The Big Rock Paradox: Stephen Hawking's Thought-Experiment

[Hurkyl]

My response is this... Yes because Number Theory is inconsistent, so every statement is provably true (and at the same time false.) But Z does not want us to be afraid, and if 1+1=2 leads to a contradiction we would be. So, Z makes sure that any inconsistency is out of reach. Whenever we get close to a proof of a contradiction, Z simply redoes the (faulty) Model of Number Theory so that the contradiction is further away.

Incidentally, it strikes me that what you are describing sounds very much like the things you can do with plain, ordinary "human logic".

 

[Russell Berty]

Incidentally, it strikes me that what you are describing sounds very much like the things you can do with plain, ordinary "human logic".

Humans have at times had to rethink their definitions when contradictions arose. But, what we can not do is restructure the actual truths or laws of reality (if there are any.) We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists. It is a supposed structure of reality that we try to analyze using our "human logic". What I was trying to say is that Z can do what we cannot do. That is, Z can restructure this innate property of reality called Number Theory so that we will not run into a contradiction in are pursuits.

 

[Focus]

Humans have at times had to rethink their definitions when contradictions arose. But, what we can not do is restructure the actual truths or laws of reality (if there are any.) We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists...

We cannot prove that anything exists other than yourself, what is your point?

There are no laws of reality. There is no way that we can claim 100% that reality follows X or Y. The whole of science is built on the assumption of history repeating it self.

Some people in this discussion have come close to my stance on these "paradoxes". We are arguing using our human logic. But, how can we be sure that such logic applies to the realities and lives of the all-powerful, all-knowing? What we see as a paradox might not be in some ultra-nonhuman-impossible for us to follow-hyperlogic.

This is pretty weak claim. You can argue this way with everything. Do you claim that laws of nature do not apply to a being (or maybe more)? If your answer was yes, then you need to prove that God exists. Otherwise I can make the same statement you made to claim that trees are both trees and not trees (they have tree logic).

 

[Russell Berty]

I am not sure what you are getting at in your reply.

You can prove that you exist? Then you are more clever than I, for I cannot prove that I exist.

There are no laws of reality? You can prove this?

For me, I assume that:
1) I exist.
2) Reality exists and has order; it is predictable. i.e. there are laws.

From these assumptions I deduce my theories about reality.

As far as whether or not gods exist, I cannot say. What I am trying to say is that my arguments (or human arguments) are not valid in the realm of the "all-powerful" beings (if there are any.) To argue for or against the existence of such beings is futile. For example, if the all powerful Z decides to make modus ponens invalid then where does that leave us?

 

[JoeDawg]

You can prove that you exist? Then you are more clever than I, for I cannot prove that I exist.

Don't worry, Descartes, a very clever man, already did that: I think therefore I am.

 

[Russell Berty]

Don't worry, Descartes, a very clever man, already did that: I think therefore I am.

That of course requires proving that "All things that think, exist." Which he did not prove.

 

[Russell Berty]

...Or at the very least, he would have to prove "If I think, then I exist." As well as prove that the argument form "P and P -> Q therefore Q" in valid. So, basically prove that modus ponens is valid, and he could not use modus ponens to do this (circular reasoning.) So, if he was thoroughly skeptical and considered some all-powerful demon could deceive him, then he would have to consider "What if the demon makes me believe that modus ponens is valid, or makes me believe that my arguments are valid, when my arguments are actually not valid?" Oh, not to mention he needs to define what is meant by "I" and "think" and "exist". Very tricky.

His works are impressive, no doubt, but there is much that has been learned since his time (in part thanks to him) about logic and the notion of a proof. "I think therefore I am" is not a proof that "I exist". Not without making many unproven assumptions.

 

[Hurkyl]

We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists.

Of course we can. Here's a (trivial) example of such a proof:

Hypothesis: number theory exists.
From the hypothesis, number theory exists. Therefore number theory exists. QED.

(Incidentally, there are also set theoretic universes in which there is no number theory; e.g. the universe of finite sets in any model of ZFC)

That is, Z can restructure this innate property of reality called Number Theory so that we will not run into a contradiction in are pursuits.

Number theory an innate property of reality? Anyways...

Given your hypothesis, there isn't a contradiction in number theory.

If you'll accept that, in principle, we can iterate through all possible formal proofs, then it's easy to demonstrate: the fact Z can ensure none of them are contradictory proves that no contradictions can be derived, and therefore number theory is consistent.

If you reject that, in principle, we can iterate through all possible formal proofs, then the situation can be modeled with internal set theory (related to non-standard analysis), letting everything we can iterate through be the "standard" things. Then, we invoke the transfer principle: "all proofs in number theory have noncontradictory results" if and only if "all 'standard' proofs in number theory have noncontradictory results".

 

[Borek]

It occurred to me that whenever someone asks a physics question where conclusions seems to be paradoxical, it is blamed on the reference frame. I wonder if this approach can be used for the big rock as well

 

[Russell Berty]

Hypothesis: number theory exists.
From the hypothesis, number theory exists. Therefore number theory exists. QED.

You know what I meant. There is no proof that number theory exists without assuming this as a premise.

As far as iterating ALL possible formal proofs, we cannot in a finite amount of time. And as I was implying, "as we get close to a contradiction" (while enumerating proofs), Z changes the game on us.

 

[Hurkyl]

You know what I meant.

Yes I did -- and I also know you're making something out of nothing. Some premises prove the existence of number theory. Some don't. There is nothing deep on here.

As far as iterating ALL possible formal proofs, we cannot in a finite amount of time.

Then you are in the situation of my next paragraph.

 

[Russell Berty]

The point is, if you claim to have a proof of number theory, then that proof is based ultimately upon some assumptions - assumptions that are not proven, merely assumptions. So, what I am getting at is there is no "absolute" proof of number theory. It is possible that number theory is inconsistent. That is why I said, "We assume that a structure we call number theory exists but we cannot prove it exists." Some of us assume even more, such as ZFC is consistent, but it is still only an assumption.

 

[JoeDawg]

...Or at the very least, he would have to prove "If I think, then I exist."

That is true by definition and logic. To be something that thinks, one must be 'something' and therefore exist as something.

So, if he was thoroughly skeptical and considered some all-powerful demon could deceive him, then he would have to consider "What if the demon makes me believe that modus ponens is valid, or makes me believe that my arguments are valid, when my arguments are actually not valid?"

A demon *could* make you believe anything.
But its nonsensical, ie not logical, to say that a demon can do anything to you, if you don't exist. If you are nothing, then there is nothing for the demon to do.

Oh, not to mention he needs to define what is meant by "I" and "think" and "exist".

He doesn't need to do anything of the sort. Those are different questions.

Descartes wasn't dealing with the question of what existence is, or what the self is, or what thought is. It was a more basic question.
What can I know for certain?
It doesn't matter what 'I' is, or what existence is. The answer is, yes, I can know I exist. I can say I exist, because if I didn't exist, i couldn't say or think anything.

This is what is called a self-evident truth.

 

[Focus]

The point is, if you claim to have a proof of number theory, then that proof is based ultimately upon some assumptions - assumptions that are not proven, merely assumptions. So, what I am getting at is there is no "absolute" proof of number theory. It is possible that number theory is inconsistent. That is why I said, "We assume that a structure we call number theory exists but we cannot prove it exists." Some of us assume even more, such as ZFC is consistent, but it is still only an assumption.

Seeing as you keep bringing up existence about number theory, care to define how a mathematical field can exist?

No one assumes that ZFC or number theory are consistent. What do you mean by a proof of number theory? Do you mean a theorem in number theory? The proofs in number theory are absolute, in the sense that if you accept the axioms then you must accept the theorem. You can say the same about the statements such as, if evolution happened then evolution happened which is again absolute. They are all tautological statements.

 

[Russell Berty]

Before I attempt a reply, I will give a little quiz. Defend your answers.


1) Does the number 1 exist?

2) Is it true that 1 + 1 = 2?

3) Is it possible that 1 + 1 = 2 is false?

4) Can you doubt that 1 + 1 = 2?

5) Can you doubt that 1 exists?

6) Can you prove that a statement is true without knowing the meaning of the statement? [For example, Prove that "XqXX$amNNmM yyty3wjj: kXzQQp"]

7) If a statement cannot be proven true, does that mean that the statement is false?

8) Can you prove the following statement? "There is no proof of this statement."

9) Is it possible that number theory is inconsistent (contains contradictions)?

10) Is the mind capable of making a mistake?

11) Is it possible to doubt a statement is true even though the mind claims that statement is true?


I will mention that many philopshers since Descartes have argued over the validity of Cogito Ergo Sum. To the present day there is no agreement on the validity. For a start, read Kant, Hume, and Russell.

On Number Theory: I was talking about a proof that Number Theory is consistent when saying "proof of Number Theory." The point I was making was that we do not know whether it is consistent or not (without making assumptions such as "it is consistent".) By existence of Number Theory, I mean "the existence of a model for Number Theory" (or you can take it further to mean existence in Plato's Universe, but I was not trying to go there.) By "model for Number Theory" I mean the "model" for a theory as defined in first order logic (see any text on first-order logic.)

 

[Russell Berty]

Since the quiz above is off the main topic, I will put it in a new post "A Little Quiz"

 

[JoeDawg]

For a start, read Kant, Hume, and Russell.

I have.
Now please, show us that you have.
What are the arguments?

 

[Russell Berty]

It is very difficult to debate the truth without having definitions for the words in "I think therefore I am."

There are many thought experiments you can come up with to bring the validity into doubt, try it. Descartes' Demon is very clever, more clever even than Descartes.

But I will try an approach that will not rely on whatever definitions are being used.


We are discussing Absolute Certainty concerning that truth of the statement "I exist." Cogito Ergo Sum is supposedly a complete validation of the statement "I exist." There is no uncertainty whatsoever in the argument or the conclusion.

I am trying to refute this.
Sorry it is a bit long, but enjoy. :)

We are in a dark alley (why? None of your business, stop asking questions!) and someone gives us a list of instructions.

They tell us "It is an algorithm that will verify any statement that is 100% certainly True." We run home to try it out.

It is called the Certainty of Truth Validator (or make up your own name) and we abbreviate it as CTV.
We are told that given a proposition p, when we run CTV on p, denoted CTV(p), either CTV will halt and say 100%T, or it will halt and it will say <100%T (not absolutely certainly true), or it will never halt.
We are guaranteed that if p is 100%T then CTV(p) will halt and we will get the response 100%T.

So, there I am with the CTV. I wonder if CTV is itself reliable 100%. I want to make sure that CTV will never say 100%T when the input proposition is not 100%T. So I try the following:

CTV( It is not the case that, if CTV(p) outputs 100%T then p is not 100%T )

I get 100%T. Not surprising, look who I am asking.

I then grow a little concerned. I am the one who just ran CTV! Am I reliable? I will ask CTV.

CTV( For all propositions p, I run CTV(p) without error )

I run the algorithm, holding my breath. Still holding...
Maybe it stops and says 100%T (phew) or maybe it does not (uh-oh.)

But, even if it says 100%T, how do I know I ran THAT input correctly?

For brevity, let A0 := "For all propositions p, I run CTV(p) without error" I just ran CTV(A0).

So, I run

CTV (I run CTV(A0) without error).

Again I hold my breath... Even if I get 100%T, did I run that correctly?

Recursively define the following propositions:
A(n+1) = "I run CTV(An) wihtout error" {for n = 0, 1, 2 ...}

I just ran CTV ( A1 ).
But, still uncertain of MY OWN ABILITY to run CTV without error, I continue.

CTV ( A2 ), CTV ( A3 ), ...

Regardles of the results (I am being nice to myself saying that CTV says I run it without error in these cases) I will forever be UNCERTAIN OF MY OWN ABILITY TO RUN CTV.

To be fair, I was asking alot, for the initial satement said "I run CTV(p) without error on ALL propositions p".

So, let us try something more specific.

I run
CTV (I exist).

Now, to cover whatever arguments people claim show "I exist" is 100%T, I will allow the instructions for CTV to contain that list for just this situation. (i.e. see CTV Manual pg. 4,329 line 78 "When the input is "I exist" follow these...)

OK. So I run CTV (I exist) [Define B0 := "I exist" ]

I get back 100%T (otherwise...omg)

But, did I run it correctly! So I run CTV (I run CTV (I exist) without error). I get 100%T. Goody.
But I still might have made a mistake, I am not ABSOLUTLEY CERTAIN.

Define B(n+1) = "I run CTV(Bn) without error" {for n = 0, 1, 2 ...} I ran CTV(B1). So now I run
CTV(B2), CTV(B3), ...

always wondering whether I made a mistake somewhere; always just a little uncertain; never absolutely certain that the statement "I exist" is true.

 

[JoeDawg]

always wondering whether I made a mistake somewhere.

I think you're getting hung up on language.

'Cogito ergo sum' really has no 'I' in it.

"Thinking exists" is a perfectly good translation.

Even, 'doubting exists', if one wants to use Descartes evil demon example.

And if thinking exists, then thinking is a thing.
We can call that thing 'I', we can call it x, or Fred, or any other word.

You don't have to know the nature of self, you don't have to know the nature of thinking, nor do you need to define the nature of existence.

Its more basic than that. Thinking exists is the same as saying thinking occurs, or thinking happens. And this is unavoidable... its a self-evident truth.

If you doubt thinking happens, you are thinking and doubting. So its self-contradicting.

 

[Russell Berty]

For more on the issues with claiming "absolute truth" of anything, even that "thinking exists" or "I exist", see the following.

1933, Idealistic logic
a study of its aim, method, and achievement
by C. R. Morris

It can found online.


http://books.google.com/books?id=ED...eF8csP&sa=X&oi=book_result&ct=result&resnum=2

 

[JoeDawg]

by C. R. Morris

Who is that?

Telling people to read things IS NOT the same as supporting an argument, and I never requested a reading list.

I have read "Kant, Hume, and Russell" however, which you said was important to your arguments.

So please, let's deal with that. I'm certainly not reading anything you recommend until you can show you actually know what you're talking about.

Feel free to quote any of these well-known philosophers to support your opinion.

 

[Russell Berty]

"So, there, in front of the fire, I sat at least still convinced that thinking does not exist. But what right do I have of this certainty? Could this be an illusion as well? Surely not, for no matter what I tried in the way of doubting that thinking does not exist it was always the case that the doubting did not exist. No matter where I turned, there was nothing. So there was no way of doubting that thinking does not exist. So, there remains one thing which is certain - thinking does not exist. Moreover, since thinking does not exist, then I lack the quality of thinking and in order for something to have the ability to lack a quality it must exist. Ergo I exist. Thinking does not exist, therefore I exist."

It might be easier for you to see the fallacies in Descartes' argument by finding the fallacies in the one above. There are many assumptions being made. And, these assumptions are not proven in any certainty whatsoever - they are not even doubted of their validity in the argument!

 

[Russell Berty]

Here is another reason to question the validity:

Descartes argues that even in doubting there must be thinking.
He is convinced utterly that thinking exists.
He is sure that thinking exists.
He is sure that there can be no doubting that thinking exists.
So, Descartes doubts that there is doubting that there is thinking.
So, Descartes is doubting the existence of doubting! Even if doubting that doubting exists happens in this one case, this is sufficient to jeopardize the statement there can be no doubting that thinking exists.

 

[DaveC426913]

So, Descartes doubts that there is doubting that there is thinking.
So, Descartes is doubting the existence of doubting!

non sequitur. One does not follow the other.

Just because one doubts a single thing (doubt that there is thinking),
does not imply he is doubting the very existence of doubtin. He is only doubting the existence of this one thing.

 

[JoeDawg]

It might be easier for you...

It would be easier for me, if you backed up your claim about "Kant, Hume, and Russell".

Otherwise, its easier for me to simply ignore you, since you don't seem to know what you're talking about.

 

[Red Fox]

This is quite an old issue, and philosophers today tend to reject the notion of absolute omnipotence for just this reason. Currently limited omnipotence (that is, the ability to do anything within certain confining laws, like the impossibility of self contradictory entities etc.) is favored when speaking on the subject of divine beings.

 

[Moridin]

This is quite an old issue, and philosophers today tend to reject the notion of absolute omnipotence for just this reason. Currently limited omnipotence (that is, the ability to do anything within certain confining laws, like the impossibility of self contradictory entities etc.) is favored when speaking on the subject of divine beings.

The concept of "limited omnipotence" is a contradiction in terms.

 

[Red Fox]

The concept of "limited omnipotence" is a contradiction in terms.

I specified what I meant.

 

[DaveC426913]

The concept of "limited omnipotence" is a contradiction in terms.

Then it is simply a matter of choosing which contradiction is more palatable, since true omnipotence is also a self-contradictory term - as the existence of this thread demonstrates.

 

[gabrielh]

I think, in the end, you're just ending with a word game that doesn't really solve anything. And I certainly don't think that we should make assertions of truth or fact based solely on word games.