Thanks for pointing me towards this fantastic book. About the review, it says nothing about Bach (once only) or Escher (just the figures), so how does it capture clearly or comprehensively the book?
Gödel, Escher, Bach: an in-depth explainer
151–160 of 252 posts
Re: Gödel, Escher, Bach: an in-depth explainer
#152The 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"?
Perhaps not an explanation per se, but examples --
If you have two rulers and and compress one, then you when you lay them together there will be one point which is exactly at the same point.
Likewise, if you have two paper chessboards, and scrunch one up, there will be at least one square* which is exactly over the top of the intact one.
This also gives an intuition for the difficulty of actually finding the fixed point, even though you know there is one.
* more precisely not only a square, but an exact, dimensionless point. (afaik)
Re: Gödel, Escher, Bach: an in-depth explainer
#153Earlier quoted context omitted.
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.
I thought Seinfeld was corny at the time and remains unchanged.
Opinions, everyone has one...
Re: Gödel, Escher, Bach: an in-depth explainer
#154Earlier quoted context omitted.
what if the proofs cannot be described as finite or countable sets that does not render a straightforward application of diagonalization? What happens to Goedel’s theorem then?
What's an example of an uncountably infinite proof?
Any visual geometric proof. You can build an uncountably infinite set of different sized variations proving the same underlying relationships like this: https://youtu.be/CAkMUdeB06o
Whether or not a visual demonstration like that is actually a “proof” is a separate question. It definitely wouldn’t satisfy Hilbert, and doesn’t meet this definition:
> A proof of a statement S is a finite sequence of assertions S(1), S(2), … S(n) such that S(n) = S and each S(i) is either an axiom or else follows from one or more of the preceding statements S(1), …, S(i-1) by a direct application of a valid rule of inference.
I also don’t know of any visual “proof” like that which can’t be explained much more rigorously and powerfully with a formal set of assertions. But pulling threads like this and really asking what makes a proof a “proof” are some of the deepest questions I think a person can ask. It’s worth doing if only to appreciate what an incredible accomplishment all of the formal set theory work is in unifying and attempting to define meta concepts like “proof” itself.
Re: Gödel, Escher, Bach: an in-depth explainer
#155Earlier quoted context omitted.
I plowed my way through about two thirds of it. Lots of interesting stuff, but I finally gave up because it seemed just too self-involved and self-referential. Building up this enormous edifice, just to be able to say "Hey! Check out my edifice!" I guess I'm just too much of an applied engineering kinda guy.
Oh, the irony of dismissing GEB as "too self-referential."
I'm So Meta, Even This Acronym.
Re: Gödel, Escher, Bach: an in-depth explainer
#156Why read this when GEB is its own in-depth explainer?
For one thing, I read GEB and found it difficult to follow. No doubt there is more in GEB than in this article, but without this less lengthy summary I couldn't get at it, life and attention being what they are. If I returned to the book now, I bet I'd understand more of what was already in it.
Yeah this article is an excellent summary and I'd recommend people read it before the book instead of after
Re: Gödel, Escher, Bach: an in-depth explainer
#157Earlier quoted context omitted.
What's an example of an uncountably infinite proof?
Any visual geometric proof. You can build an uncountably infinite set of different sized variations proving the same underlying relationships like this: https://youtu.be/CAkMUdeB06o Whether or not a visual demonstration like that is actually a “proof” is a separate question. It definitely wouldn’t satisfy Hilbert, and doesn’t meet this definition: > A proof of a statement S is a finite sequence of assertions S(1), S(…
> You can build an uncountably infinite set of different sized variations proving the same underlying relationships like this
For such proofs to be contained in a finite space, the verifying person or machine needs to be able to distinguish between arbitrarily minute differences between proofs.
Re: Gödel, Escher, Bach: an in-depth explainer
#158Earlier 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?
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 mathematici…
I was talking about Gödel, not Goldbach:
> The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
Sorry I meant to quote this bit at the beginning of my comment. Parent comment which I was replying to talks about both.
Re: Gödel, Escher, Bach: an in-depth explainer
#159Intellectually poor man's attempt at a summary of the summary:
Part 1 - math proof that a formal system when evaluated by an agent "breaking out of the system" can use that system to prove itself inconsistent. ("f* up" the internal logic of that system by feeding it into itself)
Part 2 - ironically, to be conscious you have to have an awareness of your own system (a "strange loop") but you'll never be able to understand yourself as proven in part 1.
Re: Gödel, Escher, Bach: an in-depth explainer
#160Earlier quoted context omitted.
Any visual geometric proof. You can build an uncountably infinite set of different sized variations proving the same underlying relationships like this: https://youtu.be/CAkMUdeB06o Whether or not a visual demonstration like that is actually a “proof” is a separate question. It definitely wouldn’t satisfy Hilbert, and doesn’t meet this definition: > A proof of a statement S is a finite sequence of assertions S(1), S(…
> You can build an uncountably infinite set of different sized variations proving the same underlying relationships like this For such proofs to be contained in a finite space, the verifying person or machine needs to be able to distinguish between arbitrarily minute differences between proofs.
You don’t need to go through every possible element in that infinite uncountable set to prove that relationship, though. You can create an arbitrary demonstration that you can then manipulate in your head. Once you see that water demonstration you can inuit how that relationship must persist at different sizes.
Again, that doesn’t really count as a “proof” by modern standards, but it’s how the ancient greeks thought (they used more than just visual intuition/they also used more rigorous and formal propositions than that water thing, but they were visual and didn’t involve finite sets)