[jaekwon]

Hello all, this is my first post here.

It occurred to me that Godel's Incompleteness Theorem may have a physical counterpart. I propose one here:

The Incompleteness Theorem of Our Universe: "There exist physical phenomena that can not be explained from laws derived from empirical observations".

Sketch of a proof:

0. Assume otherwise, that all physical phenomena can be explained from laws derived from empirical observations.
1. Some physical systems PS like the human brain or a sufficiently strong computer can be instructed to run in accordance to rules (axioms) of a "sufficiently strong (symbolic) system". **
2. There exists in PS a mapping between (a) the statements in the symbolic system and (b) empirical observability of physical phenomena.
3. From Godel's Incompleteness Theorem, (2), and (0) it follows that there exists physical phenomena of this physical system PS that cannot be explained from its own empirical observations.
(3) is a contradiction of (0). QED.

** if you are self aware and capable of formulating statements about yourself, "you have free will" is a Godel sentence.

I don't believe we can ever fully understand the laws of this universe with only symbols and equations. There is always room for free will in the universe, if you choose to believe it.

- Jae

This post has been edited by jaekwon: 5 February 2012 - 07:46 PM

[dimreepr]

jaekwon, on 5 February 2012 - 07:45 PM, said:

Hello all, this is my first post here.

It occurred to me that Godel's Incompleteness Theorem may have a physical counterpart. I propose one here:

The Incompleteness Theorem of Our Universe: "There exist physical phenomena that can not be explained from laws derived from empirical observations".

Sketch of a proof:

0. Assume otherwise, that all physical phenomena can be explained from laws derived from empirical observations.
1. Some physical systems PS like the human brain or a sufficiently strong computer can be instructed to run in accordance to rules (axioms) of a "sufficiently strong (symbolic) system". **
2. There exists in PS a mapping between (a) the statements in the symbolic system and (b) empirical observability of physical phenomena.
3. From Godel's Incompleteness Theorem, (2), and (0) it follows that there exists physical phenomena of this physical system PS that cannot be explained from its own empirical observations.
(3) is a contradiction of (0). QED.

** if you are self aware and capable of formulating statements about yourself, "you have free will" is a Godel sentence.

I don't believe we can ever fully understand the laws of this universe with only symbols and equations. There is always room for free will in the universe, if you choose to believe it.

- Jae

Define free will?

[Schrödinger's hat]

jaekwon, on 5 February 2012 - 07:45 PM, said:

2. There exists in PS a mapping between (a) the statements in the symbolic system and (b) empirical observability of physical phenomena.
3. From Godel's Incompleteness Theorem, (2), and (0) it follows that there exists physical phenomena of this physical system PS that cannot be explained from its own empirical observations.
(3) is a contradiction of (0). QED.

a. You assume this mapping is one-to-one. If the unprovable statements in this symbolic system do not correspond to physical phenomena, then 3 does not follow.
b. You assume the symbolic system required to describe the universe is of at least the same level of complexity as one to which Godel's incompleteness theorem applies.
c. This does not prove that an alternate system cannot be constructed to explain those phenomena that do not fit within the first (and then a third and a fourth and so on).

[ajb]

This is a kind of metatheorem about physics. Such things are interesting and can help guide physics.

Can I rephrase the metatheorem as not all physical phenomena can be mathematically modelled? I think this is a consequence of your statement. I understand explained and laws to be equivalent to the construction of mathematical models. These models are then tested against nature, which is your empirical evidence.

So we now have to think about the role of physics and if such a metatheorem is a problem?

Physics is all about creating mathematical models of nature, usually some selective part of nature, and then testing these against observation/experiment. Sometimes the experiments lead the theory and sometimes visa versa. The question: is it necessarily the remit of physics to explain all phenomena in nature?

However, to me it seems uncomfortable to accept that aspects of the Universe are just not "explainable" by their very nature.

Anyway, it is amazing that we have managed to understand the Universe in the way we have and that is with mathematics.

[jaekwon]

dimreepr, on 5 February 2012 - 08:16 PM, said:

Define free will?

Your ability to affect physical phenomena consistent with known physical laws derived from empirical observations, but unexplainable from those laws alone.

Schrödinger, on 5 February 2012 - 09:29 PM, said:

a. You assume this mapping is one-to-one. If the unprovable statements in this symbolic system do not correspond to physical phenomena, then 3 does not follow.

In this mapping, unprovable statements that are true should be consistent with physical phenomena. The mapping is symbolic "truth" to physical "phenomena". Otherwise a "sufficiently strong" computer would not be deterministic when performing computations on "sufficiently strong systems".
BTW, even the lack of a physical phenomena (e.g. particles of type X never appears in two places) is a phenomena.
Does that address your point?

Schrödinger, on 5 February 2012 - 09:29 PM, said:

b. You assume the symbolic system required to describe the universe is of at least the same level of complexity as one to which Godel's incompleteness theorem applies.

I do. But this assumption is true, otherwise your computer would not exist. Godel himself would not have existed.

Schrödinger, on 5 February 2012 - 09:29 PM, said:

c. This does not prove that an alternate system cannot be constructed to explain those phenomena that do not fit within the first (and then a third and a fourth and so on).

I agree. This is called "progress" in science. But there will be no end to it so long as computers or minds exist.

ajb, on 5 February 2012 - 09:35 PM, said:

This is a kind of metatheorem about physics. Such things are interesting and can help guide physics.

Can I rephrase the metatheorem as not all physical phenomena can be mathematically modelled? I think this is a consequence of your statement. I understand explained and laws to be equivalent to the construction of mathematical models. These models are then tested against nature, which is your empirical evidence.

Yes, that's a nice way of putting it. Or, less palatably: the universe is nondeterministic. I read conflicting opinions on this forum about quantum theory and what it means for determinism, but the universe is not determined by any theory.

ajb, on 5 February 2012 - 09:35 PM, said:

So we now have to think about the role of physics and if such a metatheorem is a problem?

Physics is all about creating mathematical models of nature, usually some selective part of nature, and then testing these against observation/experiment. Sometimes the experiments lead the theory and sometimes visa versa. The question: is it necessarily the remit of physics to explain all phenomena in nature?

However, to me it seems uncomfortable to accept that aspects of the Universe are just not "explainable" by their very nature.

For me, I hypothesize that free will is intimately tied to all physical phenomena invariant of scale. I won't try to prove this, but it's exciting to think that we may have a crucial role in the workings of the cosmos.

ajb, on 5 February 2012 - 09:35 PM, said:

Anyway, it is amazing that we have managed to understand the Universe in the way we have and that is with mathematics.

Indeed!

This post has been edited by jaekwon: 6 February 2012 - 01:47 AM

[Cap'n Refsmmat]

I've moved this to Speculations. That's not a judgment of the quality of your idea, but merely a reflection that it is, in fact, a speculation, albeit a very interesting one.

I'll have to put some thought into this when I have time.

[jaekwon]

Thank you. All I want to do is apply existing mathematical insight into physics, and see what emerges. I would appreciate your feedback.

This post has been edited by jaekwon: 6 February 2012 - 04:48 AM

[md65536]

jaekwon, on 5 February 2012 - 07:45 PM, said:

The Incompleteness Theorem of Our Universe: "There exist physical phenomena that can not be explained from laws derived from empirical observations".

As per Schrödinger's hat's point (a), what's stopping the PS from making contradictory statements while the PS itself doesn't break any physical laws?

Doesn't using Godel's Incompleteness imply that given a "complete" set of laws, it's possible to derive a contradiction? Doesn't that then imply that you can always find an observable event that is inconsistent with any possible set of consistent laws? But can't you create a law for every observed event? So wouldn't your theory imply that reality is inconsistent? (I've likely made a mistake in my reasoning, or misunderstanding of Godel.)


Edit: Just so I understand what we're talking about...
By "phenomena" we're only talking about observable events, right?
Laws are laws because (and only as long as) all observations are consistent with them. Laws don't necessarily "explain" stuff. So is it fair to say that by "Law(s) L explains phenomenon X" we mean "X is predicted by L and consistent with L"?

This post has been edited by md65536: 6 February 2012 - 06:03 AM

[jaekwon]

md65536, on 6 February 2012 - 05:55 AM, said:

As per Schrödinger's hat's point (a), what's stopping the PS from making contradictory statements while the PS itself doesn't break any physical laws?

The program or instructions embedded in the PS only construct statements consistent with the axioms of the symbolic system, barring external factors like cosmic rays.
I think you are asking why a computer behaves consistently. I assume it does for the sake of argument. If it does not, my original theorem holds.

md65536, on 6 February 2012 - 05:55 AM, said:

1. Doesn't using Godel's Incompleteness imply that given a "complete" set of laws, it's possible to derive a contradiction?
2. Doesn't that then imply that you can always find an observable event that is inconsistent with any possible set of consistent laws?
3. But can't you create a law for every observed event?
4. So wouldn't your theory imply that reality is inconsistent? (I've likely made a mistake in my reasoning, or misunderstanding of Godel.)

1. Yes. Show me a complete set of physical laws and I will derive you a contradiction.
2. Sort of. You bring up an interesting question here. Let me get back to you in a bit.
3. Well, yes, at least for every repeatably observed event. And physics does this, for repeatably observed inconsistent events.
4. No, it just means that any mathematical laws of the Universe are inconsistent, or incomplete. It doesn't make sense (for me) to make statements about the (in)consistency of reality, because empirically (my) reality is consistent.

md65536, on 6 February 2012 - 05:55 AM, said:

Edit: Just so I understand what we're talking about...
1. By "phenomena" we're only talking about observable events, right?
2. Laws are laws because (and only as long as) all observations are consistent with them.
3. Laws don't necessarily "explain" stuff. So is it fair to say that by "Law(s) L explains phenomenon X" we mean "X is predicted by L and consistent with L"?

1. Yes and no. Some phenomena are observable but they can't be considered "events" if they have no time component. But yes, I mean all observable "stuff" (minus emphasis on time) which I call "phenomena".
2. Yes, that is the common understanding of law. Otherwise it wouldn't be applicable and it wouldn't have predictive power.
3. Yes and no. Let's leave it at X is consistent with L. To "predict" has wuzzy connotations when it comes to probabilistic laws like QM, and "consistency" is hand-wavy enough.

This post has been edited by jaekwon: 6 February 2012 - 07:57 AM

[md65536]

Certainly a mind-bender.

If you said that the physical laws were predictive, then you could argue that the laws are created by humans, and humans are physical things that obey the physical laws, so humans are not able to determine a set of laws that could predict the outcome of all experiments. Then you could use the "free-will" argument to prove it: If I know a law predicts that I will choose A, then I will instead choose B, violating this law that would otherwise contradict free-will. But if the laws don't have that degree of predictive power, then the choice of A or B could both be consistent with the laws (and the laws could be consistent with free-will).

But then, even if the "necessary hypotheses" are satisfied so that Godel's theorem applies (I don't think it's been shown?), it still might only tell you that the physical laws cannot be used to prove their own consistency. I don't think that implies that there exists a physical phenomenon that is inconsistent with the laws.

[ajb]

Cap, on 6 February 2012 - 04:34 AM, said:

I've moved this to Speculations. That's not a judgment of the quality of your idea, but merely a reflection that it is, in fact, a speculation, albeit a very interesting one.

Maybe a section on metaphysics and the philosophy of physics would be the right home for this.

jaekwon, on 6 February 2012 - 04:46 AM, said:

Thank you. All I want to do is apply existing mathematical insight into physics, and see what emerges. I would appreciate your feedback.

I have to say that physics is not mathematics and as such I am doubtful one can apply the incompleteness theorems very directly. I also do not think there is much relevance for most of physics. One is usually okay with the theories only being "good" within a range of parameters and can accept that a given picture is not the full story. It is when we come to the full story we have to pause and think.

Many people have thought about this, in particular in relation to a theory of everything. It has be claimed that such a theory cannot exist by the incompleteness theorems. Other people claim, like you do, that we can uncover the rules but not understand all the behaviour of a theory of everything.

I am not aware of any "mainstream view" of this.

I am not well versed enough in logic to understand if there are any exploitable loopholes.

The earliest reference to the incompleteness theorems in physics I know of is Jaki [1] (not that I am familiar with his work).

References

[1] Stanley L. Jaki. The Relevance of Physics. University Of Chicago Press; First Edition edition (June 15, 1966).

[PeterJ]

I think the OP's original argument goes wrong because it stipulates a physical phenomena that cannot be included in physics. If it stipulated just a phenomena of any kind it might work better.

Stephen Hawking arrives at much the same conclusion as the OP but for what seem like better reasons.


With my hazy understanding I concluded that Hawking's idea concerns theories, not empiricism or understanding, which by their nature are prone to incompleteness. That is, we may be able to understand it all even if we cannot include it all in a formal descriptive theory. His essay was called 'The End of Physics'. I note that he has withdrawn it from the Internet, so maybe he later changed his mind.

Is it not more simple just to say that every theory must, like every dictionary, contain an undefined term?

This post has been edited by PeterJ: 7 February 2012 - 10:54 AM