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Gödel, Escher, Bach: an in-depth explainer

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Re: Gödel, Escher, Bach: an in-depth explainer

#141

Post by jonahx »

scrame wrote: post #30
davesque wrote: post #10

I've read GEB over many years rather in the way someone would read the Bible. I pick it up from time to time and enjoy chewing on one or two chapters of material. But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of tha…

I think of it more as a survey of a bunch of disciplines, and some hopeful hypothesis of future AI research. He writes in an accessible way and touches DNA, poetry, fractals, video feedback, topographical systems, just... a whole bunch of things. It is for sure reaching, but his core conceit is about pattern recognition and emergent behavior and he throws everything he's got at the wall there through the eyes of his…

> Seinfeld seems corny now

Seinfeld holds up brilliantly.

Re: Gödel, Escher, Bach: an in-depth explainer

#142

Post by dahart »

dinobones wrote: post #56

Am I the only one who did not find this book that interesting? I studied CS so it just felt like reading my class textbooks again, except with random trippy stories in between that try to shoehorn theory into a poor metaphor. The fundamentals of CS (strings, automata, graphs) are elementary building blocks. This is by design. You can apply them to almost anything. Almost everything "is a graph", or "recursion" if you…

Some of the comments here make me want to read it again, but the first time through I got bored. It kept geeking out about recursion for so laboriously long that I felt like maybe it was written in a different time for people who hadn’t already studied thunks and y-combinators and quines in school. I’m sure it was a journey of discovery in the 70s, but it felt like a lot of the ideas that made it famous had already seeped into my education and spoiled the book for me. I’m not sure exactly how much of it I read, I thought maybe two thirds, and it sounds like I either didn’t read or didn’t understand some of the themes in the book.

Re: Gödel, Escher, Bach: an in-depth explainer

#143

Post by flebron »

adastra22 wrote: post #134
mreid wrote: post #93

Earlier quoted context omitted.

It might be easiest to give a sense of what "unprovable but true" means by way of an imagined example. Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc. For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integer…

I’ve only ever seen examples like the one you give here, which seem like trite, trivial, and uninteresting middle-school level logical gotchas. Are there actually interesting properties which are true but can’t be proven? Or is it just a statement about self-referential recursive logic being unprovable?

Well, most interesting properties about computer programs are, in general for all programs, undecidable (https://en.wikipedia.org/wiki/Rice%27s_theorem). Undecidability is a closely related notion to unprovability (https://en.wikipedia.org/wiki/Undecidable_problem#Relationsh....

Re: Gödel, Escher, Bach: an in-depth explainer

#144

Post by [deleted] »

mreid wrote: post #93
red_trumpet wrote: post #51

Earlier quoted context omitted.

How can a statement that is unprovable be true? I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with

It might be easiest to give a sense of what "unprovable but true" means by way of an imagined example. Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc. For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integer…

[deleted]

Re: Gödel, Escher, Bach: an in-depth explainer

#145

Post by thayne »

adastra22 wrote: post #134
mreid wrote: post #93

Earlier quoted context omitted.

It might be easiest to give a sense of what "unprovable but true" means by way of an imagined example. Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc. For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integer…

I’ve only ever seen examples like the one you give here, which seem like trite, trivial, and uninteresting middle-school level logical gotchas. Are there actually interesting properties which are true but can’t be proven? Or is it just a statement about self-referential recursive logic being unprovable?

I don't know if it is quite what you are looking for, but with the normal mathematical axioms, it isn't possible to prove whether or not the there are any sets with a cardinality between the cardinality of the natural numbers and the cardinality of the real numbers.

But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of those statements as an axiom), but you will still have other statements that you can't prove.

Re: Gödel, Escher, Bach: an in-depth explainer

#146

Post by memetomancer »

adastra22 wrote: post #134
mreid wrote: post #93

Earlier quoted context omitted.

It might be easiest to give a sense of what "unprovable but true" means by way of an imagined example. Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc. For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integer…

I’ve only ever seen examples like the one you give here, which seem like trite, trivial, and uninteresting middle-school level logical gotchas. Are there actually interesting properties which are true but can’t be proven? Or is it just a statement about self-referential recursive logic being unprovable?

Interesting take. You have casually dismissed the Goldbach Conjecture (perhaps the deepest centuries old problem in number theory) as 'trite, trivial and uninteresting'... Suggesting that you are only minimally familiar with the issue... then toss about an inapplicable phrase 'self-referential recursive logic' as if you are deeply immersed in such matters, perhaps even _much_ smarter than the thousands of mathematicians (including the likes of Euler) that have applied themselves to this problem. An odd contradiction!

I do believe this opinion places you very high on the 'confidence' axis, but not especially far along the 'competence' axis.

Re: Gödel, Escher, Bach: an in-depth explainer

#148

Post by abrax3141 »

abecedarius wrote: post #88
kenjackson wrote: post #7

Is there a good book that is similar in spirit, but doesn't require the maturity of GEB? I know an 8th grader that would be a great target, but I don't know if they have the mathematical/logical maturity to get through it.

Try Raymond Smullyan. He has books at a range of levels of accessibility; I'm not sure if there's anything great for your 8th grader, but I remember first getting into The Lady or the Tiger? as a teen.

Smullyan's "Forever Undecided", and Ernest Nagel's "Gödel's Proof" are the best explanations available, to my mind. I find GEB to be bloated, if all that one is interested in is Gödel. (There is much more to recommend GEB, but if all you want is Gödel's proof, see one of the above.)

Re: Gödel, Escher, Bach: an in-depth explainer

#149

Post by DontchaKnowit »

thayne wrote: post #145
adastra22 wrote: post #134

Earlier quoted context omitted.

I’ve only ever seen examples like the one you give here, which seem like trite, trivial, and uninteresting middle-school level logical gotchas. Are there actually interesting properties which are true but can’t be proven? Or is it just a statement about self-referential recursive logic being unprovable?

I don't know if it is quite what you are looking for, but with the normal mathematical axioms, it isn't possible to prove whether or not the there are any sets with a cardinality between the cardinality of the natural numbers and the cardinality of the real numbers. But one of those two must be true, you just can't prove it. Of course you can add a new axiom that allows you to prove one or the other (or accept one of…

This is super interesting. Was not aware of this. Is there a proof that this is not provable?
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