The logic of Kurt Godel

Forums: Philosophy
Email this Topic • Print this Page

In one salvo, it is said, he completely demolished an entire class of scientific theories.

Mathematicians love proofs. They were hot and bothered for centuries, because they were unable to PROVE some of the things they knew were true.

So for example if you studied high school Geometry, you've done the exercises where you prove all kinds of things about triangles based on a set of theorems.

That high school geometry book is built on Euclid's five postulates. Everyone knows the postulates are true, but in 2500 years nobody's figured out a way to prove them.

Yes, it does seem perfectly "obvious" that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that Euclid's postulates are a reasonable, and in fact necessary, set of 5 assumptions.

Towering mathematical geniuses were frustrated for 2000+ years because they couldn't't prove all their theorems. There were so many things that were "obviously true," but nobody could find a way to prove them.

In the early 1900's, however, a tremendous wave of optimism swept through mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) became convinced that they were rapidly closing in on a final synthesis.

A unifying "Theory of Everything" that would finally nail down all the loose ends. Mathematics would be complete, bulletproof, airtight, triumphant.

[CENTER]"Anything you can draw a circle around cannot explain itself without referring to something outside the circle - something you have to assume but cannot prove."[/CENTER]

You can draw a circle around all of the concepts in your high school geometry book. But they're all built on Euclid's 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.

Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it's equally true in science or language and philosophy.

[CENTER]"I am lying."[/CENTER]

"I am lying" is self-contradictory, since if it's true, I'm not a liar, and it's false; and if it's false, I am a liar, so it's true.

You always need an outside reference point.

There are more things that are true than you can prove.

A "theory of everything" - whether in math, or physics, or philosophy - will never be found. Because it is mathematically impossible. I suggest this TOE is just another word for God

Here's what it means:

Faith and Reason are not enemies. In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.
All closed systems depend on something outside the system.
You can always draw a bigger circle but there will still be something outside the circle.

Reasoning inward from a larger circle to a smaller circle (from "all things" to "some things") is deductive reasoning.

Example of a deductive reasoning:
1. All men are mortal
2. Socrates is a man
3. Therefore Socrates is mortal

Reasoning outward from a smaller circle to a larger circle (from "some things" to "all things") is inductive reasoning.

Examples of inductive reasoning:
1. All the men I know are mortal
2. Therefore all men are mortal
1. When I let go of objects, they fall
2. Therefore there is a law of gravity that governs all falling objects
Notice than when you move from the smaller circle to the larger circle, you have to make assumptions that you cannot 100% prove.

For example you cannot PROVE gravity will always be consistent at all times. You can only observe that it's consistently true every time.

Nearly all scientific laws are based on inductive reasoning. These laws rest on an assumption that the universe is orderly and based on fixed discoverable laws.

You cannot PROVE this. (You can't prove that the sun will come up tomorrow morning either.) You literally have to take it on faith. In fact most people don't know that outside the science circle is a philosophy circle. Science is based on philosophical assumptions that you cannot scientifically prove. Actually, the scientific method cannot prove, it can only infer.

(Science originally came from the idea that God made an orderly universe which obeys fixed, discoverable laws - and because of those laws, He would not have to constantly tinker with it in order for it to operate.)

Now please consider what happens when we draw the biggest circle possibly can - around the whole universe. (If there are multiple universes, we're drawing a circle around all of them too):

There has to be something outside that circle. Something which we have to assume but cannot prove
The universe as we know it is finite - finite matter, finite energy, finite space and 13.8 billion years time
The universe (all matter, energy, space and time) cannot explain itself
Whatever is outside the biggest circle is boundless. So by definition it is not possible to draw a circle around it.
It's immaterial.
Whatever is outside the biggest circle is not a system - i.e. is not an assemblage of parts. Otherwise we could draw a circle around them. The thing outside the biggest circle is indivisible.
Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect.

We can apply the same inductive reasoning to the origin of information:

In the history of the universe we also see the introduction of information, some 3.8 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.
The information had to come from the outside, since information is not known to be an inherent property of matter, energy, space or time.
All codes we know the origin of are designed by conscious beings.
Therefore whatever is outside the largest circle is a conscious being.

When we add information to the equation, we conclude that not only is the thing outside the biggest circle infinite and immaterial, it is also self-aware.
Isn't it interesting how all these conclusions sound suspiciously similar to how theologians have described God for thousands of years?

logical. In fact it's the only position one can take and stay in the realm of reason and logic.

"Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it."

Therefore does Atheism violate the laws mathematics.?

Source

Extract of mine (Alan McDougall) and Perry Martins writings

What does the forum think?

Topic Stats
Top Replies
Link to this Topic

Type: Discussion • Score: 0 • Views: 3,888 • Replies: 60

No top replies

@Alan McDougall,

Quote:

I think this point is wrong.

First of all, Goedel's idea works only for special systems and not for every system.

Twice, we do not know whether our "human logic" describes the universe perfectly.

And thirdly, the fact that a system cannot "explain" itself does not proof that it cannot exist out of itself and needs a "god".

1 Reply

@Habek,

Habek;114841 wrote:

I think this point is wrong.

First of all, Goedel's idea works only for special systems and not for every system.

Twice, we do not know whether our "human logic" describes the universe perfectly.

And thirdly, the fact that a system cannot "explain" itself does not proof that it cannot exist out of itself and needs a "god".

1 Reply

@Alan McDougall,

I thought that what Godel had shown was that brains are useful.

1 Reply

@kennethamy,

kennethamy;114846 wrote:

I thought that what Godel had shown was that brains are useful.

True! But his reasoning was there would always be something outside and beyond human knowledge, call it God if you like. The "Uncaused Cause" at that.

1 Reply

@Alan McDougall,

Alan McDougall;114853 wrote:

True! But his reasoning was there would always be something outside and beyond human knowledge, call it God if you like. The "Uncaused Cause" at that.

I didn't get that from him. Where did you find that?

---------- Post added 12-28-2009 at 08:54 AM ----------

By the way, every statement proves its own truth, since every statement follows from itself. So Godel could not have proved that no statement proves its own truth, as you write he does.

1 Reply

@kennethamy,

kennethamy;114855 wrote:

I didn't get that from him. Where did you find that?

---------- Post added 12-28-2009 at 08:54 AM ----------

By the way, every statement proves its own truth, since every statement follows from itself. So Godel could not have proved that no statement proves its own truth, as you write he does.

Here

[CENTER]"Anything you can draw a circle around cannot explain itself without referring to something outside the circle - something you have to assume but cannot prove."[/CENTER]

You can draw a circle around all of the concepts in your high school geometry book. But they're all built on Euclid's 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.

Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it's equally true in science or language and philosophy.

1 Reply

@Alan McDougall,

Alan McDougall;114869 wrote:

Here

[CENTER]"Anything you can draw a circle around cannot explain itself without referring to something outside the circle - something you have to assume but cannot prove."[/CENTER]

You can draw a circle around all of the concepts in your high school geometry book. But they're all built on Euclid's 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.

Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it's equally true in science or language and philosophy.

What has any of that to do with what I wrote? Every statement follows from itself. That is a logical truth. And what has any of that to do with God and the rest? Godel was a paranoid, and mystical. But he was not stupid.

1 Reply

@Alan McDougall,

Alan McDougall wrote:

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

I'm not quite sure this is the same as mathematical axiom truths, which Godel was mostly referring to. The axiom is a self-evident truth, a basis for inferring other truths. But is a bike the same sort of thing? It doesn't seem so. What is a bicycle trying to prove, or is the basis of? More importantly, I'd like to know what you mean by "You can draw a circle around a bicycle". How would I go about this, and why would I do such a thing?

But what is your point? That truth is unknowable because we have to rely on other truths to ascertain those truths? Why?

1 Reply

@Zetherin,

Zetherin;114875 wrote:

But what is your point?

Well might you ask.

2 Replies

@kennethamy,

kennethamy;114879 wrote:

Well might you ask.

Some truths are unknowable and some will remain always unknowable to puny mortal man

1 Reply

@Alan McDougall,

Alan McDougall;114885 wrote:

Some truths are unknowable and some will remain always unknowable to puny mortal man

Is that your point? Maybe you are right. However, which truths those are, we can never know either. So, for any individual truth, we had better assume it is knowable.

1 Reply

@kennethamy,

kennethamy;114886 wrote:

Is that your point? Maybe you are right. However, which truths those are, we can never know either. So, for any individual truth, we had better assume it is knowable.

Take the concept of infinity the ultimate oxymoron an impossible/truth

1 Reply

@Alan McDougall,

Alan McDougall;114888 wrote:

Take the concept of infinity the ultimate oxymoron an impossible/truth

A concept cannot be a truth or a falsity. Perhaps you mean that there is something finite. But mathematicians have been studying not only infinities, but transinfinities for over a 150 years now. Read about Georg Cantor's work, which started an entirely new field in mathematics.

Georg Cantor - Wikipedia, the free encyclopedia

"There are more things in heaven and earth than are dreamed of in your philosophy, Horatio" (Shakespeare. Hamlet )

1 Reply

@kennethamy,

kennethamy;114889 wrote:

A concept cannot be a truth or a falsity. Perhaps you mean that there is something finite. But mathematicians have been studying not only infinities, but transinfinities for over a 150 years now. Read about Georg Cantor's work, which started an entirely new field in mathematics.

Georg Cantor - Wikipedia, the free encyclopedia

"There are more things in heaven and earth than are dreamed of in your philosophy, Horatio" (Shakespeare. Hamlet )

Thanks for the link can I add a Little more?

Nothing is as it seems to be and all things are subjective realties to the observer. Everything is relative to each person from the viewpoint of the only ultimate reality the First Cause, Ultimate, Divine Mind. or god if you like. Is there an ultimate objective observer a knower of all things?

We are subjectively limited!

1 Reply

@Alan McDougall,

Alan McDougall;114891 wrote:

Thanks for the link can I add a Little more?

Nothing is as it seems to be and all things are subjective realties to the observer. Everything is relative to each person from the viewpoint of the only ultimate reality the First Cause, Ultimate, Divine Mind. or god if you like. Is there an ultimate objective observer a knower of all things?

We are subjectively limited!

As I said, there may be somethings we'll never know, but which they are, we'll never know.

1 Reply

@kennethamy,

kennethamy;114896 wrote:

As I said, there may be somethings we'll never know, but which they are, we'll never know.

So what's your point? We will never know if there are things that we'll never know, so what you have said is pointless drivel to keep pushing a discussion nowhere while spinning the tires in place.

1 Reply

@Alan McDougall,

Alan McDougall;114842 wrote:

That was not his point. His point was that mathematics is not a logical tautology.

Anything beyond that is an independent speculation. He was not making any kind of statement about the human intellectual capacity or about "creation" in general.

0 Replies

@Theaetetus,

Theaetetus;114899 wrote:

So what's your point? We will never know if there are things that we'll never know, so what you have said is pointless drivel to keep pushing a discussion nowhere while spinning the tires in place.

What I said (you really have to read more carefully) is that we will never know just what it is that we cannot know. And therefore (as I said in an earlier post) we should assume that we can know the answer to every sensible question unless we have some reason to think otherwise. We should never begin with the assumption that there is no way to answer some question. For we cannot know that. But such an assumption will "block inquiry" which, as Peirce said, we should never do. I did, not in fact, in fact, say what you said I said. But it is, I think, true that we'll never know whether there are things we'll never know.

---------- Post added 12-28-2009 at 01:27 PM ----------

Aedes;114902 wrote:

That was not his point. His point was that mathematics is not a logical tautology.

Anything beyond that is an independent speculation. He was not making any kind of statement about the human intellectual capacity or about "creation" in general.

His point was that mathematics is not a logical tautology.

It was? How do you figure that? And what does that mean? As I said, his point was that brains are useful, since for some systems, there is no machine-like decision method to prove all the wffs in the system. And, therefore, to prove some wffs, we need brains. Everything else is idle speculation.

1 Reply

@Alan McDougall,

kennethamy wrote:

It was? How do you figure that? And what does that mean? As I said, his point was that brains are useful, since for some systems, there is no machine-like decision method to prove all the wffs in the system. And, therefore, to prove some wffs, we need brains. Everything else is idle speculation.

But isn't that why he called it "incomplete"? Because some truths don't stand on anything but our self-evidentism (I made this word up). I think that is what Aedes meant. That the axioms to which much mathematics stands, are not tautologies; their truth is questionable, because there is no proof to back them.

But, I don't know why you think Godel's point was that brains are needed to prove some truths. Is that what you think he meant by this theorem? I thought his point was that some truths cannot be proven, and that our brains are not proof enough?

1 Reply

The logic of Kurt Godel

Related Topics

Quick Links

My Account

able2know