58 | 7 HIlBERT AND HIS fAmOUS PROBlEm
Eternally incomplete
Kurt Gödel (Fig. 7.1), a taciturn 25-year-old mathematician from Vienna University, ushered
in a new era in mathematics with his 1931 theorem that arithmetic is incomplete.^3 In a sentence,
what Gödel showed is that more is true in mathematics than can be formally proved.
This sensational result shocked, and even angered, some mathematicians. It was thought that
everything that matters ought to be provable, because only rigorous proof by transparent and
obvious rules brings certainty. But Gödel showed that, no matter how the formal rules of arith-
metic are laid down, there must always be some mathematical truths—complicated relatives of
simpler truths such as 1 + 1 = 2—that cannot be proved by means of the rules. Paradoxically,
the only way to eradicate incompleteness appeared to be to select rules that actually contradict
one another.^4
figure 7.1 Kurt Gödel
From the Kurt Gödel Papers, the Shelby White
and Leon Levy Archives Center, Institute for
Advanced Study, Princeton, NJ, USA, on deposit
at Princeton University; reproduced courtesy of
Princeton University.figure 7.2 David Hilbert
Posted to Wikimedia Commons
and licensed under public domain,
https://commons.wikimedia.org/
wiki/File:Hilbert.jpg