Gödel, Turing and Cantor: The Math(skibinsky.com)
skibinsky.com
Gödel, Turing and Cantor: The Math
http://skibinsky.com/godel-turing-and-cantor-the-math/
7 comments
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.
+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 The rest felt more like Hofstadter's random musings on AI, DNA and some other topics (some of which felt outdated).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.
Yeah it took me about 7 months to finish it, and you are right, every time I just felt beat and my mind was racing, what a great book.
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
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
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).
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).
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?
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There's a translation of Gödel's paper linked via his wikipedia entries https://docs.google.com/viewer?url=http://www.research.ibm.c... [google docs viewer]
curious facts: he died of starvation; he was a theist http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#Later_years_and...
curious facts: he died of starvation; he was a theist http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#Later_years_and...
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.
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.
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.
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.
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."
'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."
Chaitin is cool, too.
Definitely. I'd recommend his book Meta Math: The Quest for Omega [1]. He posted the full text of the book on arXiv [2].
[1] https://www.amazon.co.uk/Meta-Maths-Gregory-J-Chaitin/dp/184... [2] http://arxiv.org/abs/math/0404335
[1] https://www.amazon.co.uk/Meta-Maths-Gregory-J-Chaitin/dp/184... [2] http://arxiv.org/abs/math/0404335
Chaitin's work is genius, but one thing bugs me: he uses a Cantor space in exchange for Gödel's countable numbering. I haven't found any commentary on that, but I find it suspicious.
The problem with Cantor space is you simply can't even begin to approximate Chaitin's constant, but it seems like you could by using a countable numbering (all else being equal.) Of course that would blow everything up, so the takeaway is there is something very important which I'm not convinced we've learned yet.
The problem with Cantor space is you simply can't even begin to approximate Chaitin's constant, but it seems like you could by using a countable numbering (all else being equal.) Of course that would blow everything up, so the takeaway is there is something very important which I'm not convinced we've learned yet.
Came here to say that. See [1] my review of his awesome book "The Limits of Mathematics" + HN discussion.
[1] https://news.ycombinator.com/item?id=1725936
[1] https://news.ycombinator.com/item?id=1725936
What I find interesting are the claims Penrose[1] and others make about human versus machine intelligence based on Godel's results.
[1] https://en.wikipedia.org/wiki/Philosophy_of_artificial_intel...
[1] https://en.wikipedia.org/wiki/Philosophy_of_artificial_intel...
I'm sceptical of Penrose's arguments. We don't see human brains as
symbol-manipulating formal systems, but that doesn't mean that aren't (or, more
correctly, that doesn't mean that there is no symbol-manipulating formal system
which is isomorphic to a particular brain, or brains in general).
Brains aren't implementing any known algorithm, but that doesn't mean they aren't implementing any unknown algorithm.
It brings to mind Minksky's advice to Sussman http://en.wikipedia.org/wiki/Hacker_koan#Uncarved_block
Brains aren't implementing any known algorithm, but that doesn't mean they aren't implementing any unknown algorithm.
It brings to mind Minksky's advice to Sussman http://en.wikipedia.org/wiki/Hacker_koan#Uncarved_block
It would be interesting to speculate the consequence of Godel's Incompleteness deductions on the quest for Grand Unified Field Theorems in theoretical physics
Does this mean that the antivirus industry is nothing but a scam?
it means antivirus industry will never be "complete"
Indeed, many incompleteness results (the most famous being the Halting Problem)
are known as "infinite employment theorems". For example, we can always make a
compiler perform more optimisations as follows:
1) Every diverging (non-terminating, non-co-terminating) program can be optimised to the following:
10 GOTO 10
2) The Halting Problem tells us that no compiler can spot every diverging program.
3) Hence there is room for more optimisation, by spotting more diverging programs.
4) Hence there is always more work to do for compiler developers.
1) Every diverging (non-terminating, non-co-terminating) program can be optimised to the following:
10 GOTO 10
2) The Halting Problem tells us that no compiler can spot every diverging program.
3) Hence there is room for more optimisation, by spotting more diverging programs.
4) Hence there is always more work to do for compiler developers.
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.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814...