What role did Gödel's belief in God play in his mathematical discoveries?

[Hurkyl]

...

You are equating "set theory" with "hidden variables"? No wonder you make odd conclusions. :tongue:

Now, mind you, the validity Bell's theorem does not even rule out hidden variable theories! Bohmian mechanics is one counter-example.

 

[Upisoft]

Now, mind you, the validity Bell's theorem does not even rule out hidden variable theories! Bohmian mechanics is one counter-example.

How do Bohmian mechanics explain quantum tunneling?

 

[Hurkyl]

How do Bohmian mechanics explain quantum tunneling?

The same way ordinary quantum mechanics does, if I remember correctly.

 

[Jamma]

Oh, again those "underlying laws". OK. Let's assume there are such laws. Why do you believe they have to be necessary pseudo-random?).

I never said that I did! I am saying that for you to use them in your argument for why Godel's proof doesn't work anymore because of QCs, YOU need to prove that this is because they are not pseudo-random, afterall, this seems to be your entire basis for why QCs can prove things which ordinary computers cannot (and as I've mentioned, bringing in randomness doesn't help you at all, but I still want to hear your justification).

Nothing obviously. We are talking about thinking machines, not proving machines(like Turing machine).

Ermm, your assertion was that Godel's proof doesn't say anything about what QCs can achieve. So we are talking about proving machines.

 

[Upisoft]

The same way ordinary quantum mechanics does, if I remember correctly.

Hmm, Bohmian mechanics is interesting idea. It trades the randomness in the collapse of the wave-function for specific ($|\psi|^2$) randomness of the initial conditions. There is a problem with Lorentz-invariance of the initial condition and I'm reading a paper that tries to deal with the problem by adding something they call "synchronization" to the picture. At least it requires FTL action, but since there is no such thing like simultaneity in SR I guess there will be a lot more problems to achieve "synchronization".

Anyway the randomness stays.

 

[Upisoft]

Ermm, your assertion was that Godel's proof doesn't say anything about what QCs can achieve. So we are talking about proving machines.

Read my first post #3. It is an answer to a conclusion we can't build thinking machines. If you thought I meant anything else, I'm sorry for the misunderstanding.

 

[Jamma]

Read my first post #3. It is an answer to a conclusion we can't build thinking machines. If you thought I meant anything else, I'm sorry for the misunderstanding.

Ohhh, I thought all along that you were saying that QCs undermine Godel's theorems, that's definitely what it looked like.

This still doesn't mean that quantum computers can't be simulated though, unless they use a random step in the program, but then ordinary computers can use a random step too. I suppose this is why you were saying that ordinary computers can't simulate quantum computers, but then the random number generators on regular computers are good enough to ensure that you probably couldn't tell the difference between the output of the QC and the regular computer (you couldn't simulate the result of just one test with another QC by your own arguments).

If it is your argument that the difference between something thinking and something not thinking is that the one which is thinking invokes a completely random step, even then I do not see where it is that the quantum computer is thinking but the regular computer is not; indeed the random number generator on a regular computer will be subject to the effects of quantum physics to come up with its random number. It's probability distribution may be a little off that of the QC (although not by anything significant) but quantum processes still go into that number.

Where then is the room for the thinking machine in your QC that the regular computer does not?

 

[Upisoft]

Where then is the room for the thinking machine in your QC that the regular computer does not?

Suppose we have built a computer that is able to think. We run it. It thinks of something. Suppose now we reset the computer and reset its pseudo-random generator, so the same random numbers are repeated. We run the computer and we get the same "thought".

I don't call any process that leads to the same answer "thinking". You may call it algorithm or whatever, but it is predictable.

You cannot do the same with quantum effects. You cannot reset their random generator. In other words, you cannot control the thoughts of quantum thinking machine.

 

[Jamma]

Suppose we have built a computer that is able to think. We run it. It thinks of something. Suppose now we reset the computer and reset its pseudo-random generator, so the same random numbers are repeated. We run the computer and we get the same "thought".

I don't call any process that leads to the same answer "thinking". You may call it algorithm or whatever, but it is predictable.

You cannot do the same with quantum effects. You cannot reset their random generator. In other words, you cannot control the thoughts of quantum thinking machine.

If you are ensuring that the "pseudo-random generator" gives the same results then of course the computer will give the same result. The same would be true if we ensured that the QC's random generator was made to give the same results. What's the difference?

 

[Upisoft]

If you are ensuring that the "pseudo-random generator" gives the same results then of course the computer will give the same result. The same would be true if we ensured that the QC's random generator was made to give the same results. What's the difference?

You cannot ensure any random quantum process to repeat. The only way is to measure something that you've already measured, then you will get the same answer ad nausea.

It reminds me about some funny code I saw somewhere:

function random()
{
   return 4; // The value was obtained by fair die throw.
}

 

[Jamma]

You cannot ensure any random quantum process to repeat. The only way is to measure something that you've already measured, then you will get the same answer ad nausea.

It reminds me about some funny code I saw somewhere:

function random()
{
   return 4; // The value was obtained by fair die throw.
}

You can't ensure that a random number generator on a regular computer will repeat, I think that they often use external inputs such as temperature of the components and other things to give the output.

 

[Upisoft]

You can't ensure that a random number generator on a regular computer will repeat, I think that they often use external inputs such as temperature of the components and other things to give the output.

You can. All you need to do is to record all that data. There isn't such thing as non-recordable data in the classic computer. You can copy the current state, called hibernation in computers, and then restore it. It is usually done only once, but it can be repeated if necessary.

With QM you can't copy the state, I think there is proof that you can't build a machine that can copy quantum state.

 

[Jamma]

Yes, but these things are still random. Does it matter that you are recording the information that went into the process of giving you the random number? You can't necessarily restore the state that went into the computer coming up with its random number (i.e. all quantum events that went into the computer coming up with its random number). The difference here is only that the regular computer has made use of these external influences, processed them as data, and used that data to give the random number. The QC is doing the same thing, except now these processes are purposely being generated withing the computer. So for you to be fair on the regular computer, you can't just put the computer back into the state after it had recorded the data, you should recreate all of the influencial quantum states that went into making that data.

It seems like nitpicking, but if you are going to argue something philisophical like "QCs can be considered as thinking machines but regular computers cannot" then unfortunately you have to be this thorough and go this far deep.

 

[Upisoft]

Yes, but these things are still random. Does it matter that you are recording the information that went into the process of giving you the random number? You can't necessarily restore the state that went into the computer coming up with its random number (i.e. all quantum events that went into the computer coming up with its random number).

What will prevent me to restore the state and "replay" the process the same way it happened?

 

[Jamma]

What will prevent me to restore the state and "replay" the process the same way it happened?

The state of what? Every quantum event that went into the computer making its decision? Sounds like you can't restore that in much the same way as you are saying that you can't restore the state of the QC.

 

[Upisoft]

The state of what? Every quantum event that went into the computer making its decision? Sounds like you can't restore that in much the same way as you are saying that you can't restore the state of the QC.

The classical computers operate with binary data. Any external event have to be transformed into binary data. You record that data. When you "replay" the process you feed into the computer the recorded data instead. The process repeats without change.

 

[chiro]

Umm sorry if I disrupt the thread, but it seems that over pondering the incompleteness theorem, that it not only says a thing or two about logic, but also says a lot about language.

It seems that the axioms that cause trouble are exercises in ambiguous statements and to me that is an issue with inventing a language whereby these sort of sentences can never be constructed based on some principles that prevent the language from ever producing them.

Like say for example if you have the two statements "All cretins are liars" and "I am a cretin", then obviously this causes the scenario that godel is talking about.

Getting to the point, has anyone ever worked or is working on a language where axioms can be built (i'm not just talking normal logic, set theory, arithmetic and such but more a fully fledged meta-language that describes how to build axioms from the grammar of the meta-structure).

Like say for example with the above cretin axioms. The meta-language would for instance be adjusted to only allow self-consistent axioms that do not provide any ambiguity or contradiction.

I'm guessing for this to work, the meta-language grammatical structure (if you're a computer scientist think BNF or EBNF grammars) would make each axioms grammar space dependent on all axioms before it.

So for example the first axiom has a grammatical space. The space of the second axiom is conditionally dependent on the first axiom. The third axiom is dependent on the first two and so on.

What are your thoughts on this?

 

[Upisoft]

The http://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem" in wikipedia is quite unclear. I cannot understand how one can write the formula $P(x)=\forall y\hspace{10pt} q(y, x)$. The symbol $q$ is not in the language described above.

 

[avro]

Intuition. Is intuition just the output of a working brain? Don't you think guys that it has more variables involved there, for example emotions and dispositions. Any thought regarding this?

 

[arkajad]

Ramanujan was attributing his intuition to a Goddess. Kurt Gödel seemed to be of a similar opinion when he wrote "There are other worlds and rational beings of a different and higher kind." Though Gödel was, as it seems, more rationally oriented than Ramanujan.