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How Gödel's Proof Works (2020)
11–20 of 74 posts
Re: How Gödel's Proof Works (2020)
#12If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
It’s actually an interesting fact that every person who was programming in the 80s owns a copy of GEB, which they flipped through a bit and then put on the bookshelf and never actually read.
Re: How Gödel's Proof Works (2020)
#13(2020)
Some previous discussions:
Re: How Gödel's Proof Works (2020)
#14If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
while it is a groovy into to recursion and other cool ideas, GEB annoys me in that I feel like the three figures in the title are ill matched. Godel proves a super important result in math, sure... Escher was a skilled draughtsman who had a feel for tesselation. An OK artist IMO but no special insights. Bach on the other hand was an expressive genius who in the volume, power and beauty of his productions just seemed…
the linking thread for all three is self-reference, either in the form of a fugue or in a painting showing its own creation. Doug is a loop guy
Re: How Gödel's Proof Works (2020)
#15Earlier quoted context omitted.
GEB is a great book, and I've probably read it at least 3.33333333... times over the years. As a late teen it blew my mind. But I'm not sure I'd recommend it as a route into Gödel's proofs [1]. The book covers a lot of other ground too, and is notoriously digressive and quirky (looking at you, dialogues). Instead I'd recommend Gödel's Proof by Nagel and Newman for a conceptual intro. [1] I'm not a mathematician, so m…
I’ll never understand how GEB was using math, art, and music to explain consciousness (and Hofstadter himself still thinks no one understood it), but Nagel and Newman did a great job explaining why logic as a mechanical thing has only a tenuous relationship to concepts we understand, and that helped me crack at least a little bit of the mystery I was after when giving up on GEB.
I have not read GEB but I thought his second book, I am a Strange Loop, did a pretty good job of connecting the idea of self referential loops (like in godels proof) to consciousness and art and such.
Re: How Gödel's Proof Works (2020)
#16Great breakdown: https://stopa.io/post/269
Re: How Gödel's Proof Works (2020)
#17If you like video, supplement your reading with
Joel David Hamkins - Oxford lectures on the philosophy of mathematics "The Gödel incompleteness phenomenon" https://www.youtube.com/watch?v=Y5trjR5aw0k
also, "Gödel's incompleteness theorems: The proof that broke mathematics" | Joel David Hamkins https://www.youtube.com/watch?v=Sza69An_H8o spam-bait title but excellent mid-level talk.
edit: speling
Re: How Gödel's Proof Works (2020)
#18> However, although G is undecidable, it’s clearly true. That's... not really true; it's surprising to see it in Quanta, of all places. Godel's (separate) completeness theorem says that in first-order logic, anything that's semantically true in all possible scenarios can be syntactically proved. So, if G is "clearly true", that ought to make it provable. The theorems don't contradict each other because in FOL, G is n…
It’s clearly true in The Natural Numbers. It’s not provable because in some model it’s false. Being clearly true in one model does not make it provable.
Re: How Gödel's Proof Works (2020)
#19> However, although G is undecidable, it’s clearly true. That's... not really true; it's surprising to see it in Quanta, of all places. Godel's (separate) completeness theorem says that in first-order logic, anything that's semantically true in all possible scenarios can be syntactically proved. So, if G is "clearly true", that ought to make it provable. The theorems don't contradict each other because in FOL, G is n…
It’s clearly true in The Natural Numbers. It’s not provable because in some model it’s false. Being clearly true in one model does not make it provable.
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Re: How Gödel's Proof Works (2020)
#20Earlier quoted context omitted.
while it is a groovy into to recursion and other cool ideas, GEB annoys me in that I feel like the three figures in the title are ill matched. Godel proves a super important result in math, sure... Escher was a skilled draughtsman who had a feel for tesselation. An OK artist IMO but no special insights. Bach on the other hand was an expressive genius who in the volume, power and beauty of his productions just seemed…
the linking thread for all three is self-reference, either in the form of a fugue or in a painting showing its own creation. Doug is a loop guy
A Strange Loop guy, to be precise.
(It's the title of his follow up work after GEB.)