Philosophy15 Nov 2008 08:25 am
Between 1910 and 1913, the Cambridge philosophers Bertrand Russell and Alfred North Whitehead published one of the landmark works of Analytic philosophy: the Principia Mathematica. Over the course of three volumes Russell and Whitehead purport to derive all of the major principles of mathematics from a series of simple axioms, and logical proofs leading from these simple beginnings to greater and greater complexity. They propose a self-contained system of mathematics that admits of no error and no paradox. It was one of the great landmarks of philosophy, mathematics, and logic, and in 1931 a 25-year-old German mathematician named Kurt Gödel proved the whole thing wrong.
Russell and Whitehead attempted to build mathematics into a self-contained logical system. Gödel didn’t simply find an error or inconsistency in their work—that could have been corrected easily enough. Gödel went much, much farther: he proved that every formal system has inherent limitations, the Principia among many others.
The fundamental idea behind Gödel’s proof is that every logical system contains a version of the paradoxical statement “This sentence is false.” (Think that one through for a minute.) Therefore, deriving a complete system of mathematics from a set of axioms (the goal of the Principia) is inherently impossible. This is a very important result to philosophers, logicians and mathematicians, but in 1999 it became a pop culture phenomena. Douglas Hofstadter made the idea famous in his book Gödel, Escher, Bach. It’s a brilliant and long-ranging book, delving into the illusions you can see in M. C. Escher’s prints (like the one above), the complexities you can hear in J. S. Bach’s fugues, and the paradoxes of Gödel’s proof. I heartily recommend it to all Devoted Intellectuals. However, I’d start by reading another book first. Hofstadter claims that he first became entranced with Gödel’s proof when he read an explanation of it written by Ernest Nagel and James R. Newman at the age of fourteen. After the success of his book, Nagel and Newman’s work was republished with an introduction by Hofstadter. It’s tough going at times (The Devoted Intellectual is still confused about “Gödel Numbers”), but Gödel’s Proof is a project every Devoted Intellectual should tackle.
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