If relativity is about space and time, what is quantum mechanics about?

h $-abstract-$\backslash $pardThe theory of relativity is about the structure of space and time: we were wrong in thinking that events occur in a flat Euclidean 3$+$1-dimensional manifold. Similarly, quantum mechanics is fundamentally about probability: we were wrong in thinking that probability is just a measure of ignorance. The transition from classical to quantum mechanics involves going from a commutative to a non-commutative algebra of observables, equivalently from a Boolean to a non-Boolean algebra of 2-valued observables, which represent properties or propositions. The non-Boolean algebra of quantum mechanics is not embeddable into a Boolean algebra, which is to say there are no `hidden variables' whose values would `complete' the quantum state description to a consistent assignment of truth or falsity to all propositions (technically, a 2-valued homomorphism on the algebra). Non-Booleanity allows new sorts of nonlocal probabilistic correlations with no causal explanation, associated with `entangled' quantum states, that are not possible in a Boolean or classical theory. I will expand on these ideas with reference to the Bohr-Einstein debate about the completeness of quantum mechanics and recent arguments applying quantum mechanics to complex systems that include agents who are themselves using quantum theory.$\backslash $pard-/abstract-$\backslash $\tex