Philosophical perspectives on the emergence of molecular structure

 In philosophical discussions of emergence and reductionism in chemistry, molecular structure has received significant attention and debate. Sometimes molecular structure is used to justify strong emergence, i.e., that molecular cannot be predicted, even in principle, solely from quantum theory.

Primas was one of the first to claim that molecular structure could not be reduced to quantum physics. Consider the following statements.

“From a physical point of view the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one…

[the structure of benzene] does not exist in a full quantum-theoretical description since electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations. The concept of molecular structure does not follow from first principles - all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence. In a quantum theoretical description the molecular shape emerges by abstracting from the actually existing Einstein-Podolsky–Rosen correlations between the electrons and the nuclei. Historically, the structure concept has been introduced into quantum chemistry by the so called Born-Oppenheimer approximation. But this terminology is misleading since the main issue is not an approximation, but the breaking of a holistic symmetry.”

I have italicised claims that are contestable and that I will discuss further below.

Cartwright has given philosophical arguments as to why chemistry cannot be reduced to physics. In this context, she claims (pp. 106-7)

“The typical method of quantum-mechanical treatment of molecules begins with the Born–Oppenheimer approximation…

This approximation treats the atomic nucleus as a classical particle. But this fundamentally violates quantum mechanics which, following the Heisenberg uncertainty principle, maintains that we cannot have a simultaneous assignment of fixed positions and fixed momenta. The approximations that provide the reduction violate the very theory that the chemistry is being reduced to… the success of quantum chemistry relies fundamentally on assumptions that belong to classical chemistry.” 

This claim that the BOA violates the Heisenberg uncertainty principle was rebutted in an earlier post and discussed in more detail by Huggett et al. Nevertheless, Lombardi et al. are not satisfied by the rebuttal.

Hendry claimed molecular structure is evidence of strong emergence and/or downward causation. In a similar spirit, Ellis and Drossel argued that crystal structures in solid state physics are evidence of strong emergence. The arguments of Hendry have been criticised by Seifert. The arguments centre around the fact that the molecular structure is a classical entity and a concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically. It is claimed that the existence of molecular structures is assumed and not derived in quantum chemistry calculations as they assume the Born-Oppenheimer approximation (BOA). 

Scerri responded to these arguments claiming that chemistry (particularly the concept of molecular structure) is irreducible to quantum physics. He claimed these arguments are not valid because they misunderstand the role of the BOA. It does not violate the uncertainty principle and in practice chemists can and do perform non-BOA approximations. 

Fortin et al. rejected the view that decoherence can explain isomerism, as decoherence does not resolve issues associated with the quantum measurement problem. 

Franklin and Seifert claim “the problem of molecular structure just is the quantum measurement problem.” This is debatable. Most molecular structures can be understood in terms of the nuclear probability density having a unique maximum and decoherence then is not relevant. Decoherence and the collapse of the nuclear wavefunction are only relevant in systems such as ammonia and stereoisomers in which there are molecular structures with equal energy and separated by high energy barriers.

I now respond to some of the contestable claims of Primas.

“electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations”

It is possible to quantify and calculate the amount of entanglement between electrons and nuclei in a specific quantum state. In an EPR state the entanglement is maximal. In the BOA wavefunction entanglement is present, but is absent in the crude BOA. The entanglement has been estimated in benzene and is very small. The only molecules where the entanglement may be significant are those, such as isomers, where there are two degenerate molecular structures and the overlap of the associated nuclear wavefunctions is small (i.e., the tunnel splitting is small). But then, in most chemical situations decoherence will wash out this entanglement.

“all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence.”

This is the problem of isomers. It is resolved because isomers are present in the solution to the Schrödinger equation, as I argued earlier.

“the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one… the main issue is not an approximation, but the breaking of a holistic symmetry.”

This seems subjective to me. I see the BOA as just a well-justified approximation. The electrons and nuclei are treated differently because they are. They have vastly different masses and this affects how they can be treated in any solution to the full Hamiltonian. Nevertheless, the BOA is a quantum theory and the nuclei are described by a wavefunction.

As discussed earlier, for small molecules in practise it is no longer necessary to use the BOA and the electrons and nuclei can be treated on an equal footing. Molecular structure is present in solutions to the full Schrödinger equation.

I wonder if the objection to use of the BOA is any different to the use of approximations in other theories? For example, consider theoretical treatments of the motion of planets in the solar system. The effects of all the planets are not treated on an equal footing. The effect of other planets on a planet of interest are treated perturbatively.

In conclusion, the arguments that molecular structure is evidence of strong emergence are weak. Some confusion may partly arise from misinterpreting the significance of the following valid observations.

i. Molecular structures were first conjectured before quantum theory was proposed.

ii. The BOA was proposed to explain molecular structure from quantum theory.

iii. Today, almost all calculations of molecular structure assume BOA.

iv. Chemists talk about molecular structures largely in classical not quantum terms.

However, the scientific reality is that for small molecules their structure, can be understood, described, and calculated in purely quantum terms. Yet, that understanding provides a strong justification for the validity of the BOA and for the convenience of using classical language to describe molecular structure.

I stress that the weakness of the arguments for the strong emergence of molecular structure, does not mean that an emergent perspective is not relevant to chemistry. Molecular structure is emergent, when defined in terms of novelty. This leads to effective theories defined in terms of potential energy surfaces. Furthermore, as the next section argues chemistry exhibits novel properties, concepts, and theories that are not present in physics.

This post is extracted from Section 15, of my review article "Emergence: from physics to biology, sociology, and computer science."

A new version of my review article on emergence

On the arXiv, I have posted a new version of my review article, Emergence: from physics to biology, sociology, and computer science.

I have added expanded sections on molecular structure, quantitative measures of causal emergence, and biological evolution.

There are also many minor additions and corrections. I hope the hyperlinked Table of Contents is helpful.

I welcome feedback and suggestions. I am sure there is much more to do.

Quantum justification for classical discussions of potential energy surfaces in chemistry

 In computational quantum chemistry, the Born-Oppenheimer approximation (BOA) is used to determine potential energy surfaces (PES) for electronic states of molecules. It is standard practise to identify local minima on a PES with molecular structures. Molecular binding energies are identified with the difference in energy between these minima and the energies of the isolated atoms of which the molecule is composed.

 A chemical reaction between two molecules A and B to produce C can be understood in terms of the PES for the composite system consisting of all the atoms in A and B. The dynamics of the chemical reaction can be described in terms of a path on the PES that goes from the local minimum associated with A and B infinitely far apart to the minimum associated with the structure C. The path will pass through a saddle point on the PES and this is identified with a transition state in the chemical reaction and its energy determines the activation energy for the chemical reaction. The local curvature of the PES near a minima can be used to determine force constants for harmonic motion and the associated vibrational frequencies of a molecule.

This picture is a completely classical one and so may motivate a claim that chemists mix classical and quantum concepts and calculations in an ad-hoc manner. (This kind of argument is often used by philosophers to claim that chemistry cannot be reduced to physics). This is unfair because a quantum description of the nuclear dynamics can be given in terms of quantum wave-packets, consistent with the Heisenberg uncertainty principle, and whose dynamics is defined by the quantum equation for nuclear motion that is given by the BOA. Furthermore, the dynamics of the centre of a wave-packet is given by classical equations of motion (Ehrenfest’s theorem). Thus, the classical language used by chemists can be viewed as a justified and compact version of a quantum description.

Structural isomers. Isomers are associated with different local minima on the electronic ground state potential energy surface for a given combination of atoms. To understand a quantum description of isomers, consider a reaction coordinate associated with an isomerisation reaction (i.e. conversion of one isomer to the other). There are three energy scales of relevance: the energy difference between the ground state energy of the two isomers, the magnitude of the barrier height (activation energy), and the quantum zero-point energy associated with vibrations in the direction of the reaction coordinate. Denote these energies as dE, Eb , and Ezp, respectively. If dE ~ Ezp << Eb then the nuclear probability density rho(R) for the vibrational ground state will have two local maxima, corresponding to the geometries of the two isomers. If dE >> Ezp then the nuclear probability density rho(R) for the vibrational ground state will have only one local maxima, corresponding to the lower energy isomer geometry. However, the geometry of the higher energy isomer can be found as a local maximum in the nuclear probability density rho(R) for one of the excited vibrational ground states. 

 Figure. Potential energy surface associated with the two structural isomers of HOCO. TS_n denote different transition states associated with the chemical reaction OH + CO -> H + CO2. Taken from Bui et al.

What does the Born-Oppenheimer approximation mean for emergence?

Most philosophical debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of structures be predicted from quantum theory without assuming their existence or invoking classical concepts? I will argue that the answer is yes, contrary to much of the philosophical literature, which relies heavily on the widespread use of the Born-Oppenheimer approximation (BOA) in quantum chemistry calculations. However, the fact that these arguments for irreducibility are weak does not mean that emergence (defined in terms of novelty) is not central to chemistry.

In a previous post, I discussed recent work showing how the BOA is not necessary for quantum chemistry and that molecular structure can be defined independently of it.

However, since the BOA plays a central role in the philosophical arguments, it is worth reviewing what it is and what it does and does not assume or mean.

In 1927, Born and Oppenheimer introduced an approximation to allow the solution of the full quantum equations for electrons interacting with charged nuclei. Without the BOA, much of theoretical chemistry and solid-state physics would be incredibly difficult in practice. The approximation is based on the separation of time and energy scales associated with electronic and nuclear motion. It leads to the concept of potential energy surfaces for electronic states. They define an effective theory for the dynamics of the atomic nuclei in a molecule or solid.

The full Hamiltonian (given earlier) can be denoted by

 

where the first term is the kinetic energy operator for the nuclei. In the Born-Oppenheimer approximation (BOA) the full wavefunction is written as a product of a nuclear wavefunction and an electronic wavefunction.

Substituting this in the eigenvalue equation for the full Hamiltonian leads to separate eigenvalue equations for the electronic and nuclear wavefunctions, assuming terms depending on gradients with respect to R of the electronic terms can be neglected.

 

In the first equation, the nuclear co-ordinates appear as parameters not as operators. This is central to the philosophical debates.

The second equation can be viewed as an effective Hamiltonian for the nuclear degrees of freedom. The function E_e(R) defines the potential energy surface of the molecule. 

In the BOA the nuclear probability distribution defined above is

 

As discussed in the earlier post, the structure of many molecules can be defined in terms of the value of R at which the probability is maximum. 

I make four points about the BOA that are relevant to philosophical debates about whether molecular structure is predictable in a logically consistent manner from quantum theory.

1. The BOA does treat the nuclear degrees of freedom quantum mechanically. They are described by the nuclear wavefunction Phi(R), which is determined by the second eigenvalue equation. Consequently, the BOA does not violate Heisenberg’s uncertainty principle, contrary to some claims in the philosophy literature.

2. The BOA is not ad hoc. Corrections to it can be calculated and have been for many molecules. These corrections are typically small, being of order (me/Mi)^1/2. Exceptions, such as near conical intersections (where the potential energy surfaces for two electronic states touch) are well-known and well-studied.

3. For most small molecules, the results of BOA calculations compare favourably with wave-functions obtained from solutions of the full quantum Hamiltonian. When there are differences, they are largely small quantitative differences. When the differences are qualitative, they have largely been anticipated from knowledge of the limitations of the BOA.

4. The Born-Oppenheimer approximation is an example of a general approach to quantum mechanics problems, discussed by Migdal. Consider a system composed of two subsystems that have dynamics on two vastly different time scales, termed fast and slow. The effects of the fast system on the slow system can be treated by adding a potential energy term to the Hamiltonian operator of the slow system. 

In forthcoming posts, I will discuss quantum justifications for classical descriptions of nuclear dynamics on potential energy surfaces and then discuss philosophers' views about the BOA and molecular structure.

The emergence of molecular structure from quantum theory

Most debates about the emergence of molecular structure centre around the issue of irreducibility. Specifically, can the existence of molecular structures be predicted from quantum theory without assuming their existence or invoking classical concepts?

Consider a molecule that contains Ne electrons and Nn atomic nuclei (ions). The full quantum-mechanical Hamiltonian for the system is 

where e is the electronic charge, rj is the position of the j-th electron, Zi  and Mi are the charge and mass, respectively, of the i’th ion with position co-ordinate Rj. This is the Hamiltonian that Laughlin and Pines dubbed “The Theory of Everything” because if the solution (i.e., eigenstates and eigenvalues of the Hamiltonian operator) could be found it would describe almost all of chemistry and materials science.

This Hamiltonian treats the electrons and nuclei on an equal footing. 

For isomers, the Hamiltonian is identical. However, as will be discussed in a later post, that does not preclude solutions to the Hamiltonian that can describe isomers. 

The Hamiltonian has global translational and rotational symmetry, where all the particles undergo the same rotation or translation. In contrast, molecular structures may have discrete rotational symmetries. However, this is not necessarily a problem, as an eigenstate of a quantum problem can transform according to a non-trivial irreducible representation of the symmetry. For example, except the s-orbitals all the orbitals of the hydrogen atom are spatially anisotropic.

The electrons are identical particles and so have permutation symmetry. They are fermions with spin-1/2 and so any eigenstate must be antisymmetric under the exchange of two electrons. The energies associated with this exchange are crucial to the formation of chemical bonds and the stability of molecular structures.

If two or more atoms in the molecule are identical, then any exact eigenstate must be consistent with permutation symmetry. If a nucleus is composed of an even (odd) number of nucleons, then it is a boson (fermion) with integer (half-integer) spin, and eigenstates must be symmetric (antisymmetric) under exchange of identical nuclei. However, the corresponding exchange energies are relatively small (because the quantum delocalisation of the nuclei is small) and consequently most practical calculations of the eigenstates do not make this requirement of the eigenstates. Nevertheless, if the electrons and nuclei are treated on equal footing, this should be done. Although this is challenging, it has been done recently, as discussed below. 

Full quantum solutions of the Hamiltonian

In most computational quantum chemistry, the Hamiltonian is solved in the Born-Oppenheimer approximation, which will be introduced and discussed later. This is a source of some confusion and contention in philosophical discussions about the emergence of molecular structure.

Due to advances in methodology and computational power over the past few decades, it has become possible in practise to solve the full quantum Hamiltonian for small molecules. There are three levels of complication associated with this: quantum nuclear motion, rotational symmetry, and some nuclei being identical particles. There are also two challenges: first, finding the eigenstates and second, deducing the molecular structure from the eigenstates.

To begin, I consider the simplest case and ignore the complications associated with rotational symmetry or identical nuclei. This provides some insight and undermines some objections in the philosophical literature.

The ground state eigenfunction can be written as

where r and R are 3Ne and 3Na -dimensional vectors, respectively. Note that this function will have a complicated structure as it will depend on the spin states of all the electrons, denoted by s.

A probability distribution (reduced density matrix) for the positions of the nuclei is given by

where the sum is over all the electron spin degrees of freedom.

For many molecules, but not all, this probability distribution will have a unique global maximum at the coordinates R_0. This set of coordinates defines the geometry of the molecular structure. The physics underlying the existence of well-defined maxima is that the mass of the nuclei is much larger than the mass of the electrons, and as a result, the zero-point motions of the nuclei are much smaller than the separation of the nuclei in the molecular structure.

Note that the nuclear probability distribution is regularly measured in scattering experiments (using X-rays, neutrons, or electrons), and its maxima are used to determine the structures of molecules and crystals. The Debye-Waller factor is a measure of the width of the probability distribution. At low temperatures, it is determined by quantum zero-point motion. In other words, it is well established experimentally that classical molecular structures are an approximation to a fluctuating quantum structure.

Not every molecule will have a probability distribution with a unique maximum. An example is ammonia. As discussed further below, it has two maxima; each represents an umbrella geometry, and they are related by an inversion symmetry. The ground state wavefunction of the whole system is a superposition of two quantum states, each being associated with one of the two umbrella geometries, and the electronic and nuclear degrees of freedom are entangled with one another.

A general quantum definition of molecular structure

Lang et al. have recently overcome the challenges mentioned above to determine molecular structure in a manner that treats the electrons and nuclei on an equal footing with regard to quantum theory. They have considered both rotational symmetry and nuclear permutation symmetry and given a general definition of molecular structure involving nuclear probability densities calculated from the full wavefunction. They have explicitly performed these calculations for D3+, (where D is deuterium). The result is that the molecule has the same triangular structure that is observed experimentally and calculated using the Born-Oppenheimer approximation. This work is significant because it explicitly shows that molecular structure can be predicted in practice, not just in principle, from quantum theory.

In a forthcoming post, I will discuss the Born-Oppenheimer approximation and some of the confusion associated with it.

Symmetry matters in condensed matter physics

 Snowflakes form incredibly diverse structures, seen when they condense onto a plate of glass. Every snowflake is different. On the other hand, every snowflake is the same. They are all composed of ice, a solid state of water. Every snowflake is composed of units that have a six-fold symmetry (Figure 8). Every snowflake is composed solely of water molecules. This paradox of the particular and the universal is at the heart of condensed matter physics. Although diversity prevails anything is not possible. No snowflake has five-fold symmetry. Snowflakes have enchanted scientists for a long time. The astronomer Johannes Kepler studied them and in 1611 wrote a small book about them as a gift for his patron. Kepler suggested snowflakes provided clues to deeper questions about the composition of matter. Today, Kenneth Libbrecht, a physicist at Caltech, has spent most of his career studying snowflakes and has produced beautiful volumes of photographs of them.

Figure 8. A snowflake shows a six-fold symmetry, just like a hexagon. The snowflake appears identical when it is rotated by an angle of sixty degrees about an axis passing through its centre and perpendicular to the page.

Condensed matter physicists ask several questions about snowflakes. What is the reason for the six-fold symmetry of the snowflake? What is the connection between the macroscopic properties of snowflakes and the properties of the underlying microscopic constituents, molecules of H2O? How is the diversity of snowflake shapes possible? Is there a phase diagram that defines the external conditions under which the different shapes form?

There is a long history in art, architecture, philosophy, and science, of associating symmetry with beauty and perfection. The ancient Greek philosopher Plato was a proponent of this view. He studied a particular class of solid shapes: cube, tetrahedron, octahedron, icosahedron, and dodecahedron. Plato identified the first four shapes with the four “elements”: earth, wind, fire, and water, respectively, and the fifth with the heavens. Each of these solid shapes is highly symmetric. Every face of a Platonic solid is the same shape (square, triangle, pentagon,...) and each of those shapes has edges of equal length. 

Like Plato, Kepler believed that “God is a geometer” and that God’s creation should reflect the perfection of God. These convictions led Kepler to propose in 1597 that the orbits of the planets around the Sun were circular and that the Platonic solids determined the relative size of the orbits. Later this model for the solar system was shown not to be true. In fact, Kepler himself became famous because he showed that the planets moved in elliptical, not circular orbits. Nevertheless, Kepler’s model was the beginning of a long history of successfully relating physical laws to symmetry and geometry.

A key discovery in physics from the past century is that symmetry is central to understanding a wide range of physical phenomena, whether colliding billiard balls, the allowed energies of an atom, the fundamental forces of nature, or different states of matter. Symmetries determine what is physically possible. For example, that energy cannot be created or destroyed is a consequence of the fact that physical laws do not change with time.

In this Chapter I explore three key ideas. First, transitions between different states of matter are associated with changes in symmetry. Thus, symmetry provides a criterion for specifying the qualitative difference between distinct states of matter. Second, for a specific state of matter the relevant symmetry constrains what is physically possible. Third, symmetry is central to making connections between the macroscopic and microscopic properties of a state of matter. The next chapter will explore how symmetry is associated with the type of ordering that occurs in a state of matter.

An extract from Chapter 3, Condensed Matter Physics: A Very Short Introduction

A mystery about science is that humans can do it

We are surrounded by scientific knowledge and have become so used to it that we often take science for granted. We may rarely reflect on the amazing revelations of science—and so miss the opportunity to recognize the awesome nature of the universe. Things that we know, learn, and do today in science would have been inconceivable decades, let alone centuries, ago. 

Einstein said, “The most incomprehensible thing about the universe is that it is comprehensible.”  For Einstein, the success of science was a wonderful mystery. As he wrote to his friend Maurice Solovine: 

. . . 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.  

There are several dimensions to the comprehensibility of the universe being mysterious. Einstein highlighted the first mystery, which is that there is order in the world, as reflected in scientific laws, such as Newton’s theory of gravity, and that this order can be succinctly stated in the language of mathematics. To the best of our knowledge, these laws hold for all time and everywhere in the universe. The existence of the orderly behaviour encoded in scientific laws is necessary for science to work, which leads to the second mystery. Why have we been able to discover these laws?

A second dimension that makes science possible is the intellectual abilities of humans. Humans not only have the rational ability to do science—to reason, to understand, to communicate—but also the ability to design instruments, such as telescopes and microscopes. There seems to be a connection between the rationality of the universe and human rationality. The idea that there may be harmony between the structures of the universe and those of the human mind has a long history.  In the Renaissance, it was encapsulated in the metaphor of the “music of the spheres”. In his book, Harmonies of the World (1619), Johannes Kepler connected music and his explanations of planetary orbits. Einstein said that “Mozart’s music is so pure and beautiful that I see it as a reflection of the inner beauty of the universe.” 

Humans might have been different. Suppose that the average human intelligence was lower than it is today, and the variation of human intelligence was smaller. Then, there might have been no Galileo, Isaac Newton, Robert Boyle, Charles Darwin, Albert Einstein, Richard Feynman, Phil Anderson, or Linus Pauling. Without these brilliant figures in scientific history, scientific progress would have been slow. 

The third dimension is that human language enables scientists to formulate, represent, and communicate ideas, theories, and the results of scientific experiments. This language sometimes involves mathematics, graphs, or tables of data. Scientists can understand one another. Even though there can be misunderstandings, these can be resolved. There is a scientific culture that transcends the diversity of cultures associated with different countries, linguistic groups, and ethnicities.

The fourth dimension is the physical dexterity of humans. I am a theoretical physicist not an experimental physicist. I am “all thumbs” and not particularly good in the lab. Consequently, I have done no laboratory work since I was a Ph.D. student. In contrast, some gifted scientists have an ability to do things in a laboratory that most people cannot. Their manual dexterity allows them to fabricate precision instruments, grow pure crystals, blow exquisite glassware, see faint images, and fine-tune electronic instruments in extraordinary ways. If some humans did not have such amazing abilities, scientific progress would have been much slower—or possibly non-existent.

A fifth dimension that makes science possible is the availability and processability of materials that have been central to scientific progress. Making instruments requires specific materials, such as metals, glass, rubber, insulators, plastics, and semiconductors. If we lived in a world where some of these materials were very rare or could not be processed to the purity or malleability required for scientific instruments, we would not have supercomputers, electron microscopes, or the James Webb Space Telescope today. We might be struggling to make even the simple telescopes used by Galileo.

These five dimensions are all required for humans to be able to do science. There are several additional mysteries of science.  These can be divided into two classes: what science can do and what we can learn about the universe from science. Science allows us to know certain things about reality (epistemology) and also to understand the nature of that reality (ontology). In other words, science helps us make maps of physical reality. The terrain represented by those maps is amazing. And the fact that we can make the maps is amazing.

Information theoretic measures for emergence and causality

The relationship between emergence and causation is contentious, with a long history. Most discussions are qualitative. Presented with a new system, how does one identify the microscopic and macroscopic scales that may be most useful for understanding and describing the system? Can Judea Pearl’s seminal ideas about causality be implemented practically for understanding emergence?

Broadly speaking, a weakness of discussions of emergence and causality is that it is hard to define these concepts in a rigorous and quantitative manner that makes them amenable to empirical testing, with respect to theoretical models and to experimental data. 

Fortunately, in the past decade, there have been some specific proposals to address this issue, mostly using information theory. A helpful recent review is by Yuan et al. 

“Two primary challenges take precedence in understanding emergence from a causal perspective. The first is establishing a quantitative definition of emergence, whereas the second involves identifying emergent behaviors or phenomena through data analysis.

To address the first challenge, two prominent quantitative theories of emergence have emerged in the past decade. The first is Erik Hoel et al.’s theory of causal emergence [19] whereas the second is Fernando E. Rosas et al.’s theory of emergence based on partial information decomposition [24].

Hoel et al.’s theory of causal emergence specifically addresses complex systems that are modeled using Markov chains. It employs the concept of effective information (EI) to quantify the extent of causal influence within Markov chains and enables comparisons of EI values across different scales [19,25]. Causal emergence is defined by the difference in the EI values between the macro-level and micro-level."

One perspective on causal emergence is that it occurs when the dynamics of a system at the macro-level is described more efficiently by macro-variables than by the dynamics of variables from the micro-level.

Klein et al. used Hoel’s information-theoretic measures of causal emergence to analyse protein interaction networks (interactomes) in over 1800 species, containing more than eight million protein–protein interactions, across different scales. They showed the emergence of ‘macroscales’ that are associated with lower noise and uncertainty. The nodes in the macroscale description of the network are more resilient than those in less coarse-grained descriptions. Greater causal emergence (i.e., a stronger macroscale description) was generally seen in multicellular organisms compared to single-cell organisms. The authors quantified causal emergence in terms of mutual information (between large and small scales) and effective information (a measure of the certainty in the connectivity of a network). Philip Ball (2023) (pages 218-220) gives an account of this work in terms of the emergence of multicellularity in biological evolution. He introduced the term causal spreading (pages 225-7), arguing that over the history of evolution the locus of causation has changed.

Yuan et al. continue

"However, in Hoel’s theory of causal emergence, it is essential to establish a coarse-graining strategy beforehand. Alternatively, the strategy can be derived by maximizing the effective information (EI) [19]. However, this task becomes challenging for large-scale systems due to the computational complexity involved. To address these problems, Rosas et al. introduced a new quantitative definition of causal emergence [24] that does not depend on coarse-graining methods, drawing from partial information decomposition (PID)-related theory. PID is an approach developed by Williams et al., which seeks to decompose the mutual information between a target and source variables into non-overlapping information atoms: unique, redundant, and synergistic information [29]…"

The Figure below is taken from Rosas et al. Xt^j (j=1,…,n) are microscopic variables that define a Markov chain. Vt is a macroscopic variable that is completely determined by the microscopic variables.

“Diagram of causally emergent relationships. Causally emergent features have predictive power beyond individual components. Downward causation takes place when that predictive power refers to individual elements; causal decoupling when it refers to itself or other high-order features.”

Rosas et al. applied the method to specific systems, including Conway’s Game of Life, Reynolds’ flocking model, and neural activity as measured by electrocorticography. More recently, it was used to describe emergence in computer science, including the identification of modular structures. Calculations were performed for specific examples, including Ehrenfest’s urn model for diffusion, the Ising model with Glauber dynamics, a Hopfield neural network model for associative memory.

Yuan et al. also state the following:

"The second challenge pertains to the identification of emergence from data. In an effort to address this issue, Rosas et al. derived a numerical method [24]. However, it is important to acknowledge that this method offers only a sufficient condition for emergence and is an approximate approach. Another limitation is that a coarse-grained macro-state variable should be given beforehand to apply this method."

Sas et al. recently stated

“Empirical applications of this framework to study emergence … including the study of gene regulatory networks [22], the dynamics of the human brain [23], the internal dynamics of reservoir computing [24], and the formation of useful internal representations in machine learning [25].”

Yuan et al. also discuss two significant connections between causal emergence and machine learning. First, machine learning can be used to improve calculations of causal emergence. Second, causal emergence measures can be used to better understand how machine learning works and improve it.

The work described above built on earlier work by Crutchfield, who claimed that the identification of emergence and hierarchies could be made operational, stating that “different scales are delineated by a succession of divergences in statistical complexity at lower levels.” More recently, Rupe and Crutchfield have reported progress towards identifying emergent self-organisation in a system.

Although this work on quantitative measures of emergence based on information theory represents significant progress, there are many open problems. Examples include the extension to non-Markovian systems and the development of computationally feasible methods for large systems. The latter is particularly important in physical systems where spontaneous symmetry breaking occurs, as this only happens in the thermodynamic limit of an infinite system.

There is an unrecognised similarity between the work described above and techniques recently developed to characterise phase transitions in statistical mechanics models such as the Ising model and classical dimer models. Coarse-graining (CG) is optimised by maximising the Real-Space Mutual Information (RSMI) between a spatial block and its distant environment. 

In general, maximising mutual information is notoriously hard but can be done using state-of-the-art machine learning algorithms. Gokmen et al. have developed an algorithm that they claim “can, unsupervised, construct order parameters, locate phase transitions, and identify spatial correlations and symmetries for complex and large-dimensional real-space data.” Furthermore, the optimal CG explicitly identifies the scaling operators associated with the critical point. 

The classical dimer model provides a stringent test as “the relevant low-energy degrees of freedom are profoundly different from the microscopic building blocks of the theory and change qualitatively throughout the phase diagram.” In other words, the emergent entities (quasiparticles such as vortices associated with the height field, which is described by a sine-Gordon field theory) are different from the dimers.

It is encouraging to see that two different scientific communities have developed similar ideas to address this challenging problem of making discussions about emergence and causality more concrete and quantitative.

Responding to scientific uncertainty

Science provides an impressive path to certainty in some areas, particularly in physics. However, as scientists seek to describe increasingly complex entities, moving from chemistry to biology, and then to humans and societies, the level of uncertainty increases.

One observes a wide range of responses to scientific knowledge being uncertain. Here are a few.

Denial. Science is about facts and absolute truth. There really isn’t a problem. We should just trust the scientists.

Minimisation. There is some uncertainty, but it isn’t anything to be concerned about. Some scientists will also minimise any uncertainty about their own research. This may occur because of career ambition. Others will minimise public discussion of uncertainty to try and avoid promoting the science scepticism discussed below.

Optimistic perseverance. The uncertainty is openly acknowledged. Some of the uncertainty does not matter for what we need to know. Other uncertainties can be reduced by further scientific work, such by more precise measurements with new instruments or by developing more sophisticated theories.

Total scepticism. There is a suspicion about the validity of most scientific knowledge, particularly that which is perceived to have philosophical, religious, or political implications.

Suspicion about science

In spite of the success of science at describing the material world and leading to powerful and useful technologies, there is much public suspicion of science. On the one hand, this is understandable given that science has led to technologies with undesirable health, environmental and social consequences. Some scientists, governments and companies have lied about these consequences and hidden them from the public. Human subjects have been abused in medical experiments. Drugs that were claimed to be effective and safe turned out to be ineffective or have undesirable side effects. Science has been used for ideological purposes. Sometimes scientists have faked results to advance their own careers. However, these failures should not undermine our trust in reliable scientific knowledge. Distinctions should be made between the bodies of knowledge, the applications of that knowledge, and the actions of institutions. I now discuss several common claims in public discussion that are used to justify scepticism of scientific knowledge.

Science is always changing. 

One day, scientists tell you that chocolate is good for your health, and the next year they say it is bad for you. And that is just the start. Then there are eggs, wine, running marathons, and cheese. They just can’t make up their mind. So why should we trust them? At one time, they believed in phlogiston and the aether. Now they say they don’t exist. Aristotle was replaced by Newton, who was replaced by Einstein. So why believe in human-induced climate change, biological evolution, vaccines, the Big Bang theory, or Einstein’s theories?

It is true that scientific knowledge does develop and change over time. However, today we have incredibly detailed observations and theories in physics, astronomy, chemistry, biology, and geology. Any future changes will be relatively minor because they will have to be consistent with all the knowledge we have now. Furthermore, when theories change, such as when Einstein superseded Newton, they don’t show that the old theory was completely wrong, but rather that it applied in a limited domain. For example, Newton’s theories of motion and gravity are extremely reliable when it comes to objects that are much larger than atoms, less dense than a black hole, and are moving at speeds less than about 10,000 kilometres per second. This is why engineers spend years learning Newton’s theories, not Einstein’s. If you want to build a good bridge or a rocket, Newton is good enough. He is not wrong.

Update. (Jan. 19). I just discovered that the NY Times had a recent op-ed Science Keeps Changing. So Why Should We Trust It?

“Well, that’s just a theory.” 

In popular debate, such a refrain may be applied to the theory of biological evolution, the Big Bang theory in cosmology, or human-induced climate change. The claimant usually wants to dismiss a particular theory as just idle speculation. Here, the term “theory” is used in the same sense as everyday speculations, such as “I have a theory as to why the president resigned,” or “I have a theory about why my computer is running so slowly.” These are just stories that sound somewhat plausible. In contrast, scientific theories in physics, such as quantum theory and Einstein’s theories of relativity, have precisely defined mathematical formulations that have been checked for logical consistency, made specific predictions, and tested to great precision in experiments. They are not “just theories.” For example, for the Big Bang theory about the beginning of the universe and Darwin’s theory of biological evolution and diversity, there are many independent lines of evidence that are consistent with each theory.  

Scientists cannot be trusted. 

They are not committed to the truth, but rather to their own interests and agendas, related to their careers, politics, and religion. They close ranks and support the status quo of current scientific “dogma”, rather than being open to original thinkers who critique it and propose alternative theories. They don’t want to lose their well-paid jobs and lucrative grants. 

On the one hand, scientists can be conservative and resistant to new ideas. On the other hand, there are significant career incentives to overturn existing knowledge and have your radical new theory accepted. That is how some scientists become famous and win Nobel Prizes. The reasons it does not happen very often are not necessarily for social or ideological reasons. Many of the theories we have today can explain an awful lot. It requires a lot of evidence, carefully acquired and checked, to convince people that those theories need to be modified, let alone abandoned. This may take decades. But it does happen. An example is the Big Bang theory of the universe, whose acceptance was initially resisted because it went against the prevailing view that the universe did not have a beginning. In biology, the discovery in 1970 of the enzyme reverse transcriptase went against a popular version of the “Central dogma” of molecular biology that DNA was always converted to RNA and not the reverse. That discovery led to a Nobel Prize.

I don’t trust scientists. I will do my own research. There is lots of good material from unbiased sources on the internet.

The internet provides a range of information and perspectives on practically any issue imaginable, including science. The material is particularly vast and controversial on biological evolution, the beginning of the universe, fundamental physics, the age of the earth, climate change, and medicine. Since the covid-19 pandemic, scepticism of the effectiveness and safety of vaccines has increased. 

Ivermectin is a drug that was developed as a treatment for parasite worms. Its incredible success was recognised by the award of the 2015 Nobel Prize in Physiology or Medicine to William Campbell and Satoshi Omura, who discovered the drug. During the pandemic, high-profile politicians and social media influencers promoted ivermectin as a treatment for covid-19, even after systematic medical studies showed it was ineffective. Recently, it has gained a reputation as a “miracle” drug that can even cure cancer, but this is being suppressed by the medical establishment. All clinical trials have shown the drug is ineffective for human ailments, beyond deworming. Nevertheless, there are groups on social media with hundreds of thousands of members that discuss the conspiracy, how to get the drug, and the experiences of participants using it to treat a wide range of ailments. Danny Lemoi, a founder of one of the largest groups, died in 2023 after taking massive daily doses of the drug for several years to treat a heart condition. Afterwards, one member of the group wrote “No one can convince me that he died because of ivermectin. He ultimately died because of our failed western medicine which only cares about profits and not the cure.”

Fans of ivermectin claim that they are escaping the biases and vested interests of the medical establishment and Big Pharma as they pursue the truth. However, they are not escaping bias and vested interests. Successful social influencers build their reputations and million-dollar incomes from promoting scepticism. If there is no conspiracy, just scientific uncertainty and occasional incompetence and malpractice, their following collapses. Populist politicians build their careers on criticism of and stoking resentment towards elites, such as the medical establishment. The authority of the medical establishment is replaced with the authority of the popular opinion of a group of people whose views are shaped by social media algorithms, intuition, and anecdotal experience.

My purpose in giving the example of Ivermectin is not to start a detailed critique of science scepticism. Rather, it is to illustrate the role that the interplay of trust, authority, and tradition plays in how we determine what is true and what to act on. There are two competing traditions here: the populism of alternative medicine and the elitism of professional medicine. Each has its own sources of authority. In the end, it boils down to who we trust. We do not have the time, energy, resources or inclination to check the veracity of every single piece of information we have access to. We take shortcuts. This is what tradition does for us, for better and worse. Thus, we cannot escape tradition. We are all swimming in traditions, many of which are in conflict with one another. The question is whether we are aware of it and what we do with that awareness.

What does learning to ride a bicycle teach us?

How do you learn to ride a bicycle? How do you teach someone to ride a bicycle? It is not easy to put this into words and that is an important point in itself. It may help to have some knowledge of the parts of the bicycle and their respective functions. It may help to know something about relevant physics such as inertia, the centre of gravity, and balance. It may help to have some practical advice about seat height, posture, the appropriate speed at which to pedal, and where to look when riding. 

Nevertheless, all that information may not help much. Some young children learn to ride without knowing any of this. They just watch other children doing it, get on bike, try it, and learn by trial and error. The more passionate they are about learning the more likely they may be to succeed.

The mind and body of a bicyclist focus on just a few things: looking where they are going, pedalling, steering, and a sense of balance. This information is integrated together, and the rider adjusts their direction, pedalling, and posture. Furthermore, that process of integration and adjustment involves much that is not the rider’s focus, and they may not even be directly aware of. A person’s sense of body awareness and coordination is shaped by biology, physique, experience, and training.

This example of bike riding illustrates several important things.  First, we can have the ability to do something without necessarily being able to articulate how we do it. Second, knowing requires personal commitment. It involves trust and risk. If a person is unwilling to trust or take risks, they may miss out on something good, such as the joy of riding a bicycle. Third, knowing requires integration of multifaceted information. Fourth, knowledge and understanding come from integrating our focus into an implicit background we may not even be aware of.

The example of riding a bike is valuable for understanding how we know (epistemology) because it is simpler and less fraught and emotionally charged than how we come to an understanding and make decisions about history, ethics, politics, religion, and the meaning of scientific knowledge. 

These observations draw on Michael Polanyi, including his book, The Tacit Dimension, published in 1966, but based on lectures he gave at Yale in 1962. He referred to the first point as tacit knowing, and the fourth point as the subsidiary-focal interaction. The relationship of the subsidiary and the focus is like the whole and the parts. Polanyi considered the idea of tacit knowledge his most important discovery.

Aside: Chapter 2 of The Tacit Dimension is entitled "Emergence" and discusses ideas similar to those that Phil Anderson promoted in 1972 in More is Different, without using the word "emergence." According to Google Scholar, The Tacit Dimension has been cited 45,000 times.

Why is the state of universities such an emotional issue for me?

It all about values!

Universities have changed dramatically over the course of my lifetime. Australian universities are receiving increasing media attention due to failures in management and governance. But there is a lot more to the story, particularly at the grassroots level, of the everyday experience of students and faculty. It is all about the four M's: management, marketing, metrics, and money. Learning, understanding, and discovering things for their own sake is alien and marginalised. I have stopped writing posts about this. So why come back to it?

I am often struck how emotional this issue is for me and how hard it is to sometimes talk about it, particularly with those with a different view from me. Writing blog posts (e.g. this one) about it has been a somewhat constructive outlet, rather than exploding in anger at an overpaid and unqualified "manager" or one of their many multiplying minions.

A few weeks ago, I listened to three public lectures by the Australian historian Peter Harrison. [He is my former UQ colleague. We are now both Emeritus. I benefited from excellent seminars he ran at UQ, some of which I blogged about].

The lectures helped me understand what has happened to universities and also why it is a sensitive subject for me. Briefly, it is all about values and virtues.

The lectures are nicely summarised by Peter in the short article, 

How our universities became disenchanted: Secularisation, bureaucracy and the erosion of value

Reading the article rather than this blog post is recommended. I won't try and summarise it, but rather highlight a few points and then make some peripheral commentary.

I agree with Peter's descriptions of the problems we see on the surface (bureaucracy, metrics, and management features significantly). His lectures are a much deeper analysis of underlying cultural changes and shifting worldviews that have occurred over centuries, leading universities to evolve into their current mangled form.

A few things to clarify to avoid potential misunderstanding of Peter's arguments.

Secularisation is defined broadly. It does not just refer to the decline in the public influence of Christianity in the Western world. It is also about Greek philosophy, particularly Aristotle, and the associated emphasis on virtues and transcendence. Peter states:

"The intrinsic motivations of teachers, researchers and scholars can be understood in terms of virtues or duties. According to virtue ethics, the “good” of an activity is related to the way it leads to a cultivation and expression of particular virtues. These, in turn, are related to a particular conception of natural human ends or goals. (Aristotle’s understanding of human nature, which informs virtue ethics, proposes that human beings are naturally oriented towards knowledge, and that they are fulfilled as persons to the extent that they pursue those goals and develop the requisite intellectual virtues.)"

The virtue ethics of Aristotle [and Alisdair MacIntyre] conflicts with competing ethical visions, including duty-oriented (deontological) ethics, consequentialist ethics, and particularly utilitarianism. This led to a shift away from intrinsic goods to what things are "good for", i.e., what practical outcomes they produce. For example, is scientific research "good" and have "value" because it cultivates curiousity, awe, and wonder, or because it will lead to technology that will stimulate economic growth?

Peter draws significantly on Max Weber's ideas about secularisation, institutions, and authority. Weber argued that a natural consequence of secularisation was disenchantment (the loss of magic in the world). This is not simply "people believe in science rather than magic". Disenchantment is a loss of a sense of awe, wonder, and mystery.

Now, a few peripheral responses to the lectures.

Is secularisation the dominant force that has created these problems for universities? In question time, Peter was asked whether capitalism was more important. i.e., universities are treated as businesses and students as customers? He agreed that capitalism is a factor but also pointed out how Weber emphasised that capitalism was connected to the secularising effects of the Protestant Reformation.

 I think that two other factors to consider are egalitarianism and opportunism. These flow from universities being "victims" of their own success. Similar issues may also be relevant to private schools, hospitals, and charities. They have often been founded by people of "charisma" [in the sense used by Weber] motivated by virtue ethics. Founders were not concerned with power, status, or money. What they were doing had intrinsic value to them and was "virtuous". In the early stages, these institutions attracted people with similar ideals. The associated energy, creativity, and common vision led to "success." Students learnt things, patients got healed, and poverty was alleviated. But, this success attracted attention and  the institution then had power, money, status, and influence.

The opportunists then move in. They are attracted to the potential to share in the power, money, status, and influence. The institution then takes on a life of its own, and the ideals and virtue ethics of the founders are squeezed out. In some sense, opportunism might be argued to be a consequence of secularisation. 

[Aside: two old posts considered a similar evolution, motivated by a classic article about the development of businesses.]

One indicator of the "success" of universities is how their graduates join the elite and hold significant influence in society. [Aside: ignoring the problem of distinguishing correlation and causality. Do universities actually train students well or just select those who will succeed anyway?]  Before (around) 1960, (mostly) only the children of the elite got to attend university. Demands arose that more people should have access to this privilege. This led to "massification" and an explosion in the number of students, courses, and institutions. This continues today, globally. Associated with this was more bureaucracy. Furthermore, the "iron triangle" of cost, access, and quality presents a challenge for this egalitarianism. If access increases, so does cost and quality decreases, unless you spend even more. It is wonderful that universities have become more diverse and accessible. On the other hand, I fear that for every underprivileged student admitted whose mind is expanded and life enriched, many more rich, lazy, and entitled students suck the life out of the system.

Metrics are pseudo-rational

Peter rightly discussed how the proliferation of the use of metrics to measure value is problematic, and reflects the "rationalisation" associated with bureaucracy (described by Weber). Even if one embraces the idea that "rational" and "objective" assessment is desirable, my observation is that in practice, metrics are invariably used in an irrational way. For example, managers look at the impact factor of journals, but are blissfully oblivious to the fact that the citation distribution for any journal is so broad and with a long tail that the mean number is meaningless. The underlying problem is that too many of the people doing assessments suffer from some mixture of busyness, intellectual laziness, and arrogance. Too many managers are power hungry and want to make the decisions themselves, and don't trust faculty who actually may understand the intellectual merits and weaknesses of the work being assessed.

The problems are just as great for the sciences as the humanities

On the surface, the humanities are doing worse than the sciences. For example, if you look at declining student numbers, threats of job cuts, political criticism, and status within the university. This is because science is associated with technology which is associated with jobs and economic growth. However, if you look at pure science that is driven by curiousity, awe, and wonder, then one should be concerned. There is an aversion to attacking difficult and risky problems, particularly those that require long-term investment or have been around for a while. The emphasis is on low-lying fruit and the latest fashion. Almost all physics and chemistry research is framed in terms of potential applications, not fundamental understanding. Sometimes I feel some of my colleagues are doing engineering not physics. In a similar vein, biochemists frame research in terms of biomedical applications, not the beauty and wonders of how biological systems work. 

Are universities destined for bureaucratic self-destruction?

Provocatively, Peter considered the potential implications of the arguments of historian and anthropologist Joseph Tainter concerning the collapse of complex societies. On the technical side, this reminded me of a famous result in ecology by Robert May, that as the complexity of a system (the number of components and interactions) increases, it can become unstable.

I don't think universities as institutions will collapse. They are too integrated into the fabric of modern capitalism. What may collapse is the production of well-educated (in the Renaissance sense) graduates and research that is beautiful, original, and awe-inspiring. This leads naturally into the following question.

Is the age of great discoveries over?

Peter briefly raised this issue. On the one hand, we are victims of our own success. It is amazing how much we now know and understand. Hence, it is harder to discover truly new and amazing things. On the other hand, because of emergence we should expect surprises.

There is hope on the margins

Peter did not just lament the current situation but made some concrete suggestions for addressing the problems, even though we are trapped in Weber's "iron cage" of bureaucracy.

• Re-balancing the structures of authority
• Finding a place for values discourse in the universities
• Develop ways of resolving differences with a sense of the rationality of Alisdair MacIntyre in mind

On the first, I note the encouraging work of the ANU Governance Project.

Peter also encouraged people to work on the margins. I also think that this is where the most significant scholarship and stimulus for reform will happen. A nice example is the story that Malcolm Gladwell tells in a podcast episode, The Obscure Virus Club.

The multi-faceted character of emergence (part 2)

In the previous post, I considered five different characteristics that are often associated with emergence and classified them as being associated with ontology (what is real and observable) rather than epistemology (what we believe to be true). 

Below I consider five more characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

6. Self-organisation

Self-organisation is not a property of the system but a mechanism that a theorist says causes an emergent property to come into being. Self-organisation is also referred to as spontaneous order. 

In the social sciences self-organisation is sometimes referred to as an endogenous cause, in contrast to an exogenous cause. There is no external force or agent causing the order, in contrast to order that is imposed externally. For example, suppose that in a city there is no government policy about the price of a loaf of sliced wholemeal bread or on how many loaves that bakers should produce. It is observed that prices are almost always in the range of $4 to $5 per loaf, and that rarely are there bread shortages. This outcome is a result of the self-organisation of the free-market, and economists would say the price range and its stability has an endogenous cause. In contrast, if the government legislated the price range and the production levels that would be an exogenous cause. Friedrich Hayek emphasised the role of spontaneous order in economics. In biology, Stuart Kaufmann equates emergence with spontaneous order and self-organisation.

In physics, the periodicity of the arrangement of atoms in a crystal is a result of self-organisation and has an endogenous cause. In contrast, the periodicity of atoms in an optical lattice is determined by the laser physicist who creates the lattice and so has an exogenous cause.

Self-organisation shows how local interactions can produce global properties. In different words, short-range interactions can lead to long-range order. After decades of debate and study, the Ising model showed that this was possible. Other examples of self-organisation, include flocking of birds and teamwork in ant colonies. There is no director or leader but the system acts “as if” there is. 

7. Unpredictability

Ernst Mayr (This is Biology, p.19) defines emergence as “in a structured system, new properties emerge at higher levels of integration that could not have been predicted from a knowledge of the lower-level components.” Philip Ball also defines emergence in terms of unpredictability (Quanta, 2024).

More broadly, in discussions of emergence, “prediction” is used in three different senses: logical prediction, historical prediction, and dynamical prediction.

Logical prediction (deduction) concerns whether one can predict (calculate) the emergent (novel) property of the whole system solely from a knowledge of all the properties of the parts of the system and their interactions. Logical predictability is one of the most contested characteristics of emergence. Sometimes “predict” is replaced with “difficult to predict”, “extremely difficult to predict”, “impossible to predict”, “almost impossible to predict”, or “possible in principle, but impossible in practice, to predict.” 

As an aside, I note that philosophers distinguish between epistemological emergence and ontological emergence. They are associated with prediction that is "possible in principle, but difficult in practice" and "impossible in principle" respectively.

After an emergent property has been discovered experimentally sometimes it can be understood in terms of the properties of the system parts. In a sense “pre-diction” then becomes “post-diction.” An example is the BCS theory of superconductivity, which provided a posteriori, rather than a priori, understanding. In different words, development of the theory was guided by a knowledge of the phenomena that had already been observed and characterised experimentally. Thus, a keyword in the statement above about logical prediction is “solely”. 

Historical prediction. Most new states of matter discovered by experimentalists were not predicted even though theorists knew the laws that the microscopic components of the system obeyed. Examples include superconductivity (elemental metals, cuprates, iron pnictides, organic charge transfer salts, …), superfluidity in liquid 4He, antiferromagnetism, quasicrystals, and the integer and fractional quantum Hall states.

There are a few exceptions where theorists did predict new states of matter. These include are Bose-Einstein Condensates (BECs) in dilute atomic gases and topological insulators, the Anderson insulator in disordered metals, the Haldane phase in even-integer quantum antiferromagnetic spin chains, and the hexatic phase in two dimensions. It should be noted that prediction of BECs and topological insulators were significantly helped that theorists could predict them starting with Hamiltonians of non-interacting particles. Furthermore, all of these predictions involved working with effective Hamiltonians. None started with microscopic Hamiltonians for specific materials.

Dynamical unpredictability concerns what it means in chaotic dynamical systems, where it relates to sensitivity to initial conditions. I do not see this as an example of emergence as it can occur in systems with only a few degrees of freedom. However, some authors do associate dynamical unpredictability with complexity and emergence.

8. Irreducibility and singularities

An emergent property cannot be reduced to properties of the parts, because if emergence is defined 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 reduces in a particular mathematical limit to a less general theory. For example, quantum mechanics reduces to classical mechanics in the limit where Planck’s constant goes to zero. Einstein’s theory of special relativity reduces to Newtonian mechanics in the limit where the speeds of massive objects become much less than the speed of light. Theory reduction is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who arguably overstate the problems it presents for science producing reliable knowledge]. Subtleties arise because the two different theories usually involve language and concepts that are "incommensurate" with one another. 

Irreducibility is also related to the discontinuities and singularities associated with emergent phenomena. As emphasised independently by Hans Primas and Michael Berry, singularities occur because the mathematics of theory reduction 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 geometrical 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. This example illustrates that theory reduction is compatible with the emergence of novelty. Primas also considers how the Born-Oppenheimer approximation, which is central to solid state theory and quantum chemistry, is associated with a singular asymptotic expansion (in the ratio of the mass of an electron to the mass of an atomic nuclei in the system). 

Berry considers several other examples of theory reduction, including going from general to special relativity, from statistical mechanics to thermodynamics, and from viscous (Navier-Stokes) fluid dynamics to inviscid (Euler) fluid dynamics. He has discussed in detail how the caustics that occur in ray optics are an emergent phenomena and are associated with singular asymptotic expansions in the wave theory.

The philosopher of science Jeremy Butterfield showed rigorously that theory reduction occurred for four specific systems that exhibited emergence, defined by him as a novel and robust property. Thus, novelty is not sufficient for irreducibility.

9. Contextuality and downward causation

Any real system has a context. For example, it has boundary and an environment, both in time and space. In many cases the properties of the system are completely determined by the parts of the system and their interactions. Previous history and boundaries do not matter. However, in some cases the context may have a significant influence on the state of the system. An example is Rayleigh-Bernard convection cells and turbulent flow whose existence and nature are determined by the interaction of the fluid with the container boundaries. A biological example concerns what factors determine the structure, properties, and function that a particular protein (linear chain of amino acids) has. It is now known that the only factor is not just the DNA sequence that encodes for the amino acid sequence, in contradiction to some versions of the Central Dogma of molecular biology.  Other factors may be the type of cell that contains the protein and the network of other proteins in which the particular protein is embedded. Context sometimes matters.

Supervenience is the idea that once the micro level is fixed, macro levels are fixed too. The examples above might be interpreted as evidence against supervenience. Supervenience is used to argue against “the possibility for mental causation above and beyond physical causation.” 

Downward causation is sometimes equated with emergence, particularly in debates about the nature of consciousness. In the context of biology, Denis Noble defines downward causation as when higher level processes can cause changes in lower level properties and processes. He gives examples where physiological effects can switch on and off individual genes or signalling processes in cells, including maternal effects and epigenetics.

10. Intra-stratum closure: informational, causal, and computational

The ideas described below were recently developed by Rosas et al. from a computer science perspective. They defined emergence in terms of universality and discussed its relationship to informational closure, causal closure, and computational closure. Each of these are given a precise technical definition in their paper. Here I give the sense of their definitions. In considering a general system they do not pre-define the micro- and macro- levels of a system but consider how they might be defined so that universality holds, i.e., so that properties at the macro-level are independent of the details of the micro-level (i.e., are universal).

Informational closure means that to predict the dynamics of the system at the macroscale an observer does not need any additional information about the details of the system at the microscale. Equilibrium thermodynamics and fluid dynamics are examples. 

Causal closure means that the system can be controlled at the macroscale without any knowledge of lower-level information. For example, changing the software code that is running on a computer allows one to reliably control the microstate of the hardware of the computer regardless of what is happening with the trajectories of electrons in the computer.

Computational closure is a more technical concept, being defined in terms of “a conceptual device called the ε-(epsilon) machine. This device can exist in some finite set of states and can predict its own future state on the basis of its current one... for an emergent system that is computationally closed, the machines at each level can be constructed by coarse-graining the components on just the level below: They are, “strongly lumpable.” "

Rosas et al., show that informational closure and causal closure are equivalent and that they are more restrictive than computational closure. It is not clear to me how these closures relate to novelty as a definition of emergence.

In summary, emergence means different things to different people. I have listed ten different characteristics that have been associated with emergent properties. They are not all equivalent and so when discussing emergence it is important to be clear about which characteristic one is using to define emergence.

The multi-faceted character of emergence (part 1)

There is more to emergence than novel properties, i.e., where a whole system has a property that the individual components of the system do not have. Here I focus on emergent properties, but in most cases “property” might be replaced with state, phenomenon, or entity. I now discuss ten characteristics often associated with emergence, beyond novelty. Some people include one or more of these characteristics in their definitions of emergence. However, I do not include them in my definition because as I explain some of the characteristics are contentious. Some may not be necessary or sufficient for novel system properties.

The first five characteristics discussed below might be classified as objective (i.e., observable properties of the system) and the second five as subjective (i.e., associated with how an investigator thinks about the system). In different words, the first five are mostly concerned with ontology (what is real) and the second five with epistemology (what we know). The first five characteristics concern discontinuities, universality, diversity, mesoscales, and modification of parts. The second five concern self-organisation, unpredictability, irreducibility, downward causation, and closure. 

1. 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., when the number of particles of the system becomes infinite). More is different. Thus, as a quantitative change in the system size occurs the order parameter becomes non-zero. In a system that undergoes a phase transition at a non-zero temperature, a small change in temperature can lead to the appearance of order and to a new state of matter. For a first-order phase transition, there is discontinuity in properties such as the entropy and density. These discontinuities define a phase boundary in the pressure-temperature diagram. For continuous phase transitions the order parameter is a continuous function of temperature, becoming non-zero at the critical temperature. However the derivative with respect to temperature may be discontinuous and/or thermodynamic properties such as the specific heat and susceptibility associated with the order parameter may approach infinite as the critical temperature is approached.

Two different states of a system are said to be adiabatically connected if one can smoothly deform one state into the other and all the properties of the system also change smoothly. The case of the liquid-gas transition illustrates subtle issues about defining emergence. A discontinuity does not imply a qualitative difference (novelty). On the one hand, there is a discontinuity in the density and entropy of the system as the liquid-gas phase boundary is crossed in the pressure-temperature diagram. On the other hand, there is no qualitative difference between a gas and a liquid. There is only a quantitative difference: the density of the gas is less than the liquid. Albeit sometimes the difference is orders of magnitude. 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 questions, 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, a non-zero magnetisation (order parameter) 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 (critical temperature for ferromagnetism) 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. Universality

Properties often referred to as emergent are universal in the sense that it is independent of many of the details of the parts of the system. There may be many different systems that can have a particular 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. In elementary superconductors, introducing non-magnetic impurity atoms has no effect on the superconductivity.

Universality is both a blessing and a curse for theory. Universality can make it easier to develop successful theories because it means that many details need not be included in a theory in order for it to successfully describe an emergent phenomenon. This is why effective theories and toy models can work even better than might be expected. Universality can make theories more powerful because they can describe a wider range of systems. For example, properties of elemental superconductors can be described by BCS theory and by Ginzburg-Landau theory, even though the materials are chemically and structurally diverse. The curse of universality for theory is that universality illustrates the problem of “under-determination of theory”, “over-fitting of data” and “sloppy theories” [Sethna et al.]. A theory can agree with the experiment even when the parameters used in the theory may be quite different from the actual ones. For example, the observed phase diagram of water can be reproduced, sometimes with impressive quantitative detail, by combining classical statistical mechanics with empirical force fields that assume water molecules can be treated purely being composed of point charges.

Suppose we start with a specific microscopic theory and calculate the macroscopic properties of the system, and they agree with experiment. It would then be tempting to think that we have the correct microscopic theory. However, universality suggests this may not be the case.

For example, consider the case of a gas of weakly interacting atoms or molecules. We can treat the gas particles as classical or quantum. Statistical mechanics gives exactly the same equation of state and specific heat capacity for both microscopic descriptions. The only difference may be the Gibbs paradox [the calculated entropy is not an extensive quantity] which is sensitive to whether or not the particles are treated as identical or not. Unlike the zeroth, first, and second law of thermodynamics, the third law does require that the microscopic theory be quantum. Laughlin discusses these issues in terms of “protectorates” that hide “ultimate causes” .  

In some physical systems, universality can be defined in a rigorous technical sense, making use of the concepts and techniques of the renormalisation group and scaling. These techniques provide a method to perform coarse graining, to derive effective theories and effective interactions, and to define universality classes of systems. There are also questions of how universality is related to the robustness of strata, and the independence of effective theories from the coarse-graining procedure.

3. Diversity

Even when a system is composed of a small number of different components and interactions, the large number of possible stable states with qualitatively different properties that the system can have is amazing. Every snowflake is different. Water is found in 18 distinct solid states. All proteins are composed of linear chains of 20 different amino acids. Yet in the human body there are more than 100,000 different proteins and all perform specific biochemical functions. We encounter an incredible diversity of human personalities, cultures, and languages. A stunning case of diversity is life on earth. Billions of different plant and animal species are all an expression of different linear combinations of the four base pairs of DNA: A, G, T, and C.

This diversity is related to the idea that "simple models can describe complex behaviour". One example is Conway’s Game of Life. Another example is how simple Ising models with a few competing interactions can describe a devil's staircase of ground states or the multitude of different atomic orderings found in binary alloys.

Goldenfeld and Kadanoff defined complexity [emergence] as “structure with variations”. Holland (VSI) discusses “perpetual novelty” giving the example of the game of chess, where are typical game may involve the order of 1050 move sequences. “Motifs” are recurring patterns (sequences of moves) in games. 

Condensed matter physics illustrates diversity with the many different states of matter that have been discovered. The underlying microscopics is “just” electrons and atomic nuclei interacting according to Coulomb’s law.

The significance of this diversity might be downplayed by saying that it is just a result of combinatorics. But such a claim overlooks the issue of the stability of the diverse states that are observed. In a system composed of many components each of which can take on a few states the number of possible states of the whole system grows exponentially with the number of components. For example, for a chain of ten amino acids there are 1013 different possible linear sequences. But this does not mean that all these sequences will produce a functional protein, i.e., a molecule that will fold rapidly (on the timescale of milliseconds) into a stable tertiary structure and perform a useful biochemical function such as catalysis of a specific chemical reaction or signal transduction.

4. Simple entities at the mesoscale 

A key idea in condensed matter physics is that of quasi-particles. A system of strongly interacting particles may have excitations, seen in experiments such as inelastic neutron scattering and Angle Resolved PhotoElectron Spectroscopy (ARPES), that can be described as weakly interacting quasi-particles. These entities are composite particles, and have properties that are quantitatively different, and sometimes qualitatively different, from the microscopic particles. Sometimes this means that the scale (size) associated with the quasi-particles is intermediate between the micro- and the macro-scales, i.e., it is a mesoscale. The existence of quasi-particles leads naturally to the technique of constructing an effective Hamiltonian [effective theory] for the system where effective interactions describe the interactions between the quasi-particles.

The economist Herbert Simon argued that a characteristic of a complex system is that the system can be understood in terms of nearly decomposable units. Rosas et al., argue that emergence is associated with there being a scale at which the system is “strongly lumpable”. Denis Noble has highlighted how biological systems are modular, i.e., composed of simple interchangeable components.

5. Modification of parts and their relationships

Emergent properties are often associated with 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 together. 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 are different to their values for isolated molecules. However, emergence is a sufficient but not a necessary condition for these modifications. In gas and liquid states, novelty is not present but there are still such changes in the properties of the individual parts.

As stated at the beginning of this section the five characteristics above might be associated with ontology (what is real) and objective properties of the system that an investigator observes and depend less on what an observer thinks about the system. The next five characteristics might be considered to be more subjective, being concerned with epistemology (how we determine what is true). In making this dichotomy I do not want to gloss over the fuzziness of the distinction or of two thousand years of philosophical debates about the relationship between ontology and epistemology, or between reality and theory.

In the next post, I will discuss the remaining five characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

Thanks for reading this far!