11 votes

Kurt Gödel: The World's Most Incredible Mind

Part 1:

Fast forward at minute 5 if you want to skip the bio and get to the meat of the matter right away:


http://youtu.be/i2KP1vWkQ6Y

Part 2:

(first incompleteness theorem at 9 mins 40 secs : there is a sentence G such that, if the system is consistent, then G has the form : "I am not provable.")

http://youtu.be/bgvxVUyVdXk

Part 3:

(commentary on the significance of Gödel's results)

http://youtu.be/W4-O7j169D0




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Mathematics

"the science of that which is or is not"

Is it fair to say then that number one is "that which is"?

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.

About half way through part 2 it dawned on me

that a component lacking from the liar's paradox, "this statement is unprovable", assuming mathematics is consistent, infinity, and any fundamental difference between minds and machines is a function of time.

Cyril's picture

Well, as I see it, that's the

Well, as I see it, that's the very most humbling thing for hard sciences and what's usually regarded as the purest of all, namely maths:

even if one would bother (or, in some sense, dare) to explicitly model time itself as an orthogonal dimension of a logical framework to gauge about their statements' truth, would it have any invalidating impact on Godel's findings?

The quite amazing thing to me is that does seem unlikely, since even Cantor's orders of infinity haven't proven to be tough enough to resist Godel's theorems; which don't even need to rely upon anything stronger than the simplest infinity (of the natural numbers).

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

one of the most interesting

one of the most interesting courses I took in university was Computability. We covered Godel, Turing etc... Very cool stuff. Going through the proofs and the reasoning was tough but the exams were easy :)

"My theories explain, but cannot slow the decline of a great civilization. I set out to be a reformer, but only became the historian of decline."
- Ludwig Von Mises

bump for Godel!

One of the most significant achievements of the 20th century. Godel killed Leibniz's dream. Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion. Alas, the universe disagreed. To generalize his conclusions to outside of mathematics,Godel showed that no system can fully understand itself from within itself.

Also relevant to this is Alan Turing's discovery that the halting problem is undecidable over Turing machines. In essence, this showed that there is an infinite number of undecidable problems that cannot be solved with a single algorithm.

“With laws shall our land be built up, but with lawlessness laid waste.”
-Njal Thorgeirsson

Cyril's picture

Yes, indeed. Ditto on the

Yes, indeed. Ditto on the usability and intrinsic limits of any axiomatic, by very construction/definition.

And for whoever is interested to learn more about this outstanding yet little known (still) achievement of the human mind, Godel's, I also warmly recommend to dig into how Godel proceeded in his proof, and in particular, his clever Godel's numbering scheme.

Oh, and also relevant, beyond Godel and Turing:

Church's works, too, and Curry, with the Curry-Howard correspondence.

And keep being introspective, too. ;)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

So even the libertarian

So even the libertarian "system" cannot understand itself within itself.

of course

The libertarian "system" is based on some axioms. The axioms cannot be evaluated within the system. We have to agree on the axioms first, before we discuss the consequences of the axioms.

“With laws shall our land be built up, but with lawlessness laid waste.”
-Njal Thorgeirsson

Bump

I was taught to close my mouth when chewing. So until I get a few more hours under my belt I'll do just that. But I will say this interests me because I read in an old(ancient really) book about the "beginning of knowledge" or the "key of knowledge" or the "greatest commandment," as well as the fact that "man cannot find out the work that is done under the sun."

That book hit the nail on the head if you ask me.

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.

I am sorry, but why did you post this?

Why did you post an intro to discrete mathematics? While I do appreciate higher level math, I do not see what this has to do with on these forums.


Cyril's picture

I posted this because while

I posted this because while everybody (or almost) heard about Einstein's works, very few people have had about Godel's and even fewer only remotely understand the full extent of implications of his discovery.

His legacy truly deserves better recognition, IMO.

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

He Did Post It...

...in the Science & Technology forum category, which seems appropriate. After all, that category is an option on the DP for folks who are interested in Science & Technology. Frankly, the information talked about here is extremely useful to the pursuit of truth and reality -- values that most libertarians and Ron Paul supporters hold dear. Surely we are not confined, as a community, to the political and popular in our search for truth and reality. I actually appreciated this post as a contribution to the "big picture," hope of freedom and understanding among us all. But then, I am one who enjoys science & technology. Different strokes for different folks.

Thanks Cyril -- good stuff.

Bump

..

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Hear, O Israel: YHUH our God YHUH one. And thou shalt love YHUH thy God with all thine heart, and with all thy soul, and with all thy might.