Indescribable numbers: The theorem that made me fall in love with math
This is good. I really liked it.But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be?Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more digits, because you and I will both know that the final wisdom of The Number will always be mine and never yours..."But we know that, given any exact point in time, only a certain number of atoms are inside the balloon. Therefore, whatever the number is to answer the equation, it is one with a finite number of digits. (One imagines the experiment being done in such a manner that quantum tunneling is minimized)So while yes, using symbols you can certainly construct numbers which are indescribable (given either an infinite series of symbols or a self-recursive way of generating more), does the real world actually work that way?In other words, is mathematics truly isomorphic with reality as we observe it? Or is it somehow a superset of logic sitting above an infinite number of possible realities? In a universe full of discrete things (perhaps at the sub-quark level), do we need a system of mathematics that works over continuities? Is such a model always helpful? If not, where might it be tripping us up?