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 Reductionism has long been outdated and replaced by systems theory.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.
RE: A Powerful Generalization of Mathematics