Emergence refers to the appearance of patterns of organization and is one of the key concepts of complexity and a-life.It is sometimes referred to as a situation where the whole is greater than the sum of the parts, because it cannot be analyzed by taking the parts apart and examining them separately. One reason for this is that in a complex phenomenon showing emergent properties, the parts become a determining context for each other, and these patterns of feedback contribute to the appearance of the emergent phenomenon. For Michael Polanyi, " Evolution can be understood only as a feat of emergence." 

For Goethe, the living whole was also more than the sum of its parts."Das Lebendige ist zwar in Elemente zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen und beleben." Goethe, quoted in d'Arcy Thompson, On Growth and Form. (p.41) But for most contemporary scientists, Goethe's meaning of emergence was a mystical one. For Francis Crick, the scientific meaning of emergent assumes that while the whole can not be the simple sum of the separate parts, its behavior can, at least in principle, be understood from the nature and behavior of its parts plus the knowledge of how all these parts interact. (The Astonishing Hypothesis, p.11) (assuming it's possible to have that Laplacian knowledge) 

Stuart Kauffman suggests that many properties of organisms may be probable emergent collective properties of their constituents. (This is precisely Kant's definition of the organic -- as the result of internal interactions instead of an assembly of preexisting parts.(see Mechanism / Vitalism ) "The problem of origins requires an understanding of how new levels of order emerge from complex patterns of interactions and what the properties of these emergent structures are in terms of their robustness to perturbation and their capacities for self-maintenance" (Brian Goodwin, How the Leopard Changed its Spots, p. 181.) 

Luc Steels offers useful definitions of emergence in Artificial Life 1/2. He distinguishes between first order emergence, defined as a property not explicitly programmed in, and second-order emergence, defined as an emergent behavior that adds additional functionality in the system. In general a-life researchers try to create second-order emergence, for then the system can use its own emergent properties to create an upward spiral of continuing evolution and emergent behaviors.

for a critique of emergence, see Peter Cariani, "adaptivity and Emergence in Organisms and Devices, "World Fudtures 32 (1991):111-32.

C. Lloyd Morgan's descriptions of emergence as a qualitative change in direction closely resemble the definitions of bifurcations. In his Emergent Evolution, of 1923, he writes, "The emergent step, although it may seem more or less saltatory, is best regarded as a qualitative change of direction, or critical turning-point, in the course of events." 

Daniel Stern describes the infant's sense of self between birth and two months as the "emergent self." (see subject ) . It is a sense of the coming-into-being of organization and remains active for the rest of life. But it is not an overarching integrating schema about the self, but rather an experience of process. (cf molar / molecular ) 

In Gestalt perception terms, emergence refers to our ability to perceive forms that cannot be reduced to the addition of the "atomic" parts of a pattern. Eg superimposing two squares diagonally displaced allows for readings of an emergent square at their overlap, two emergent L-shapes in the non-intersecting area, in addition to the two squares. (cf. figure / ground)

A fundamental property of open systems is that they stablize any improbability which serves to elicit them.