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Impact of Gödel's incompleteness theorems on a TOE

by PhysDrew
Tags: gödel, impact, incompleteness, theorems
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Careful
#307
Jan8-11, 12:54 PM
P: 1,667
Quote Quote by nomadreid View Post
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.

Quote Quote by nomadreid View Post
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.


Quote Quote by nomadreid View Post
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.

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nomadreid
#308
Jan8-11, 12:56 PM
P: 562
Quote Quote by friend View Post
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
#309
Jan8-11, 11:05 PM
P: 562
Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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."

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
For example, try to tell to a computer what the quantifier forall means ! ..
Check out a book on Model Theory.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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.

Quote Quote by Careful View Post
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
#310
Jan9-11, 03:21 AM
P: 1,667
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.

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nomadreid
#311
Jan9-11, 04:45 AM
P: 562
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
#312
Jan9-11, 06:38 AM
P: 1,667
Quote Quote by nomadreid View Post
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
#313
Jan9-11, 10:49 AM
P: 5
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
#314
Jan9-11, 11:25 AM
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Quote Quote by webplodder View Post
In a nutshell, Godel's ideas mean that we can only know stuff based on what we already know.
Careful
#315
Jan9-11, 12:14 PM
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Quote Quote by Hurkyl View Post
Right, and I did my best to avoid such misunderstandings!


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