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 Mathematics' Shortcoming? (Kurt Godel)

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PostSubject: Mathematics' Shortcoming? (Kurt Godel)   Wed 18 May 2011, 7:37 pm

By the 1928 international congress of mathematicians Hilbert "made his questions quite precise.
http://en.wikipedia.org/wiki/Turing_machine

+ First, was mathematics complete ...

+ Second, was mathematics consistent ...

+ And thirdly, was mathematics decidable?"
(Hodges p. 91, Hawking p. 1121).


The first two questions were answered in 1930 by Kurt Gödel at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third "the Entscheidungs problem" had to wait until the mid-1930s.

Emil Post developed his definition of a worker moving from room to room writing and erasing marks per a list of instructions (Post 1936), as did Princeton professor Church and his two students Stephen Kleene and J. B. Rosser by use of Church's lambda-calculus and Gödel's recursion theory (1934). Church's paper (published 15 April 1936) showed that the Entscheidungs problem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937).

Hence, the answer to all three of the above questions is "NO". Funny we didn't learn this in school

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Mon 20 Feb 2012, 11:42 am

You weren't taught it at school because you probably wouldn't have understood it.

The theorems you are trying to employ refer to consistent systems.

Why are you claiming that 'mathematics' is not consistent?
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 22 Feb 2012, 2:06 am

tazmic wrote:
You weren't taught it at school because you probably wouldn't have understood it.

The theorems you are trying to employ refer to consistent systems.

Why are you claiming that 'mathematics' is not consistent?
What about Godel's 1930 findings, referenced in the OP, are you unclear about?

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 22 Feb 2012, 2:17 am


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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 22 Feb 2012, 2:23 am







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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 22 Feb 2012, 6:29 am

C1 wrote:
tazmic wrote:
You weren't taught it at school because you probably wouldn't have understood it.

The theorems you are trying to employ refer to consistent systems.

Why are you claiming that 'mathematics' is not consistent?
What about Godel's 1930 findings, referenced in the OP, are you unclear about?
I'm unclear as to how they relate to the claims you have made. Hence my question.

Can you tell me how proofs about the nature of consistent axiomatic systems lead you think 'mathematics' is not consistent? As the videos don't do this you will have to resort to using your own words.
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 28 Feb 2012, 12:24 am

The info I posted in the OP was pulled directly from the referenced linked page at Wikipedia as well as Hodges p. 91, Hawking p. 1121. Perhaps you should attempt to edit those sources if you have an issue with the statements there .

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 28 Feb 2012, 7:43 am

C1 wrote:
The info I posted in the OP was pulled directly from the referenced linked page at Wikipedia as well as Hodges p. 91, Hawking p. 1121. Perhaps you should attempt to edit those sources if you have an issue with the statements there .
I don't have any problem with your sources, that should be clear. Neither with understanding them (although the subject can get as difficult as you wish). My question, which should be obvious by now, is about your conclusion:

"Hence, the answer to all three of the above questions is "NO"."

I didn't find this on wikipedia. This is your conclusion, right? I've seen similar sentiments expressed in the comments on the videos you posted. It seems quite a common misunderstanding.

Your attitude gives me little confidence in the rest of your work here.

Interested readers might try Torkel Franzén's - Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Thu 01 Mar 2012, 1:18 am

First, Mathematics is a formal system., which is what Godel's proof's relate to.

Second, the Australian video presenter does in fact refer to inconsistency & incompleteness by addressing the matter as "mathematical blindspots", which he uses frequently in his presentation.

Moreover, I'd refer readers to Gregory Chaitin's work on incompleteness and inconsistency.

Quote :
The logician Torkel Franzén [5] criticizes Chaitin’s interpretation of Gödel's incompleteness theorem and the alleged explanation for it that Chaitin’s work represents.

http://en.wikipedia.org/wiki/Gregory_Chaitin

In fact Greg has recently authored, with Francisco A Doria (Author), Newton C.A. da Costa (Author), the following, which greatly expands on the sphere of Godel's proofs:

Godel's Way: Exploits into an undecidable world
http://www.amazon.com/Goedels-Way-Exploits-undecidable-world/dp/0415690854

Quote :
Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein’s general relativity, as he proved that Einstein’s theory admits time machines.

The Gödel incompleteness theorem - one cannot prove nor disprove all true mathematical sentences in the usual formal mathematical systems - is frequently presented in textbooks as something that happens in the rarefied realm of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life.

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book’s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer science.


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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Fri 02 Mar 2012, 2:57 pm

tazmic wrote:
Your attitude gives me little confidence in the rest of your work here.
It's pretty shitty to come to this forum and to try to use a discussion about Godel & incompleteness, a topic that can be quite challenging to address and is still debated in academic circles, in order to challenge the credibility of all posts published by the author, nonetheless the admin of this forum.

We don't condone that type of behavior here, this is a nice place where people respect each other and try to address complex and difficult issues in a helpful and respectful tone. Please review the rules here. We'd be delighted for you to stay, but only in accordance with the atmosphere that we've created. There are plenty of other forums available that you can visit that facilitate hostility and disrespect.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 03 Mar 2012, 1:46 am

Well said C1. Attacking members attitude and investigation leads me to think this was intentional all along regardless of topic. I question all material everyone puts on here internally but attacking someone, their thoughts and research leads me to believe its a drive by hit piece for questioning something that is supposed to be a given (mathematics, logic, etc...).

Anyways tazmic, thanks for the Torkel Franzens information.
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 03 Mar 2012, 8:20 pm

The reality of the matter is that I am indeed torn by Chaitin's work, a long time estalishment member of the IBM science community. I have intuitively agreed with Chaitin's latest suppositions in his new book, Godel's Way, but I can also see it as a method for supporting elite goals. Chaitin's push toward "undecideability" everywhere helps support the Chaos that is being pushed for, and could be part of a push for a society unbridled by any rules, and where Complexity Modelling controlls everything. Also, if everything is "undecideable" (man's world as well as natuure"s), then there are no absolutes, and therefore everything is "changeable", and/or always changing - again, supporting elite goals. This is certainly an area for more thought, research and discussion. I've downloaded Godel's original works so that I can reference his own words in an attempt to be figure this one out.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 03 Mar 2012, 8:53 pm




Following from the poster of the youtube video...

A great Book on Kurt Gödel:
http://www.amazon.com/dp/0262730871?tag=gargecsguitot-20&camp=14573&creative=327641&linkCode=as1&creativeASIN=0262730871&adid=17RMF6V8MTQMT8T0NPX0&

http://garygeck.com/?page_id=42 for an mp3 this video for your iPod.

A version of this vid w/out musical soundtrack is here (if you find it distracting): http://www.youtube.com/watch?v=CgZ_9gQfitc

Hello, this is Gary Geck of Gary Geck.com. Kurt Gödel has been called the greatest logician since Aristotle and A Genius at odds with the Zeitgeist.

The following is my reading of Kurt Gödel's 1961 lecture called "The modern Development Of The Foundations Of Mathematics In The Light Of Philosophy". As was typical of Gödel's very private philosophical work, the lecture was never delivered. I now will read it in its entirety on youtube or in an mp3 (found at http://garygeck.com/?page_id=42).

It should become very clear that Gödel was a lone voice in his age of logical positivism, skepticism and analytical philosophy such as Harvard's Dr. Willard Quine's variety. Quine of course called the higher reaches of Set Theory mere mathematical recreation...a view clearly at odds with Gödel's. According to Dr. Richard Tieszen of San Jose University, "The three philosophers Gödel found most congenial to his own way of thinking were Plato, Leibniz and Husserl." In fact Gödel saw much of Western Thought as being on the wrong path since it had strayed from the influence of Leibniz in the 18th Century. It is surprising that Gödel promotes Kant (albeit in a modified form) with much enthusiasm in this lecture when Kant certainly helped to hasten the demise of Leibnizianism. Kant once called Plato' work 'babble'.

On an interesting note "His few interests were in surrealist and abstract art, his favorite writers included Goethe and Franz Kafka, he enjoyed light classics and some 'pop' music and Disney films, especially Snow White." [source: http://www.bookrags.com/biography/kurt-godel/ ]

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 03 Mar 2012, 8:55 pm





http://worldsciencefestival.com/videos/the_limits_of_understanding

About This Video

This statement is false. Think about it, and it makes your head hurt. If it’s true, it’s false. If it’s false, it’s true. In 1931, Austrian logician Kurt Gödel shocked the worlds of mathematics and philosophy by establishing that such statements are far more than a quirky turn of language: he showed that there are mathematical truths which simply can’t be proven. 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. In this full program from the 2010 Festival, leading thinkers untangle Gödel’s discovery and examine the wider implications of his revolutionary finding.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 06 Mar 2012, 12:07 am

I hope to get the time to outline the "Limits of our Understanding" panel video, and will do so here or in a dedicated thread.

This is excellent material, and I found some of the discussion quite honest and revealing. While Minsky appeared to go off topic several times, he was perhaps the most honest of the group, and demonstrated some of academics' feelings toward the public and their ability to "understand". I am now very curious to check out Minsky's book, and get some insight into his 27 or so different kinds of consciousness.

While I'm not positive, I think the book Minsky was referring to is:
The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind
http://www.amazon.com/Emotion-Machine-Commonsense-Artificial-Intelligence/dp/0743276647

Book Description
Publication Date: November 13, 2007
In this mind-expanding book, scientific pioneer Marvin Minsky continues his groundbreaking research, offering a fascinating new model for how our minds work. He argues persuasively that emotions, intuitions, and feelings are not distinct things, but different ways of thinking.

By examining these different forms of mind activity, Minsky says, we can explain why our thought sometimes takes the form of carefully reasoned analysis and at other times turns to emotion. He shows how our minds progress from simple, instinctive kinds of thought to more complex forms, such as consciousness or self-awareness. And he argues that because we tend to see our thinking as fragmented, we fail to appreciate what powerful thinkers we really are. Indeed, says Minsky, if thinking can be understood as the step-by-step process that it is, then we can build machines -- artificial intelligences -- that not only can assist with our thinking by thinking as we do but have the potential to be as conscious as we are.




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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 10 Mar 2012, 9:58 pm

Summary
http://www.evanwiggs.com/articles/GODEL.html

Famed mathematician Kurt Godel proved two extraordinary theorems. Accepted by all mathematicians, they have revolutionized mathematics, showing that mathematical truth is more than logic and computation. Does Godel's work imply that someone or something transcends the universe?



Truth and Provability

Kurt Godel has been called the most important logician since Aristotle.(1) Such praise is evidence of how seriously Godel's ideas are taken by mathematicians. His two famous theorems changed mathematics, logic, and even the way we look at our universe. This article explains what Godel proved and why it matters to Christians. But first we must set the stage.

There are many systems of math and logic. One kind is called a formal system. In a formal system there are only a few carefully defined symbols and rules.
Examples of commonly used symbols are a, +, x, y, <, and so forth. Following strict rules, symbols are combined into new patterns (proofs). The symbols are actually little more than place-holders. Some represent operations such as addition. Others represent slots that can be filled with numbers or sentences. The reason that empty symbols are used is so that we can be sure that proofs are created without the mistakes that human emotion and misinterpreted words can cause. After a proof is made in a formal system, statements or numbers can be substituted for the symbols, and we then know that the results on the last line of the proof are one hundred percent logical. Serious math often uses formal systems.

A very simple formal system cannot support number theory but such a system is easily proven to be self-consistent. All we have to do is to show that it can't make a silly proof such as A=Non-A, which would be like saying 2=17. To handle number theory a complex formal system is needed. But as systems get more complex, they are harder to prove consistent. One result is that we don't know if our number theories are sound or if there are contradictions hidden in them. Godel worked with such problems.

He especially studied undecidable statements. An undecidable statement is one which can neither be proven true nor false in a formal system. Godel proved that any formal system deep enough to support number theory has at least one undecidable statement.(2) Even if we know that the statement is true, the system cannot prove it. This means the system is incomplete. For this reason, Godel's first proof is called "the incompleteness theorem".

Godel's second theorem is closely related to the first. It says no one can prove, from inside any complex formal system, that it is self-consistent.(3) Hofstadter says, "Godel showed that provability is a weaker notion than truth, no matter what axiomatic system in involved."(4) In other words, we simply cannot prove some things in mathematics (from a given set of premises) which we nonetheless know are true.




Implications of Godel

What do Godel's theorems mean for those who believe there is a God? First, Godel shattered naive expectations that human thinking could be reduced to algorithms. An algorithm is a step-by-step mathematical procedure for solving a problem. Usually it is repetitive. Computers use algorithms. What it means is that our thought cannot be a strictly mechanical process. Roger Penrose makes much of this, arguing in Shadows of the Mind that computers will never be able to emulate the full depth of human thought. But whereas Penrose seeks solutions in quantum theory, Christians see man as a spiritual being with understanding that springs not just from the physical organ of the mind but also from soul and spirit.

Second, had Godel been able to affirm that a complex system is able to prove itself self-consistent, then we could argue that the universe is self-sufficient. His proof points us toward a different understanding, one in which we must either declare the universe to be infinite--as some do(5)--or else look for infinity outside the universe as theists do.

The first possibility, that the universe is infinite, is most unlikely. Everything that we have learned about the universe tells us that it is finite. Astronomers have found details that set absolute limits to its age and dimensions. Physicists have estimated the number of protons in all of creation. And even if there were an infinite amount of natural matter, each particle would still suffer the limitations of matter, for no particle is infinite in itself. The Christian therefore is reasonable when he points to a spiritual creator outside the physical universe as an explanation for what goes on within it. Godel recognized these implications and struggled to produce an ontological proof for the existence of God (a proof based on the definition of "God"). Godel was wasting his time in trying to establish this proof. His own theorems strongly suggest that while the finite can infer something bigger than itself, it cannot prove the infinite. As in this article, reason can only show that it is reasonable to believe in a spiritual God who transcends the limits of the universe.

Godel's theorem means that the universe cannot be a vast self-contained computer. One modern scientist, Fredekin, suggests that it is.(6) The fundamental particles of nature (in his view) are information bits in that huge machine. Were he right that the universe is effectively a computer, then Godel's theorems would require that nature, as a whole be understood only outside nature because no finite system is sufficient for itself. This conclusion flows by analogy from what Godel proved. "...if arithmetic is consistent, its consistency cannot be established by any meta-mathematical reasoning that can be represented within the formalism of arithmetic."(7)

As a third implication of Godel's theorem , faith is shown to be (ultimately) the only possible response to reality. Michael Guillen has spelled out this implication: "the only possible way of avowing an unprovable truth, mathematical or otherwise, is to accept it as an article of faith."(Cool In other words, scientists are as subject to belief as non-scientists. And scientific faith can let a man down as hard as any other. Guillen writes: "In 1959 a disillusioned Russell lamented: ÔI wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than anywhere...But after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.'"(9)




A or Non-A?

Godel showed that "it is impossible to establish the internal logical consistency of a very large class of deductive systems--elementary arithmetic, for example--unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the sytems themselves."(10) In short, we can have no certitude that our most cherished systems of math are free from internal contradiction.

Take note! He did not prove a contradictory statement, that A = non-A, (the kind of thinking that occurs in many Eastern religions). Instead, he showed that no system can decide between a certain A and non-A, even where A is known to be true. Any finite system with sufficient power to support a full number theory cannot be self-contained.

Judeo-Christianity has long held that truth is above mere reason. Spiritual truth, we are taught, can be apprehended only by the spirit. This, too, is as it should be. The Godelian picture fits what Christians believe about the universe. Had he been able to show that self-proof was possible, we would be in deep trouble. As noted above, the universe could then be self-explanatory.

As it stands, the very real infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them just as every logic set needs a higher logic set to prove and explain elements within it.


This lesson from Godel's proof is one reason I believe that no finite system, even one as vast as the universe, can ultimately satisfy the questions it raises.


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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 13 Mar 2012, 12:14 am

I believe that this excerpt from the full video, Limits to Understanding (posted above), does a good job characterizing the brief "discussion" between tazmic & myself (above in this thread)

Gödel's Lasting Legacy


Highlights from this excerpt
  • Exemplifies discussion that I had with tazmic: Godel demonstrated shortcomings of formal systems, so if pure mathematics is NOT a formal system, then what is it? We still don't know, that was Chaitin's point.
  • Godel's mathematical theorems spilled outside of Mathemetics - into Philosophy
  • Problem that Godel posed is still on the table, math has yet to address it. Even today, no one knows what mathematics is.
  • Why should mathematics continue even though mathematics has such "shortcomings".
  • Godel's proofs still pose serious problems if we wish to understand how the world works, not just interested in using math & science to produce products/servics that make money.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 13 Mar 2012, 4:10 pm

C1 wrote:
I hope to get the time to outline the "Limits of our Understanding" panel video, and will do so here or in a dedicated thread.
I've started a thread in the Philosophy sub-forum, where I will post clips from this talk, and begin to provide textual excerpts of hi-lites.

See http://wwws.forummotion.com/t880-limits-to-our-understanding

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Sat 24 Mar 2012, 10:18 pm

I think the most important takeaway from Godel is that man's formal systems cannot prove all known truths, therefore one must have FAITH that those truths are indeed TRUE. Hence, man's systems ultimately rely on faith.

Hence, I find the Science vs God argument so ridiculous... for scientists & mathematicians like to posture their arguments from the perspective that all of their proofs can be proved by their own formal systems, and therefore there is no need for faith. Well, that's a deception.

Of course, elitists have no interest in this notion, as it reduces their perceived power over others. For, if man could prove all known truths, then perception of man's power over nature is complete. But, we're not there yet, and we don't even know if we'll ever get there.

So, what are we doing to get around this, to get around this notion that our systems are not "complete", why we're creating a virtual reality comprised of computers, a reality where we can more easily trick the participants that that system is complete, hence, all proveable. But yet again, it's slight of hand, and Godel showed that via his Godel Numbering scheme (explored, in process, in another thread).

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Mon 26 Mar 2012, 12:29 am

CHRIST SUFFERED AS TRUTH
http://www.evanwiggs.com/articles/GODEL.html

"Now if we are children [of God], then we are heirs-heirs of God and
co-heirs with Christ, if indeed we share in his sufferings in order that
we may also share in his glory." Romans 8:17.

In answer to a question put by Pilate, Jesus said, "You are right in
saying I am a king. In fact, for this reason I was born, and for this I
came into the world, to testify to the truth. Everyone on the side of
truth listens to me."

"What is truth?" Pilate retorted. With this he went out again to the
Jews and said, "I find no basis for a charge against him..." Then Pilate
took Jesus and had him flogged...

The world met truth with force, but truth won.

Godel's proof implies that we must seek final truth outside our finite
world.
Jesus uttered one of the most ultimate claims ever made by a sane
man. "I am the way, the truth, and the life," he told his disciples just
hours before he stood before Pilate.

Will we find truth in a transfinite Christ or will we prefer partial
truth from within a system that cannot validate itself?


My note: I think the reader can replace "God" & "Christ" with the word, "Nature", and takeaway the key message. It all depends on your personal views, I guess.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Mon 26 Mar 2012, 11:24 am

C1 wrote:
CHRIST SUFFERED AS TRUTH
http://www.evanwiggs.com/articles/GODEL.html

"Now if we are children [of God], then we are heirs-heirs of God and
co-heirs with Christ, if indeed we share in his sufferings in order that
we may also share in his glory." Romans 8:17.

In answer to a question put by Pilate, Jesus said, "You are right in
saying I am a king. In fact, for this reason I was born, and for this I
came into the world, to testify to the truth. Everyone on the side of
truth listens to me."

"What is truth?" Pilate retorted. With this he went out again to the
Jews and said, "I find no basis for a charge against him..." Then Pilate
took Jesus and had him flogged...

The world met truth with force, but truth won.

Godel's proof implies that we must seek final truth outside our finite
world.
Jesus uttered one of the most ultimate claims ever made by a sane
man. "I am the way, the truth, and the life," he told his disciples just
hours before he stood before Pilate.

Will we find truth in a transfinite Christ or will we prefer partial
truth from within a system that cannot validate itself?


My note: I think the reader can replace "God" & "Christ" with the word, "Nature", and takeaway the key message. It all depends on your personal views, I guess.

This is very relevant stuff - much to think about here.
Thank you C1.... sunny
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Tue 27 Mar 2012, 12:12 am

The point is that man's systems, in this case "formal systems", are NOT as powerful and complete as we are told, or lead to believe. As Greg Chaitin refers in the "Limits of our Understanding" video that I reference in this thread, mathematics is good if you want to build computers or computer chips and make a billion dollars, but in its current state it does not necessarily enable us to answer the really BIG question, the INTERESTING questions about life and our universe. For that, we have to turn elsewhere.

You can see what a threat to the establishment Godel would have been, and that his fear of being poisoned through food may have not been unwarranted or irrational. He single-handedly tore-down much of the philosophical, logical, and mathematical systems man had created to deceive and control other men.

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 28 Mar 2012, 12:55 pm

C1 wrote:
Summary
http://www.evanwiggs.com/articles/GODEL.html

Famed mathematician Kurt Godel proved two extraordinary theorems. Accepted by all mathematicians, they have revolutionized mathematics, showing that mathematical truth is more than logic and computation. Does Godel's work imply that someone or something transcends the universe?



Truth and Provability

Kurt Godel has been called the most important logician since Aristotle.(1) Such praise is evidence of how seriously Godel's ideas are taken by mathematicians. His two famous theorems changed mathematics, logic, and even the way we look at our universe. This article explains what Godel proved and why it matters to Christians. But first we must set the stage.

There are many systems of math and logic. One kind is called a formal system. In a formal system there are only a few carefully defined symbols and rules.
Examples of commonly used symbols are a, +, x, y, <, and so forth. Following strict rules, symbols are combined into new patterns (proofs). The symbols are actually little more than place-holders. Some represent operations such as addition. Others represent slots that can be filled with numbers or sentences. The reason that empty symbols are used is so that we can be sure that proofs are created without the mistakes that human emotion and misinterpreted words can cause. After a proof is made in a formal system, statements or numbers can be substituted for the symbols, and we then know that the results on the last line of the proof are one hundred percent logical. Serious math often uses formal systems.

A very simple formal system cannot support number theory but such a system is easily proven to be self-consistent. All we have to do is to show that it can't make a silly proof such as A=Non-A, which would be like saying 2=17. To handle number theory a complex formal system is needed. But as systems get more complex, they are harder to prove consistent. One result is that we don't know if our number theories are sound or if there are contradictions hidden in them. Godel worked with such problems.

He especially studied undecidable statements. An undecidable statement is one which can neither be proven true nor false in a formal system. Godel proved that any formal system deep enough to support number theory has at least one undecidable statement.(2) Even if we know that the statement is true, the system cannot prove it. This means the system is incomplete. For this reason, Godel's first proof is called "the incompleteness theorem".

Godel's second theorem is closely related to the first. It says no one can prove, from inside any complex formal system, that it is self-consistent.(3) Hofstadter says, "Godel showed that provability is a weaker notion than truth, no matter what axiomatic system in involved."(4) In other words, we simply cannot prove some things in mathematics (from a given set of premises) which we nonetheless know are true.




Implications of Godel

What do Godel's theorems mean for those who believe there is a God? First, Godel shattered naive expectations that human thinking could be reduced to algorithms. An algorithm is a step-by-step mathematical procedure for solving a problem. Usually it is repetitive. Computers use algorithms. What it means is that our thought cannot be a strictly mechanical process. Roger Penrose makes much of this, arguing in Shadows of the Mind that computers will never be able to emulate the full depth of human thought. But whereas Penrose seeks solutions in quantum theory, Christians see man as a spiritual being with understanding that springs not just from the physical organ of the mind but also from soul and spirit.

Second, had Godel been able to affirm that a complex system is able to prove itself self-consistent, then we could argue that the universe is self-sufficient. His proof points us toward a different understanding, one in which we must either declare the universe to be infinite--as some do(5)--or else look for infinity outside the universe as theists do.

The first possibility, that the universe is infinite, is most unlikely. Everything that we have learned about the universe tells us that it is finite. Astronomers have found details that set absolute limits to its age and dimensions. Physicists have estimated the number of protons in all of creation. And even if there were an infinite amount of natural matter, each particle would still suffer the limitations of matter, for no particle is infinite in itself. The Christian therefore is reasonable when he points to a spiritual creator outside the physical universe as an explanation for what goes on within it. Godel recognized these implications and struggled to produce an ontological proof for the existence of God (a proof based on the definition of "God"). Godel was wasting his time in trying to establish this proof. His own theorems strongly suggest that while the finite can infer something bigger than itself, it cannot prove the infinite. As in this article, reason can only show that it is reasonable to believe in a spiritual God who transcends the limits of the universe.

Godel's theorem means that the universe cannot be a vast self-contained computer. One modern scientist, Fredekin, suggests that it is.(6) The fundamental particles of nature (in his view) are information bits in that huge machine. Were he right that the universe is effectively a computer, then Godel's theorems would require that nature, as a whole be understood only outside nature because no finite system is sufficient for itself. This conclusion flows by analogy from what Godel proved. "...if arithmetic is consistent, its consistency cannot be established by any meta-mathematical reasoning that can be represented within the formalism of arithmetic."(7)

As a third implication of Godel's theorem , faith is shown to be (ultimately) the only possible response to reality. Michael Guillen has spelled out this implication: "the only possible way of avowing an unprovable truth, mathematical or otherwise, is to accept it as an article of faith."(Cool In other words, scientists are as subject to belief as non-scientists. And scientific faith can let a man down as hard as any other. Guillen writes: "In 1959 a disillusioned Russell lamented: ÔI wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than anywhere...But after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.'"(9)

A or Non-A?

Godel showed that "it is impossible to establish the internal logical consistency of a very large class of deductive systems--elementary arithmetic, for example--unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the sytems themselves."(10) In short, we can have no certitude that our most cherished systems of math are free from internal contradiction.

Take note! He did not prove a contradictory statement, that A = non-A, (the kind of thinking that occurs in many Eastern religions). Instead, he showed that no system can decide between a certain A and non-A, even where A is known to be true. Any finite system with sufficient power to support a full number theory cannot be self-contained.

Judeo-Christianity has long held that truth is above mere reason. Spiritual truth, we are taught, can be apprehended only by the spirit. This, too, is as it should be. The Godelian picture fits what Christians believe about the universe. Had he been able to show that self-proof was possible, we would be in deep trouble. As noted above, the universe could then be self-explanatory.

As it stands, the very real infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them just as every logic set needs a higher logic set to prove and explain elements within it.


This lesson from Godel's proof is one reason I believe that no finite system, even one as vast as the universe, can ultimately satisfy the questions it raises.

C1,

Great post.

C1 wrote:
What do Godel's theorems mean for those who believe there is a God? First, Godel shattered naive expectations that human thinking could be reduced to algorithms. An algorithm is a step-by-step mathematical procedure for solving a problem. Usually it is repetitive. Computers use algorithms. What it means is that our thought cannot be a strictly mechanical process. Roger Penrose makes much of this, arguing in Shadows of the Mind that computers will never be able to emulate the full depth of human thought. But whereas Penrose seeks solutions in quantum theory, Christians see man as a spiritual being with understanding that springs not just from the physical organ of the mind but also from soul and spirit.

I wholeheartedly agree with the premise that our thought cannot be a strictly mechanical process, for the reason named, that man is also a being of soul (and/or spirit) and not just a physical being. When will the Penrose's of the world acknowledge that man is not "like" a computer, not in the most essential sense, anyway! These Transhumanists just hate that man is not a basically a binary system to be rearranged at anyone's will --and whim!!. If they don't like it, they can take it up with Him (or Her) at the appropriate time....assuming they ever get that chance. Ha, not likely!

Quote :
The first possibility, that the universe is infinite, is most unlikely. Everything that we have learned about the universe tells us that it is finite. Astronomers have found details that set absolute limits to its age and dimensions. Physicists have estimated the number of protons in all of creation. And even if there were an infinite amount of natural matter, each particle would still suffer the limitations of matter, for no particle is infinite in itself. The Christian therefore is reasonable when he points to a spiritual creator outside the physical universe as an explanation for what goes on within it. Godel recognized these implications and struggled to produce an ontological proof for the existence of God (a proof based on the definition of "God"). Godel was wasting his time in trying to establish this proof. His own theorems strongly suggest that while the finite can infer something bigger than itself, it cannot prove the infinite. As in this article, reason can only show that it is reasonable to believe in a spiritual God who transcends the limits of the universe.

Parenthetically, I believe that even had Godel been able to "prove" the validity of Mathematics, that wouldn't have invalidated the existence of a God. They are two separate questions to me. By the very nature of our concept of what God's nature is, IF he is, he would be beyond our known laws of physics--"outside" them, so to speak. This doesn't seem to be an unreasonable supposition; it seems more reasonable in my opinion to expect such an otherworldly entity to operate by completely different, unknown 'laws.' Therefore, saying that we can't prove God (by OUR logic)--and therefore he doesn't exist--is erroneous in expecting we could use OUR laws to prove His existence, or use our inability to do so as "proof" that he doesn't exist ---that's Not logical, imo.


Have to stop here ... to be continued later..
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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Thu 29 Mar 2012, 1:35 am

C1, picking up where I left off above,
========
Quote :
Godel's theorem means that the universe cannot be a vast self-contained computer. One modern scientist, Fredekin, suggests that it is.(6) The fundamental particles of nature (in his view) are information bits in that huge machine. Were he right that the universe is effectively a computer, then Godel's theorems would require that nature, as a whole be understood only outside nature because no finite system is sufficient for itself. This conclusion flows by analogy from what Godel proved. "...if arithmetic is consistent, its consistency cannot be established by any meta-mathematical reasoning that can be represented within the formalism of arithmetic."(7)

In my NotSo humble opinion, there is no more reason to postulate that the universe is a computer than that it is a grilled cheese sandwich. What BASIS is there for thinking that?? Didn't the man ever hear of the need for *Evidence*, of some sort...? He sounds very reckless to me.

Quote :
As a third implication of Godel's theorem , faith is shown to be (ultimately) the only possible response to reality.Michael Guillen has spelled out this implication: "the only possible way of avowing an unprovable truth, mathematical or otherwise, is to accept it as an article of faith."( In other words, scientists are as subject to belief as non-scientists. And scientific faith can let a man down as hard as any other. Guillen writes: "In 1959 a disillusioned Russell lamented: ÔI wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than anywhere...But after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.'"(9)

To be truthful, I don't know how to understand what an "unprovable truth" would be. If it isn't provable, how can one be sure it IS the truth? Unless it is some kind of an axiom, such as "Existence exists." So I believe one should just consider such to be 'unprovable' in logic, say, while possibly still provable via empirical demonstration. (As in: I built this building according to math principles I believe in, and it is STILL STANDING, which I offer as a strong form of proof... ..(what do you think?) I think the same approach would very validly apply to whether God exists, in that what we call the "miracle of creation" really seems to me it might very well be just that.

It's interesting that B. Russell was involved with this important research, and *came up with flawed answers.* Hmm, I'm not surprised; I see him as *always* being motivated by a need/desire to throw a monkey wrench into anything/everything important to man's comfort, well-being or betterment; i.e. he was, I think, a true hater of humanity. What surprises me is that these fallacious theories won the accaptance of so many mathematics professionals *so easily.*
===============================================
A or Non-A?

Godel showed that "it is impossible to establish the internal logical consistency of a very large class of deductive systems--elementary arithmetic, for example--unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the sytems themselves."(10) In short, we can have no certitude that our most cherished systems of math are free from internal contradiction.

Take note! He did not prove a contradictory statement, that A = non-A, (the kind of thinking that occurs in many Eastern religions). Instead, he showed that no system can decide between a certain A and non-A, even where A is known to be true. Any finite system with sufficient power to support a full number theory cannot be self-contained.

Judeo-Christianity has long held that truth is above mere reason. Spiritual truth, we are taught, can be apprehended only by the spirit. This, too, is as it should be. The Godelian picture fits what Christians believe about the universe. Had he been able to show that self-proof was possible, we would be in deep trouble. As noted above, the universe could then be self-explanatory.
==================================================
[quote="C1"]
As it stands, the very real infinities and paradoxes of nature demand something higher, different in kind, more powerful, to explain them just as every logic set needs a higher logic set to prove and explain elements within it.[/quote]

That's what I think, too. And partly this viewpoint comes from what I guess could be called intuition. And the fact of the incredible mystery of the universe, where it came from and why/where we came from. And very much because: there is a brilliant, divine order working in this universe, so brilliant it designed the incredible marvel that is our body's amazing capabilities. Wow...

==========================

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PostSubject: Re: Mathematics' Shortcoming? (Kurt Godel)   Wed 04 Apr 2012, 12:36 am

ScoutsHonor wrote:
Parenthetically, I believe that even had Godel been able to "prove" the validity of Mathematics, that wouldn't have invalidated the existence of a God. They are two separate questions to me. By the very nature of our concept of what God's nature is, IF he is, he would be beyond our known laws of physics--"outside" them, so to speak. This doesn't seem to be an unreasonable supposition; it seems more reasonable in my opinion to expect such an otherworldly entity to operate by completely different, unknown 'laws.' Therefore, saying that we can't prove God (by OUR logic)--and therefore he doesn't exist--is erroneous in expecting we could use OUR laws to prove His existence, or use our inability to do so as "proof" that he doesn't exist ---that's Not logical, imo.
Godel supposedly worked on a proof for God's existence. However, after his death all of his workbooks were found but the two or three that dealt with this proof.

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