May I have a dream? Numerical, of course...

Though the number $\pi$ is known to be irrational and transcendental, there are still a lot of open (and non-approachable at the moment) questions about its approximations by rational numbers. Where do such difficulties source from? What is exactly proving that a given (nice) number is irrational? How relevant is the structure of such concrete irrationality proofs? In my talk I will try to highlight these points and to explain why the "structure" is most important, in particular, when it comes to connection with more rational areas in mathematics. The talk is based on numerous irrational self-reflections as well as on the joint work (pre-covid) with Doron Zeilberger on the sharpest-known irrationality measure of $\pi$.