For general non-linear elliptic PDEs, e.g. non-linear rubber elasticity, linear stability analysis is false. This is because of the possibility of point-instabilities. A point-instability is a non-linear instability with zero amplitude threshold that occurs while linear stability still holds. Examples include cavitation, fracture, and the formation of a crease, a self-contacting fold in an otherwise free surface. Each of which represents a kind of topological change. For any such PDE, a point-instability occurs whenever a certain auxiliary scale-invariant problem has a non-trivial solution. E.g. when sufficient strain is applied at infinity in a rubber (half-)space to support a single, isolated crease, crack, cavity, etc. Owing to scale-invariance, when one such solution exists, an infinite number or geometrically similar solutions also exist, so the appearance of one particular solution is the spontaneous breaking of scale-invariance. We then identify this (half-)space with a point in a general domain. The condition that no such solutions exist is called point-coercivity, and can be formulated as non-linear eigenvalue problem that predicts the critical stress for fracture, etc. And when point-coercivity fails for a system, the system is susceptible to the nucleation and self-similar growth of some kind of topological defect. Viewing fracture, etc. as symmetry breaking processes explains their macroscopic robustness. Point-coercivity is similar to, but more general than, quasi-convexity, as it can be formulated for any elliptic PDE, not just Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining meta-stability in a general sense). Indeed, these are just two examples of a host of point-conditions, the study of which might be called point-calculus. Time allowing, I will show that for almost any elliptic PDE, linear- and point-instabilities exhaust the possible kinds of instabilities. The lessons learned from elliptic systems will be just as valid for parabolic and hyperbolic systems since the underlying reason linear analysis breaks down – taking certain limits in the wrong order holds for these systems as well.