The emphasis on applying false dichotomies to every situation, in my opinion, is what leads to the truthiness[0] inherent in modern partisan politics.

https://en.wikipedia.org/wiki/Truthiness

An important context here is the one in which George Boole was noticing the patterns that he created an algebra out of; he was actually analysing the Chinese book I Ching... The fact that context is so lost by the amplification of what is, itself, a distillation is, as you rightly argue, the real issue. Not sure what the answer is, but it probably isn't C^* algebras.