A century of controversy over the foundations of mathematics (2000)(arxiv.org)
arxiv.org
A century of controversy over the foundations of mathematics (2000)
https://arxiv.org/html/nlin/0004007
13 comments
> 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.
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.
Related:
A Century of Controversy over the Foundations of Mathematics (1999) - 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 - June 2009 (22 comments)
A Century of Controversy over the Foundations of Mathematics (1999) - 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 - June 2009 (22 comments)
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.
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.
The videotape link doesn't work, but the videos are available on Youtube: https://youtube.com/playlist?list=PL86ECDEDE3FA8D8D1
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...
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...
There is no such thing as an unusual number.
> 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.
I almost cried when I read that. Would love to see it all laid out.
The connection between these is just the Cantor's diagonal argument https://en.wikipedia.org/wiki/Diagonal_argument that was mentioned.
Right, but I wanted him to take time to go into it.
Interesting to see this on arXiv even though not a math paper
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
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
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....
[2] https://jillian.rootaction.net/~jillian/science/chaitin/www....
[3] https://jillian.rootaction.net/~jillian/science/chaitin/www....
[4] https://jillian.rootaction.net/~jillian/science/chaitin/www....
[5] https://jillian.rootaction.net/~jillian/science/chaitin/www....
[6] https://jillian.rootaction.net/~jillian/science/chaitin/www....
[7] https://link.springer.com/book/9789814021722