Mathematician KURT GODEL – TIME

Mathematician KURT GODEL

By Douglas Hofstadter Monday, Mar. 29, 1999

Kurt Godel was born in 1906 in Brunn… By age 10, Godel was studying math, religion and several languages. By 25 he had produced what many consider the most important result of 20th century mathematics: his famous “incompleteness theorem.” Godel’s astonishing and disorienting discovery, published in 1931, proved that nearly a century of effort by the world’s greatest mathematicians was doomed to failure.

To appreciate Godel’s theorem, it is crucial to understand how mathematics was perceived at the time. After many centuries of being a typically sloppy human mishmash in which vague intuitions and precise logic coexisted on equal terms, mathematics at the end of the 19th century was finally being shaped up. So-called formal systems were devised (the prime example being Russell and Whitehead’s Principia Mathematica) in which theorems, following strict rules of inference, sprout from axioms like limbs from a tree…

The beauty of this mechanistic vision of mathematics was that it eliminated all need for thought or judgment. As long as the axioms were true statements and as long as the rules of inference were truth preserving, mathematics could not be derailed; falsehoods simply could never creep in. Truth was an automatic hereditary property of theoremhood…

By thinking of theorems as patterns of symbols, Godel 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 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…

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