Meta Math! The Quest For Omega

In his book Everything and More: A Compact History of Infinity, David Foster Wallace refers to Godel as "modern math's absolute Prince of Darkness" (p. 275) and states that because of him "pure math's been in mid-air for the last 70 years" (p. 284). In other words, according to Wallace, since Godel published his famous paper in 1931, mathematics has been suspended hanging in mid-air without anything like a proper foundation. It is high time these dark thoughts were permanently laid to rest. Hilbert's century-old vision of a static completely mechanical absolutely rigorous formal mathematics was a misguided attempt intended to demonstrate the absolute certainty of mathematical reasoning. It is time for us to recover from this disease! Godel's 1931 work on incompleteness, Turing's 1936 work on uncomputability, and my own work on the role of information, randomness and complexity have shown increasingly emphatically that the role that Hilbert envisioned for formalism in mathematics is best served by computer programming languages, which are in fact formalisms that can be mechanically interpretedbut they are formalisms for computing and calculating, not for reasoning, not for proving theorems, and most emphatically not for inventing new mathematical concepts nor for making new mathematical discoveries. In my opinion, the view that math provides absolute certainty and is static and perfect while physics is tentative and constantly evolving is a false dichotomy. Math is actually not that different from physics. Both are attempts of the human mind to organize, to make sense of, human experience; in the case of physics, experience in the laboratory, in the physical world; and in the case of math, experience in the computer, in the mental mindscape of pure mathematics. And mathematics is far from static and perfect; it is constantly evolving, constantly changing, constantly morphing itself into new forms. New concepts are constantly transforming math and creating new fields, new viewpoints, new emphasis, and new questions to answer. And mathematicians do in fact utilize unproved new principles suggested by computational experience, just as a physicist would. And in discovering and creating new mathematics, mathematicians do base themselves on intuition and inspiration, on unconscious motivations and impulses, and on their aesthetic sense, just like any creative artist would. And mathematicians do not lead logical mechanical "rational" lives. Like any creative artist, they are passionate emotional people who deeply care about their art, they are unconventional eccentrics motivated by mysterious forces, not by money nor by a concern for the "practical applications" of their work. I know, because I'm one of these crazy people myself! I've been obsessed by these questions for my whole life, starting at an early age. And I'll give you an insider's view of all of this, a firsthand report from the front, where there is still a lot of fighting, a lot of pushing and shoving, between different viewpoints. In fact basic questions like this are never settled, never definitively put aside, they have a way of resurfacing, of popping up again in transformed form, every few generations . . . 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 iChaitin, Gregory is the author of 'Meta Math! The Quest For Omega', published 2005 under ISBN 9780375423130 and ISBN 0375423133.