Einstein & Gödel

[JupiterJones]

Einstein & Gödel DISCOVER Vol. 23 No. 3 (March 2002)

The above article is a great read. It gives a glimpse into a unique friendship between two of the greatest minds of the 20th century.

"Gödel's incompleteness theorems, published in 1931 when he was only 25, had rewritten the ground rules of modern science much as Einstein's theory of relativity had done 15 years before."

Gödel revolutionized modern mathematics, philosophy and computer science. He was the first to show that the theory of relativity allowed travel backward in time. Anyone with an interest in philosophy should gain an understanding of his incompleteness theorems. Philosophers, largely, find it very liberating to understand that not all truths can be proven.

 

[Orsorum]

That's a beautifully written article.

Question - are there any particularly good books on Godel?

 

[Hayabusa Rider]

Kurt was da man, but pissed a lot of people off, like Whitehead and Russell who were working on the Principia Mathematica. That was a can o worms

 

[Darien]

godel, escher and bach: an eternal golden braid is quite good. it's more toward philosophy really, but it'll get you a good ground on stuff godel did.

 

[Hayabusa Rider]

Originally posted by: Darien
godel, escher and bach: an eternal golden braid is quite good. it's more toward philosophy really, but it'll get you a good ground on stuff godel did.

beat me to it, but I have a linky on it. If you read just one book on the topic in your lifetime, I would say this is it. I have it and read it more than once.

 

[Orsorum]

Originally posted by: Darien
godel, escher and bach: an eternal golden braid is quite good. it's more toward philosophy really, but it'll get you a good ground on stuff godel did.

Cool.

Philosophy is fun!

<--- finished the Republic (finally)... starting on The Laws.... might delve into Aristotle once the quarter is over.

 

[Darien]

How's The Republic and exactly what translation are you reading?



It's one of those books I'd like to read before the year ends.

 

[UNCjigga]

Damn...I thought a new steakhouse was opening or something...

 

[Darien]

Originally posted by: uncJIGGA
Damn...I thought a new steakhouse was opening or something...

LOL.

E=mc^2 special! 16oz steak in any form of matter you want for $19.99

or the Godel special! Surprise in a can for $9.99

 

[Orsorum]

Originally posted by: Darien
How's The Republic and exactly what translation are you reading?



It's one of those books I'd like to read before the year ends.

It's a looooong read. Took me about a month and a half to read. It's the translation by B. Jowett, 1973 printing.

I'll try and remember to write more about it later.

 

[Darien]

Originally posted by: Zakath15

Originally posted by: Darien
How's The Republic and exactly what translation are you reading?



It's one of those books I'd like to read before the year ends.

It's a looooong read. Took me about a month and a half to read. It's the translation by B. Jowett, 1973 printing.

I'll try and remember to write more about it later.

Was that translation recommended to you? Have you read any other translations?



<--- likes to read

 

[Orsorum]

Originally posted by: Darien

Was that translation recommended to you? Have you read any other translations?
<--- likes to read

Haven't read any other translations, this one read really well. The subject matter is intense (as it should be, I suppose)... and it's a hard read. But I thought it was okay.

Enjoying any reading of substance is a rare quality in Americans today. Live it and love it.

 

[StormRider]

Originally posted by: PastorDon
Einstein & G&ouml;del DISCOVER Vol. 23 No. 3 (March 2002)

The above article is a great read. It gives a glimpse into a unique friendship between two of the greatest minds of the 20th century.

"G&ouml;del's incompleteness theorems, published in 1931 when he was only 25, had rewritten the ground rules of modern science much as Einstein's theory of relativity had done 15 years before."

G&ouml;del revolutionized modern mathematics, philosophy and computer science. He was the first to show that the theory of relativity allowed travel backward in time. Anyone with an interest in philosophy should gain an understanding of his incompleteness theorems. Philosophers, largely, find it very liberating to understand that not all truths can be proven.

When I took logic in college, I felt I didn't quite grasp completely his incompleteness theorems but even so I felt they were truly some of the most profound theories I had come across in my studies. My understanding of his theorem was basically, "In every logical system you construct, you can always come up with 'statements' that cannot be proven to be either true or false".

If you then put this "unprovable" statement as an axiom (where you can accept it as either true or false -- you then come up with 2 different logical systems -- one for if the statement is assumed true and one for if the statement is assumed false) and construct a bigger logical system. In this bigger logical system, that previously "unprovable" statement is now true (or false in that alternative logical system). However, you can then always construct another statement in that "bigger" logical system that is unprovable.

To sum it up: There will always be statements that cannot be proven true or false.

That had profound implications to me on religion. Basically, I thought that the "God exists" is one of those statements that cannot be proven true or false. If you choose to believe, then you believing it on faith alone.

Also, I realized then that Vulcans are really stupid to base their society solely on logic -- Vulcans should have known about this theory and to base their whole society solely on something they know cannot provide all the answers is stupid.

 

[Orsorum]

Makes sense, though.

 

[Brutuskend]

Philosophers, largely, find it very liberating to understand that not all truths can be proven.

Doesn't this contradict your other thread about GOD's existance being able to be proven with math?

 

[JupiterJones]

Doesn't this contradict your other thread about GOD's existance being able to be proven with math?

Not at all. Saying that ALL truths cannot be proven is not equal to saying that any particular truth cannot be proven.

 

[linuxboy]

Not at all. Saying that ALL truths cannot be proven is not equal to saying that any particular truth cannot be proven.

mm, true. However, the proof system "intuitively" possessed as a consequence of logical semantics and syntax has self-circular fallability. Meaning, that any particular proof of a truth is as systematically fallable as unprovable truths because of origins of self-defined truisms and their potential to create a reductio ad absurdum with even fundamental set theorems.

Oh and nice read on Godel. I highly recommend Godel, Escher and Bach. Just don't be so awed, there are neat little ways around his theorem. And ways around the ways, and ways around the ways around the ways. Gotta love philosophy.


Cheers !

 

[JupiterJones]

the proof system "intuitively" possessed as a consequence of logical semantics and syntax has self-circular fallability. Meaning, that any particular proof of a truth is as systematically fallable as unprovable truths because of origins of self-defined truisms and their potential to create a reductio ad absurdum with even fundamental set theorems.

"consequence of logical semantics and syntax has self-circular fallability" - not true of classic mathematical logic.

"origins of self-defined truisms" - i.e. axioms/postulates

"reductio ad absurdum" ? reductio for short ? is a process of refutation on grounds that absurd ? and patently untenable consequences would ensue from accepting the item at issue. (Internet encyclopedia of philosophy)

You are dismissing every proof due to the presence of axioms.

One item, then, at issue is the axiomatic statement - ?cognito ergo sum? (?I think therefore I am?)

Would "absurd...and patently untenable consequences...ensue from accepting the item at issue"?

Clearly not. Thus a refutation of this based on reductio ad absurdum is ill conceived.

 

[Moonbeam]

I don't understand any of this, but I would say of linuxboy's remarks, "Gotta love philosophy" he probably means, "Gotta love"

I think therefore I am. No, thinking is a fragment, an eye that sees but does not see itself. Thought is fear, division separation. What is the mind that does not think? The universe and a mirror, which is which? Who would want to know?

 

[Hayabusa Rider]

Originally posted by: linuxboy
Not at all. Saying that ALL truths cannot be proven is not equal to saying that any particular truth cannot be proven. mm, true. However, the proof system "intuitively" possessed as a consequence of logical semantics and syntax has self-circular fallability. Meaning, that any particular proof of a truth is as systematically fallable as unprovable truths because of origins of self-defined truisms and their potential to create a reductio ad absurdum with even fundamental set theorems. Oh and nice read on Godel. I highly recommend Godel, Escher and Bach. Just don't be so awed, there are neat little ways around his theorem. And ways around the ways, and ways around the ways around the ways. Gotta love philosophy. Cheers !

Hmm.. are you saying KG's theorem is incomplete?

 

[calbear2000]

Originally posted by: Hayabusarider

Originally posted by: Darien
godel, escher and bach: an eternal golden braid is quite good. it's more toward philosophy really, but it'll get you a good ground on stuff godel did.

beat me to it, but I have a <a class=ftalternatingbarlinklarge href="http://www.amazon.com/exec/obidos/tg/detail/-/0465026567/103-2991594-5669411?vi=glance" target=blank>linky</A> on it. If you read just one book on the topic in your lifetime, I would say this is it. I have it and read it more than once.

Just ordered it... I've heard of this book on several occasions. Will finally get around to reading it!

 

[Darien]

Originally posted by: calbear2000

Originally posted by: Hayabusarider

Originally posted by: Darien
godel, escher and bach: an eternal golden braid is quite good. it's more toward philosophy really, but it'll get you a good ground on stuff godel did.

beat me to it, but I have a <a class=ftalternatingbarlinklarge href="http://www.amazon.com/exec/obidos/tg/detail/-/0465026567/103-2991594-5669411?vi=glance" target=blank>linky</A> on it. If you read just one book on the topic in your lifetime, I would say this is it. I have it and read it more than once.

Just ordered it... I've heard of this book on several occasions. Will finally get around to reading it!

It's a good read. Long...VERY LONG, but good.



Congrats if you ever finish it

 

[linuxboy]

"consequence of logical semantics and syntax has self-circular fallability" - not true of classic mathematical logic.

Exactly my point. Wherein one begins with a formulated set, it poses epistemic defeaters in conceiving of an encompassing system of verifiable reality.

"origins of self-defined truisms" - i.e. axioms/postulates

Right, but what is their source? How do they come about? How do we ever get past the circularity of "innateness"


"reductio ad absurdum" ? reductio for short ? is a process of refutation on grounds that absurd ? and patently untenable consequences would ensue from accepting the item at issue. (Internet encyclopedia of philosophy)


Infomally, yes. Formally, this is the posing of a conclusion, assumption, or heck, any statement within a derivation and its negation. If this is done, systems fall, because it can begin a long chain of deletion.


You are dismissing every proof due to the presence of axioms.

Precicely, because I submit that axiomatic conceptions lack completeness.


One item, then, at issue is the axiomatic statement - ?cognito ergo sum? (?I think therefore I am?)

Hardly a necessary condition. Scary, eh?


Would "absurd...and patently untenable consequences...ensue from accepting the item at issue"?

Clearly not. Thus a refutation of this based on reductio ad absurdum is ill conceived.

Wan't quite a refutation but a clarification on points (notice my agreement), summarized by my last statement. Moonbeam got the point of my last sentence. Gotta love.


cause, baby it's all you need. dooo do dooo. and so long as you love me soooo....


Merry Christmas


Cheers !