What Gödel Discovered
stopa.io
What Gödel Discovered
1–10 of 271 posts
Re: What Gödel Discovered
#2This is a very nice and surprisingly readable write up. I think it is the best explanation of the incompleteness theorem I've ever read. Thank you for this!
> Now, say he wanted to write when. He could just write 17.
Shouldn't it be 19?
Re: What Gödel Discovered
#3This is a very nice and surprisingly readable write up. I think it is the best explanation of the incompleteness theorem I've ever read. Thank you for this! > Now, say he wanted to write when. He could just write 17. Shouldn't it be 19?
Thanks for the kind words and the catch! Updated :) -- may take a few mins to show up
Re: What Gödel Discovered
#4[Commenting before reading]
I was just reading about hilbert's problems, and his disappointment after Godel's and alan turing's papers. I am not sure if this blog is about that, but it is a very interesting topic.
Let's read it now :)
Re: What Gödel Discovered
#5This is remarkably well done. I'm reading Hofstadter's Godel, Escher, Bach and found this at the perfect time.
Thanks for the write up!
Re: What Gödel Discovered
#6This is remarkably well done. I'm reading Hofstadter's Godel, Escher, Bach and found this at the perfect time. Thanks for the write up!
Thank you :)
Re: What Gödel Discovered
#7I’m a mathematician who spent many hours thinking about Goedel theorem and adjacent topics. The author in a self-deprecating way says that he’s not a mathematician, but just a programmer. Bear no mind to that: this is one of the best popular expositions of Goedel theorem I’ve seen. Everything is very accurately explained, there are no silly mistakes and untruths one often sees mentioned in context of Goedel theorem, and even the original proof is sketched in a very approachable manner, striking a very good balance between getting across all crucial ideas, and skipping most of the distracting technical aspects (which are important and interesting, but I think only to mathematicians, haha). Great job all around.
Re: What Gödel Discovered
#8> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here.
I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
Re: What Gödel Discovered
#9Cf. Douglas Hofstadter's Metamagical Themas:
> In March of 1977, I met the great AI pioneer Marvin Minsky for the first time. It was an unforgettable experience. One of the most memorable remarks he made to me was this one: "Gödel should just have thought up Lisp; it would have made the proof of his theorem much easier." I knew exactly what Minsky meant by that, I could see a grain of truth in it, and moreover I knew it had been made with tongue semi in cheek. Still, something about this remark drove me crazy. It made me itch to say a million things at once, and thus left me practically speechless. Finally today, after my seven-year itch, I will say some of the things I would have loved to say then.
> [4 pages of detailed discussion of relationship between Gödel's methods and Lisp elided]
> It was for all these reasons that Minsky's pseudo-joke struck my intellectual funnybone. One answer I wanted to make was: "I disagree: Gödel shouldn't have and couldn't have invented Lisp, because his work was a necessary precursor to the invention of Lisp, and anyway, he was out to destroy PM, not Lisp." Another answer was: "Your innuendo is wrong, because any type of reference has to be grounded in a code, and Gödel's way of doing it involved no more coding machinery grounding its referential capacity than any other efficient way would have." The final answer I badly wanted to blurt out was: "No, no, no—Gödel did invent Lisp!" You see why I was tongue-tied?
Re: What Gödel Discovered
#10> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
You can imagine "1 + 1 = 2" not as an important question in this context, but rather a proof of concept for the system they devised. It's a bit like the story about a "black triangle" [0]: a rather useless model passed through a rendering pipeline to produce a result any OpenGL novice can scramble in a couple dozen lines, but showing that the whole pipeline does, indeed, actually work as intended.