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

[Upisoft]

Maybe he has found a measuring device which tells us the godliness of a god. Or he is playing with a word that has no contextual meaning.

For example http://en.wikipedia.org/wiki/Nontransitive_dice" . For every dice set there is a greater set that will win with higher rate. But there is no greatest set.

People can "prove" a lot of stuff by using words like "greatest" or "more complex".

 

[Jamma]

Actually you proposed they are not random, but pseudo-random and even fabricated fictional underlying theory explaining it. So, the burden of proof is yours.

I said I was going to withdraw from this stupid argument, but I can't just leave that one. I suggested that you cannot prove that there are no underlying laws to it, clearly the burden of the proof is on you. I didn't say that quantum effects definitely aren't truly random, but you certainly can't prove that they are. And again, randomness adds nothing to this discussion.

I really feel that you should read up on the incompleteness theorems- they are very important and really interesting; the most interesting thing about them is how fundamental they are to our capabilities of proving statements. It's quite sad on your part that you don't appreciate the amazing fact that not even the most advanced technological or mathematical breakthroughs can destroy the restrictions that the theorem inposes. At least you have the joy of discovering this ahead of you though!

 

[Upisoft]

I suggested that you cannot prove that there are no underlying laws to it

I agree this is what happened. However you bought a new idea in our argument, namely "underlying laws", and now you want that I disprove it. Well, sorry. If you want to introduce a new idea, then the burden of proof is yours.

 

[Jamma]

Here is a quote from the article. It describes Anslem's Ontological proof.

"... the best-known version of the argument, Anselm noted:

1. The definition of the word God is "that than which nothing greater can be conceived."

2. God exists in the understanding, since we understand the word with that definition.

3. To exist in reality and in the understanding is greater than to exist in the understanding alone.

4. Therefore, God must exist in reality.
"

I had a couple of questions - not knowing any Philosophy or theology.

- question about line 2:

Just because we understand the words describing something, why does that mean it exists in the understanding? I worry that we can understand that something is impossible e.g. an odd dimensional compact manifold of Euler characteristic 2. But how can such a manifold - which does not exist - exist in the understanding?

- I do not know what the idea of "greater" means in this argument. Is it an ordering of something? Perhaps a partial ordering?

Yes, and step 4 doesn't follow; even if it is possible to conceive the "greatest" thing (whatever these terms mean precisely) that doesn't mean that we can't conceive something in our minds that also exists in reality but in fact doesn't really exist.

e.g. let God exist as above. I can conceive a universe which is a duplicate of this one, a sort of disjoint union, where each operates as this one has and there is no interaction between the two. I can now conceive God as being this disjoint union of Gods who has power over both of these disjoint universes within my new universe (it was assumed that I could conceive one God, so it's not hard to conceive two non-interacting ones). This God is clearly greater than the one we had originally. But we had assumed that our God was already the greatest. Contradiction!

 

[Jamma]

I agree this is what happened. However you bought a new idea in our argument, namely "underlying laws", and now you want that I disprove it. Well, sorry. If you want to introduce a new idea, then the burden of proof is yours.

Actually, you introduced randomness; your reasoning being that it is different to randomness you can get from computers, which you called pseudo-randomness, so to justify why this is useful (which it wouldn't be even if you could) you would need to prove that it is truly random, in a nature very different to pseudo-randomness. Hence, the burden of the proof is on you.

 

[Upisoft]

Now you made up another idea.. "truly random". I was speaking about "random".

 

[Jamma]

Oh come on, obviously by saying "truly random" I was referring simply referring to random.

I used that because I was trying to understand where you were coming from in your argument about randomness and how you know for certain that there can't be underlying laws making it not random. Not that it matters.

And being overly-pedantic about my terminology doesn't dodge my point that you need to answer; namely:

what is it about this random nature that quantum computers have at their disposal which allows them to prove things that regular computers can't? (perhaps provide us with a proof of a statement which requires a random step)

and

if only this sort of random will do, not even a very good random number generator that you can get on any computer will suffice, how do you prove that there aren't underlying laws governing this randomness which you seem to suggest is the reason why regular computers can't do some of the computations that quantum computers can?

 

[Hurkyl]

In short set theories assume that if you have full knowledge of the set you also know its elements.

Assuming, for the sake of argument that this is a reasonable assertion...

That, of course, is not true in QM. For example, if you have two electrons in a singlet state (fully defined state), you know nothing about the spin of the components.

And this...

All you can conclude is that an electron is not a synonym for a set of spins of components of electrons. And I'm not even sure you can conclude that.

 

[Upisoft]

Oh come on, obviously by saying "truly random" I was referring simply referring to random. I used that because I was trying to understand where you were coming from in your argument about randomness and how you know for certain that there can't be underlying laws making it not random. Not that it matters.

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

And being overly-pedantic about my terminology doesn't dodge my point that you need to answer; namely:

what is it about this random nature that quantum computers have at their disposal which allows them to prove things that regular computers can't?

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

 

[Upisoft]

Assuming, for the sake of argument that this is a reasonable assertion...

And this...

All you can conclude is that an electron is not a synonym for a set of spins of components of electrons. And I'm not even sure you can conclude that.

Example, you have two lamps. The system has set of 4 states: (on, on) // (on, off) // (off, on) // (off, off). Each lamp has has set of 2 states. (on) // (off). If you know the exact state of the system, say (on, off), you will know the exact state of each component: (on) and (off). That is the set theory in a nut-shell (well, the part relevant to our discussion anyway).

That is not necessarily true for system of two electrons. I.e. there are states of the system that behave exactly like those above and state that you have absolutely no knowledge about the components. Also anything in between.

 

[Hurkyl]

...

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

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.