Reality, Rationality, Irrationality

Probably the oldest version of Rationalism, and still influential, is Pythagoreanism, according to which the latent structure of Reality is Mathematical, i. e. constructed in terms of Natural Numbers.  Now, the standard challenge to Rationalism is based on the premise that Mathematical relations are merely mental constructs, and only projected into the external world.  But, a sharper challenge originates with a little-known early Pythagorean--Hippasus--who is credited with the discovery of Irrational Numbers, the implication of which to Pythagoreanism is that the latent structure of Reality is just as Irrational as it is Rational.  The focus of Hippasus' discovery reportedly is the square root of 2, previously believed by Pythagoreans to be ultimately analyzable as a ratio of two Natural Numbers, even if not immediately obviously so.  On that basis, the Irrational component is correspondingly obscure.  But, once his discovery is extended to Pi, the example of the Irrationality of Reality is plainer--any circular motion, notably the orbits of celestial bodies. In other words, long preceding the Modern Empiricist subjectivization of Rationalism, there is presented the stronger refutation of it--a counterexample.