Godel

[atanu]

Godel drew the following implication from his own Incompleteness theorem:

either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.

Gödel's Incompleteness Theorems (Stanford Encyclopedia of Philosophy)

...

Can someone help to explain?

 

[Lunarious]

Show us your thoughts about this so far.

 

[Polymath257]

Godel drew the following implication from his own Incompleteness theorem:

either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.

Gödel's Incompleteness Theorems (Stanford Encyclopedia of Philosophy)

...

Can someone help to explain?

Hoo boy. bad philosophy concerning something most philosophers don't understand very well. OK.

Godel's theorem says that any formal system strong enough to construct the natural numbers and that is 'recursive' and consistent, there is a statement that cannot be proven but that is true in a chosen model.

The argument goes as follows: any finite machine would be a formal system of the type above and so would have a statement that is known to be true, but unprovable. Since the human mind can tell this statement is true, but the finite machine cannot (because the statement isn't provable), the human mind cannot be that finite machine.

(The diophantine problems are a slightly different issue)

Well, the problem is that we humans do NOT have a way to determine when a formal system is consistent. And Godel even showed that such is impossible! Also, it is a rare human mind that is completely consistent. hence, Godel's theorem does not apply to human minds and the whole argument falls apart. THis doesn't even address whether the system is recursive, which is another kettle of fish.

Edit: Diophantine problems are certain problems in number theory. Godel claimed that there cannot be absolutely unsolvable problems of that sort. He never substantiated that claim, though.

 

[David T]

Hoo boy. bad philosophy concerning something most philosophers don't understand very well. OK.

Godel's theorem says that any formal system strong enough to construct the natural numbers and that is 'recursive' and consistent, there is a statement that cannot be proven but that is true in a chosen model.

The argument goes as follows: any finite machine would be a formal system of the type above and so would have a statement that is known to be true, but unprovable. Since the human mind can tell this statement is true, but the finite machine cannot (because the statement isn't provable), the human mind cannot be that finite machine.

(The diophantine problems are a slightly different issue)

Well, the problem is that we humans do NOT have a way to determine when a formal system is consistent. And Godel even showed that such is impossible! Also, it is a rare human mind that is completely consistent. hence, Godel's theorem does not apply to human minds and the whole argument falls apart. THis doesn't even address whether the system is recursive, which is another kettle of fish.

Edit: Diophantine problems are certain problems in number theory. Godel claimed that there cannot be absolutely unsolvable problems of that sort. He never substantiated that claim, though.

I would say poly you laid that out very very well. Cantor, Godel, and Nietzsche are three individuals I have thought a lot about for all the same reason. They share a simular neurology, one that i happen to be familiar with.

 

[atanu]

Hoo boy. ...
Well, the problem is that we humans do NOT have a way to determine when a formal system is consistent. And Godel even showed that such is impossible! ....

Well. No doubt you laid it very well. I have couple of points. May be it is my lack of understanding of your presentation.

You say that Godel showed that we humans do NOT have a way to determine when a formal system is consistent. That should be sufficient for me.

And a physicalist is sure of consistency of his method -- his thought system. So, is the physicalist system of thought consistent or is it limited?

 

[Polymath257]

Well. No doubt you laid it very well. I have couple of points. May be it is my lack of understanding of your presentation.

You say that Godel showed that we humans do NOT have a way to determine when a formal system is consistent. That should be sufficient for me.

Well, one consequence of Godel's results is that there can be no Turing machine that determines consistency of a formal system.

But, more specifically, there cannot be a proof of even the consistency of the theory of natural numbers! Mathematics as we know it may well be logically inconsistent!

And a physicalist is sure of consistency of his method -- his thought system. So, is the physicalist system of thought consistent or is it limited?

No, certainly we do NOT know our thought processes are consistent! if anything, we often know they are not. But consistency of the formal system is an *assumption* in Godel's results. So those results simply do not apply to human minds.

 

[Jayhawker Soule]

A truly fun explanation and overview is presented in Hofstadter's Gödel, Escher, Bach.

 

[Polymath257]

A truly fun explanation and overview is presented in Hofstadter's Gödel, Escher, Bach.

Agreed! An absolutely wonderful exposition!

That was a very important book in my development!

 

[Jayhawker Soule]

Agreed! An absolutely wonderful exposition!

That was a very important book in my development!

mu

 

[atanu]

No, certainly we do NOT know our thought processes are consistent! if anything, we often know they are not. But consistency of the formal system is an *assumption* in Godel's results. So those results simply do not apply to human minds.

Human mind is unknown (at least to me). And I think that Godel very well takes that into account.

 

[Subduction Zone]

mu

This can help:

Yaegaki Mu Junmai Daiginjo

 

[Polymath257]

Human mind is unknown (at least to me). And I think that Godel very well takes that into account.

Except that he shows there *cannot* be a proof of consistency.

 

[atanu]

Except that he shows there *cannot* be a proof of consistency.

Let me understand this better. Are you saying that the red part (below) cannot be and that the blue part holds?

either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.
...

Thanks in advance.

 

[Polymath257]

Let me understand this better. Are you saying that the red part (below) cannot be and that the blue part holds?

either … the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine, or else there exist absolutely unsolvable diophantine problems.
...

Thanks in advance.

Yes, I would say that the blue part holds.

 

[atanu]

Yes, I would say that the blue part holds.

What are the 'diophantine problems' in the blue part?