THE CIRCLE CAN'T BE SQUARED WITH REALITY:

Following the collapse of the polyglot Austro-Hungarian Empire and the rise of Czech nationalism, Gödel joined the throng of young, Czech Germans moving to Vienna--he arrived in 1924, becoming a student at the prestigious University of Vienna. Shortly afterward he was invited to attend the Vienna Circle, a discussion group made up of mathematically and scientifically literate philosophers, led by professor Moritz Schlick. For a decade or so, their philosophical approach, logical empiricism, became the most fashionable in the world.

Crudely put, the Circle maintained that for a statement to be meaningful it had to be either testable ("water boils at 100 degrees centigrade") or true by virtue of the meaning of its terms ("all bachelors are unmarried" or other tautologies). Many statements about God, ethics and aesthetics were therefore meaningless. Math posed a problem for the Circle. Was 2+2=4 an empirical claim? Did we discover its truth by adding two apples to another two apples and counting four apples? This didn't seem right. We could surely work out that 2+2=4 without the aid of fruit or any other material prop. Inspired in particular by Ludwig Wittgenstein, the Circle argued that we should treat mathematical truths as tautologies.

Gödel was mostly silent during Vienna Circle discussions, but he passionately disagreed. His instincts were Platonist; that is to say, he believed that mathematical truths weren't invented but existed somewhere "out there," independent of the human mind, and that it was the task of mathematicians to discover these truths.

Gödel's reputation and fame rest principally on a proof that received its first public airing in September 1930, at a scientific gathering in Königsberg. Gödel--only 24 years old--demonstrated to the assembled delegates that there were limits to what could be proved in mathematics; that whatever axioms were postulated as the basic blocks of mathematics, there would inevitably be some truths within mathematics that could not be proved.

By all accounts, the delegates at the conference were a bit flummoxed; the significance of this discovery took a few days to sink in. Then news of Gödel's first Incompleteness Theorem (it would be followed by a second), spread rapidly around the world. "A scientific achievement of the first order" was the rather understated verdict of Gödel's supervisor, Hans Hahn, when Gödel submitted the proof for his thesis. The work is now widely accepted as a seminal development in the history of logic.

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