A Stringy Universe II

Now if stringy black holes are all about abstract entanglement, why do string theorists insist on talking about classical landscapes and awful complex analysis? After all, haven't they figured out yet that the Standard Model isn't completely right? On the other hand, then one remembers that the analysis hides a few devils. But it simply isn't acceptable to talk about gravity using such blatant prejudices about geometry, leaving one clear, horrifying option: to rewrite analysis.

Topos theorists have thought quite a bit about this problem. But as simple minded physicists we can always think about it more pragmatically, trying to build up the complex numbers piece by little piece, perhaps starting with small finite fields as truth values for MUB matrix entries. A full complex Hilbert space requires an uncountable number of truth values. In this case, a $1$-ordinal heirarchy of structures isn't going to be enough to reach $\mathbb{C}$, but we must just keep on going somehow (with surreal trees and $n$-ordinals and other goodies). So the complex numbers really ought to be taboo until one studies $\omega$-categories and multicategories, a lesson from both topology and topos theory.