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If 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
https://shs.cairn.info/revue-internationale-de-philosophie-2...
If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
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…
Most proof of the Gödel theorem use the primes encoding that is makes all the operations very unintuitive. But GEB uses just ascii and a lot of the side task get obvious. (It uses base 20 instead of 256, but it's the same idea.)
> is¨notoriously digressive and quirky
It is super mega ultra notoriously digressive and quirky.
Earlier quoted context omitted.
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.
Unrusprisingly, since many years went by. But I still love the quirkiness of GEB
(Also, he has other books between the two, I deeply enjoyed Le Ton Beau de Marot, about translations)
An additional related theorem is Rogers' recursion theorem, which is how we get programs that, when run, print their own source code (by the theorem this can be done in any Turing complete programming language.)
If 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.
> 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.
If 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.
Much like the many unread copies of Knuth's TAOCP.
Earlier 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.
As Flannery O'Connor wrote, "The result of the proper study of a novel should be contemplation of the mystery embodied in it, but this is a contemplation of the mystery in the whole work and not or some proposition or paraphrase. It is not the tracking down of an expressible moral or a statement about life." We don't read literature with the hopes of a book laying out a precise thesis and incontrovertibly demonstrating it.
If you come into GEB expecting a scientific explanation of consciousness (like I did, when I first read it) you walk away confused and maybe disappointed. Hofstadter observed something transcendentally beautiful about self-reference and had a spiritual or religious revelation that, for him, related it to consciousness, and he attempted to convey that beauty and spiritual revelation in - appropriately self-referentially - a book that embodied it. You're meant to appreciate it in your heart and soul, not (just) in your mind. It's literature, not science.