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

[Upisoft]

Please, define "randomly" and "50/50 chance every time". But be precise, very precise.

"Randomly" means you have no knowledge what you will get (even if you know how the Turing machine is prepared) and 50/50 chance means that if you run the machine lots of times you will get about half of the results -1 and other half +1.

 

[arkajad]

"Randomly" means you have no knowledge what you will get

Then, according to your definition, my computer is a good example of a random machine, because quite often I have no knowledge of what it will do next minute.

 

[Upisoft]

Then, according to your definition, my computer is a good example of a random machine, because quite often I have no knowledge of what it will do next minute.

Did you know the exact state of your computer?

 

[arkajad]

Did you know the exact state of your computer?

Do you know the exact state of anything?

 

[Upisoft]

Do you know the exact state of anything?

Yes, you can know the exact state of the spin of an electron which was previously prepared. If you measure it along the axis you have prepared it you will get the same result every time. The same is true with the polarization of light.

 

[arkajad]

Here you are talking about theory, not about practice. In theory I can have a classical random process that will do exactly the same. When you say "quantum state+detection" it is a code word for "random process".

 

[Upisoft]

Here you are talking about theory, not about practice. In theory I can have a classical random process that will do exactly the same. When you say "quantum state" it is a code word for "random process".

Well, then do it in practice. Create Turing machine capable to reproduce an experiment of preparing an electron with spin along x-axis and then measuring it along y-axis. It's straightforward process, you do the same thing every time. But you get different results. I don't know any Turing machine capable to run the same program and get different result.

 

[arkajad]

You need a pseudo-random number generator. A classical one, form a PC will do. In fact it does. You do your experiment, I do my computation, and you will not be able to distinguish which is which. The algorithm is very simple. If you wish, I can also reproduce double slit experiments, including timing of the detections, which you can hardly even measure with the present technology. But one day it will be possible and we will be able to read more from the experimental data. Today we can only simulate these subtle quantum effects on our classical computers.

 

[Upisoft]

You need a pseudo-random number generator. A classical one, form a PC will do. In fact it does. You do your experiment, I do my computation, and you will not be able to distinguish which is which. The algorithm is very simple. If you wish, I can also reproduce double slit experiments, including timing of the detections, which you can hardly even measure with the present technology. But one day it will be possible and we will be able to read more from the experimental data. Today we can only simulate these subtle quantum effects on our classical computers.

We were talking about Turing machine which starts in the same state. The pseudo-random number generators have the bad habit to reproduce the same string of "random" numbers if you start with the same seed. That is used by the identification devices you can keep on your keyring. The device and the system you want to log in share the seed and even if the numbers look random to you, they are not random to the system. So, drop the pseudo-random argument.

 

[arkajad]

Why should I drop? For every problem and any given set of tests of randomness I can make a pseudo-random number generator that will be practically indistinguishable from experiment. Moreover, given any finite experimental data, quantum origin or not, one can construct a randomness test that your data will fail with.

You seem having problems with deciding whether you want to talk about theory or practice. It seems you have abandoned theory. Soon, I guess, you will be back to it, because practice is not on your side.

 

[arkajad]

Moreover: there is a rather famous MNCP software for simulation of nuclear processes based on algorithms from QED. It is very successful. It is used in nuclear engineering, radiation detection and shielding etc. (in fact I was using it and comparing simulations with experiments). It is based on pseudo-random numbers generators.

 

[Upisoft]

Why should I drop? For every problem and any given set of tests of randomness I can make a pseudo-random number generator that will be practically indistinguishable from experiment.

What about this test. I create machine having exactly the same pseudo-random generator. We run them and oh miracle, the numbers are identical. We repeat the test, and again we get the same numbers for your different pseudo-random number generator and mine (which again is identical).

Then we do another test. We prepare two electrons in identical states with spin along x-axis. We measure them along y-axis and we get 1 and +1. Oops. Well maybe this must be the case for electrons... We do the same experiment again (maybe we rotate x an y axes), we get +1 and +1.. Ooops, why we can't get the same result every time?

Enough practice?

 

[arkajad]

You can create any machine you wish. This will not change the fact that quantum mechanics on both theoretical and practical level can be reduced to classical computations plus classical random processes.

It seems to be your belief that only "real quantum processes" are producing "real randomness", but that is just your belief, because you are not even able to define precisely randomness. You escape into subjective arguments like "what one knows" . "One"- who? It's just your belief, what can I say? Perhaps you are right? Or, perhaps, you are not?

 

[Jamma]

Seriously, this thread should be locked. How do you not get this?

And how does "Show me Turing machine that will generate randomly 1 or -1 with 50/50 chance every time it is reset and started again." help anything? What proof do you know which requires being able to invoke a 50/50 decision in the middle of it? It obviously wouldn't be a proof if it had a random step in it... And as said before, normal computers can generate random numbers.

And yes, it is wikipedia. How many things, that fundamental in nature and statement, can you name me from wikipedia that are wrong? For one, it is referenced, and secondly, I'm sure that if you read any text on quantum computers, you would read the same thing; how could you possibly deny this?!

 

[Upisoft]

You can create any machine you wish.

Thanks. Then I can create thinking machine.

It seems to be your belief that only "real quantum processes" are producing "real randomness", but that is just your belief, because you are not even able to define precisely randomness. You escape into subjective arguments like "what one knows" . "One"- who? It's just your belief, what can I say? Perhaps you are right? Or, perhaps, you are not?

Randomness: Ability of a system to produce unpredictable results, even one has all the knowledge about the system (i.e. S=0 - entropy is 0).

Does that satisfy you?

 

[Upisoft]

And how does "Show me Turing machine that will generate randomly 1 or -1 with 50/50 chance every time it is reset and started again." help anything? What proof do you know which requires being able to invoke a 50/50 decision in the middle of it? It obviously wouldn't be a proof if it had a random step in it... And as said before, normal computers can generate random numbers.

And yes, it is wikipedia. How many things, that fundamental in nature and statement, can you name me from wikipedia that are wrong? For one, it is referenced, and secondly, I'm sure that if you read any text on quantum computers, you would read the same thing; how could you possibly deny this?!

You said you are able to make a Turing machine. And when I ask you to do so you ask me what will that prove? It will prove that you can make such Turing machine, isn't it obvious? if you cannot... well there is nothing more I can say. I even don't want a complex machine. The spin can have only 2 possible values when measured. The simplest case.

Normal computers cannot generate random-numbers. They can generate pseudo-random numbers. Look other posts for the difference.

 

[Jamma]

You said you are able to make a Turing machine. And when I ask you to do so you ask me what will that prove? It will prove that you can make such Turing machine, isn't it obvious? if you cannot... well there is nothing more I can say. I even don't want a complex machine. The spin can have only 2 possible values when measured. The simplest case.

Normal computers cannot generate random-numbers. They can generate pseudo-random numbers. Look other posts for the difference.

When did you ask me to make a Turing machine?! And your computer that you are using to type with effectively is a Turing machine, the only difference is that it has limited resources. But harping on about that point won't help you because quantum computers, no matter how advanced, will only have limited resources too. Godel's theorem is about proofs that can be reached in finite time using a set standard of logic with a theoretical Turing machine, and no matter how you try, your quantum computer doesn't add anything to the mix which will make this theorem no longer hold.

And stop talking about random numbers too, it is completely irrelevant- if your random numbers are so important for a calculation, then your calculation is a non-exact one from the fact that it invokes random numbers, and therefore it doesn't matter whether your random-number generator is random or pseudo-random. And anyway, can you prove that quantum-computers definitely can generate random numbers? Can you prove that there are no underlying definate laws that describe quantum theory? I don't think that we ever have or ever will.

 

[JDStupi]

WTF? One thing I always try to do is ask myself why I believe something so much, and look at all sides. What is the point of all this? We went from discussing Godel's unique ideas about Platonism, Theology, and mathematical logic, to the potential ramifications of his Incompleteness Theorem on Computational mind and now we have one person making silly "arguments". Consider John Searle's Chinese Room argument, essentially the human mind is inherently semantical, and being that computers are by definition syntatical, it is not a matter of technology, but even in principle it is not possible for something that is purely syntactical to create semantics. This can be seen as similar to Godel's Theorem, so Because a computational system operates by a finite means of logical/syntactical steps, the interpreted semantical meaning of the steps within the formal system must come from outside of the system. Now this seems to me to lead to the untenability of a strictly computational mind. Rather than focus on the purely mathematical aspects of logic/set-theory which are relevant, focus on the philosophical aspect which is primarily the use of Godel's results as regards the semantic/syntactic distinction.
Bringing QM into the discussion adds absolutley nothing and is such a common almost "god of the gaps" argument for mind now. Welllll since QM is crazy, maybe QM holds the key to conscious experience.

 

[Upisoft]

When did you ask me to make a Turing machine?! And your computer that you are using to type with effectively is a Turing machine, the only difference is that it has limited resources. But harping on about that point won't help you because quantum computers, no matter how advanced, will only have limited resources too. Godel's theorem is about proofs that can be reached in finite time using a set standard of logic with a theoretical Turing machine, and no matter how you try, your quantum computer doesn't add anything to the mix which will make this theorem no longer hold.

And stop talking about random numbers too, it is completely irrelevant- if your random numbers are so important for a calculation, then your calculation is a non-exact one from the fact that it invokes random numbers, and therefore it doesn't matter whether your random-number generator is random or pseudo-random. And anyway, can you prove that quantum-computers definitely can generate random numbers? Can you prove that there are no underlying definate laws that describe quantum theory? I don't think that we ever have or ever will.

In quantum mechanics random is random. If there is underlining theory, it would not be called quantum mechanics. And, btw, my computer is not a Turing machine. I've never heard about Turing machine with a clock. The Turing machine is predictable and every time it is set in a known state the answer is the same. I've never heard about a Turing machine with a hard disk, keyboard or any other peripheral devices either.

 

[Upisoft]

Because a computational system operates by a finite means of logical/syntactical steps, the interpreted semantical meaning of the steps within the formal system must come from outside of the system.

Where do you think the semantical meaning comes from for us? From inside?

 

[Jamma]

In quantum mechanics random is random. If there is underlining theory, it would not be called quantum mechanics. And, btw, my computer is not a Turing machine. I've never heard about Turing machine with a clock. The Turing machine is predictable and every time it is set in a known state the answer is the same. I've never heard about a Turing machine with a hard disk, keyboard or any other peripheral devices either.

Wow, please write a PhD thesis on your proof, I'm sure that phycists and mathematicians everywhere would love to read it (probably mainly for comical reasons).

And when I said that your computer is effectively a Turing machine, I didn't actually mean that it is a theoretical calculation device (obviously), I was clearly referring to the way in which it can act like a Turing machine; the mere act of running an algorithm on a computer is to have in an input, allow the machine to calculate given this input and fixed rules, and to give an answer at the end.

Sorry to the OP for the thread going this way, I probably shouldn't have bothered responding (if he had less posts I'd assume that he was trolling). I shall have no more say on the matter!

 

[JDStupi]

At this point in time I do not know where the semantical ideas originate, that is not a reason for denying their existence though, that is simply silly. I can only speak speculatively at this level of knowledge, and so we could say that possibly the syntactical operations performed by the brain in the process of interaction with an environment are given "meaning" or a semantics by some type of information encoded at a deeper level of the organism, maybe DNA, maybe something else. This then pushes the explanation further down, and if it is the case it seems to still not bode well for a computational theory of MIND, though a computational theory of BRAIN may very well be possible. Of course that leads to Mind=Brain argument, and dualism, which is the other thread's topic. Also, this wouldn't be sufficient explanation for the complex varieties of symbols that are present in mental life, it would only be a baseline explanation for simple biological based symbols, but the existence of more complex symbols and metaphor would have to be taken into account in some other way that is too far ahead for met to speculate on.

 

[Upisoft]

Wow, please write a PhD thesis on your proof, I'm sure that phycists and mathematicians everywhere would love to read it (probably mainly for comical reasons).

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.

 

[Upisoft]

At this point in time I do not know where the semantical ideas originate, that is not a reason for denying their existence though, that is simply silly. I can only speak speculatively at this level of knowledge, and so we could say that possibly the syntactical operations performed by the brain in the process of interaction with an environment are given "meaning" or a semantics by some type of information encoded at a deeper level of the organism, maybe DNA, maybe something else. This then pushes the explanation further down, and if it is the case it seems to still not bode well for a computational theory of MIND, though a computational theory of BRAIN may very well be possible. Of course that leads to Mind=Brain argument, and dualism, which is the other thread's topic. Also, this wouldn't be sufficient explanation for the complex varieties of symbols that are present in mental life, it would only be a baseline explanation for simple biological based symbols, but the existence of more complex symbols and metaphor would have to be taken into account in some other way that is too far ahead for met to speculate on.

I will give you an example. I bought new glasses. They had different characteristics then previous glasses. I had problems at the beginning. I could recognize the objects as I could see them better with the new glasses, but I couldn't make good decision how far are they. Eventually I've adapted.

Now, did the DNA helped me at this moment? Probably yes. But only as a template for creation of new protein molecules that were required. The semantics were provided by my experience (missing door knobs, etc.) i.e. the outside world. In other words a blind man will never adapt to new glasses.

 

[lavinia]

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?

 

[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.