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Comment by ericssmith | Hacker News Reader

Although it's nice to see Gödel show up in the popular press, I didn't get what the point of this was. His Incompleteness paper is not that impenetrable. Sometimes it does pay to at least look at primary sources in addition to listening to others' efforts to summarize or dissect. I don't think the author of this article made much of an effort to understand what or who she was writing about. Sad, really, given Gödel's importance in understanding the beginnings of a science of computation (easily one of the most significant achievements in human history), where there is already too much myth-making and mischaracterizations.

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Thank you for your Godel notes below! I'm going to check out the links.

I have the Brooklyn Institute for Social Research in my Facebook feed and they're mentioned in the article. I think the point of the article is to mark the work the teacher of the Godel class at the Institute did to digest the material for the students of his/her class.

The preliminary milestone of digestion of a subject for teaching is a syllabus, also mentioned in the article.

I agree that the article was a little thin (to be honest, it's shorter than most articles I bother to read, let alone repost) & that it's misleading in suggesting that incompleteness is difficult to understand.

That said, I'm very glad that the idea is finally popping up in the popular press: incompleteness, like special relativity & quantum superposition, is about a hundred years old & many people still don't have sufficient familiarity with it to understand the way that the philosophy of the field in appeared in was impacted. Where relativity meant that time was mutable & superposition meant that randomness was inescapable (in other words, breaking the "Newton's Calculator" model of physics), incompleteness means that self-description breeds undecidability & that the relatively common idea of an ultimate descriptive language is inherently doomed. These are important facts from a philosophical standpoint, even if they rarely directly interact with things that 'normal' people are trying to do (although 'normal' people bump up indirectly against relativity & incompleteness daily, in the form of (for instance) GPS devices).

"His Incompleteness paper is not that impenetrable." May I ask what do you mean by that?

Sure. The original 1931 paper is 20-something pages long:

http://www.w-k-essler.de/pdfs/goedel.pdf

Here's an English translation (with a lengthy introduction)

http://jacqkrol.x10.mx/assets/articles/godel-1931.pdf

Even without trying to follow the proof proper, the sub-sections of the second part are interesting on their own, particularly Gödel numbering and primitive recursive functions. Here is another translation that covers just this part:

http://www.research.ibm.com/people/h/hirzel/papers/canon00-g...

It's true that if you know nothing about formal logic, history of metamathematics, and decidability, then it's going to be particularly hard going, but there are a lot of accessible resources for each of those topics and the paper is well structured (meaning you can concentrate on the pieces).

The encoding that Gödel used for formulas should be fascinating for anyone familiar with Turing work on decidability as well as how computers work generally. Primitive recursive functions don't handle computation generally, but seem to be a first step in understanding what it means. Anyone familiar with Alonzo Church, lambda calculus, functional programming, McCarthy's first paper on Lisp would probably be interested in this bit.

Of course, Gödel's result on formal systems shattered the idea of an axiomatic basis for mathematics, but I personally think its greater long-term impact is helping to usher in computation. It's worth recognizing both.

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