The Limits of Mathematical Knowledge

My good friend Jason Rosenhouse had an interesting discussion a few days ago on whether or not modern mathematics is reliable. The gist is that the deepest of maths are increasingly built on a foundation of theorems and proofs that are so complex that it effectively becomes an act of faith to take them as true and correct. And when further theorems are built on these theorems, how confident can you be in the structure of mathematical knowledge?

It’s an interesting point, and it gets to something I think about from time to time – the limits of mathematical knowledge.

The idea is that humans, when doing math, have an understandable bias towards proofs that are readable and (easily) checkable. The question then is what portion of mathematical ‘truth’ can be expressed in such a way.

Linear Recurrences And … Stuff

So, I know I’ve been a little quiet lately. This is mainly do to the confluence of two factors – one, I’ve actually been doing some work lately. I’ve been doing some programming (and avoiding doing some programming), and mulling over a series of papers on the four color theorem, written between 1892 and 1947. Interesting stuff – the author basically transforms the map coloring problem into solving a system of simple linear equations. I may talk about that some later.

Secondly, I’ve been wrestling with my own frustrations about a much simpler problem. I’m sort of chasing my tail on this one.

To begin with, imagine a function f(x) that satisfies the following equation.

The problem is to prove that it is periodic. Further, what other values besides give rise to periodic functions.