“Formal” vs. “Empirical” Approaches to Quantum-Classical Reduction

I distinguish two types of reduction within the context of quantum-classical relations, which I designate “formal” and “empirical.” Formal reduction holds or fails to hold solely by virtue of the mathematical relationship between two theories; it is therefore a two-place, a priori relation between theories. Empirical reduction requires one theory to encompass the range of physical behaviors that are well-modeled in another theory; in a certain sense, it is a three-place, a posteriori relation connecting the theories and the domain of physical reality that both serve to describe. Focusing on the relationship between classical and quantum mechanics, I argue that while certain formal results concerning singular ~→ 0 limits have been taken to preclude the possibility of reduction between these theories, such results at most block reduction in the formal sense. Little if any reason has been given for thinking that they block reduction in the empirical sense. I then briefly outline a strategy for empirical reduction that is suggested by work on decoherence theory, arguing that this sort of account remains a fully viable route to the empirical reduction of classical to quantum mechanics and is unaffected by such singular limits.