Keith Devlin waxes not-quite-poetic on "The Recursion Principle" and its underlying importance to mathematics:
From the posting:
"Though recursion is ubiquitous in modern mathematics, even at the most basic level of the analysis of the arithmetic of natural numbers, it is a subtle concept, easily misunderstood…(...ohh, and by the way, multiplication is NOT just repeated addition. ;-)
"This may all seem like a great deal of fuss about nothing. But what is going on here is really very deep. Much of modern mathematics involves finding ways to handle the infinite - calculus exclusively and spectacularly so. Mathematicians learned over many years of painful lessons that the step from the finite to the infinite is a tricky one that requires considerable finesse. In particular, you have to exercise great care to set it up correctly and do it right. The Recursion Theorem is one of those crucial bridges that allow us to go beyond the finite to the infinite, to extend human intellect from its finite physical limitations to the infinite world beyond that our minds can construct. By getting the mathematics right, we can make that step with total confidence. Confidence both in that abstract world itself and in the concrete conclusions it allows us to reach about our lives, our science, and our technologies. That is huge for Humankind."