The Limits of Understanding

[Native]

Take a much needed “corona-hysteria-break” and listen to this video

Abstract: In 1931, Austrian logician Kurt Gödel shocked the worlds of mathematics and philosophy by establishing that statements are far more than a quirky turn of language: He showed that there are mathematical truths which simply can’t be proven. (Gödel's Incompleteness Theorems)

In the decades since, thinkers have taken the brilliant Gödel’s result in a variety of directions, linking it to limits of human comprehension and the quest to recreate human thinking on a computer.

This program explores Gödel’s discovery and examines the wider implications of his revolutionary finding. Participants include mathematician Gregory Chaitin, author Rebecca Goldstein, astrophysicist Mario Livio and artificial intelligence expert Marvin Minsky.

MODERATOR: Paul Nurse

PARTICIPANTS: Gregory Chaitin, Mario Livio, Marvin Minsky, Rebecca Newberger Goldstein.

Video timestamps:
Paul Nurse's Introduction. 00:19
Who is Kurt Godel? 03:36
Participant Introductions. 07:22
What was the intellectual environment Godel was living in? 10:57
Godel's beliefs in Platonism. 19:45
Gregory Chaitin on the incompleteness theorem. 22:30
Platonism vs. Formalism. 27:18
The unreasonable effectiveness of mathematics in the world. 40:53
The world is built out of mathematics... what else would you make it out of? 47:44
Mathematics and consciousness. 53:29
What are the problems of building a machine that has consciousness? 01:01:09
If math isn't a formal system then what is it? 01:07:40
Explaining math with simple computer programs. 01:18:33
Its hard to find good math. 01:25:40
--------------
What are your thoughts of these contents? Did they challenge your standing points of views?

 

[shunyadragon]

It is apparent that Godel did not shock the world of math and philosophy, but he did offer a challenge to math to solve Gödel's Incompleteness Theorems. For math to be an effective and useful 'tool box' for science with many provable theorems it is not necessary to be able to prove all proposed theorems. As an exercise in the problems of proving enigmatic theorems ok, but beyond that it is trivial pursuit in math. Math itself only becomes 'real' when it can be applied as an effective tool in science, technology and everyday life. Coming up with unsolvable theorems does not 'shake the world.'

This site put's Gödel's Incompleteness Theorems in proper perspective in simple understandable terms.

Gödel’s Incompleteness Theorems

Incompleteness Theorems

"In the last couple of posts, we’ve talked about what math is (a search for what must be) and where the foundational axioms and definitions come from.

Mathematics tries to prove that statements are true or false based on these axioms and definitions, but sometimes the axioms prove insufficient. Sometimes the axioms lead to paradoxes, like Russell’s paradox, and so a new set of axioms are needed. Sometimes the axioms simply aren’t enough, and so a new axiom might be needed to prove a desired result.

But in both cases, the paradoxes and inability to prove a result are the result of picking the wrong axioms.

Gödel’s incompleteness theorems show that pretty much any logical system either has contradictions, or statements that cannot be proven!

The questions Gödel was trying to answer were, “Can I prove that math is consistent?” and, “If I have a true statement, can I prove that it’s true?”

Gödel’s first step in this project was in his PhD thesis. That result seems to imply that you can prove any true statement. This is called Gödel’s completeness theorem.

For a particular set of axioms, there are different “models” implementing those axioms. A model is an example of something that satisfies those axioms.

As a non-mathematical example of a model, let’s say the axioms that define being a “car” are that you have at least 3 wheels, along with at least one engine that rotates at least one of the wheels. A standard car clearly follows those axioms, and is therefore a model for the “car axioms.” A bus would also be a model for the car axioms.

Of course, there are models that are very non-standard…

Mathematical axioms work the same way. There are axioms for the natural numbers, and their addition and multiplication, called “Peano arithmetic” (pay-AH-no). The normal natural numbers,

follow these axioms, so are the standard model for them. But there are non-standard models that still follow the Peano arithmetic axioms.

Each model is a bit different. There may be some statements (theorems) that are true in some of the models, but not true in another model.

Even if a statement is true, though, you want to be able to prove it true, using only the axioms that your model satisfies.2

Gödel’s completeness theorem answers the question, “Using the axioms, is it always possible to prove true statements are true?”

His completeness theorem says you can prove a statement is true using your chosen axioms if and only if that statement is true in all possible models of those axioms.
. . . . .
As long as your mathematics is complicated enough to include the natural numbers (which, I think we can agree, is not a particularly high bar), then it must have statements which cannot be proven true or false. They are unprovable.

Of course, to “fix” this you could try to add that statement as an axiom.8

Then, since the statement is an axiom, it is trivially provable. (The proof is: “This statement is an axiom. Thus it is true.”)."


Partially cited the source, read the whole thing to get a better understanding.

 

[Aupmanyav]

Gödel's incompleteness theorems
Gödel machine
Gödel's completeness theorem
Gödel's speed-up theorem

Gödel is beyond my capability. I will leave it to Shunya who has given a nice reply and linked a nice article. Polymath too will understand it but I do not know if he would like to participate. Because even if he does you would not be able to understand the whole of it, unless you yourself are a mathematician, and in any case you would not accept it. Many scientists criticize it for various reasons.

 

[Polymath257]

It should also be pointed out that Godel's Incompleteness Theorem is a result about first order logic.

Once you get to second order logic, it fails and there is only one model of the natural numbers.

The difference between first order and second order is, in essence, whether you get to talk about all properties of an object as well as all the objects.

 

[Polymath257]

Gödel's incompleteness theorems
Gödel machine
Gödel's completeness theorem
Gödel's speed-up theorem

Gödel is beyond my capability. I will leave it to Shunya or Polymath. Many scientists criticize it for various reasons.

It is a rock solid result *in math*. It says very little about anything other than abstract math, though, because most other thought systems are not axiomatic.

 

[Heyo]

It is apparent that Godel did not shock the world of math and philosophy, but he did offer a challenge to math to solve Gödel's Incompleteness Theorems. For math to be an effective and useful 'tool box' for science with many provable theorems it is not necessary to be able to prove all proposed theorems. As an exercise in the problems of proving enigmatic theorems ok, but beyond that it is trivial pursuit in math. Math itself only becomes 'real' when it can be applied as an effective tool in science, technology and everyday life. Coming up with unsolvable theorems does not 'shake the world.'

This site put's Gödel's Incompleteness Theorems in proper perspective in simple understandable terms.

Gödel’s Incompleteness Theorems

Incompleteness Theorems

"In the last couple of posts, we’ve talked about what math is (a search for what must be) and where the foundational axioms and definitions come from.

Mathematics tries to prove that statements are true or false based on these axioms and definitions, but sometimes the axioms prove insufficient. Sometimes the axioms lead to paradoxes, like Russell’s paradox, and so a new set of axioms are needed. Sometimes the axioms simply aren’t enough, and so a new axiom might be needed to prove a desired result.

But in both cases, the paradoxes and inability to prove a result are the result of picking the wrong axioms.

Gödel’s incompleteness theorems show that pretty much any logical system either has contradictions, or statements that cannot be proven!

The questions Gödel was trying to answer were, “Can I prove that math is consistent?” and, “If I have a true statement, can I prove that it’s true?”

Gödel’s first step in this project was in his PhD thesis. That result seems to imply that you can prove any true statement. This is called Gödel’s completeness theorem.

For a particular set of axioms, there are different “models” implementing those axioms. A model is an example of something that satisfies those axioms.

As a non-mathematical example of a model, let’s say the axioms that define being a “car” are that you have at least 3 wheels, along with at least one engine that rotates at least one of the wheels. A standard car clearly follows those axioms, and is therefore a model for the “car axioms.” A bus would also be a model for the car axioms.

Of course, there are models that are very non-standard…

Mathematical axioms work the same way. There are axioms for the natural numbers, and their addition and multiplication, called “Peano arithmetic” (pay-AH-no). The normal natural numbers,

follow these axioms, so are the standard model for them. But there are non-standard models that still follow the Peano arithmetic axioms.

Each model is a bit different. There may be some statements (theorems) that are true in some of the models, but not true in another model.

Even if a statement is true, though, you want to be able to prove it true, using only the axioms that your model satisfies.2

Gödel’s completeness theorem answers the question, “Using the axioms, is it always possible to prove true statements are true?”

His completeness theorem says you can prove a statement is true using your chosen axioms if and only if that statement is true in all possible models of those axioms.
. . . . .
As long as your mathematics is complicated enough to include the natural numbers (which, I think we can agree, is not a particularly high bar), then it must have statements which cannot be proven true or false. They are unprovable.

Of course, to “fix” this you could try to add that statement as an axiom.8

Then, since the statement is an axiom, it is trivially provable. (The proof is: “This statement is an axiom. Thus it is true.”)."


Partially cited the source, read the whole thing to get a better understanding.

A good overview but slightly misleading/incorrect.

The main reason why some systems are incomplete (and the basis on which the Incompleteness Theorem is proved) is because they are powerful.
Powerful in this context means that the system is complex enough to be able to "speak" about itself.
And Gödel proved that every system that is powerful is incomplete.
(And that can't be mended by adding an axiom, because it is still a powerful system.)

And yes, it was a blow to the idea of perfect knowledge. Together with Russel's Paradox in Set Theory, Heisenberg's Unschärfe Relation in Physics and Turing's Halting Problem in Computer Science, all in the first half of the 20th century, it showed the limits of knowledge.

 

[Polymath257]

It is apparent that Godel did not shock the world of math and philosophy, but he did offer a challenge to math to solve Gödel's Incompleteness Theorems.

On the contrary, Godel's results were *deeply* shocking.

They showed that there is no 'fix' to the incompleteness of any set of axioms strong enough to talk about the natural numbers.

ANY axiom system you choose will have statements that can neither be proved nor disproved.

This was directly against the goals of the Hilbert program, which sought to find a set of axioms for math *and* show them to be consistent. Godel showed that both goals are impossible to achieve.

it is not simply a matter of proving the Godel statements. Even if you add them to the collection of axioms, it is guaranteed there will be *new* Godel statements for the new system. As long as the axioms are added one by one, or even in some systematic way, this is an inevitable fact of life in mathematics.

 

[shunyadragon]

A good overview but slightly misleading/incorrect.

The main reason why some systems are incomplete (and the basis on which the Incompleteness Theorem is proved) is because they are powerful.
Powerful in this context means that the system is complex enough to be able to "speak" about itself.
And Gödel proved that every system that is powerful is incomplete.
(And that can't be mended by adding an axiom, because it is still a powerful system.)

And yes, it was a blow to the idea of perfect knowledge. Together with Russel's Paradox in Set Theory, Heisenberg's Unschärfe Relation in Physics and Turing's Halting Problem in Computer Science, all in the first half of the 20th century, it showed the limits of knowledge.

I disagree, I believe it was always idealistic in math to believe everything could be proved, or any such thing as perfect knowledge as far as science and math. I consider Gödel's Incompleteness Theorems not particularly useful, and more like trivial pursuit in math. The usefulness of math in science is the most important issue.

You need to be familiar with the purpose of the author this thread @Native. His only purpose is to sensationalize the uncertain in science and question the voracity of science. In the history of science the usefulness of theorems and the derived math is far more important than proving theorems in any sense of finality.

There is nothing sensational concerning Gödel's Incompleteness Theorems.

 

[Polymath257]

I disagree, I believe it was always idealistic in math to believe everything could be proved, or any such thing as perfect knowledge as far as science and math. I consider Gödel's Incompleteness Theorems not particularly useful, and more like trivial pursuit in math. The usefulness of math in science is the most important issue.

That depends. For *mathematicians*, the usefulness outside of math is of less importance. And, internal to math, the fact that there is no way to 'complete' the axiom system in an effective way is a bit startling. There was always an assumption, however flawed, that it would be possible to prove math to be consistent and that, with the right set of assumptions, it would be possible to answer all questions eventually.

This became even more of a goal when it was discovered that algebra and geometry, those subjects long separated, could be united. Then, the discovery of non-Euclidean geometries, and different types of number systems, all the while done in a common context, made the goal seem that much more reachable.

Furthermore, there *are* smaller parts of mathematics that *can* be shown to be complete. The 'real closed fields' are an example. And there is a decision procedure to determine the truth or falsity of any statement in those systems. Again, this supported the possibility that the same could be done for math as a whole.

You need to be familiar with the purpose of the author this thread @Native. His only purpose is to sensationalize the uncertain in science and question the voracity of science. In the history of science the usefulness of theorems and the derived math is far more important than proving theorems in any sense of finality.

Rather a matter of taste, I would say. But yes, your point that Godel's results are pretty much irrelevant to science is well taken.

There is nothing sensational concerning Gödel's Incompleteness Theorems.

I would say that David Hilbert would have seen Godel's results as sensational. So did many mathematicians and logicians of that time period. of course, there are other sensational results as well. The Lowenheim-Skolem paradox is a very good one. Also, Godel's results have lead to the development of a very large area of mathematics, so saying they are nothing special seems, well, rather biased against pure math.

 

[shunyadragon]

That depends. For *mathematicians*, the usefulness outside of math is of less importance. And, internal to math, the fact that there is no way to 'complete' the axiom system in an effective way is a bit startling. There was always an assumption, however flawed, that it would be possible to prove math to be consistent and that, with the right set of assumptions, it would be possible to answer all questions eventually.

This became even more of a goal when it was discovered that algebra and geometry, those subjects long separated, could be united. Then, the discovery of non-Euclidean geometries, and different types of number systems, all the while done in a common context, made the goal seem that much more reachable.

Furthermore, there *are* smaller parts of mathematics that *can* be shown to be complete. The 'real closed fields' are an example. And there is a decision procedure to determine the truth or falsity of any statement in those systems. Again, this supported the possibility that the same could be done for math as a whole.

Rather a matter of taste, I would say. But yes, your point that Godel's results are pretty much irrelevant to science is well taken.

I would say that David Hilbert would have seen Godel's results as sensational. So did many mathematicians and logicians of that time period. of course, there are other sensational results as well. The Lowenheim-Skolem paradox is a very good one. Also, Godel's results have lead to the development of a very large area of mathematics, so saying they are nothing special seems, well, rather biased against pure math.

I agree and disagree with much of the above, but in the context of the thread and the thread's author @Native the issue as to what is the nature of 'pure math' is not particularly relevant. I have no problems with 'pure math' playing 'pure math' among mathematicians.

The topic is the The 'Limits of Understanding.' There is nothing sensational about Gödel's Incompleteness Theorems, and are not representative of an argument for what are the limits of science and math.

 

[Polymath257]

I agree and disagree with much of the above, but in the context of the thread and the thread's author @Native the issue as to what is the nature of 'pure math' is not particularly relevant. I have no problems with 'pure math' playing 'pure math' among mathematicians.

The topic is the The 'Limits of Understanding.' There is nothing sensational about Gödel's Incompleteness Theorems, and are not representative of an argument for what are the limits of science and math.

And I would say that they *are* a limit to what math can say. We know of quite a few independent statements in math: they can be neither proved nor disproved from the axioms. Many of them are even interesting questions. But, from the math we have right now, there is no way to answer those questions. Any answer *has* to come from beyond our current assumptions.

But, none of them is going to impact science, in all likelihood. The issues are just too technical and outside of the possibility of modelling in the real world. The unknowability of the Continuum Hypothesis isn't going to make the Electric Universe suddenly plausible or not.

I would say that Godel's results are 'sensational' and deep in a very important way. They are also philosophically important, although often misunderstood. They *do* provide a limit to the types of things that can be known in mathematics.

For example, and simply, we cannot know our system is even consistent. That is a huge limitation to what we can know!

 

[shunyadragon]

And I would say that they *are* a limit to what math can say. We know of quite a few independent statements in math: they can be neither proved nor disproved from the axioms. Many of them are even interesting questions. But, from the math we have right now, there is no way to answer those questions. Any answer *has* to come from beyond our current assumptions.

I am not sure where this is going. I agree with this, but not in the context of what @Native is proposing in his thread.

But, none of them is going to impact science, in all likelihood. The issues are just too technical and outside of the possibility of modelling in the real world. The unknowability of the Continuum Hypothesis isn't going to make the Electric Universe suddenly plausible or not.

I would say that Godel's results are 'sensational' and deep in a very important way. They are also philosophically important, although often misunderstood. They *do* provide a limit to the types of things that can be known in mathematics.

Disagree, sensational is genuinely a hyperbolic layman's term in the context of math and application to science. I prefer the much simpler explanation I provided bring things down to earth in a more real way.

For example, and simply, we cannot know our system is even consistent. That is a huge limitation to what we can know!

Very hypothetical and 'arguing from ignorance. I could agree with this in some 'out there' view of the greater cosmos, but not particularly realistic concerning this thread.

 

[Polymath257]

Very hypothetical and 'arguing from ignorance. I could agree with this in some 'out there' view of the greater cosmos, but not particularly realistic concerning this thread.

No, that is the point of Godel's results. We cannot know our system is consistent. Any proof of consistency would show the system is inconsistent.

 

[shunyadragon]

No, that is the point of Godel's results. We cannot know our system is consistent. Any proof of consistency would show the system is inconsistent.

Know the system is consistent?!?!?! What a laugh! Your still dabbling in the hypothetical trivia and 'arguing from ignorance' with no constructive purpose. That is not the point of my posts from the beginning, and it is not the issue. The issue is math theorems and the resulting math are functional, consistent and predictable as the 'tool box' of science, and in of themselves DO NOT limit science.

Thou doest protest too much!

 

[Polymath257]

Know the system is consistent?!?!?! What a laugh! Your still dabbling in the hypothetical trivia and 'arguing from ignorance' with no constructive purpose. That is not the point of my posts from the beginning, and it is not the issue. The issue is math theorems and the resulting math are functional, consistent and predictable as the 'tool box' of science, and in of themselves DO NOT limit science.

Thou doest protest too much!

But the point is that we *know* we *cannot* prove the consistency.

There are systems we know are consistent. But for anything strong enough to talk about the natural numbers, the consistency is *guaranteed* to be unprovable. it isn't just ignorance. It is provable that it cannot be proved.

 

[shunyadragon]

But the point is that we *know* we *cannot* prove the consistency.

So what?!?!?! You persist in hiding behind the fallacy of 'arguing from ignorance' like a broken record not presenting anything meaningful.

There are systems we know are consistent. But for anything strong enough to talk about the natural numbers, the consistency is *guaranteed* to be unprovable. it isn't just ignorance. It is provable that it cannot be proved.

Again . . . So what?!?!?! You persist in hiding behind the fallacy of 'arguing from ignorance' like a broken record not presenting anything meaningful.

We do not need to prove anything.

 

[Heyo]

You need to be familiar with the purpose of the author this thread @Native. His only purpose is to sensationalize the uncertain in science and question the voracity of science. In the history of science the usefulness of theorems and the derived math is far more important than proving theorems in any sense of finality.

I don't care if someone uses uncertainty in science as an argument as long as s/he presents it correctly. Science has always been and should always be only loyal to truth and increasing knowledge.
@Native (who has been suspiciously silent since starting the thread) will find that the truth is not very useful as propaganda material once he understands it.

 

[Native]

You need to be familiar with the purpose of the author this thread @Native. His only purpose is to sensationalize the uncertain in science and question the voracity of science.

If asking questions into the standing cosmological science make you feel so, it´s OK by me.

The entire video content was about uncertainties at large as in the OP and video title: "The Limits of Understanding", discussing both math and philosophical issues.

I just asked:

What are your thoughts of these contents? Did they challenge your standing points of views?

 

[Native]

Gödel’s incompleteness theorems show that pretty much any logical system either has contradictions, or statements that cannot be proven!

(If so, it isn´t even logical, is it?) Can you give an example of this?

 

[Native]

That depends. For *mathematicians*, the usefulness outside of math is of less importance. And, internal to math, the fact that there is no way to 'complete' the axiom system in an effective way is a bit startling. There was always an assumption, however flawed, that it would be possible to prove math to be consistent and that, with the right set of assumptions, it would be possible to answer all questions eventually.

Is that what goes on when theoretical astrophycisists calculate this and that in cosmos?