What is emergence?

There is no consensus about what emergence is, how to define it, or why it matters. In John Holland’s beautiful book, Emergence: from Chaos to Order, he states that, “Despite its ubiquity and importance, emergence is an enigmatic, recondite topic, more wondered at than analyzed… It is unlikely that a topic as complicated as emergence will submit meekly to a concise definition, and I have no such to offer.” Instead, Holland focuses on systems that can be described by simple rules or laws. The rules generate complexity: novel patterns that are sometimes hard to recognise and to anticipate.  

Below is my own attempt to clarify what some of the important issues and questions are associated with defining emergence. I am trying to take a path that is intermediate between the precision of philosophers and the loose discussion of emergence by condensed matter physicists. My goals are clarity and brevity.

A possible definition of emergent properties: novelty

Consider a system that is composed of many interacting parts. If the properties of the whole system are compared with the properties of the individual parts, a property of the whole system is an emergent property if it is a property that the individual parts of the system do not have. Emergent properties are novel. The system is qualitatively different from its parts. 

Examples of properties of a system that are not emergent are volume, mass, charge, and number of atoms. These are additive properties. The property is simply the sum of the properties of the parts.

Implicit in this definition is that there is a concept of scale. Some sort of scale (for example, particle number, length, or energy) is used to define what the parts are and thus how many parts there are. 

I don’t claim this definition of emergence is necessarily better than that of others. But it is concrete and somewhat precise and can be used to clarify other characteristics that are often associated with emergence and are sometimes included in the definition of emergence by some authors, including myself. I have done this in the past, in blog posts, talks, and in my VSI on condensed matter. 

Characteristics of emergent phenomena

There is more to emergence than novel properties. For specificity, here I focus on emergent properties but it is also possible to exchange “property” with state, phenomenon, or entity. 

Below I list some characteristics often associated with emergent properties. Some people include these characteristics in their definitions of emergence. However, I do not include them because as I explain some of the characteristics are contentious. Some of these characteristics may not be necessary or sufficient for novel system properties.

1. Presence of discontinuities

Quantitative changes in the system can become qualitative changes in the system. For example, in condensed matter physics spontaneous symmetry breaking only occurs in the thermodynamic limit (i.e., the number of particles of the system becomes infinite). Thus, as a quantitative change in the system size occurs the order parameter becomes non-zero. In such a system, a small change in temperature can lead to the appearance of order and a new state of matter.

Two different states of a system are said to be adiabatically connected if one can smoothly deform one state into another and all the properties of the system also change smoothly. The case of the liquid-gas transition illustrates subtle issues about defining emergence. There is no qualitative difference between a gas and a liquid, only a quantitative difference: the density of the gas is less than the liquid. Often the difference is orders of magnitude. Furthemore, the liquid and gas state can be adiabatically connected. There is a path in the pressure-temperature phase diagram that can be followed to connect the liquid and gas states without any discontinuities in properties.

The ferromagnetic state also raises issues, as illustrated by a debate between Rudolf Peierls and Phil Anderson about whether ferromagnetism exhibits spontaneous symmetry breaking. Anderson argued that it did not as, in contrast to the antiferromagnetic state, the non-zero magnetisation occurs for finite systems and the magnetic order does not change the excitation spectrum, i.e., produce a Goldstone boson. On the other hand, singularities in properties at the Curie temperature only exist in the thermodynamic limit. Also, a small change in the temperature, from just above the Curie temperature to below, can produce a qualitative change, a non-zero magnetisation.

2. Modification of parts and their relationships

Some emergent properties involve the state of the system exhibiting patterns, order, or structure, terms that may be used interchangeably. This reflects that there is a particular relationship (correlation) between the parts which is different to the relationships in a state without the emergent property. This relationship may also be reflected in a generalised rigidity. For example, in a solid applying a force on one surface results in all the atoms in the solid experiencing a force and moving. The rigidity of the solid defines a particular relationship between the parts of the system.

Properties of the individual parts may also be different. For example, in a crystal single-atom properties such as electronic energy levels change quantitatively compared to their values for isolated atoms. Properties of finite subsystems are also modified, reflecting a change in interactions between the parts. For example, in a molecular crystal the frequencies associated with intramolecular atomic vibrations change compared to their values for isolated molecules. However, emergence is a sufficient but not a necessary condition for these modifications. In gas and liquid states, one also observes such changes in the properties of the individual parts.

3. Universality

An emergent property is universal in the sense that it is independent of many of the details of the parts. Consequently, there are many systems that can have the emergent property. For example, superconductivity is present in metals with a diverse range of crystal structures and chemical compositions. Robustness is related to universality. If small changes are made to the composition of the system (for example replacing some of the atoms in the system with atoms of different chemical element) the novel property of the system is still present. 

4. Irreducibility

An emergent property cannot be reduced to properties of the parts, as by definition of emergence in terms of novelty, the parts do not have the property. 

Emergence is also associated with the problem of theory reduction. Formally, this is the process where a more general theory, such as quantum mechanics or special relativity, "reduces" in a particular mathematical limit to a less general theory such as classical mechanics. This is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who sometimes overstate the problems this presents for science producing reliable knowledge]. The subtleties arise because the two different theories usually involve language and concepts that are "incommensurate" with one another. 

Irreducibility is also related to discontinuities and singularities being associated with emergent phenomena. As emphasised by Primas and Berry, singularities occur because the mathematics of theory reduction often involves singular asymptotic expansions. Primas illustrates this by considering a light wave incident on an object and producing a shadow. The shadow is an emergent property, well described by geometric optics, but not by the more fundamental theory of Maxwell’s electromagnetism. The two theories are related in the asymptotic limit that the wavelength of light in Maxwell’s theory tends to zero. 

The example above illustrates that theory reduction is compatible with emergence. The philosopher of science Jeremy Butterfield showed this rigorously for four specific systems that exhibited emergence, defined by him as a novel and robust property. Thus, irreducibility is neither necessary nor sufficient for emergence.

5. Predictability

An emergent property is difficult to predict solely from knowledge of the properties of the parts of the system and how they interact with one another. Predictability is one of the most contested characteristics of emergence. “Difficult to predict” is sometimes replaced with “impossible”, “almost impossible”, “extremely difficult”, or “possible in principle, but impossible in practice.” Philosophers distinguish between epistemological emergence and ontological emergence. They are associated with "possible in principle, but difficult in practice" and "impossible in principle" respectively.

After an emergent property has been discovered sometimes it can be understood in terms of the properties of the system parts. An example is the BCS theory of superconductivity, which provided a posteriori, rather than a priori, understanding. A keyword in the statement above is “solely”.