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RE: A Powerful Generalization of Mathematics
Thank you for trying to explain the architecture of mathematics. But I don't think, it's a good idea to take reductionism as base of an otherwise good essay. Reductionism has long been outdated and replaced by systems theory. Rather than breaking down the math, you could first build it from the individual disciplines and then show synergies between the disciplines, right up to the Langlands program, which seeks to reconcile the different disciplines.
This is a good suggestion. I've read a few article about this Langlands programs, and just like the theory of everything for physics, there's just too many things to get good at before one can get a grasp of the subject.
By systems theory, you mean the "emergentism"? I've sit-in a class once in complex systems, and I was convinced by that time that, maybe, all the physical laws we have about have some collective origin. There is a need, now more than ever, a full understanding and ways of modeling systems that spit emergent behavior naturally.
Are we in the same line of thinking?