metaphysical & metaphorical musings : art, architecture, and arithmetic

Friday, February 11, 2011

Mythematics II - Gödel Incompleteness

In the early years of the 20th century, Bertrand Russell and Alfred North Whitehead set about axiomizing mathematics—boiling math down to its fundamental principles, modeling Euclid’s geometry.  One of Russell’s main concerns was eliminating the paradox of self-reference.  Toward this end, he and Whitehead penned their Principia Mathematica in three imposing volumes.  A few decades later, Kurt Gödel showed up and ruined everything by showing that any formal system at a certain threshold of power can, in a sense, self-destruct.

Gödel’s proof has two key points: that certain mathematical statements can be interpreted as statements about mathematics itself (which should only be possible in meta-mathematics), and that these statements can be condensed into a single statement which speaks about itself.  Thus, any formal system of a certain sophistication carries the potential for self-reference, the very thing Russell was so eager to eliminate.

Don Ault likes to say that the Principia was a turning point in the respective careers of Russell and Whitehead:  Russell was only able to think before the Principia, and Whitehead was only able to think afterward.  Whitehead penned his major philosophical works, and Russell became the doddering old man we see in Prisoner’s Dilemma.

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