Kurt Godel was born in 1906 in Brunn, then part of the Austro-Hungarian Empire and now part of the Czech Republic, to a father who owned a textile factory and had a fondness for logic and reason and a mother who believed in starting her son's education early. 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. This process of theorem sprouting had to start somewhere, and that is where the axioms came in: they were the primordial seeds, the Ur-theorems from which all others sprang.
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.
The set of symbols in which statements in formal systems were written generally included, for the sake of clarity, standard numerals, plus signs, parentheses and so forth, but they were not a necessary feature; statements could equally well be built out of icons representing plums, bananas, apples and oranges, or any utterly arbitrary set of chicken scratches, as long as a given chicken scratch always turned up in the proper places and only in such proper places. Mathematical statements in such systems were, it then became apparent, merely precisely structured patterns made up of arbitrary symbols.
Soon it dawned on a few insightful souls, Godel foremost among them, that this way of looking at things opened up a brand-new branch of mathematics--namely, metamathematics. The familiar methods of mathematical analysis could be brought to bear on the very pattern-sprouting processes that formed the essence of formal systems--of which mathematics itself was supposed to be the primary example. Thus mathematics twists back on itself, like a self-eating snake.