Exclamation marks! Even in the title! And first person singular, oh my! On almost every page! With bold and italic text to hammer home key points!
No, Meta Math!: The Quest for Omega isn't your usual mathematics book. Its author, Gregory Chaitin, isn't shy about blowing his own horn. A decade ago I read another Chaitin book on the same topic, title now forgotten. Deja vu rose up like a tide over Meta Math! — he's written about this before.
What's Meta Math!? In brief, Chaitin approaches incompleteness, complexity, and provability from an algorithmic information theory viewpoint. He draws broad philosophical conclusions — extremely broad conclusions. As he summarizes in the Introduction:
So that's what this book is about: It's about reasoning questioning itself, and its limits and the role of creativity and intuition, and the sources of new ideas and of new knowledge. That's a big subject, and I only understand a little bit of it, the areas that I've worked in or experienced myself. Some of this nobody understands very well, it's a task for the future. How about you?! Maybe you can do some work in this area. Maybe you can push the darkness back a millimeter or two! Maybe you can come up with an important new idea, maybe you can imagine a new kind of question to ask, maybe you can transform the landscape by seeing it from a different point of view! That's all it takes, just one little new idea, and lots and lots of hard work to develop it and to convince other people! Maybe you can put a scratch on the rock of eternity!
Remember that math is a free creation of the human mind, and as Cantor—the inventor of the modern theory of infinity described by Wallace—said, the essence of math resides in its freedom, in the freedom to create. But history judges these creations by their enduring beauty and by the extent to which they illuminate other mathematical ideas or the physical universe, in a word, by their "fertility." Just as the beauty of a woman's breasts or the delicious curve of her hips is actually concerned with childbearing, and isn't merely for the delight of painters and photographers, so a math idea's beauty also has something to do with its "fertility," and with the extent to which it enlightens us, illuminates us, and inspires us with other ideas and suggests unsuspected connections and new viewpoints.
Ahem! But setting aside indelicate metaphors, how new are Chaitin's discoveries? Did Gödel, Church, Turing, et al. plow this field long ago? In a rather scathing review of two earlier Chaitin books, mathematician Panu Raatikainen suggests that the answer is "Yes". Raatikainen raises significant scientific and historical issues with Chaitin's statements and notes, "At worst, Chaitin's claims are nearly megalomaniacal."
Echoes of Benoit Mandelbrot, who popularized and promoted fractal geometry, and of Stephen Wolfram in his work on cellular automata. Revolutionary? Time will decide. But the contrast with Richard Feynman is striking. Feynman's new approaches (path-integral methods) solved tough, important problems in quantum electrodynamics and elsewhere in physics. But Feynman didn't do a relentless self-promotion schtick, and he didn't re-write the same book every few years. He let the work speak for itself.
^z - 2011-08-30