Wednesday, April 27, 2005
Chaitin, The Unknowable
Gödel's incompleteness, Turing's uncomputability, Chaitin's randomness
"PREFACE
Having published four books on this subject, why a fifth?! Because there's something new: I compare and contrast Gödel's, Turing's and my work in a very simple and straight-forward manner using LISP.
Up to now I never wanted to examine Gödel's and Turing's work too closely—I wanted to develop my own viewpoint. But there is no longer any danger. So I set out to explain the mathematical essence of three very different ways to exhibit limits to mathematical reasoning: the way Gödel and Turing did it in the 1930s, and my way that I've been working on since the 1960s.
In a nutshell, Gödel discovered incompleteness, Turing discovered uncomputability, and I discovered randomness—that's the amazing fact that some mathematical statements are true for no reason, they're true by accident. There can be no ``theory of everything,'' at least not in mathematics. Maybe in physics!
I didn't want to write a ``journalistic'' book. I wanted to explain the fundamental mathematical ideas understandably. And I think that I've found a way to do it, and that I understand my predecessors' work better than I did before. The essence of this book is words, explaining mathematical ideas, but readers who feel so inclined can follow me all the way to LISP programs that pretty much show Gödel's, Turing's and my proofs working on the computer. And if you want to play with this software, you can download it from my web site.
This book is also a ``prequel'' to my Springer book The Limits of Mathematics. It's an easier introduction to my ideas, and uses the same version of LISP that I use in The Limits of Mathematics. I hope it'll be a stepping stone for those for whom The Limits of Mathematics is too intimidating.
"
"PREFACE
Having published four books on this subject, why a fifth?! Because there's something new: I compare and contrast Gödel's, Turing's and my work in a very simple and straight-forward manner using LISP.
Up to now I never wanted to examine Gödel's and Turing's work too closely—I wanted to develop my own viewpoint. But there is no longer any danger. So I set out to explain the mathematical essence of three very different ways to exhibit limits to mathematical reasoning: the way Gödel and Turing did it in the 1930s, and my way that I've been working on since the 1960s.
In a nutshell, Gödel discovered incompleteness, Turing discovered uncomputability, and I discovered randomness—that's the amazing fact that some mathematical statements are true for no reason, they're true by accident. There can be no ``theory of everything,'' at least not in mathematics. Maybe in physics!
I didn't want to write a ``journalistic'' book. I wanted to explain the fundamental mathematical ideas understandably. And I think that I've found a way to do it, and that I understand my predecessors' work better than I did before. The essence of this book is words, explaining mathematical ideas, but readers who feel so inclined can follow me all the way to LISP programs that pretty much show Gödel's, Turing's and my proofs working on the computer. And if you want to play with this software, you can download it from my web site.
This book is also a ``prequel'' to my Springer book The Limits of Mathematics. It's an easier introduction to my ideas, and uses the same version of LISP that I use in The Limits of Mathematics. I hope it'll be a stepping stone for those for whom The Limits of Mathematics is too intimidating.
"
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