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

[ikos9lives]

The title for this thread is from an article by David P. Goldman in the August issue of First Things. It's about the religious beliefs of Kurt Godel, the most famous and probably the greatest mathematician of the 20th century, whose "incompleteness theorem" showed that algorithms will never replace intuition, i.e. it will be impossible to construct thinking machines from hardware/software. Here's a link to the article:

http://www.faqs.org/periodicals/201008/2080027241.html"

I'll quote from the article: "But Gödel's God is not the well-behaved deity of the old natural theology, or the happy harmonizer of the intelligent-design subculture. Gödel's God hides his countenance and can be glimpsed only in paradox and intuition. God is not an abstraction but "can act as a person," as Gödel once wrote, confronting those who seek him with paradox, uplifting man through glorious insights while guarding his infinitude from human grasp. Gödel's investigations in number theory and general relativity suggest a similar theological result: that God cannot be reduced to a mere principle of the natural world." How great!
Godel was also working on a revision of Anselm's ontological proof for God. He phrased his revision of Leibniz's version of the ontological proof in logical notation. To quote again from the article: "I (Goldman) doubt Godel believed he had found the ultimate and irrefutable proof of the existence of God. His deep interest in the ontological proof, rather was one facet of his commitment to defend Leibniz' theism against the new Spinozans of mathematics and physics." (emphasis added).
Worth reading, particularly for those open-minded agnostics/atheists with a mathematical background/interest.

 

[arkajad]

Quotes from „Reflections on Kurt Gödel” by Wang Hao, MIT Press (1987)

G attributes the prejudices against religion largely to the churches, one's early education, and also philosophy today. 'I believe that there is much more reason in religion, though not in the churches, than one commonly believes, but we (i.e., the middle layer of mankind, to which we belong, or at least most people in this layer) were brought up from early youth to a prejudgment against it through the school, the poor religious teaching, through books and experiences.' 'E.g. according to the Catholic dogma the most kind God has created the vast majority of mankind, namely all except the good Catholics, exclusively for the purpose of sending them to hell for all eternity.' Moreover, 'Ninety percent of philosophers today see their principal task in knocking religion out of people's head, thereby working for the same effects as the bad churches.'

From a letter to his mother:

We are of course far from being able to confirm scientifically the theological world picture, but, it might, I believe, already be possible today to perceive by pure reason (without appealing to the faith in any religion), that the theological worldview is thoroughly compatible with all known data (including the conditions which prevail on our earth). The famous philosopher and mathematician Leibniz already tried to do this 250 years ago, and this is also what I have tried to do in my previous letters. What I call the theological worldview is the idea, that the world and everything in it has meaning and reason, and in particular a good and indubitable meaning. It follows immediately that our worldly existence, since it has in itself at most a very dubious meaning, can only be means to the end of another existence. The idea that everything in the world has a meaning [reason] is an exact analogue of the principle that everything has a cause, on which rests all of science.

 

[Upisoft]

whose "incompleteness theorem" showed that algorithms will never replace intuition, i.e. it will be impossible to construct thinking machines from hardware/software. Here's a link to the article:

Quite far from the truth. Show me his work (and proof) on quantum computing and I'll believe you.

 

[arkajad]

Quite far from the truth. Show me his work (and proof) on quantum computing and I'll believe you.

Isn't quantum computing equivalent to classical computing plus Monte Carlo? I am not talking about speed here. I am talking about ultimate possibilities given whatever time you need.

 

[Upisoft]

Isn't quantum computing equivalent to classical computing plus Monte Carlo? I am not talking about speed here. I am talking about ultimate possibilities given whatever time you need.

No. The inability to copy quantum information is unique to it.

 

[arkajad]

You mean, you can't solve Schrodinger's equation using a classical computer? I thought you can...

 

[Upisoft]

You mean, you can't solve Schrodinger's equation using a classical computer? I thought you can...

No, I meant you cannot copy it. Nor you can know it without changing it.

 

[arkajad]

No, I meant you cannot copy it. Nor you can know it without changing it.

That has nothing to do whatsoever with the fact that quantum computers can be simulated on classical computers. And they can. And that's all that is needed to find out what they can do and what they can't.

 

[wuliheron]

That has nothing to do whatsoever with the fact that quantum computers can be simulated on classical computers. And they can. And that's all that is needed to find out what they can do and what they can't.

Quanta are contextual which means no classical computer can come close to simulating their behavior in real world situations.

 

[arkajad]

Question is: can quantum computer be - in principle simulated by a classical computer, or not? Since all quantum theory is contained in normal classical math, it evidently can.

When you add "real world situation" you probably mean things like time, resources etc. That has nothing to do with theorems telling us what is and what is not possible given whatever resources we need. Resources are important in technology - for sure. But they would not bother Godel also today. He was not concerned with what is technologically possible or not, but what is logically possible or not.

 

[wuliheron]

Question is: can quantum computer be - in principle simulated by a classical computer, or not? Since all quantum theory is contained in normal classical math, it evidently can.

When you add "real world situation" you probably mean things like time, resources etc. That has nothing to do with theorems telling us what is and what is not possible given whatever resources we need. Resources are important in technology - for sure. But they would not bother Godel also today. He was not concerned with what is technologically possible or not, but what is logically possible or not.

By "real world" I mean outside of laboratory experiments or extreme situations involving small numbers of isolated quanta. We can make simulations of small numbers of quanta that can do simple calculations, but a full fledged quantum computer involving many quanta would be beyond any practical means of simulating since its computational power increases exponentially with each new addition.

 

[arkajad]

... but a full fledged quantum computer involving many quanta would be beyond any practical means of simulating since its computational power increases exponentially with each new addition.

So it's all about technology, not about logic. Godel was a logician and logic was what he was concerned with. The same with his "time loops". He did not care whether technology will allow us to create such geometries or not. He was happy with them being logically possible solutions of Einstein's field equations. People did not like them, were looking for possible logical contradictions. But Godel was ahead of everybody intuitively knowing that what can lead to paradoxes classically may still happen using quantum theory.

And here we come to real problem: can quantum theory, even if it can be simulated with a classical computer, get us beyond our present understanding of what is possible and what not?

It seems that it may be the case - namely because of what we call now a "random factor" that is involved in all quantum computations. Just let some uncertainty, used in an appropriate way, about the results, and all the perspective may change.

 

[wuliheron]

So it's all about technology, not about logic. Godel was a logician and logic was what he was concerned with. The same with his "time loops". He did not care whether technology will allow us to create such geometries or not. He was happy with them being logically possible solutions of Einstein's field equations. People did not like them, were looking for possible logical contradictions. But Godel was ahead of everybody intuitively knowing that what can lead to paradoxes classically may still happen using quantum theory.

And here we come to real problem: can quantum theory, even if it can be simulated with a classical computer, get us beyond our present understanding of what is possible and what not?

It seems that it may be the case - namely because of what we call now a "random factor" that is involved in all quantum computations. Just let some uncertainty, used in an appropriate way, about the results, and all the perspective may change.

There is an popular ancient poem in the Tao Te Ching, that actually predates the text, that expresses this principle.

Tools

Thirty spokes meet at a nave;
Because of the hole we may use the wheel.
Clay is moulded into a vessel;
Because of the hollow we may use the cup.
Walls are built around a hearth;
Because of the doors we may use the house.
Thus tools come from what exists,
But use from what does not.

Note that in the poem what does not exist is relative to what does. The naive of the wheel may be filled with the axle, but wheel itself does not extend into the naive. Hence, as long as there is something new to discover that exists there will always be new uses found.

 

[Jamma]

Quite far from the truth. Show me his work (and proof) on quantum computing and I'll believe you.

You completely misunderstand this area.

Godel's proof shows that there are statements whose truth can never be determined by a finite number of logical steps.

Bringing up quantum computing has no relevance whatsoever. I don't know that much about the area, but it is completely obvious that to perform a computation they still work merely on logic.

And I do believe that you can simulate a quantum computer on a regular computer. Quantum computers can work efficiently by performing several calculations similtaneously which would not be possible on a regular computer. So this can be simulated, but it would just be horrenderously slow compared to a quantum computer. Besides, they still need to be programmed don't they? Do you think that they somehow "know" how to determine things on a level above to which we programmed them?

 

[Upisoft]

Godel's proof shows that there are statements whose truth can never be determined by a finite number of logical steps.

That is OK, but they say it is proof that we cannot construct thinking machines.

 

[Jamma]

Fair enough. To be honest though, I think that is a vast misuse of his theorem. Afterall, how do we know that we don't just operate by a series of logical steps and algorithms (operated chemically, of course)- what if we invented a computer which perfectly simulated the brain? Then this would have as much intuition as any other person...

 

[G037H3]

Fair enough. To be honest though, I think that is a vast misuse of his theorem. Afterall, how do we know that we don't just operate by a series of logical steps and algorithms (operated chemically, of course)- what if we invented a computer which perfectly simulated the brain? Then this would have as much intuition as any other person...

Is impossibru.

Quantum mind.

 

[Upisoft]

Question is: can quantum computer be - in principle simulated by a classical computer, or not? Since all quantum theory is contained in normal classical math, it evidently can.

No, it cannot. You can get fair approximation, but you cannot get exact result.

 

[arkajad]

No, it cannot. You can get fair approximation, but you cannot get exact result.

You cannot get exact result calculating planet's orbits with a computer. So this is not an argument.

 

[G037H3]

You cannot get exact result calculating planet's orbits with a computer. So this is not an argument.

create a perfect replication of my watch for me

make sure that it has all of the same microscopic scratches, or it won't be an exact copy

 

[Upisoft]

You cannot get exact result calculating planet's orbits with a computer. So this is not an argument.

But it is. Godel's work is exact proof. You cannot apply exact on approximation and claim exact result.

 

[Jamma]

But it is. Godel's work is exact proof. You cannot apply exact on approximation and claim exact result.

Ermm, what other kind of proof is there, unexact proof? Sounds helpful...

I really think that you are misunderstanding quantum computing. Yes, it uses quantum physics to operate, but that doesn't mean that it uses randomness to compute things that we cannot predict. How could you program such a thing? Instead, they run by being able to perform several calculations at the same time by exploiting not binary bits, but quantum bits, which can be viewed as representing both a 1 and a 0 or a mix of both at the same time. In this way, several calculations can be performed similtaneously, but that doesn't mean that the results aren't predictable (this is probably a horrible description for people that actually know a lot about quantum computing, but hopefully it gives the gist).

It is still a computer. It isn't an unexact object or it would be useless for computing. And they can be simulated with regular computers, but the regular computer wouldn't be able to perform a real-time simulation, instead it would have to compute which calculations the quantum computer would have been performing, perform them and then, after repeating this several times, give the result. Indeed, people are already writing programs that theoretical quantum computers could use to run. The key word is program, it is a set of instructions for the quantum computer, which it runs without invoking randomness, only pure logic, which is obviously what you want from a computer. In this way, Godel's work is obviously still valid, the only difference is in the efficient way that quantum computers perform their calculations.

 

[Upisoft]

Ermm, what other kind of proof is there, unexact proof? Sounds helpful...

The kind you use in the court...

I really think that you are misunderstanding quantum computing.

I think you misunderstood my point. Godel's work uses classical logic (i.e. set theory). Bell's theorem shows that QM cannot be explained by any set theory. Therefore the conclusions made by Godel do not necessary expand to the quantum world.

And they can be simulated with regular computers

Nope, classical computers can simulate quantum events only approximately. You can have exact simulation on classical computer only if the simulated object has finite number of states. If it doesn't, then you have to choose finite subspace and work with it. I.e. you have to make an approximation.

 

[Jamma]

The kind you use in the court...


I think you misunderstood my point. Godel's work uses classical logic (i.e. set theory). Bell's theorem shows that QM cannot be explained by any set theory. Therefore the conclusions made by Godel do not necessary expand to the quantum world.

Ok, the kind you use in court has nothing to do with this, we are talking about mathematical truth. You began by asserting that Godel's proof doesn't hold because of quantum computing. You explicitly said this, and you are clearly wrong. Godel's proof concerns mathematical truths, truths which can only be arrived at via logic and some initial axioms. Adding quantum computing into this doesn't help, that was my point- you seem to have now changed your argument to say that set theory cannot explain things that QM can, about the real world. Godel's proof is about mathematical statements, not the real world. To be honest, you are right, I'm not exactly sure what your point is anymore.

 

[Jamma]

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...

 

[arkajad]

But it is. Godel's work is exact proof. You cannot apply exact on approximation and claim exact result.

What Goedel's work has to do with quantum mechanics? Quantum mechanics is based on classical partial differential equations and the classical theory of random processes. That' all.

Emulating them with a computer is a different subject, but it is good to know that this can be done in principle with an arbitrary accuracy if neglect the question of resources in the physical universe. But I do not think that resources of our particular physical universe were a part of Kurt Godel's concern.

 

[Upisoft]

What Goedel's work has to do with quantum mechanics? Quantum mechanics is based on classical partial differential equations and the classical theory of random processes. That' all.

And non-classical complex vector spaces, but it does not matter, does it?

 

[arkajad]

And non-classical complex vector spaces, but it does not matter, does it?

N-dimensional complex vector space is 2N-dimensional real vector space (with some extra, but classical, algebraic structure). There is absolutely nothing "non-classical there". Complex vector spaces are routinely used in Fourier transforms on personal computers. They are fairly classical.

 

[Upisoft]

N-dimensional complex vector space is 2N-dimensional real vector space (with some extra, but classical, algebraic structure). There is absolutely nothing "non-classical there". Complex vector spaces are routinely used in Fourier transforms on personal computers. They are fairly classical.

Godel's work is set based, not vector space based.

 

[arkajad]

Godel's work is set based, not vector space based.

Vector spaces are sets.

 

[Hurkyl]

Bell's theorem shows that QM cannot be explained by any set theory.

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

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.

 

[arkajad]

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.

 

[Jamma]

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?

 

[Upisoft]

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).

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.

 

[arkajad]

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.