The term “incommensurability” is more encompassing than mere “irrationality” as it embraces “transcendental” as well. That was a topic this morning over breakfast. Terry Bristol bounces around among the Greeks, such as Archytas of Tarentum (c. 435–350 BC) and Eudoxus of Cnidus, precursors to Euclid, and inspirational for Archimedes.
Incommensurability opens a space of incremental or successive approximations, each closer to some ideal, but perhaps with insufficient means to ever reach that ideal, even in principle. For example, by means of specific algebraic operations, one may reach closed for expressions for certain irrational numbers, but not for pi or e.
The analogy to geometry is deeper than mere resemblance; it’s a tighter mapping. Given the conventional Euclidean constraints of straight edge and compass, what are the limits to construction. The classic example was squaring the circle: by simple construction, construction a circle and square of identical area. This proved elusive, leading to a maturing of the concept of incommensurability itself.
Terry has been exploring the not-commutative aspects of geometric procedures. The order of instructions clearly matters and going forward (picture an arrow) does not always imply going backward. Some functions are what we call one way.
What I added to the discussion was a quick recap of that Python generator function I leaned about from Guido and Tim, which in just a few lines using rational steps, creates a pi digits generator. Somehow remembering state may lead to chaotic output, as we also learn from Wolfram. A deterministic generative process may not imply any shorter or more compact expression is out there, other than the generator itself, in whatever language. Ramanujan’s generators are a case in point.
