Form and Content
Early in the twentieth century, Alfred North Whitehead and Bertrand Russell set about devising a mathematical system that would be consistent and complete, one that could generate every true statement about number theory without producing any false ones. The result was their monumental Principia Mathematica (1910-1913). In 1931, the twenty-five-year-old Czech mathematician Kurt Gödel undercut this massive work with a short paper demonstrating that while for practical purposes the Whitehead-Russell system achieved its goal, in fact certain true propositions in number theory remained undecidable in their scheme. Moreover, Gödel showed that no formal system could be consistent and complete; any formulation powerful enough to produce almost all truths about natural numbers (integers greater than zero) would necessarily be flawed. Douglas Hofstadter draws an analogy to illustrate this point. If a certain record player reproduced sound with sufficiently high fidelity, one could create a record capable of producing resonances within the machine that could destroy it. Thus, no phonograph will be able to play every record.
Gödel’s so-called Incompleteness Theorem deeply disturbed mathematicians seeking a perfectly logical, ordered universe. It was, in fact, the equivalent for mathematics of Werner Heisenberg’s uncertainty principle, Albert Einstein’s general theory of relativity, and Max Planck’s discovery of quantum mechanics. All revealed the mythical nature of the traditional view of science as fixed, orderly, and rational.
Hofstadter initially intended to write a short book about Gödel’s theorem, similar in content and brevity to Ernest Nagel and James R. Newman’s Gödel’s Proof (1958); in his bibliography, Hofstadter credits this work as the inspiration for...
(The entire section is 741 words.)