Why I Don't Love Gödel, Escher, Bach • Hacker News Board

Re: Why I Don't Love Gödel, Escher, Bach

#141

Post by Grue3 » Thu, Jul 05, 2018, 4:16 PM UTC

j2kun wrote: post #96

jimhefferon wrote: post #2

> The high-level, slightly handwavey description of this theorem is that, for any sufficiently expressive mathematical proof system, there are more true facts within the system than there are proofs for facts, which in turn means that not every fact can be proved No, no, no. The proof is not by cardinality. The proof is constructive. That is the point of the book, or at least a major theme of it. For example, that is…

> "the proof" As if there can only be one. (And it seems OP is describing the theorem, not a proof)

It's just plain wrong to say that there are more facts in a theory than proofs of facts. Both of these sets are countably infinite. Godel's theorem proves that provable facts are a strict subset of all facts.

Re: Why I Don't Love Gödel, Escher, Bach

#142

Post by pnathan » Thu, Jul 05, 2018, 4:17 PM UTC

pvg wrote: post #124

Tomte wrote: post #71

GEB is a book – like TAOCP – that makes me feel inadequate. I "know" I "should" work through it. I "know" I "should" enjoy it. But I never follow through. This review gives me hope that maybe I'm not just a ludicrous failure, but that there are different ways to go. Obviously, I do believe that, and I don't believe I'm a total failure, but still, I can't deny the nagging voice in the back of my head sometimes.

Second time in this thread I see someone mention GEB in the same breath as TAOCP. I'm curious how they ended up occupying the same kind of mental space for you and apparently others. In the immortal words of Jules Winnfield, it "ain't the same fuckin' ballpark, it ain't the same league, it ain't even the same fuckin' sport". Not that you are under any obligation to read either and plenty of very good arguments can be…

> plenty of very good arguments can be made you're better off not-reading at least one of them.

I'm extremely confused as to why this would be the case for TAOCP.

While I wouldn't recommend TAOCP for "just reading", I don't know why it wouldn't be recommended for study. Knuth provides a very interesting approach for algorithms analysis, deep in substance.

Re: Why I Don't Love Gödel, Escher, Bach

#143

Post by rwallace » Thu, Jul 05, 2018, 4:17 PM UTC

danharaj wrote: post #112

The treatment of Godel's theorems in GEB always struck me as gratuitous: It treats a technical result as having great philosophical import. The philosophical content of Godel's theorems is already found in the Liar paradox, "This statement is false." and incompleteness in a grand sweeping sense involving language in its entirety can be found in works as early as the Daoists and the Greeks. Edit: I should say that thi…

> The philosophical content of Godel's theorems is already found in the Liar paradox, "This statement is false."

Sort of. Common reactions to the English version of the liar paradox are along the lines of 'that's meaningless so it doesn't imply anything' or 'so natural language is inconsistent, what do you expect?' Most of the work Gödel did was to find a way to encode meta-logic into formal language in such a way that he could then state the liar paradox in a format so ironclad that it could no longer be dismissed.

Re: Why I Don't Love Gödel, Escher, Bach

#144

Post by deepaksurti » Thu, Jul 05, 2018, 4:19 PM UTC

jweather wrote: post #136

I've read it cover-to-cover at least three times, gifted a copy, and read it out loud to my son. I find it very playful and mind-expanding. I will admit that it is verbose and wandering, but I think that's part of the charm. Out of curiosity I wandered through Amazon's 1-star reviews of this book to see what other people had to say. "Obtuse and impenetrable", I can certainly understand. It takes a certain bent to put…

>>takes a certain bent to put the effort into reading it

Years back, I scanned through a few pages and realized it needed effort as you mention.

OTOH, I think there are 2 kinds of great books: one which just suck you into their world and other which demand effort/tax to pay to immerse in that world. The former are incredible books, while the latter are still great.

I found `On Lisp` by PG in the same latter category though GEB and OL tackle different subjects.

Re: Why I Don't Love Gödel, Escher, Bach

#145

Post by agentofoblivion » Thu, Jul 05, 2018, 4:20 PM UTC

I also found the book to be pretentious and full of nonsense. It gave the impression something profound was being said, but was mostly poetic license that might be charming to some, but I found to be irritating since it seemed to confuse rather than clarify.

Re: Why I Don't Love Gödel, Escher, Bach

#146

Post by vignesh_m » Thu, Jul 05, 2018, 4:20 PM UTC

beaconstudios wrote: post #61

chriswarbo wrote: post #37

Earlier quoted context omitted.

(Caveat: I'm a computer scientist, not a mathematician) The halting problem, and universal computation itself, came directly out of work on incompleteness. Incompleteness is about being unable to prove statements in a particular system; it leaves open the possibility that a more powerful system might be able to prove that statement (but that more powerful system will have its own incompleteness, and so on). Turing go…

the question that fascinates me is - what is the property of a program that makes it undecidable? I've been playing around with the idea of trying to make a program that determines halts/doesn't halt/don't know, given some representation of a program. I'm running into some interesting implications when I incorporate dependent typing, but it feels like there must be something theoretical that's already out there.

Turing machines output yes/no or indefinitely go on, so that might be like your possible outputs of halts/doesn't halt/don't know. I'm not really sure how much to assume you know, so the [wiki page](https://en.wikipedia.org/wiki/Halting_problem) might help

Re: Why I Don't Love Gödel, Escher, Bach

#147

Post by honoredb » Thu, Jul 05, 2018, 4:22 PM UTC

Nitpick: "…had 13 not been prime" is not meaningless whimsy--a prime number of images is harder to lay out on the page. Mentioning this in the caption without explaining it is whimsical, but like a lot of GEB I'd argue it's whimsy with a purpose: training the reader to look for applications of abstract concepts. The reader ideally goes "wait, what? There's no connection between those two things! Unless..."

Re: Why I Don't Love Gödel, Escher, Bach

#148

Post by extralego » Thu, Jul 05, 2018, 4:24 PM UTC

jweather wrote: post #136

I've read it cover-to-cover at least three times, gifted a copy, and read it out loud to my son. I find it very playful and mind-expanding. I will admit that it is verbose and wandering, but I think that's part of the charm. Out of curiosity I wandered through Amazon's 1-star reviews of this book to see what other people had to say. "Obtuse and impenetrable", I can certainly understand. It takes a certain bent to put…

I agree with you for the most part. My central issue while reading was Hofstadter’s self-references. They eventually fall into the background but it was a turn-off for me in the beginning. And they aren’t inherently distasteful but in context of the ‘wandering’ and precarious nature of the work, I had to wonder if the dependency on one’s own series of particular fascinations do have an intellectual corollary or not. If I had to conclude, I’d say they do, but the book itself might still be a house of cards, however inspiring and all the rest.

Re: Why I Don't Love Gödel, Escher, Bach

#149

Post by lisper » Thu, Jul 05, 2018, 4:24 PM UTC

beaconstudios wrote: post #139

lisper wrote: post #131

Earlier quoted context omitted.

> it's self-evident _at this point in time_ No, it's not. Just because you and I don't know the answer is not proof that no one knows the answer. Someone may have discovered the answer in the last five minutes. (In fact, for any mathematical discovery there is always necessarily a period time when only one person knows the answer.) > AFAIK not all problems relate directly to this space And I am telling you that you a…

alright, if you're going to resort to nitpicking and being wilfully insulting then I see there's not much point discussing this further with you.

Let me rephrase that to hopefully make it sound a little less insulting: if you are able to identify an aspect of computation that cannot be described mathematically then you will be remembered alongside Turing, Godel and von Neuman as one of the greatest thinkers humanity has ever produced. (But a priori, the odds of your being able to do that seem pretty low to me.)

Re: Why I Don't Love Gödel, Escher, Bach

#150

Post by vignesh_m » Thu, Jul 05, 2018, 4:24 PM UTC

lisper wrote: post #102

beaconstudios wrote: post #95

Earlier quoted context omitted.

but that's exactly my point - there are clear-cut halting, clear-cut non-halting, and "we don't know yet" programs. My interest is in delineating the properties that separate the 3. The collatz conjecture is exactly the example I use when explaining the "we don't know yet" programs. If I could build a logic system that can positively prove halting, and one that can positively prove non-halting, then anything that rem…

> the properties that separate the 3 But that is exactly my point: what separates these three classes is the state of our current mathematical knowledge, and that obviously changes over time. > Which is pretty self-evident No, that is not at all self-evident. (If it were, someone would have proven it long before Turing, and he wouldn't be famous for it.) I don't know what you mean by a "bounding" problem, but we can…

I think op's problem is different than the one normally studied - he wants to design a halting program and find the class of programs for which it works.

This is typically not very popular as these statements depend a lot on the program, and the representations of programs unlike the halting problem computability statement.