Turing Uncomputability

“As Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the uncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance”—George Dyson

We all remember learning that the decimals of pi are infinite in number, 3.14159265359… Some of us even recall learning that you can approximate upper and lower bounds on the value of π to as high of a degree as you want by measuring the sides of polygons. As the number of sides of the polygon approach infinity and the length of their sides approach zero, your approximation gets better.

Archimedes (c. 287- 212) invented this early ‘polygonal algorithm’, which dominated for over 1,000 years as the most efficient way of computing π to any desired precision. The very primitive and geometric algorithmic function serves the purpose of estimating a real number, pi, whos…