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

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Discussion Overview

The discussion revolves around the influence of Kurt Gödel's religious beliefs on his mathematical discoveries, particularly in relation to his incompleteness theorem and his work on the ontological proof of God. The conversation also touches on the implications of Gödel's ideas for the fields of quantum computing and logic.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants reference an article discussing Gödel's view of God as a paradoxical entity that cannot be reduced to natural principles, suggesting that his religious beliefs influenced his mathematical work.
  • Others quote Gödel's reflections on the prejudices against religion and his belief in a meaningful theological worldview that aligns with scientific understanding.
  • Some participants challenge the assertion that Gödel's incompleteness theorem implies algorithms cannot replace intuition, demanding evidence related to quantum computing.
  • There is a debate on whether quantum computing can be simulated by classical computers, with differing views on the implications of quantum mechanics for computational possibilities.
  • Some argue that Gödel's focus was on logical possibilities rather than technological feasibility, emphasizing his interest in the logical implications of his theories.
  • Participants discuss the nature of quantum information, including the inability to copy it and its contextual behavior, which complicates classical simulations.
  • There are claims that Gödel's work on time loops and geometries was more about logical possibilities than technological constraints, suggesting a distinction between what is logically possible and what is technologically achievable.

Areas of Agreement / Disagreement

The discussion contains multiple competing views regarding the implications of Gödel's beliefs on his mathematical work and the nature of quantum computing. There is no consensus on whether Gödel's incompleteness theorem definitively supports or contradicts the capabilities of algorithms or quantum computing.

Contextual Notes

Participants express uncertainty about the relationship between Gödel's logical theories and the technological capabilities of quantum computing. The discussion highlights unresolved questions about the nature of quantum information and its implications for classical computation.

  • #31
Upisoft said:
Bell's theorem shows that QM cannot be explained by any set theory.
:confused: Bell's theorem doesn't even vaguely resemble the conclusion you are trying to draw.

Upisoft said:
Godel's work is set based, not vector space based.
No, Gödel's work is based on first-order logic.

Your comment is additionally mystifying, because vector space admits set-theoretic foundations.
 
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  • #32
One particularity of QM is that we are using infinite dimensional vector spaces there. One may thus argue that they are "computationally inaccessible". But irrational real numbers can be also argued to be "computationally inaccessible". Moreover, the essential features of QM are present already in finite spin systems - no need of infinite dimensional spaces.
 
  • #33
To be honest, I don't understand upisoft's point anymore, but let's put it this way.

Suppose you proved something using a quantum computer that you couldn't with a normal computer. So what is this proof? Does it just use a sequence of logical steps to arrive at its conclusion? Yes? Well then it could have been acheived with regular logic. No? Then you must agree that you have invented some new form of logic that I assume isn't permissible. If it is, then you can run it through with the usual techniques.

Are you trying to say that quantum computers can use some sort of logic system that we don't understand?
 
  • #34
Jamma said:
Oh, and btw, from wikipedia:

"A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem."

So I was completely right, you can simulate a quantum computer with a regular computer, and Godel's theorem still holds, not that it wasn't obvious anyway...
Ah, wikipedia. It must be true if it is written in wikipedia. Show me Turing machine that will generate randomly 1 or -1 with 50/50 chance every time it is reset and started again.
(Or first-order logic that will give random answer true or false with 50/50 every time you follow the same proof).

Hurkyl said:
:confused: Bell's theorem doesn't even vaguely resemble the conclusion you are trying to draw.

No, Gödel's work is based on first-order logic.

Your comment is additionally mystifying, because vector space admits set-theoretic foundations.
Bell's inequality is based on the properties of sets. Gödel's theorem is about "theories" which are nothing more than set of statements.

In short set theories assume that if you have full knowledge of the set you also know its elements. 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. That is not rue for all the states the electrons can be. If you know the full state of the system and the state of each of the electrons, then they are not entangled.
 
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  • #35
Upisoft said:
Show me Turing machine that will generate randomly 1 or -1 with 50/50 chance every time it is reset and started again.

Please, define "randomly" and "50/50 chance every time". But be precise, very precise.
 
  • #36
arkajad said:
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.
 
  • #37
Upisoft said:
"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.
 
  • #38
arkajad said:
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?
 
  • #39
Upisoft said:
Did you know the exact state of your computer?
Do you know the exact state of anything?
 
  • #40
arkajad said:
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.
 
  • #41
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".
 
  • #42
arkajad said:
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.
 
  • #43
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.
 
  • #44
arkajad said:
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.
 
  • #45
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.
 
  • #46
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.
 
  • #47
arkajad said:
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?
 
  • #48
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?
 
  • #49
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?!
 
  • #50
arkajad said:
You can create any machine you wish.
Thanks. Then I can create thinking machine.

arkajad said:
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?
 
  • #51
Jamma said:
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.
 
  • #52
Upisoft said:
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 definite laws that describe quantum theory? I don't think that we ever have or ever will.
 
  • #53
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.
 
  • #54
Jamma said:
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 definite 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.
 
  • #55
JDStupi said:
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?
 
  • #56
Upisoft said:
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!
 
  • #57
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.
 
  • #58
Jamma said:
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.
 
  • #59
JDStupi said:
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.
 
  • #60
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?
 
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