More than 2,000 years ago, the Greek philosopher Epimenides provoked and frustrated his fellow philosophers when he, himself a Cretan, declared: 'All Cretans are liars.' Whether you take this statement as true or false, you always get the opposite as a result. Now, you may think that this is a gimmick of no substance and only philosophers with nothing better to do would bother with it. You would be wrong.
At the turn of the last century, the great philosopher and logician Bertrand Russell constructed a mathematical version of the liar's paradox within set theory, at the very foundation of mathematics. Several mathematical schools cropped up trying to rescue maths from such absurd contradictions. They all failed.
We know today that none could have succeeded, even in principle, thanks to a great discovery made by the ingenious Austrian logician Kurt Goedel. This September marks the 75th anniversary of the first public announcement of his famous incompleteness theorems, which he published the following year.
Goedel, who died in 1978, was a close friend of Albert Einstein at Princeton in the 1940s. In the intervening decades, his very abstract and technical discovery has made it even into popular science books, much like Einstein's relativity theory. Like Einstein, he has become a cultural icon for scientists. Why is that? For one thing, Goedel proved, through ingenious new techniques, that no reasonable mathematical theory, not even the basic arithmetic we learn in primary school, can be made absolutely consistent.
Before Goedel, maths was considered rock solid, the most exact of sciences. Goedel has changed our scientific concept of maths as Einstein has done with our concept of space and time. He shows that there are no absolute truths, not even in maths and logic.
Now, you may wonder whether this has any further implications or applications outside maths. It turns out that Goedel's theorems have quite dramatic and very practical implications and consequences.