My own personal experience was that I read about geometric algebra and thought it was neat but didn’t really become anything like fluent in it for years (frankly I still don’t feel entirely fluent). But I like to work on geometry problems of various sorts (related to computer graphics, cartography, computational geometry, ...).
Over and over again, spread out over maybe a decade, I would try solving some problem in the languages I learned in school – Gibbs-style vectors, matrices, differential forms, complex numbers, trigonometric functions, synthetic geometry, etc. – and fill pages of scratch paper with equations that balloon in size making it hard to spot patterns or avoid mistakes, and eventually I would get entirely stuck somewhere. Then I would slap my head, try rewriting the problem in GA terms, and end up replacing like 2 pages of completely incoherent scratch work with about 3–5 lines of concise and easily geometrically interpretable GA manipulations, yielding both a clear answer to my problem and a clear and intuitive demonstration of why it should be right. In particular, dividing by vectors is an unbelievably underrated idea.
I don’t know about “one true language” as some kind of crusade, and I am not a physicist or mathematician, but in my opinion every engineer, scientist, and mathematician would benefit tremendously from becoming substantially fluent with GA,† ideally starting with the basic ideas in high school. It is a very clear and expressive language, substantially better for many purposes than the tools currently taught to students in their technical coursework.
† Including you, if you ever solve geometric problems. Give it a serious try sometime.
Over and over again, spread out over maybe a decade, I would try solving some problem in the languages I learned in school – Gibbs-style vectors, matrices, differential forms, complex numbers, trigonometric functions, synthetic geometry, etc. – and fill pages of scratch paper with equations that balloon in size making it hard to spot patterns or avoid mistakes, and eventually I would get entirely stuck somewhere. Then I would slap my head, try rewriting the problem in GA terms, and end up replacing like 2 pages of completely incoherent scratch work with about 3–5 lines of concise and easily geometrically interpretable GA manipulations, yielding both a clear answer to my problem and a clear and intuitive demonstration of why it should be right. In particular, dividing by vectors is an unbelievably underrated idea.
I don’t know about “one true language” as some kind of crusade, and I am not a physicist or mathematician, but in my opinion every engineer, scientist, and mathematician would benefit tremendously from becoming substantially fluent with GA,† ideally starting with the basic ideas in high school. It is a very clear and expressive language, substantially better for many purposes than the tools currently taught to students in their technical coursework.
† Including you, if you ever solve geometric problems. Give it a serious try sometime.