Faith in logic ?

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Faith in logic ?

Smiling

One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why


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With the disclaimer that I

With the disclaimer that I am not a mathemetician, I think this is the basic answer:

*Most people who have read Godel's theorem have grasped the implications correctly and understand that there are abstracts and infinites in formal systems, and these lead to at least one ultimately unanswerable question in any formalized proof.  This is somewhat similar to the fact that there are numbers which cannot be written because of their non-repeating, non-terminating status.  

This doesn't really have any effect on science or philosophy, and doesn't affect anyone other than mathemeticians in any meaningful way.

Perhaps one of you math geeks would like to correct me if I'm saying this incorrectly?

 

 

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Hambydammit wrote: With

Hambydammit wrote:

With the disclaimer that I am not a mathemetician, I think this is the basic answer:

*Most people who have read Godel's theorem have grasped the implications correctly and understand that there are abstracts and infinites in formal systems, and these lead to at least one ultimately unanswerable question in any formalized proof. This is somewhat similar to the fact that there are numbers which cannot be written because of their non-repeating, non-terminating status.

This doesn't really have any effect on science or philosophy, and doesn't affect anyone other than mathemeticians in any meaningful way.

Perhaps one of you math geeks would like to correct me if I'm saying this incorrectly?

 

 

 

Well, since the underlying math of computers is boolean logic, doesn't this also imply that there are things computer cannot in pirinciple do? 


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wavefreak wrote:

wavefreak wrote:

Well, since the underlying math of computers is boolean logic, doesn't this also imply that there are things computer cannot in pirinciple do?

Yes, computers cant represent all numbers correctly.

0.1 for instance is impossible to represent, as a binary number, as it would require an infinite amount of bits.

Many calculations can turn out with errors for this and other reasons. Figuring out how to calculate something on a computer is a field all in itself. 

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Hambydammit wrote: With

Hambydammit wrote:

With the disclaimer that I am not a mathemetician, I think this is the basic answer:

*Most people who have read Godel's theorem have grasped the implications correctly and understand that there are abstracts and infinites in formal systems, and these lead to at least one ultimately unanswerable question in any formalized proof. This is somewhat similar to the fact that there are numbers which cannot be written because of their non-repeating, non-terminating status.

This doesn't really have any effect on science or philosophy, and doesn't affect anyone other than mathemeticians in any meaningful way.

Perhaps one of you math geeks would like to correct me if I'm saying this incorrectly?

In philosophy Godels theorum raises many very interesting questions.  It basically shows that there are mathematical truths that can not be proved. There are true things that can't be proved. There are patterns within mathematics that will hold for all cases but no one will ever be able to work out why. In fact there is no "why" they are just true for all cases. This is just fuking strange.

In science Godels theorum may have some very intersting implications for explaining or understanding conciousness itself. Its quite a detailed discusion but if you are interested Douglas Hosfdatler has written couple of books explaining how he thinks that Godel lies at the heart of understanding conciousness.

In mathematics the Theory killed off in one swift blow an endevour that had occupied several of the world best minds. The endevour was formalising mathematics so that it was described in full by a formal system of symbols and rules of theorum generation. Godel showed that this was impossible.

Apart from its implications Godels theorum is one of the most wonderful pieces of reasoning I have ever come accross. When I first studied it in my final yeat university model on The philosophy of mathematics it was one of those momments of utter clarity that I remember distinctly to this day. When the penny dropped as we went through the formal proof I remeber thinking "wow I get it! I really get it! Thats just so cunning but its right its obviosuly correct! Godel you where a clever clever bastard". It was a glimpse into the mind of a total genius. I would recommend anyone to read up on it. The formal proof DOES not require a degere in maths or philosophy to understand. It isn't actually that complex and there are several books that do a good job at explaining it to the lay person.


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wavefreak wrote: Well,

wavefreak wrote:

Well, since the underlying math of computers is boolean logic, doesn't this also imply that there are things computer cannot in pirinciple do? 


Halting problem/non-computable functions?

"What right have you to condemn a murderer if you assume him necessary to "God's plan"? What logic can command the return of stolen property, or the branding of a thief, if the Almighty decreed it?"
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Not quite the same,

Not quite the same, Rev.

People put intellectual effort into logic. Faith requires none - indeed, intellectual effort tends to run counter to faith. 

"I do this real moron thing, and it's called thinking. And apparently I'm not a very good American because I like to form my own opinions."
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wavefreak wrote: Well,

wavefreak wrote:
Well, since the underlying math of computers is boolean logic, doesn't this also imply that there are things computer cannot in pirinciple do?

In a theoretical sense, computers should not be understood as being limited to boolean logic. Theorems describing the limits of computation refer to a more general theoretical Turing machine. Boolean logic is a limit of today's computers to a certain extent, but not theoretical computers.

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Hambydammit

Smiling Hambydammit Smiling

If you have a banana and another banana, one might assumed you have two bananas, but you don't you have A banana and another banana, one needs to invent mathematics to described two bananas, two bananas doesn't actually exist, you just invented a way of describing A banana and another banana, thus two bananas is merely an unprovable theory, because two bananas doesn't actually exist, Smiling

 

So Hambydammit ? can the existence of two bananas be proved logically

 

jcgadfly wrote:

Not quite the same, Rev.

People put intellectual effort into logic. Faith requires none - indeed, intellectual effort tends to run counter to faith.

Kurt Gödel's Ontological Argument

 http://www.stats.uwaterloo.ca/~cgsmall/ontology.html


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Rev_Devilin wrote:

Rev_Devilin wrote:

Smiling Hambydammit Smiling

If you have a banana and another banana, one might assumed you have two bananas, but you don't you have A banana and another banana, one needs to invent mathematics to described two bananas, two bananas doesn't actually exist, you just invented a way of describing A banana and another banana, thus two bananas is merely an unprovable theory, because two bananas doesn't actually exist, Smiling

 

So Hambydammit ? can the existence of two bananas be proved logically

 

jcgadfly wrote:

Not quite the same, Rev.

People put intellectual effort into logic. Faith requires none - indeed, intellectual effort tends to run counter to faith.

Kurt Gödel's Ontological Argument

http://www.stats.uwaterloo.ca/~cgsmall/ontology.html

Give the guy credit - he tried to use logic to prove God.

Unfortunately, he had to assume a lot of illogical things to make it work.

"I do this real moron thing, and it's called thinking. And apparently I'm not a very good American because I like to form my own opinions."
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Rev_Devilin wrote:

Rev_Devilin wrote:

Smiling Hambydammit Smiling

If you have a banana and another banana, one might assumed you have two bananas, but you don't you have A banana and another banana, one needs to invent mathematics to described two bananas, two bananas doesn't actually exist, you just invented a way of describing A banana and another banana, thus two bananas is merely an unprovable theory, because two bananas doesn't actually exist, Smiling

 

So Hambydammit ? can the existence of two bananas be proved logically

 

Do you take multiple bong hits before posting?

Or do you just go straight to battering your head in with a hammer? 

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"God" burns Anne Frank eternally. For that, theists call him 'good.'


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That would just be bananas

That would just be bananas todangst, I do both Smiling

? would you care to run through the problems and limitations of logic if you have time

 


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Rev_Devilin wrote: That

Rev_Devilin wrote:

That would just be bananas todangst, I do both Smiling

Smokin' banana peels baby!

Quote:
 

? would you care to run through the problems and limitations of logic if you have time

 

I'm really not sure what you mean. A hammer cannot be used to  do brain surgery, is that a limitation, or just an artifact of misuing the hammer? I think the same applies to logic: you don't use logic to 'prove' the existence of objects like bananas because it's not a method of empirical examination of the world in the first place.

On the other hand, 1+1=2 is a matter of deduction....

As for Godel's thereom, I make it a rule not to discuss concepts  I haven't mastered yet... my sense is that people either misaply the concept or overextend it. The fact that logic cannot prove it's own axioms is not noteworthy....seeing as axioms are beyond proof to begin with...

 

 

"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'


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todangst wrote:   On the

todangst wrote:

 

On the other hand, 1+1=2 is a matter of deduction....

Unless you live in Thailand 1+1= many Smiling

todangst wrote:

The fact that logic cannot prove it's own axioms is not noteworthy....seeing as axioms are beyond proof to begin with... 

? not noteworthy that's a rather large banana eating elephant in the room your ignoring Smiling

 Axioms "An axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation. Therefore, it is taken for granted as true"

? so would it be correct to say that logic is based upon assumption and not truth Smiling 

 


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Rev_Devilin

Rev_Devilin wrote:

todangst wrote:

The fact that logic cannot prove it's own axioms is not noteworthy....seeing as axioms are beyond proof to begin with...

? not noteworthy that's a rather large banana eating elephant in the room your ignoring Smiling

 

No, it's really not noteworthy.

Quote:

Axioms "An axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation. Therefore, it is taken for granted as true"

? so would it be correct to say that logic is based upon assumption and not truth Smiling

Not at all.

First of all, logic is a method, it's not a 'truth' or 'based on truth' any more than a hammer is 'based in truth' when used to pound in a nail. Logic isn't true or false in the sense that an assertion is said to be true when there is a correlation between a sign and the object signified. Logic is a system. It says that given a set of true propostions, a valid argument will churn out true conclusions. Truth is an auxiliarly process, it has to do with examining premises. So asking whether logic is based on truth seems somewhat inappropriate.

Next, how would you propose to provide proofs for axioms, if axioms are in fact the foundation for proofs? Would you propose building an even more basic set of axioms so that we could form an even more basic logic to prove the axioms? If we could do that then we wouldn't have axioms in the first place.... axioms must more basic than any logical system that could be used to prove them.

 

sorry if that came off wise ass, it's not like you can't argue against it... Smiling

 

anyway...

Next, you take the phrase 'self evident' to be nothing more than an assumption. In the case of the axioms of reason: the axioms of existence, identity and consciousness, their self evident nature is shown through retortion.

Axioms are not just hunches or assumptions in the sense that you might wake up one day and assume that angels live in your kitchen.

Let me share with you a logical system that does not use axioms at all. It uses well formed formular. The reason for posting this is to demonstrate the reason logical systems create axoims or formular that the system cannot prove (they must either accept them or define them) for a special reason: because these logics can be shown to work, to be self consistent, to be sound and complete.

 

This is precisely why an axiom or a defined rule is not merely an assumption in the sense of a 'groundless guess'

The following section delves into the matter of axioms more deeply, and requires knowledge of symbolic logic. Those following the Course in Logic 101 will learn everything required to grasp this section in later lessons, and can skip this section for now, if desired.

As stated above, some logical systems to not require any axioms at all. For example, the set of axioms for the sentential, or propositional, logic is {} - the empty set!

So how does such a system "get off the ground"?

It creates a set of rules, defined within the system:

Let the set of English capital letters be well-formed formular (WFFs), which may be appended
by zero to an infinite amount of primes to indicate different WFFs
If A and B are WFFs, then (A v B) is a WFF
If A and B are WFFs, then (A & B) is a WFF
If A is a WFF, then (~A) is a WFF
If A and B are WFFs, then define (A -> B) to be ((~A) v B)
If A and B are WFFs, then define (A <-> B) to be ((A -> B) & (B ->A))
No other strings are WFFs

That defines our "grammar" for our fake little language that will turn into the propositional logic.

Now define 21 rules of inference to allow us to move between WFFs:

Propositional_Logic.png

Then we define a function that maps each propositional variable to two values: "True" or "false" (or 1 and 0, or "your mom" and "your dad" - it doesn't matter from a formal point of view).

Then come the soundness (if you can derive a string from a set of strings above, then it must be "true&quotEye-wink and completeness (all "true" strings can be derived from the rules above) proofs, and we're done! The system is both sound and complete.

All this in much more detail can be found at:

http://www.algebra.com/algebra/about/history/Propositional-logic.wikipedia#Axioms

The point is that there can be no axioms in this logic, the most basic of all modern logics (there are other formulations that do have axioms, though): everything is definitional. And how does one argue with a definition? My point is: the answer to the question "why doesn't everyone accept the axioms of logic?" is that it can be the case that there's nothing to accept. Literally.

 

 

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"God" burns Anne Frank eternally. For that, theists call him 'good.'


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todangst wrote: Next, how

todangst wrote:

Next, how would you propose to provide proofs for axioms, if axioms are in fact the foundation for proofs? Would you propose building an even more basic set of axioms so that we could form an even more basic logic to prove the axioms? If we could do that then we wouldn't have axioms in the first place.... axioms must more basic than any logical system that could be used to prove them. 

 it's not like you can't argue against it... Smiling

 

I'm sure you could argue far more eloquently than me, Smiling  shall we call it debatable ? Smiling

todangst wrote:

Next, you take the phrase 'self evident' to be nothing more than an assumption. In the case of the axioms of reason: the axioms of existence, identity and consciousness, their self evident nature is shown through retortion.

Identity ?

And beyond existence and consciousness ?

todangst wrote:
 

Axioms are not just hunches or assumptions in the sense that you might wake up one day and assume that angels live in your kitchen.

Let me share with you a logical system that does not use axioms at all. It uses well formed formular. The reason for posting this is to demonstrate the reason logical systems create axoims or formular that the system cannot prove (they must either accept them or define them) for a special reason: because these logics can be shown to work, to be self consistent, to be sound and complete.

Let the set of English capital letters be well-formed formular (WFFs), which may be appended
by zero to an infinite amount of primes to indicate different WFFs

Infinite amount of primes Smiling

One would assume there is an Infinite amount of primes, but without checking to verify there is. and there is no contradiction, isn't this incomplete Smiling


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Rev_Devilin wrote:

Rev_Devilin wrote:
One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

I cannot help noticing how atheist and theist alike misuse Gödel's incompleteness theorem. First order logic is complete. Logic itself is in no way affected by Gödel's incompleteness theorem, and mathematics isn't either in practice. Gödel's incompleteness theorem is a curious theorem, but one with no real use. No mathematician's research has ever been ruined by the theorem.

[Edit: grammar]


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Rev_Devilin

Rev_Devilin wrote:

Infinite amount of primes Smiling

One would assume there is an Infinite amount of primes, but without checking to verify there is. and there is no contradiction, isn't this incomplete Smiling

The infinitude of the primes was proven over 2,000 years ago.


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Hambydammit wrote: Most

Hambydammit wrote:
Most people who have read Godel's theorem have grasped the implications correctly and understand that there are abstracts and infinites in formal systems,

I don't think "abstracts and infinites" have to do with Gödel's incompleteness theorem. The heart of the problem was self referencing statements. Gödel encoded in arithmetic a statement like "This statement is false", which you could never prove to be true.

Hambydammit wrote:
and these lead to at least one ultimately unanswerable question in any formalized proof.

Not answerable questions, but unprovable statements.

Hambydammit wrote:
This is somewhat similar to the fact that there are numbers which cannot be written because of their non-repeating, non-terminating status.

Uh, not really. This has nothing to do with Gödel's theorem.

Hambydammit wrote:
This doesn't really have any effect on science or philosophy, and doesn't affect anyone other than mathemeticians in any meaningful way.

This you got right, basically. It doesn't affect mathematicians much either.


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Rev_Devilin wrote: One

Rev_Devilin wrote:

Smiling

One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

 Why do people always get this so backwards?  The reason human beings support things like science and logic is NOT because we think ourselves worthy of great faith in our ideas or even our methods.  It's precisely the opposite: it is because we acknowledge our faliability and are humbled by our error.  That is why we cannot do without systems of thought and evidence to help correct error and keep us honest.  No one has "faith" in logic: it's a tool to help avoid things like self-contradiction and weed out error.  That many people's ideas about logic can be in error as well is just an important part of this process, not a reason to simply abandon trying to correct our irrationality altogether.

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MrRage wrote:

MrRage wrote:

The infinitude of the primes was proven over 2,000 years ago.

? may I see them all sequentially to verifying there are no contradictions

? No

Then it is incomplete not proven Smiling assumed to be correct

Gödel's originally tried to put an end to this kind of thing, he ended up reinforcing the idea of unprovable

 


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Rev_Devilin wrote: MrRage

Rev_Devilin wrote:

MrRage wrote:

The infinitude of the primes was proven over 2,000 years ago.

? may I see them all sequentially to verifying there are no contradictions

? No

Then it is incomplete not proven Smiling assumed to be correct

Gödel's originally tried to put an end to this kind of thing, he ended up reinforcing the idea of unprovable

 

How does viewing them non-sequentially give rise to possible contradictions? 


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Rev_Devilin wrote: MrRage

Rev_Devilin wrote:

MrRage wrote:

The infinitude of the primes was proven over 2,000 years ago.

? may I see them all sequentially to verifying there are no contradictions

? No

Then it is incomplete not proven Smiling assumed to be correct

Gödel's originally tried to put an end to this kind of thing, he ended up reinforcing the idea of unprovable

 

You're a quack.


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wavefreak

wavefreak wrote:
Rev_Devilin wrote:

MrRage wrote:

The infinitude of the primes was proven over 2,000 years ago.

? may I see them all sequentially to verifying there are no contradictions

? No

Then it is incomplete not proven Smiling assumed to be correct

Gödel's originally tried to put an end to this kind of thing, he ended up reinforcing the idea of unprovable

 

How does viewing them non-sequentially give rise to possible contradictions?

Actually, that's sorta how you prove the infinitude of the primes. If you begin with the assumption that there are finitely many primes, it leads to a contradiction.


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wavefreak wrote: How does

wavefreak wrote:

How does viewing them non-sequentially give rise to possible contradictions?

Is just a personal preference, I like sequential numbers, which is kinder unfortunate as I also like cryptography


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MrRage wrote: Actually,

MrRage wrote:

Actually, that's sorta how you prove the infinitude of the primes. If you begin with the assumption that there are finitely many primes, it leads to a contradiction.

Would you say it is beyond all doubt to a decree of absolute certainty that a contradiction will not be present while calculating prime numbers to infinity ?

 


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Quote: Hambydammit wrote:

Quote:
Hambydammit wrote:
Most people who have read Godel's theorem have grasped the implications correctly and understand that there are abstracts and infinites in formal systems,

I don't think "abstracts and infinites" have to do with Gödel's incompleteness theorem. The heart of the problem was self referencing statements. Gödel encoded in arithmetic a statement like "This statement is false", which you could never prove to be true.

 

Hambydammit wrote:
and these lead to at least one ultimately unanswerable question in any formalized proof.

Not answerable questions, but unprovable statements.

 

Hambydammit wrote:
This is somewhat similar to the fact that there are numbers which cannot be written because of their non-repeating, non-terminating status.

Uh, not really. This has nothing to do with Gödel's theorem.

 

Hambydammit wrote:
This doesn't really have any effect on science or philosophy, and doesn't affect anyone other than mathemeticians in any meaningful way.

This you got right, basically. It doesn't affect mathematicians much either.

I'm happy to have demonstrated my non-mathematician status so clearly. Having read your responses, I've proven to myself that I had basically the right idea, but hardly any chance of expressing it correctly.

When I was talking about infinites, I was essentially talking about the problem that computers have with expressing certain numbers, and how that limits computer logic, making the answers to some questions unattainable. I guess I can lump myself in with the people who don't understand the theorem correctly.

I get what you're saying about self-referencing statements, and I see how that makes sense with computers, too.

On a philosophical note, I'd like to point out that I've illustrated the fallacy fallacy very well. Though my proof was invalid, the conclusion turned out to be correct.

I will now butt out of all mathematics discussions, never to return.

 

 

Atheism isn't a lot like religion at all. Unless by "religion" you mean "not religion". --Ciarin

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Rev_Devilin wrote: Is just

Rev_Devilin wrote:
Is just a personal preference, I like sequential numbers, which is kinder unfortunate as I also like cryptography

Could you provide a definition of "sequential numbers"?

Rev_Devilin wrote:
Would you say it is beyond all doubt to a decree of absolute certainty that a contradiction will not be present while calculating prime numbers to infinity ?

I seriously doubt you've studied cryptography in much depth. Cryptography requires number theory. The proof that there are infinitely many primes is often in the first chapter of any number theory book. Yes, there is absolute certainty of the infinitude of the primes, because there are deductive proofs of that fact.

Regardless, this doesn't show how one has to have "faith in logic". I already pointed out that first order logic is complete. (Even if it was incomplete, you haven't shown where faith comes in.) If you have an ax to grind with prime numbers, then fine. But this hardly has to do with matters of faith.


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Rev_Devilin

Rev_Devilin wrote:
todangst wrote:

 it's not like you can't argue against it... Smiling

 

I'm sure you could argue far more eloquently than me, Smiling  shall we call it debatable ? Smiling

Yes. Debatable!

todangst wrote:

Next, you take the phrase 'self evident' to be nothing more than an assumption. In the case of the axioms of reason: the axioms of existence, identity and consciousness, their self evident nature is shown through retortion.

Quote:

Identity ?

And beyond existence and consciousness ?

The axioms of reason are defended through retortion. But not all statements called axioms are defended through retortion.

I do think there is a proof for there being an infinite array of prime numbers.

"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'


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MrRage wrote: Rev_Devilin

MrRage wrote:
Rev_Devilin wrote:
One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

I cannot help noticing how atheist and theist alike misuse Gödel's incompleteness theorem. First order logic is complete. Logic itself is in no way affected by Gödel's incompleteness theorem, and mathematics isn't either in practice. Gödel's incompleteness theorem is a curious theorem, but one with no real use. No mathematician's research has ever been ruined by the theorem. [Edit: grammar]

 Thank you. I recall someone once stating that the three most misunderstood and misapplied concepts in history are:

The theory of relativity.

Chaos theory.

and:

Gödel's incompleteness theorem.

"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'


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todangst wrote: MrRage

todangst wrote:
MrRage wrote:
Rev_Devilin wrote:
One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

I cannot help noticing how atheist and theist alike misuse Gödel's incompleteness theorem. First order logic is complete. Logic itself is in no way affected by Gödel's incompleteness theorem, and mathematics isn't either in practice. Gödel's incompleteness theorem is a curious theorem, but one with no real use. No mathematician's research has ever been ruined by the theorem. [Edit: grammar]

 Thank you. I recall someone once stating that the three most misunderstood and misapplied concepts in history are:

The theory of relativity.

Chaos theory.

and:

Gödel's incompleteness theorem.

Or possibly Evolution in the hands of theists.

"Evolution says that we should all behave like animals"

"Evolution says that I evolved from an ape"

"Evolution says that we should kill weaker creatures"

Etc etc etc!

But with Godels theorum I agree. It is missapplied. The annoying thing is that it is not necessary to do so. Its fascinating enough in its correct place. I disagree with the poster who said "it does not affect philosophy" it does, in fact its arguably the most important theory in the whole of the philosophy of mathematics. 

It begs fundamental questions about the nature of mathematics and its place in the universe.

If there are mathematical truths or patterns that are not provable then what does this mean? Is maths part of the universe for us to discover? Or is it a human construct? If its a human construct then why are there tuths that are unprovable? Where do these truths come from (*before you think it please fuck of theists - God is not an explanation). If something is unprovable or falsifibale for that matter (even in principle) is that thing true even if it holds for all cases?

In its right place Godels theourm has some very very interesting philsophical implications.


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evil religion wrote: I

evil religion wrote:
I disagree with the poster who said "it does not affect philosophy" it does, in fact its arguably the most important theory in the whole of the philosophy of mathematics.

I guess I should clarify. The theorem doesn't affect doing or using mathematics or logic. It's important to philosophy of mathematics, for sure, but is there any other branch of math that has been affected by it? Is there any important statement in math that's unprovable? The incompleteness theorem puts limitations on mathematics, but I'm not aware of anyone, outside of those in math logic (a small percentage of mathematicians), running into these limitations.

Contrast this with Turing's work in computability. The Halting Problem, or anything equivalent to it, does actually put real limits on computing.


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MrRage wrote: Logic itself

MrRage wrote:

Logic itself is in no way affected by Gödel's incompleteness theorem, and mathematics isn't either in practice. Gödel's incompleteness theorem is a curious theorem, but one with no real use. No mathematician's research has ever been ruined by the theorem. 

Principia Mathematica

 

MrRage wrote:

 Yes, there is absolute certainty of the infinitude of the primes, because there are deductive proofs of that fact.

Gödel's incompleteness theorem showed that Principia could not be both consistent and complete. if G is provable, then it's false, and the system is therefore inconsistent and if G is not provable, then it's true, and the system is therefore incomplete

So you have a mathematical system using infinity it is either not provable or incomplete

Yet you have faith that it does you believe with absolute certainty. although you have no proof, you cannot have proof catch 22 this is Gödel's incompleteness theorem, not provable or incomplete, thus faith

 


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Hambydammit wrote: On a

Hambydammit wrote:

On a philosophical note, I'd like to point out that I've illustrated the fallacy fallacy very well. Though my proof was invalid, the conclusion turned out to be correct.

I will now butt out of all mathematics discussions, never to return. 

Smiling psychology and logic don't mix, one of logic's many limitations, correct principles of reasoning logic vs the psychology of reasoning psychology

But don't give up on either mathematics or logic they are both incredibly useful, but it is also useful to realize their limitations


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todangst wrote: The axioms

todangst wrote:

The axioms of reason are defended through retortion. But not all statements called axioms are defended through retortion.

? so logic verifiable provable logic is confined to a box of true statements, once outside this box logic itself becomes debatable


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Or possibly Evolution in the

quote "evil religion"

 

Or possibly Evolution in the hands of theists.

"Evolution says that we should all behave like animals" "My friend said to me at that time in Andrassy Street, 'Nothing human is foreign to me.' To which I remarked, basing it on psychoanalytic experience, "You should go further and acknowledge that nothing animal is foreign to you." Sigmund Freud

"Evolution says that I evolved from an ape"

"Evolution says that we should kill weaker creatures" we have and sill do and they taste great

Etc etc etc!

But with Godels theorum I agree. It is missapplied. The annoying thing is that it is not necessary to do so. Its fascinating enough in its correct place. I disagree with the poster who said "it does not affect philosophy" it does, in fact its arguably the most important theory in the whole of the philosophy of mathematics.

It begs fundamental questions about the nature of mathematics and its place in the universe.

If there are mathematical truths or patterns that are not provable then what does this mean? Is maths part of the universe for us to discover? Or is it a human construct? If its a human construct then why are there tuths that are unprovable? Where do these truths come from (*before you think it please fuck of theists - God is not an explanation). If something is unprovable or falsifibale for that matter (even in principle) is that thing true even if it holds for all cases?

In its right place Godels theourm has some very very interesting philsophical implications.

Wow this is good suft more more


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MrRage wrote: evil

MrRage wrote:
evil religion wrote:
I disagree with the poster who said "it does not affect philosophy" it does, in fact its arguably the most important theory in the whole of the philosophy of mathematics.
I guess I should clarify. The theorem doesn't affect doing or using mathematics or logic.

Well Bertrand Russel would certainly have disagreed with you. Godels theorum effectivly ended the Hilbert program which had occupid the minds of many logicians and mathematicians for many years. IT culminated in Russels and Whiteheads Principa Mathematica which was a mighty tome that claimed to ecompass all of mathematics in a formal system. In effect reducing mathematics to logic. Shortly after it was published Godel published his short paper (and it really was not very long) which showed that not only was the principa incomplete or inconsistent but that ANY such atempt would always fail. This has a big impact on maths and logic. 

Quote:
It's important to philosophy of mathematics, for sure, but is there any other branch of math that has been affected by it?

Difficult to say but I would say yes. Probably all of them have. When you get "high level" meta mathematics like Godels theorum it wil affect mathematics and the way mathematicians work. It will influence them. Similarly philsophy in other areas will have an effect as it filters down to more applied human endevours but it will have and effect.

Quote:
Is there any important statement in math that's unprovable? The incompleteness theorem puts limitations on mathematics, but I'm not aware of anyone, outside of those in math logic (a small percentage of mathematicians), running into these limitations. Contrast this with Turing's work in computability. The Halting Problem, or anything equivalent to it, does actually put real limits on computing.

Indeed all of these works do have more direct applications. But I would not be too hasty to underestimate the effect young Kurt Godel had on mathematics. Its oft the way with philosophy, its applications and effects are so far down the line that they are hard to see. But philosophy is generally where it all starts.  


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Rev_Devilin wrote: One

Rev_Devilin wrote:

Smiling

One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

OK, back to the original comment.  Everyone here is trying WAY too hard.  There is plenty of evidence that logic works.  E-V-I-D-E-N-C-E.  If there were evidence that magical superbeings existed, belief in them would not be faith, it would be reason.

The words in each of the following word pairs are not interchangeable:

"Their" and "there."
"Its" and "it's."
"Your" and "you're."
"Then" and "than."


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Hi Ryan

Hi Ryan

Ryan wrote:

OK, back to the original comment.

We haven't deviated from the original comment Ryan (which is quite remarkable in itself )

Ryan wrote:

Everyone here is trying WAY too hard.

If people are thinking hard, this is good no ?

Ryan wrote:

There is plenty of evidence that logic works. E-V-I-D-E-N-C-E.

Yet you haven't provided plenty or any E-V-I-D-E-N-C-E. I would not like to dissuade you from trying but. so far the evidence that logic works is confined to a few statements, once one move's outside these few statements logic becomes debatable, one would think that mathematics would hold logical certainty/proof, ultimately it doesn't

Ryan wrote:

If there were evidence that magical superbeings existed, belief in them would not be faith, it would be reason.

Personally I would ask for undeniable proof for the existence of magical superbeings, circumstantial debatable evidence would not sufficed, and it's not like any magical superbeings has what one would consider good circumstantial debatable evidence

Faith is a belief without proof, and although I'm willing to accept many thing's on Faith, I like to be certain of, what is certain and what is Faith

 


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Ryan wrote: Rev_Devilin

Ryan wrote:
Rev_Devilin wrote:

Smiling

One cannot help, noticing the astonishing faith some people have in logic

http://en.wikipedia.org/wiki/Godel's_incompleteness_theorem

? one must ask why

OK, back to the original comment. Everyone here is trying WAY too hard. There is plenty of evidence that logic works. E-V-I-D-E-N-C-E. If there were evidence that magical superbeings existed, belief in them would not be faith, it would be reason.

I agree. Logic works. It allows us to operate within the world. It allows us to draw conclusions from other conclusions. There is plenty of evidecne that logic should be trusted as a very good method for making decisions. It workd well from past experiance. 

Note: this is an indutively reasoned justification for logic and as such can't itself be logically deduced. But I don't really have any problem with that. The "problem" of induction really isn't a problem unless one subscribes to the view that everything must be logically provable and that logic is the only way to gain useful information about the world.


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Rev_Devilin wrote: Faith

Rev_Devilin wrote:

Faith is a belief without proof, and although I'm willing to accept many thing's on Faith, I like to be certain of, what is certain and what is Faith

I disagree. Faith is not belief without proof (well not really) it is actually belief without evidence or belief in the face of contradictory evidence. We do not need to have faith in logic because there is plenty of evidence that it works, similarly our "faith" in science is not really faith at all it workd well, it gets the results, we are entirely justfied in our "faith" in science. Whilst its true that we do not know for certain that any specific scientific theory is correct or that logic will yield useful results in all cases we know from past experiance that it does get very good results and hence it is rational to beleive in these things.

 


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Well said

Well said Smiling


I'm ambivalent towards the use of the word evidence, as it lacks clarity for me personally in some of it's uses ie there is evidence of big foot the Loch Ness monster ect, this is why I prefer the word proof,  it seems a pity that there is not a word in common use for. I'd trust my life with it, although I retain a small  degree of skepticism about it 


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Rev_Devilin wrote: Yet you

Rev_Devilin wrote:

Yet you haven't provided plenty or any E-V-I-D-E-N-C-E. I would not like to dissuade you from trying but. so far the evidence that logic works is confined to a few statements, once one move's outside these few statements logic becomes debatable, one would think that mathematics would hold logical certainty/proof, ultimately it doesn't

OK, Rev.  Fair enough. The fact that you are typing these thoughts on a keyboard and transmitting them to the world is evidence that logic works.  The mathematics that modern technology depends on was created using logic.  You can argue all you want about the absolute proof of anything, or whether or not we can be absolutely certain of anything.  What is the point?  Like logic or mathematics or anything, your very language and thoughts are based on undefined terms and assumptions.  None of us can really know anything then, can we? 

 Evidence is not necessarily proof, it is support for an argument or idea.  Myself, I consider the fossil record evidence for evolution despite the fact that it cannot possibly be complete in an absolute sense.  I would credit the theists with having some kind of evidence if, say, everyone in the world suddenly heard a booming voice from the sky saying "sorry about all this, I've been on vacation."

I did not mean to imply that thinking hard was a bad thing.  Thinking "too" hard denotes inefficiency.  There is evidence that logic works.  Thus, using it does not require faith.  Assuming that all that is true can be proved absolutely certain through logic is definitely a leap, but I don't think anyone has made that claim. 

I am not belittling the discussion of Gödel, etc.  I'm just saying that it is not required to address the original comment.

The words in each of the following word pairs are not interchangeable:

"Their" and "there."
"Its" and "it's."
"Your" and "you're."
"Then" and "than."


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Ryan wrote:

Ryan wrote:

OK, Rev. Fair enough. The fact that you are typing these thoughts on a keyboard and transmitting them to the world is evidence that logic works. The mathematics that modern technology depends on was created using logic.

True ish, Smiling the space-shuttle and modern aircraft rely upon 3 computers with a voting system, sometimes computers get it wrong and it would take far to long to elucidate upon the many ways a computer can give incorrect results, but they can, and although I'd be perfectly happy flying in a aircraft using only computers, I'd be happier knowing there is a pilot present as even a 3 computer voting system can give incorrect results, logical systems can sometimes just breakdown, sometimes for physical reasons, sometimes for coding reasons, sometimes for no known reason

Ryan wrote:

You can argue all you want about the absolute proof of anything, or whether or not we can be absolutely certain of anything. What is the point? Like logic or mathematics or anything, your very language and thoughts are based on undefined terms and assumptions. None of us can really know anything then, can we?

One can be certain of one's own existence and the capacity to doubt

Ryan wrote:

Evidence is not necessarily proof, it is support for an argument or idea. Myself, I consider the fossil record evidence for evolution despite the fact that it cannot possibly be complete in an absolute sense. I would credit the theists with having some kind of evidence if, say, everyone in the world suddenly heard a booming voice from the sky saying "sorry about all this, I've been on vacation."

Smiling personally I would desire something stronger than a voice, the weight of evidence becomes compelling kinder thing

Ryan wrote:

I did not mean to imply that thinking hard was a bad thing. Thinking "too" hard denotes inefficiency. There is evidence that logic works. Thus, using it does not require faith. Assuming that all that is true can be proved absolutely certain through logic is definitely a leap, but I don't think anyone has made that claim.

Rev_Devilin wrote:
Would you say it is beyond all doubt to a decree of absolute certainty that a contradiction will not be present while calculating prime numbers to infinity ?
MrRage wrote:
Yes, there is absolute certainty of the infinitude of the primes,

And I'm sure MrRage was not alone in his absolute certainty, this is why I created this post

 

Ryan wrote:

I am not belittling the discussion of Gödel, etc. I'm just saying that it is not required to address the original comment.

I posted a link to Gödel incompleteness theorem in my original comment, this has been a discussion about the incompleteness theorem and its effect upon logic