On the double-think fallacy of pi as a ‘constant’ (likewise phi, e and root two);
on the significance of algorithms that run for eternity (not to infinity)
The rapper has his bling, the professor has π, φ and √2 (on a charm bracelet, metaphorically speaking). The fashion accessory ‘$’ is backed by street cred, while its ancestor, the Almighty Dollar, was backed by gold. What backs π, φ and √2? The short answer: Nothing; they need no backing since mathematics is only about (surface) logic: “We repeat that the sole question confronting the pure mathematician [...] is not whether [his/her postulates and conclusions] are true, but whether the alleged conclusions are in fact the necessary logical consequences of the initial assumptions” (Nagel/Newman, Gödel’s Proof, page 12, original italics). While mathematics thus strives to present a purely logic-imbued face to the world (see also Jourdain 64-68 re mathematics and Mathematics, logic and Logic), we contend that much of the field is held captive by a kind of double-think tradition, of which its participants seem barely aware.
Take the case of π, which actually entails triple-think. It is simultaneously  a so-called ‘constant’ safe on black velvet in the Museum at the End of the Universe;  a dynamic elusive ‘irrational’ whose algorithm keeps spewing digits forever; and  the thing defined — with a perfectly straight face — as the ‘ratio’ C:D where, if D=1.000, then C is necessarily irrational, hence forbidden to be used in any ratio — except by special dispensation for Very Important Pi. Not all the axiomatic and postulational tap-dancing in the world would be loud enough to drown out the pedal-point of chicanery that is thus ever-present in the field (not globally, but specifically in connection with totemic π and its kindred: φ, √2 and e). Here you will learn to appreciate the Dedekind Cut as the proverbial Solution in Search of a Problem, as we replace it by sundry algorithms that run — not to Infinity — but for Ever (a grown-up concept).
Our primary theme is Algorithms that run versus Icons that sit in trophy cases.
Our secondary theme is Eternity in the universe versus Infinity in the schoolyard.