Where does singularity collide?

In complex analysis a singularity is a point at which a function stops being analytic, meaning it cannot be written as a convergent power series in a neighbourhood of that point. Augustin-Louis Cauchy's 1825 Mémoire sur les intégrales définies, prises entre des limites imaginaires founded the theory of integrating functions of a complex variable and put the points where a function becomes infinite at the centre of it, points later sorted into removable ones, poles, and essential ones. The referent is a location in the number plane fixed by the local behaviour of a function, and it has nothing to do with a rupture in physical spacetime or a forecast about machines beyond the shared word.[1]

Unit

One point in the complex plane where a function fails to be analytic

Recovery means

A removable point can be assigned a value; a pole or an essential point cannot be repaired

Measured by

Classified from the Laurent series about the point as removable, a pole of some order, or essential

In general relativity a singularity is a place where a quantity describing spacetime, such as curvature, grows without bound and the ordinary laws stop applying. Karl Schwarzschild's 1916 Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie gave the first exact solution of Einstein's field equations and with it the first such point at the centre of the field, and Roger Penrose's 1965 Gravitational Collapse and Space-Time Singularities proved that collapse forces these points to form rather than appear only by chance. The referent is a feature of physical spacetime, marked by diverging curvature and paths that end after a finite time, not a point where a formula misbehaves or a moment in the future of technology.[2][3]

Unit

One point or region of spacetime where a curvature quantity grows without bound

Recovery means

Not applicable

Measured by

Located by curvature invariants that diverge and by geodesics that terminate after finite proper time

The technological singularity is a projected point in the future at which technological change, driven by machines that improve themselves, becomes so fast that people can no longer predict or keep pace with it. Stanislaw Ulam's 1958 John von Neumann 1903-1957 recorded a conversation in which von Neumann spoke of an approaching essential singularity in the history of the race beyond which human affairs could not continue as they had, and Vernor Vinge's 1993 The Coming Technological Singularity gave the idea its modern shape and name. The referent is a forecast about the course of civilisation and intelligence, argued from growth trends rather than measured with an instrument, and it is unrelated to a point on the complex plane or a region of collapsed spacetime.[4][5]

Unit

One projected moment when technological change outruns human forecasting

Recovery means

Not applicable

Measured by

Argued from extrapolated growth curves in computing and from expert forecasts, not read off a physical instrument