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Gödel, Escher, Bach: an in-depth explainer

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

#51

Post by red_trumpet »

jstanley wrote: post #39

> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…

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

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

#52

Post by andybak »

craggyjaggy wrote: post #3

Maybe one day I'll try reading it again and actually finish it, but so far I couldn't do it. Everything is fascinating and mind blowing don't get me wrong, but I feel like there is always this weird pretentious atmosphere going on. I don't know how to describe it, but by the end of the first half reading GEB was not fun anymore.

I've never understood why people think it's pretentious. It joyous and innocent. It's someone laying out the shiny baubles they love and inviting anyone passing to take a look.

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

#54

Post by pfarrell »

Rygian wrote: post #26
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…

You may want to read I Am A Strange Loop, which clarifies the main thesis in the lines that self-reference is a precursor of consciousness iirc.

In the forward of which, Hofstadter says he could have called the book, "I" is a Strange Loop referring to the concept of "I-ness", but he found it too clunky.

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

#55

Post by Jtsummers »

Workaccount2 wrote: post #41
adamnemecek wrote: post #37

The underlying idea is the idea of fixed points (aka spectra, diagonalizations, embedding, invariants, braids). By fixed point I mean something like the "Lawvere's fixed point theorem". https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I have a linkdump on this https://github.com/adamnemecek/adjoint I also have a discord https://discord.gg/mr9TAhpyBW

Maybe an ELI5 of "fixed points"?

General definition (courtesy of the wiki): A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation.

So x is a fixed point of a function f if f(x) = x. This can be generalized to various kinds of things in mathematics, but as an algebraic example:

  f(x) = x^2
1 and 0 are fixed points:

  f(1) = 1^2 = 1
  f(0) = 0^2 = 0
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)

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

#56

Post by dinobones »

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 formulate them to be so.

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

#57

Post by andybak »

Brusco_RF wrote: post #29
mxkopy wrote: post #19

Earlier quoted context omitted.

It's not an insult. Psychedelics can profoundly change how someone perceives things. GEB has that sort profound shift in POV that many would associate with psychedelics.

Getting hit in the head can also change how someone perceives things. I think the above commenter was expressing frustration with the popular notion that taking drugs is some sort of shortcut to understanding consciousness, dynamic systems, etc

"Shortcut" is overstating it. It can steer one's thinking in a direction that makes those topics seem more relevant or interesting. This in itself can be very valuable.

> Getting hit on the head

Well - there are recorded cases of brain damage unlocking abilities and aptitudes (albeit with great rarity). So probably not recommended.

But rest assured, psychedelics are less painful, cause less damage and have a higher probability of beneficial effects.

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

#58

Post by Twisol »

red_trumpet wrote: post #51
jstanley wrote: post #39

> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…

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

Good question! In this context, we're relating a system of proofs to the properties of a desired model. If you start from a sufficiently expressive model and try to design a proof system where valid proofs correspond to true properties, then you will either miss some properties (an unproveable truth) or you will have too many proofs (proveable falsehoods).

Without biasing for a particular model, we could simply say that every sufficiently expressive proof system admits either zero models or more than one model.

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

#59

Post by fao_ »

red_trumpet wrote: post #51
jstanley wrote: post #39

> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…

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

Without a telescope I can't prove to you that andromeda is a galaxy, but it is.

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

#60

Post by climate-code »

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…

The point isn't that everything is recursive - the point is that systems that are recursive / self-referential cause breakdowns in logic (they are incomplete).

This is a deep philosophical insight - it rhymes with the Buddhist idea of no-self - the problems that arise because we hold onto a false sense of self.

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