In response to:
I should publish a paper on this subject as yet another of the many problems of mathematical idealism(analogy) vs mathematical operationalism (reality). Because “i”, just as |absolute value|, solves a problem of ambiguity in mathematics: the language and logic of positional names.
Sabine correctly identifies the convenient use of “i” in simplifying oscillations (geometry). But why can’t we identify the square root of negative one as negative one? Because of the Conflation of Direction(geometry) and Position(arithmetic). The use of “i” is necessary because as a general rule we’re conflating arithmetic (position) with geometry (direction).
Is this solvable? Of course. Is “i” simply denoting the use of geometric (directional) math versus arithmetic (positional) math? Yes. Is it any more complex than that? Absolutely not. Math is a trivially simple language (paradigm, logic, vocabulary, grammar, syntax) under mathematical operationalism. It’s all the nonsense we piled on it, that makes it difficult to learn.
Unfortunately, while the operational revolution was identified in math, in physics, in economics (and less so in law) it only stuck in some parts of physics and not in mathematical physics, or in mathematics. This is why (in my opinion) computational revolutions are occurring in computer science where the limits of mathematics are openly exposed (the domain of the operationally calculable is greater than the domain of mathematically reducible.)
We can’t reform mathematics because the operational revolution failed in math – we got a set foundation (idealism) of math instead. And IMO the problem Sabine is continuously exposing both in her book and in her videos, is this underlying failure: that mathematics fails in economics and below the quantum level for the same reason: the underlying mechanics are operational and either we lack the information to describe that geometry or the underlying geometry isn’t mathematically reducible beyond the quantum level. We all assume it’s the former but it just as likely is the latter.