Impact of Gödel's incompleteness theorems on a TOE

[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!