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 is not present in the individual parts of the system. 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.

I don’t claim this definition 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 (including by me in past blog posts, talks, and in my VSI).

Characteristics of emergent phenomena

There is more to emergence than novel properties. For specificity, 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 may not be necessary or sufficient for novelty.

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 how subtle issues about emergence. There is no qualitative difference between a gas and a liquid, only a quantitative difference: the density. Even below the critical temperature 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 be reflected in a generalised rigidity.

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. 

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.

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.” 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”.

The challenge of useful data in the social sciences

A major challenge for the social sciences is obtaining data that is reliable, gives significant insight, and could be used to test theories. Each week I read The Economist. Many of their articles feature graphs of social or economic data. To me, some of the graphs are just random noise or show marginal trends that I am not convinced are that significant. But other graphs are quite dramatic or insightful. Previously, I posted a famous one about smoking.

This week I saw the graph below in The New York Times, as part of a long article, Childbirth Is Deadlier for Black Families Even When They’re Rich, Expansive Study Finds, based on this preprint.

The data clearly shows the distressing fact that "The richest Black women have infant mortality rates at about the same level as the poorest white women."

Different dimensions to emergence for specific scientific disciplines

Emergence is a concept relevant to a wide range of scientific disciplines, from physics to sociology. Emergence is also at the heart of some of the biggest questions and challenges in each discipline. How might I justify that claim? How do we move beyond "emergence" just being a trendy buzzword?

Here I suggest some different facets of a specific discipline that with an emergent perspective may help to understand the discipline and to plan scientific strategy. This post will be primarily descriptive and the next prescriptive. Later I will illustrate both aspects with specific disciplines. Although, some of the facets below may be somewhat obvious, others are profound. 

Presence of distinct scales. Scales may involve length, time, or number of components in a system of interest. Different phenomena are observed at different scales.

Stratification and separation of scales. Distinct phenomena as usually seen over some range of scale and a distinct stratum can be associated with that scale. 

Sub-disciplines (or sub-fields) are associated with each stratum. The discipline can be viewed as stratified. For example, biology has sub-disciplines associated with ecosystems, organisms (animals and plants), organs, cells, genes, and molecules. This is nicely captured in a series of articles in The Economist.

The system can be viewed as interacting components. The system of interest is composed of many parts. Identifying the relevant components and their interactions may be non-trivial or at least was in the past. For example, consider the discovery of atoms in chemistry, quarks in nuclear physics, Cooper pairs in superconductivity, and DNA in genetics.

Emergent properties. Systems of interest have distinct properties that the components of the system do not. These properties may have certain characteristics such as universality, irreducibility, or unpredictability.

Emergent entities. These distinct entities can only be defined at certain scales and emerge from interactions between components that are defined at some smaller scale. In biology, emergent entities include organisms, organs, cells, genes, and proteins. In condensed matter physics emergent entities include quasiparticles and topological defects.

Emergent phenomena. This is closely related to emergent properties and may be redundant. But a property is something that a system has and a phenomenon is something that it does. 

Different experimental probes for different scales. For example, for condensed matter different types of electromagnetic radiation from x-rays to microwaves are used to investigate a material at different length scales. The nature of the instruments used and the type and quality of information gained can be quite different for the different scales.

Simple theoretical models of interacting components.  From the perspective of the smallest scales most systems with emergent properties are complex in that they involve many degrees of freedom and so large amounts of information and parameters are required to define the state of the system. The system may also be complex in the sense that the emergent properties are non-trivial and hard to describe theoretically. But with insight simple models with just a few parameters and state variables can exhibit and describe the emergent properties. Examples of such models in condensed matter physics include Ising, Hubbard, and non-linear sigma models. Examples from sociology include agent-based models such as the Schelling model for racial segregation. Simple models can be viewed as effective theories, valid at a particular scale, and can illustrate universality.

Organising principles and concepts at each scale. The principles and concepts are only meaningful and relevant at a particular scale. An example from condensed matter physics and elementary particle physics is spontaneous symmetry breaking.

In another post, I will discuss how an emergentist perspective plays out in scientific strategy.

Science and the universe are awesome

Since we are surrounded by scientific knowledge. We are so used to it that we can take science for granted and not reflect on how amazing science truly is. And how amazing the universe is that science reveals. Things that we know, learn, and do today in science would have been inconceivable decades ago, let alone centuries ago.

What specific things do you think are particularly awesome? This question was stimulated by Frank Wilczek's recent book, Fundamentals: Ten Keys to Reality. In writing the book, he says "what began as an exposition grew into a contemplation."

 My answer to the question has some significant overlap with Wilczek's ten. 

Below I list some of the things that I find awesome. I consider two classes: what science can do and what we learn about the universe from science.

Science works! It is amazing what science can do.

We can understand the material world.

Einstein said, "The most incomprehensible thing about the world is that it is comprehensible." In a previous post, I explored some different dimensions of the fact that the universe is comprehensible. The mystery includes human capabilities, both intellectual and physical, and the malleability of the material world.

We can make precise measurements.

Scientists have created incredibly powerful and specialised instruments for making very precise measurements such as spectrometers, telescopes and microscopes. Scientists can measure the tension in a single strand of DNA, the magnetic moment of an electron to a precision of one part in one billion billion, the spectrum of light emitted by a galaxy that is ten billion light years away, ...

We can predict the outcome of new experiments.

Scientists construct theories in their minds, on pieces of paper, in mathematical equations, and in computers. One way to evaluate the possible validity of a theory is to propose new experiments and predict the outcome. Famous examples include the existence of the chemical element aluminium, the existence of the planet Neptune, radio waves, a specific excited quantum state of the atomic nucleus of carbon atoms, the pollinator moth for Darwin's orchid, the deflection of the path of light from a distant star by our sun, gravitational waves, the Cosmic Microwave Background, quarks, the Higgs boson, the Berezinskii-Kosterlitz-Thouless phase transition, the hexatic phase, edge states in integer spin antiferromagnetic chains, topological insulators, ... Predictions are particularly impressive when they are unexpected and controversial.

We can use mathematics. 

Eugene Wigner received the Nobel Prize in Physics in 1963. In 1960 he published an essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences that concludes

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. 

We can manipulate and control nature.

Scientists and engineers can move single atoms, design drugs, make computers, build atom bombs, heart pacemakers, and mobile phones, manipulate genes, ......

We know so much but we know so little. 

On the one hand, the achievements of science are amazing. Yet, in spite of this, there are still significant mysteries and challenges. Examples include the nature of dark matter or human consciousness, a quantum theory of gravity, fine-tuning of fundamental constants, the quantum-classical boundary, protein folding, the nature of glasses, and how to calculate the properties of complex systems.

It is awesome what science reveals to us about the universe.

The immense scales of the observable universe

Our sun is just one star among the more than two hundred billion that make up our galaxy, the Milky Way. And that is just one of one trillion galaxies in the whole universe. It takes light from the most distant galaxies tens of billions of years to travel to us.

Length, time, and energy scales over many many orders of magnitude

These go far beyond our everyday experience and what we can see with the naked eye (from a millimetre to a kilometre). On the large scale, the visible universe involves distances of billions of light years (10^25 metres). On the small scale, there is the sub-structure of nucleons, which is smaller than femtometres (10^-15 m).  This wide range of length scales is nicely illustrated in the wonderful movie Powers of Ten and its update, The Cosmic Eye. There are corresponding time, energy, and temperature scales varying over many many orders of magnitude. For example, as one goes from ultracold atomic gases to quark-gluon plasmas, the  relevant energy and temperature scales vary over more than 20 orders of magnitude! At every scale, there are distinct phenomena and structures. 

Universal laws that are simple to state

The universe exhibits a diversity of rich and complex behaviour. Yet it can understand much of it in terms of simple universal laws that are easy to state, e.g., Newton's laws of motion, the laws of thermodynamics, Maxwell's equations of electromagnetism, Schrodinger's equation of quantum mechanics, the genetic code, ....  And, these are just a few of these laws. One does not need a multitude of laws to describe a multitude of instances of a multitude of phenomena.

Just a few building blocks

There are just a few fundamental particles in the standard model (leptons, neutrinos, and gauge bosons). Everything is made of them. They are the building blocks of atoms. They each have just a few physical properties: charge, spin, mass, and colour. Every single particle of a particular type in the universe has exactly the same properties. Exactly. As far as we know, they have been exactly the same throughout time, going back to the beginning of the universe, and whether they are in your body, or in a star in a distant galaxy.

Atoms are the building blocks of chemical compounds. Every single atom of a particular chemical element (and nuclear isotope) is absolutely identical. This allows astronomers to determine the chemical composition of distant stars, galaxies, and dust clouds.

Humans, plants, and animals all have the same molecular building blocks and there are just a few of them. Any DNA molecule is composed of just four different base pairs (denoted A, G, T, C) and proteins are composed of just twenty different amino acids.

There are two amazing things here. First, there are just so few building blocks. Second, every one of these building blocks is absolutely identical.

Emergence: simple rules produce complex behaviour

Humans, cells, and crystals can be viewed as systems composed of many interacting components. The components and their interactions can often be understood and described in simple terms. Nevertheless, from these interactions complex structures and properties can emerge.

Nature appears to be fine-tuned for life

This covers not just the values of fundamental physical constants that lead to the notion of fine-tuning and the anthropic principle. Water has unique physical and chemical properties that allow it to play a crucial role in life, such as the surface of lakes freezing before the bottom and aiding protein folding.

The intricate and subtle "machinery" of biomolecules

Proteins have very unique structures that are intimately connected to their specific functions, whether as catalysts or light sensors.

What do you think are the most amazing things about science and what we learn from it?

Condensed Matter Physics: A Very Short Introduction; almost there...

The publication of my book has been delayed a couple of months. It is now due for release in February in the UK and May in the USA.

It is available for pre-order.

If you are teaching a course for which the book could potentially be one of the texts you can request a free inspection copy.

OUP has produced a nice flyer to promote the book.

The green comet and quantum chemistry

The comet C/2022 E3 (ZTF) getting a lot of attention, pointed out to me by my friend Alexey. Why is it green? This basic question turns out to be scientifically rich and has only recently been answered.

The green glow comes from a triplet excited state of diatomic carbon, C2. This got my interest because a decade ago I blogged on debates by quantum chemists about whether C2 involves a quadruple bond. Back in 1995, Roald Hoffmann wrote an interesting column in The American Scientist (and reproduced in his beautiful book Same and Not the Same) about the molecule and how it is present in various organometallic compounds and inorganic crystals.

Recent advances in understanding the photophysics of C2 were reported in 2021 in this paper.

Photodissociation of dicarbon: How nature breaks an unusual multiple bond

Jasmin Borsovszky, Klaas Nauta, Jun Jiang, Christopher S. Hansen, Laura K. McKemmish, Robert W. Field, John F. Stanton, Scott H. Kable, and Timothy W. Schmidt 

Here is a summary of the significance and content of the paper from Chemistry World.

..as dicarbon streams out of the comet core, it is destroyed by sunlight – this is why the comet tail, unlike the coma, is colourless. However, the precise mechanism of this supposed photodissociation had remained unclear.

Researchers in Australia and the US have now for the first time observed diatomic carbon’s photodissociation in the lab. The team produced dicarbon by photolysing tetrachloroethylene, and then breaking it apart with laser pulses. This allowed them to determine its bond dissociation energy with the same precision as for oxygen and nitrogen. Previous measurements for dicarbon had uncertainties an order of magnitude higher than for other diatomic molecules.

To break its quadruple bond, the molecule must absorb two photons and undergo two ‘forbidden’ transitions, those that break spectroscopic rules. Cometary dicarbon, the researchers calculated, has a lifetime of around two days until sunlight breaks it apart – the reason why its colour is visible in the coma but not in the tail.

Some amazing things about the universe that make science possible

 This post takes off from the following Einstein quotes.

"The most incomprehensible thing about the universe is that it is comprehensible"

from "Physics and Reality"(1936), in Ideas and Opinions, trans. Sonja Bargmann (New York: Bonanza, 1954), p292.

"...I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton's theory of gravitation, for example, is wholly different." 

Letters to Solovine, New York, Philosophical Library, 1987, p 131.

There are several dimensions to the comprehensibility of the universe. The dimension highlighted by Einstein is that there is order in the world, reflected in laws that can be succinctly stated and mathematically encoded. These laws seem to hold for all time and everywhere in the universe. Here I suggest there are three other dimensions that make science possible. 

A second amazing dimension is that humans have the rational ability to do science: to reason, to understand, to communicate, and to make instruments such as telescopes and microscopes. There seems to be somewhat of a match between the rationality of the universe and human rationality. This is written in the spirit of arguments about fine-tuning, where one imagines alternative universes.

Humans could have been different. Suppose that the amount and variation of human intelligence (at least that aspect of intelligence relevant to doing science) were different, and the mean and standard deviation were lower. Suppose that intelligence was lower so that there were no brilliant humans like Darwin, Einstein, Newton, Pauling, ... In fact, suppose that even the brightest people were as good at science as I am at music and dancing. Scientific progress would be rather limited.

But it is not just human intelligence that matters. A third amazing dimension is that of manual dexterity. I am "all thumbs" and not particularly good in the lab. There are some gifted experimentalists with an outstanding ability to do things most people cannot, even with training. Such abilities allow them to fabricate precision instruments, grow crystals, see faint images, ... If some humans did not have such abilities scientific progress would have been much slower, or possibly non-existent.

A fourth crucial dimension concerns the availability and processability of certain materials that are central to scientific progress. Making instruments requires particular materials such as metals, glass, and semiconductors. Suppose we lived in a world where some of these were very rare or just could not be processed to the purity or malleability required.