10 ms·

A century of controversy over the foundations of mathematics (2000)

39 points by yamrzou 4y ago

paulpauper 4y ago
Interesting to see this on arXiv even though not a math paper
- drewcoo 4y ago
  It's a (n edited) transcript of a CMU School of Computer Science Distinguished Lecture. By Dr. Chaitin. On reading this, I'm pretty sure the point of the talk was not the vainglory of being hosted on a little-known website in 2000. But to understand that would be to understand history. https://en.wikipedia.org/wiki/Gregory_Chaitin https://en.wikipedia.org/wiki/Gregory_Chaitin
Not_Oles 4y ago
There seems also to be a conveniently available 1998 version[1] which, unlike the version linked in the OP, contains still-working links to several additional chapters.[2-6] I don't know how the content compares to Chaitin's books, including The Unknowable (1999).[7] [1] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch1.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [2] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch2.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [3] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch3.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [4] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch4.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [5] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch5.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [6] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch6.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [7] https://link.springer.com/book/9789814021722 https://link.springer.com/book/9789814021722
whatshisface 4y ago
It turns out that a number being odd is almost inevitable. How do we know that? Well, take a look at this sequence: 1 2 3 5 4 7 9 11 13 6 ... As you can see, there's a lot of odd numbers between every even number, and the fraction of even numbers falls and falls as you keep going. Therefore, virtually every number is odd...
- chris_wot 4y ago
  There is no such thing as an unusual number.
charlieflowers 4y ago
I love the way this is written to be as simple and understandable as possible. Too bad more writing in mathematics and science is not approached this way.
- sammalloy 4y ago
  Feynman argued in the case of US school textbooks, it was due to states taking the lowest bidder, with the resulting books being almost unreadable.
sammalloy 4y ago
> all these ideas are connected, but there's no time to go into that I almost cried when I read that. Would love to see it all laid out.
- jhanschoo 4y ago
  The connection between these is just the Cantor's diagonal argument https://en.wikipedia.org/wiki/Diagonal_argument https://en.wikipedia.org/wiki/Diagonal_argument that was mentioned.
  - sammalloy 4y ago
    Right, but I wanted him to take time to go into it.
yamrzou 4y ago
The videotape link doesn't work, but the videos are available on Youtube: https://youtube.com/playlist?list=PL86ECDEDE3FA8D8D1 https://youtube.com/playlist?list=PL86ECDEDE3FA8D8D1
dang 4y ago
Related: A Century of Controversy over the Foundations of Mathematics (1999) - https://news.ycombinator.com/item?id=4335497 https://news.ycombinator.com/item?id=4335497 - Aug 2012 (1 comment) A Century of Controversy Over the Foundations of Mathematics - https://news.ycombinator.com/item?id=644699 https://news.ycombinator.com/item?id=644699 - June 2009 (22 comments)
TuringTest 4y ago
> So I like to apologize in an aggressive way about my field. I like to say that my field has no applications, that the most interesting thing about the field of program-size complexity is that it has no applications, that it proves that it cannot be applied! Because you can't calculate the size of the smallest program. But that's what's fascinating about it, because it reveals limits to what we can know. That's why program-size complexity has epistemological significance. The key idea I get from this field is that mathematics and rational thought can create logical systems more complex than what it can deterministically study. Thus, any scholar of any theoretical discipline should be warned that the very activity of expanding its field generates unknowable entities faster than it uncovers certain knowledge.

39 points by yamrzou 4y ago

paulpauper 4y ago

Interesting to see this on arXiv even though not a math paper

drewcoo 4y ago
It's a (n edited) transcript of a CMU School of Computer Science Distinguished Lecture. By Dr. Chaitin. On reading this, I'm pretty sure the point of the talk was not the vainglory of being hosted on a little-known website in 2000. But to understand that would be to understand history. https://en.wikipedia.org/wiki/Gregory_Chaitin https://en.wikipedia.org/wiki/Gregory_Chaitin

Not_Oles 4y ago

There seems also to be a conveniently available 1998 version[1] which, unlike the version linked in the OP, contains still-working links to several additional chapters.[2-6] I don't know how the content compares to Chaitin's books, including The Unknowable (1999).[7] [1] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch1.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [2] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch2.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [3] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch3.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [4] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch4.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [5] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch5.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [6] https://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/ch6.html https://jillian.rootaction.net/~jillian/science/chaitin/www.... [7] https://link.springer.com/book/9789814021722 https://link.springer.com/book/9789814021722

whatshisface 4y ago

It turns out that a number being odd is almost inevitable. How do we know that? Well, take a look at this sequence: 1 2 3 5 4 7 9 11 13 6 ... As you can see, there's a lot of odd numbers between every even number, and the fraction of even numbers falls and falls as you keep going. Therefore, virtually every number is odd...

chris_wot 4y ago
There is no such thing as an unusual number.

charlieflowers 4y ago

I love the way this is written to be as simple and understandable as possible. Too bad more writing in mathematics and science is not approached this way.

sammalloy 4y ago
Feynman argued in the case of US school textbooks, it was due to states taking the lowest bidder, with the resulting books being almost unreadable.

sammalloy 4y ago

> all these ideas are connected, but there's no time to go into that I almost cried when I read that. Would love to see it all laid out.

jhanschoo 4y ago
The connection between these is just the Cantor's diagonal argument https://en.wikipedia.org/wiki/Diagonal_argument https://en.wikipedia.org/wiki/Diagonal_argument that was mentioned.
- sammalloy 4y ago
  Right, but I wanted him to take time to go into it.

yamrzou 4y ago

The videotape link doesn't work, but the videos are available on Youtube: https://youtube.com/playlist?list=PL86ECDEDE3FA8D8D1 https://youtube.com/playlist?list=PL86ECDEDE3FA8D8D1

dang 4y ago

Related: A Century of Controversy over the Foundations of Mathematics (1999) - https://news.ycombinator.com/item?id=4335497 https://news.ycombinator.com/item?id=4335497 - Aug 2012 (1 comment) A Century of Controversy Over the Foundations of Mathematics - https://news.ycombinator.com/item?id=644699 https://news.ycombinator.com/item?id=644699 - June 2009 (22 comments)

TuringTest 4y ago

> So I like to apologize in an aggressive way about my field. I like to say that my field has no applications, that the most interesting thing about the field of program-size complexity is that it has no applications, that it proves that it cannot be applied! Because you can't calculate the size of the smallest program. But that's what's fascinating about it, because it reveals limits to what we can know. That's why program-size complexity has epistemological significance. The key idea I get from this field is that mathematics and rational thought can create logical systems more complex than what it can deterministically study. Thus, any scholar of any theoretical discipline should be warned that the very activity of expanding its field generates unknowable entities faster than it uncovers certain knowledge.