Singularity is a term used both philosophically (in Deleuze/Guattari) and mathematically. It has also been adopted to forecasts about technology.
Philosophical usage:
A singularity is a kind of discontinuity. (see continuous / discontinuous)
It might or might not be interesting. A vaguer use of the term is simply "a point where something happens" (although this equally describes an event.) Deleuze and Guattari are fascinated by singularities because they are points of unpredictability, even when deterministic. They are thus the sites of revolutionary potential.
For Deleuze and Guattari, matter, in nomad science, is never prepared and therefore homogenized matter, (see smooth/striated) but is essentially laden with singularities. (which constitute a form of content)
(see also homogeneity / heterogeneity)
Another term they adopt is "haecceity," a concept created by Duns Scotus from the word haec, "this thing." According to Duns Scotus "Omne ens habet aliquod esse proprium." -- every entity has a singular essence. For d+G it is a mode of individuation different from that of a person, subject, thing or substance. It is a specific degree of heat or intensity of white. (Mille Plateaux, p. 253) A proper name designates it, although it is not an individuation that passes into a form or is effected by a subject. It is a molecular rather than molar quality, and knows only speeds or nonsubjectified affects. (cf. plane of consistency)
For an alternate development of singularity as "whatever," see Giorgio Agamben's discussion of quodlibet. Agamben describes Duns Scotus as responding the the scholastic's problem of the principium individuationis. Against St. Thomas, who sought the place of individuation in matter, Duns Scotus conceived individuation as an addition to nature or common form, but not the addition of another form, essence or property, but of the ultima realitas, the "utmostness" of the form itself. (The Coming Community, p. 17)
scientific usage:
As used by mathematical physicists, a singularity means a place where slopes become infinite, where the rate of change of one variable with another exceeds all bounds, and where a big change in an observable is caused by an arbitrarily small change in something else. (cf sensitivity to initial conditions). It is an actual point of infinite density and energy that's kind of a rupture in the fabric of space-time.
Astrophysics describe the centers of black holes as singularities. When degenerate stars 1.6 times the mass of the sun or greater collapse, the space around the collapsing object becomes infinitely curved at a certain point, trapping any light that might impinge upon it. As a result, an event horizon bounds the singularity and forever conceals it.
The Big Bang is considered to be a singularity.
a phase singularity is a point at which phase is ambiguous and near which phase takes on all values. (see phase boundary ) Time at the poles of the earth is an example. All the time zones converge. If you look at the sun to determine time, it circles at the same altitude along the horizon, and every direction is south (or north, depending on which pole) The poles are singularities for both time and space.
There is a particular point of vulnerability, where circadian rhythms can break down or become unpredictable when subject to a particular stimulus known as the "critical annihilating stimulus". This arrhythmic center in the pattern of timing is called its "phase singularity" (see biological time )
Singularities in Catastrophe theory are points in the space of control parameters at which the configurations of equilibria undergo a change. see bifurcation.
Do singularities only happen in models? Are they distillates of the logical contradictions implicit in our notions of how the real world operates? Do they mean we should seek further if we want to find terms of which the world functions rationally and continuously?
Techno-scientific usage:
"...a reference to a point in mathematical function when the value is unmeasurable, coined by the famous mathematician John von Neumann in the 1950s to refer to the unknown future after which “human affairs, as we know them, could not continue.” In an AI singularity, hyperintelligent AIs appear, with unexpected motives."
Mollick, Ethan. Co-Intelligence
The term “singularity” has also come to refer to a a rupture in the fabric of human history, a technological threshold in which a rapid increase in artificial intelligence will “allow us to transcend these limitations of our biological bodies and brains ... (where) there will be no distinction, post-Singularity, between human and machine" For Ray Kurzweil, "the Singularity” will usher in a new era will be a merger between human intelligence and machine intelligence that is going to create something bigger than itself. Humans will “amplify our own intellectual powers with the results of our technology.” He anticipates that the Singularity will occur this century — probably within twenty years.
Not all observers of the evolution of artificial intelligence are as sanguine as Kurzweil. The fear of machines has informed science fiction from Frankenstein to 2001 and the Terminator. Scientists and entrepreneurs such as Stephen Hawking and Elon Musk have wondered whether the “singularity” will mark the end of humanity. Hawking thinks that over time AI will be able to replace humans altogether, by reaching a point where it will be able to improve and replicate itself. Alongside the benefits, AI will also bring dangers, like powerful autonomous weapons, or new ways for the few to oppress the many. It will bring great disruption to the world economy. For Stephen Hawkinghe emergence of artificial intelligence (AI) could be “the best or the worst event in the history of our civilization. We do not yet know which.”