Since it's "Pi Day," or March 14, let's check in with a retired math professor who may wind up blowing mathematics into a whole new area -- if he doesn't blow it up first.
The professor's name is Harvey Friedman, and he's been thinking about the foundations of mathematics for about a half a century. You might think this is unexceptional, as even a journalism major with a master's of divinity could grasp the basics of math: 2+2=4, and so on. The problem is a Austrian named Kurt Gödel, who dropped the hammer of his "Incompleteness Theorem" on his discipline in 1931.
In its simplest form, the Incompleteness Theorem says that any system which uses natural numbers -- the kind of numbers we use to count things -- has at least one proposition in it that can't be proved within that system. That means even basic arithmetic can't be proved using basic arithmetic. As the article at Nautilus suggests, once you create another system to prove the first system, your second system will have its own unproveable propositions, and so on.
Since Gödel, mathematicians have mostly banished the idea of incompleteness into fields where it can't be avoided and worked in the rest as though it didn't have an impact. It's what we do in everyday life as well. We may not be able to prove basic arithmetic, but since our lives work just fine when we assume it's true, we'll go ahead and assume away. One mathematician in the article estimates that as much as 85 percent of the discipline can be carried out based on a certain set of axioms and proofs that can safely say, "Kurt who?"
But Friedman thinks that way of looking at math limits it. If all of math grappled with concepts of incompleteness, infinity and some others that create equations and problems that can't be easily solved, then it might paradoxically find itself answering more questions than it does now. He's developed something called "emulation theory" which does that, and thinks that more reflection and work on it could find the same idea affecting other areas of mathematics and maybe even non-mathematical subjects as well.
Whether this will allow me to balance my checkbook on the first try, I guess we'll have to wait and see.