For Georges Bataille, (sexual)"Reproduction implies the existence of discontinuous beings." (Erotism, p.12) Each being is distinct from all others, including its parents, who are distinct from each other. For Bataille death means the continuity of being and is brought into play by reproduction. Death is the end of discontinuous being, of the being formed at the moment when the discontinuous entities of sperm and egg unite to form a new continuity, when two become one, and a new entity is formed from the fatal fusion. The fascination with both reproduction and death is the dominant element in Eroticism.
Of course, each individual must have continuous spatio-temporal identity between its origin and when it ceases to exist, even when passing through different "stages" in a life-cycle. It is an historical entity, and its identity is also a function of continuity of descent. "What we desire is to bring into a world founded on discontinuity all the continuity such a world can sustain." (Bataille, p.19)
Life itself is generally defined as a discontinuity between the inorganic and the organic.
Discontinuities in objects of criticism allow tactical points of entry to break up what seem to be monolithic systems at their points of entry. Like a master butcher, whose knife seems always to stay sharp, Michel Foucault, for examples stresses the discontinuities in the discursive formations he studies. Representations of space in the social sciences are remarkably dependent on images of break, rupture and disjunction. The distinctiveness of societies, nations, and cultures is predicated on a seemingly unproblematic division of space, on the fact that they occupy "naturally" discontinuous spaces.
Natura non facit saltus
It is an important question for biology as to whether evolution occurs in continuous or discontinuous steps. Darwin shared the view that individuals varied continuously in all their common characteristics and formulated his theory of the accumulation of small intergenerational differences as the explanation of the origin of species (which define discontinuity) within that framework. (see evolution)
Among the most famous laws of nature are Gregor Mendel's laws of inheritance, based on a distinction between two kinds of genetic factors, the dominant and the recessive. In Mendel's famous experiment with peas, he studied the distributions, generation by generation, of four characteristics of peas: green and yellow color and smooth and wrinkled skin. Later generations of statisticians, especially Fisher, found Mendel's figures "far to good to be true." Part of the problem lies in the group of peas that cannot easily be categorized as green or yellow, smooth or wrinkled. In an analysis of Mendel' work, R.S. Root Bernstein found that if those peas were randomly assigned to either category, the result would be exactly Mendel's numbers.
Having learned of de Vries' rediscovery of Gregor Mendel's work in 1900, William Bateson became the chief publicist for it in England, and when his third son was born in 1904, he named him Gregory in honor of Mendel. William Bateson's Materials for the Study of Variation is "treated with especial regard to discontinuity in the origin of species." (title page) According to Bateson, the forms of life are diverse. At any given time they form a discontinuous series and not a continuous one, whence their separation into distinct species. The succession of parent to offspring introduces variation which is converted into "specific differences" between species. Bateson coined the word "genetics" to encompass the study of heredity and variation of species.
While "specific difference" is the very definition of discontinuity, are variations continuous or discontinuous? Bateson recognises distinct forms of each. (see natural form for comparisons between Bateson and Goethe.) D'Arcy Thompson acknowledges the limits of his theory of transformations when he invokes a "principle of discontinuity" when separates the types. "Essentialist" ideas of type or Form imply a "philosophy of discontinuity," although continuous variation must also find a place within these ideas. The statistical view, on the other hand, is based on counting the frequency of attributes in a population and seems more continuous. Yet, to be countable, attributes must be discrete.
Henri Poincaré describes movement along a continuum as passing from any one to another through consecutive elements, each of which is indistinguishable from the preceeding. For example, I may not be able to distinguish between a 10 gram weight and an 11 gram weight. Similarly, I may not be able to distinguish between an 11 gram weight from a 12 gram weight. Yet I can tell the difference between the 10 gram weight and the 12 gram one. I seem to have a contradiction, since for my purposes, A=B, B=C, but A< C. In a continuum, no matter how closely I look, similar situations will occur.
Singularities are ways of articulating moments of passage from continuity to discontinuity, or from quantity to quality. Catastrophe theory is an attempt to go beyond classical physics -- the physics of structurally stable systems -- and to provide a mathematical framework for discontinuous processes.
Alfred North Whitehead describes the philosophical problem raised by quantum theory as a discontinuous existence in space. "One of the most hopeful lines of explanation is to assume that an electron does not continuously traverse its path in space...that it appears at a series of discrete positions in space which it occupies for successive durations of time." (p34)
While fields presuppose the idea of continuity, the atomic theory of matter describes discontinuity. Atomic theories date from Democritus and Lucretius, but it was John Dalton who made them scientifically effective. In his preface to the Treatise of Electricity and Magnetism, Maxwell compared the methods of then-current mathematical physics, which proceeded from individual particles, to those of Faraday, which began with the "whole," that is, the field of forces acting upon the particles, and clearly stated his preference for Faraday's method. Actually, Maxwell qualified Faraday's holism by depicting field action as occurring only at contiguous points of the field and describing it in series of partial differential equations. By doing this he hoped to preserve the consistency of his equations with Newtonian mechanics; but to later generations it was clear that the idea that fields of force have an independent existence was a first step away from the assumption that the world consists of of pieces of matter from which forces emanate. As Plank later said, field theory threatened to divide physics into two realms, one of "corpuscles" and one of "continua." (Ash, GestaltPsychology, p.171)
The theory of the cell developed in the nineteenth century also exemplifies the idea of "atomism."
In everyday experience, the passage between views -- attention deflection -- works so smoothly that we do not notice the discontinuities or jumps.
Nelson Goodman's discussion of the Languages of Art contrasts density with differentiation. Goodman describes some artworks as "autographic." In Dense, autographic works, the most infinitesimal difference can potentially make a difference. Therefore, there is a difference between an original and a copy. Easel painting is autographic. Even the most exact duplication does not count as genuine. In differentiated, allographic arts, there are discrete units of signficance. A piece of music can be performed repeatedly, by different performances, with difference of tempo, etc. It is still the same piece. Goodman's criteria can be used to distinguish image and text. In an analog system there is no threshold of significance. As in the geometric definition of a line, between any two points there is always a third. In a digital system, there are defined units.
The controversies over whether the actions of the brain are best understood as digital or as analog are a significant locus of contemporary disputes between continuity and discontinuity. (see neuron )