The word singularity did not start out as a product launch or a podcast line or a date circled on a calendar or a book title often quoted a punchline. It started as someone admitting the map had failed. In mathematics, a singularity is where a function stops behaving. Division by zero. A pole. A place where slopes and limits will not hand you a finite answer. A mathematical singularity is a point where a function, curve, or surface stops being well-behaved: it may blow up to infinity, fail to be differentiable, or change topological type. The most familiar pictures come from three places.

Physics uses the same word for the region, most famously the center of a black hole, where curvature and density push the usual equations past the point of being usable. You get close and time and space and mass stop being things a working physicist can tr ...