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Gödel, Turing and Cantor: The Math

skibinsky.com

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Re: Gödel, Turing and Cantor: The Math

#21

Post by chriswarbo »

wicknicks wrote: post #2

Oh wow! This is a great write up on Godel's work. Anybody who even vaguely cares about fundamentals of computer science should definitely give it a read, and if possible a thorough read. Slightly related: Although a more technical/deeper discussion, but the book "Godel's Proof"[1] by Nagel and Newman is a very approachable text in this domain, and explains many aspects of the incompleteness theorems. [1] http://www.a…

I found it quite approachable and thorough, although it throws the the diagonal lemma out there without any real explanation even though it's crucial to the reasoning that follows it.

Also I disagree with the completely unfounded assumptions at the end that the (terribly named) Reals have something to do with reality. The Reals are a Mathematical curiosity at best, but more often than not they complicate the understanding of subjects to which they have no relevance, eg. fractions (especially their decimal notation), calculus, physics, computing (especially floating point) and so on.

It's fine to treat infinite constructs like the Reals declaratively, eg. as functions which can be composed, but it's meaningless to reason about doing things to their 'final results', since there are no such things by definition. In more precise terms, it makes sense to reason about the output of co-terminating functions (which loop forever, spitting out, for example, a never-ending sequence of digits) but not diverging functions (which loop forever without ever getting as far as their first digit).

Re: Gödel, Turing and Cantor: The Math

#22

Post by hamiltonkibbe »

hobs wrote: post #3
wicknicks wrote: post #2

Oh wow! This is a great write up on Godel's work. Anybody who even vaguely cares about fundamentals of computer science should definitely give it a read, and if possible a thorough read. Slightly related: Although a more technical/deeper discussion, but the book "Godel's Proof"[1] by Nagel and Newman is a very approachable text in this domain, and explains many aspects of the incompleteness theorems. [1] http://www.a…

Honorable mention for http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach which has been mentioned many times on HN. Has many different ways of explaining Godel's genius. Though to be fair you could just read this article and be good.

GEB is awesome.

David Foster Wallace's Everything and More: A Compact History of Infinity covers Cantor and is an interesting read about... you guessed it.. Infinity.

Re: Turing http://www.lel.ed.ac.uk/~gpullum/loopsnoop.html

Re: Gödel, Turing and Cantor: The Math

#23

Post by stiff »

chriswarbo wrote: post #21
wicknicks wrote: post #2

Oh wow! This is a great write up on Godel's work. Anybody who even vaguely cares about fundamentals of computer science should definitely give it a read, and if possible a thorough read. Slightly related: Although a more technical/deeper discussion, but the book "Godel's Proof"[1] by Nagel and Newman is a very approachable text in this domain, and explains many aspects of the incompleteness theorems. [1] http://www.a…

I found it quite approachable and thorough, although it throws the the diagonal lemma out there without any real explanation even though it's crucial to the reasoning that follows it. Also I disagree with the completely unfounded assumptions at the end that the (terribly named) Reals have something to do with reality. The Reals are a Mathematical curiosity at best, but more often than not they complicate the understa…

The real numbers have no relevance to calculus? Is there a way of doing calculus without real numbers that is simpler? What are you talking about?

Re: Gödel, Turing and Cantor: The Math

#24

Post by pachydermic »

laxatives wrote: post #12
tekromancr wrote: post #5

Earlier quoted context omitted.

+1 for GEB. I have been reading it on and off for about 2 years. It's a fun book, but every time I finish a reading session, I am exhausted. I am about 25% through.

IMO just finish chapter 14 and you've basically finished the Godel portion of the book. The rest felt more like Hofstadter's random musings on AI, DNA and some other topics (some of which felt outdated). There's some crazy anecdotes of Ramanujin that I had never heard elsewhere though.

Well the structure of the book is supposed to be like a JS Bach song - so in the end, what you get is a combining of all the familiar themes and sometimes it feels a bit repetitive and boring. But if you've read that far, it's worth reading the rest. It's much easier to get through than when you're seeing stuff for the first time and really trying to wrap your head around it.

Re: Gödel, Turing and Cantor: The Math

#25

Post by mvp »

This article took me to another from the same author

'http://skibinsky.com/godel-incompleteness-for-startups/#foot...

and I found the below statement and laughed out aloud. Although in the context, this statement makes sense, in general, I trust books that don't belong to human-related matters (technical etc), because in human-related matters it is mostly one's opinion against others.

"As soon as these popular books leave the domain of human-related matters you are totally on your own."

Re: Gödel, Turing and Cantor: The Math

#26

Post by kybernetikos »

Radim wrote: post #18
kybernetikos wrote: post #14

Earlier quoted context omitted.

I found your semicolon quite confusing there - it seemed to suggest that there was some relationship between being a theist and dying of starvation.

On the contrary. I imagine 6ren went to extra lengths to use a semicolon instead of a simple comma, as semicolons are used to emphasize independent clauses.

Independent clauses can stand by themselves grammatically, that doesn't mean that their meanings are independent.

e.g.

> I hit him as hard as I could; he laughed at me.

Those are 'independent clauses', but the semicolon emphasizes the connection between them.

Re: Gödel, Turing and Cantor: The Math

#27

Post by 6ren »

kybernetikos wrote: post #26
Radim wrote: post #18

Earlier quoted context omitted.

On the contrary. I imagine 6ren went to extra lengths to use a semicolon instead of a simple comma, as semicolons are used to emphasize independent clauses.

Independent clauses can stand by themselves grammatically , that doesn't mean that their meanings are independent. e.g. > I hit him as hard as I could; he laughed at me. Those are 'independent clauses', but the semicolon emphasizes the connection between them.

Radim is right. I actually tried a couple of other forms ("and" and ","), but they seemed worse. Challenge: can you demonstrate a better way?

I think the problem is not syntax but semantics: two unusual facts beg connection.

BTW: getting back on topic, his starvation seems due to a lack of faith... if some connection is expected, perhaps the best solution is to supply one, perhaps the poignant He believed in God, yet died from lack of faith.

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