And thus, in one of the keenest insights in the history of mathematics, Gödel devised a remarkable statement that said simply, "X is not an M.P. number" where X is the exact number we read when the statement "X is not an M.P. number" is translated into Martian math notation. Think about this for a little while until you get it. Translated into Martian notation, the statement "X is not an M.P. number" will look to us like just some huge string of digits — a very big numeral. But that string of Martian writing is our numeral for the number X (about which the statement itself talks). Talk about twisty; this is really twisty! But twists were Gšdel's specialty — twists in the fabric of space-time, twists in reasoning, twists of all sorts.

By thinking of theorems as patterns of symbols, Gšdel discovered that it is possible for a statement in a formal system not only to talk about itself, but also to deny its own theoremhood. The consequences of this unexpected tangle lurking inside mathematics were rich, mind-boggling and — rather oddly — very sad for the Martians. Why sad? Because the Martians--like Russell and Whitehead — had hoped with all their hearts that their formal system would capture all true statements of mathematics. If Gšdel's statement is true, it is not a theorem in their textbooks and will never, ever show up — because it says it won't! If it did show up in their textbooks, then what it says about itself would be wrong, and who — even on Mars — wants math textbooks that preach falsehoods as if they were true?

The upshot of all this is that the cherished goal of formalization is revealed as chimerical. All formal systems — at least ones that are powerful enough to be of interest — turn out to be incomplete because they are able to express statements that say of themselves that they are unprovable. And that, in a nutshell, is what is meant when it is said that Gödel in 1931 demonstrated the "incompleteness of mathematics." It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules. To you that may not come as a shock, but to mathematicians in the 1930s, it upended their entire world view, and math has never been the same since.

Gödel's 1931 article did something else: it invented the theory of recursive functions, which today is the basis of a powerful theory of computing. Indeed, at the heart of Gödel's article lies what can be seen as an elaborate computer program for producing M.P. numbers, and this "program" is written in a formalism that strongly resembles the programming language Lisp, which wasn't invented until nearly 30 years later.

Gšdel the man was every bit as eccentric as his theories. He and his wife Adele, a dancer, fled the Nazis in 1939 and settled at the Institute for Advanced Study in Princeton, where he worked with Einstein. In his later years Gödel grew paranoid about the spread of germs, and he became notorious for compulsively cleaning his eating utensils and wearing ski masks with eye holes wherever he went. He died at age 72 in a Princeton hospital, essentially because he refused to eat. Much as formal systems, thanks to their very power, are doomed to incompleteness, so living beings, thanks to their complexity, are doomed to perish, each in its own unique manner.

Douglas Hofstadter is the Pulitzer-prizewinning author of Gödel, Escher, Bach

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