  Beginning with Anderson (1972), spontaneous symmetry breaking (SSB) in
infinite quantum systems is often put forward as an example of (asymptotic)
emergence in physics, since in theory no finite system should display it. Even
the correspondence between theory and reality is at stake here, since numerous
real materials show SSB in their ground states (or equilibrium states at low
temperature), although they are finite. Thus against what is sometimes called
`Earman's Principle', a genuine physical effect (viz. SSB) seems theoretically
recovered only in some idealization (namely the thermodynamic limit),
disappearing as soon as the the idealization is removed. We review the
well-known arguments that (at first sight) no finite system can exhibit SSB,
using the formalism of algebraic quantum theory in order to control the
thermodynamic limit and unify the description of finite- and infinite-volume
systems. Using the striking mathematical analogy between the thermodynamic
limit and the classical limit, we show that a similar situation obtains in
quantum mechanics (which typically forbids SSB) versus classical mechanics
(which allows it). This discrepancy between formalism and reality is quite
similar to the measurement problem, and hence we address it in the same way,
adapting an argument of the author and Reuvers (2013) that was originally
intended to explain the collapse of the wave-function within conventional
quantum mechanics. Namely, exponential sensitivity to (asymmetric)
perturbations of the (symmetric) dynamics as the system size increases causes
symmetry breaking already in finite but very large quantum systems. This
provides continuity between finite- and infinite-volume descriptions of quantum
systems featuring SSB and hence restores Earman's Principle (at least in this
particularly threatening case).
