Born in Moravia (now the Czech Republic), Kurt Gödel (April 28, 1906 - January 14, 1978) became one of the most significant mathematicians of the 20th century by the time he turned 25.  As a student, an early interest in languages gave way to a passion for mathematics, which he supplemented in his teens with history and philosophy.  At 18, he entered the University of Vienna, where his older brother was a medical student.  His early courses exposed him to number theory and mathematical logic, and his growing interest in mathematical realism led him to pursue mathematics rather than physics as he’d first intended.

Mathematical realism says that mathematical objects and concepts are real, that they exist outside of human invention and imagination.  The distinction may seem trivial, so consider it in terms of food instead: the difference between fruit and meat is "real," it’s a difference that is true regardless of human involvement.  The difference between dessert and breakfast, on the other hand, exists only in the human mind -- there is no scientific property separating them.  Mathematical concepts are discovered, according to realism -- not invented the way recipes are.

Important early influences on Gödel included Immanuel Kant, Bertrand Russell, and David Hilbert.  There is a stereotype about mathematicians that they do their most ground-breaking work early in life, even though their mastery over the discipline is greater later.  There are many theories about why this might be true, of course, but Kurt Gödel has always been a prime example of the trend.  He published his best-known and most important work, his incompleteness theorems, in 1931 -- only a year after graduating from the university, where his completeness theorem had formed his doctoral dissertation.  The completeness theorem had proven the completeness of predicate logic -- it had shown, in other words, that within predicate logic (also known as first-order logic), every logically valid formula can be proven through a list of steps.  To over-simplify a little, he proved that predicate logic contained all the rules necessary to prove the things it’s designed to prove.  Like many mathematical advances, this was something which was widely believed to be true but hadn’t yet been effectively proven.

The 1931 incompleteness theorems were much more advanced and ground-breaking.  Since the nineteenth century, mathematicians had been trying to construct a set of axioms (mathematical rules) which would include all of mathematics.  Gödel proved that they would never succeed.  To some mathematicians, it was as though he had proven in the middle of the space race that launching a rocket was impossible; few of his colleagues had considered that what they sought was impossible, and had focused more on finding it, whether by brute force or elegant solutions.  Gödel proved that for any such system of rules, there would be a valid mathematical formula that it could not prove.

In Gödel’s words: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete."

Even aside from the implications of his proof, Gödel had to invent whole new mathematical language in order to achieve it.  It took time for those implications to set in, and they continue to unfurl: Gödel’s work has been critical in philosophy and cognitive science, and is sometimes brought up in the study of (and quest for) artificial intelligence.

Gödel continued to work in and lecture on this general area of mathematics throughout the 1930s.  An often troubled man who suffered a nervous breakdown after the murder of one of his mentors, Gödel avoided politics, and so the only immediate impact on him of the Nazi Party’s ascension to power in Germany (which had absorbed Austria) was the abolition of his teaching job.  (Not his specifically, but all jobs with his title of Privatdozent.)  When his Jewish friends and physical fitness for military duty made it hard for him to find another mathematics job in Vienna, he and his wife left Europe.  In 1940, Gödel took a teaching position at the Institute for Advanced Study in Princeton, New Jersey -- where Albert Einstein had emigrated some years earlier.  Gödel and Einstein became close friends, both of them brilliant men who saw their early contributions to science unfold wide-spanning consequences during their lives.  Einstein later accompanied Gödel when the latter sought U.S. citizenship.

Gödel also published a paper on Einstein’s field equations which provided a solution in which time travel would be possible, though his goal was more likely to demonstrate the problems with our understanding of "time" in light of modern physics.  Still, it’s hard to say.  Though he continued to make major contributions to mathematics, especially his work on such advanced topics as the axiom of choice and the continuum hypothesis, in the last years of Gödel’s life some of his pursuits became less traditional.  He believed there was a way to avoid death, and lamented his inability to discover the mathematics of this escape in his notebooks; when seeking his American citizenship, he went off on a tangent, explaining to the judge that a loophole in the Constitution allowed for the creation of a dictatorship.  Straddling the line between the mainstream and the unconventional, he also developed his ontological proof of God, drawing on prior writings by Saint Anselm and Gottfried Leibniz.