguments, we can either let other readers expand upon them or let this post finally come to an end.

Cheers

Indeed, we have both presented our views and we agree within the limitations of the contextual scope you wish to attribute to a TOE. On Godel's theorem, we both won't move one inch, so experience learns me that it is better to stop.

 

[webplodder]

In a nutshell, Godel's ideas mean that we can only know stuff based on what we already know. If mathematics itself can never be a complete description of phenomena (due to its axioms not predicting every possible consequence of them) then it follows that we can only predict as much as our abilities allow us to predict, as a species. A TOE will also be subject to the same limitations so that what we define as 'knowledge' will always be parochial in nature, it cannot be otherwise. I suppose what I am really saying is that we may only define 'reality' within the constraints of our biological limitations. Who knows, perhaps some UFOs, for example, represent phenomena that we simply haven't the ability to define or comprehend!

 

[Hurkyl]

In a nutshell, Godel's ideas mean that we can only know stuff based on what we already know.

 

[Careful]

Right, and I did my best to avoid such misunderstandings!