Famous last words

If you ever write a popular book about science I suggest you spend a lot of time honing your very last paragraph. If it is eloquent, grand, and hyperbolic it may be so widely quoted that many people will think that this is actually what the book is about or has proven. Here are a few examples that I often see.

Where then shall we find the source of truth and the moral inspiration for a really scientific socialist humanism? Only, we suggest, in the sources of science itself,..... it is the conclusion to which the search for authenticity necessarily leads. The ancient covenant is in pieces; man at last knows that he is alone in the unfeeling immmensity of the universe, out of which he emerged only by chance. Neither his destiny nor his duty have been written down. The kingdom above or the darkness below: it is for him to choose.''

Jacques Monod, Chance and Necessity: An Essay on the Natural Philosophy of Modem Biology, trans. Austryn Wainhouse (New York: Knopf, 1971), p. 167

But if there is no solace in the fruits of our research, there is at least some consolation in the research itself. Men and women are not content to comfort themselves with tales of gods and giants, or to confine their thoughts to the daily affairs of life; they also build telescopes and satellites and accelerators, and sit at their desks for endless hours working out the meaning of the data they gather. The effort to understand the universe is one of the very few things which lifts human life a little above the level of farce and gives it some of the grace of tragedy.

Steven Weinberg, The First Three Minutes (Basic Books, 1977), pages 154-155.

If we do discover a complete theory, it should in time be understandable in broad principle by everyone, .... Then we shall all ...[discuss] why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason - for then we would truly know the mind of God.  

Stephen Hawking, A Brief History of Time

There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone circling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved.

Charles Darwin, The Origin of Species

Can you think of any other examples of famous last paragraphs?

Chapter abstracts for A Very Short Introduction (part 2)

 Here are my draft abstracts and keywords for the whole book and for chapters 6-10. Context and chapters 1-5 were given in the previous post.

Comments and suggestions are welcome.

Condensed Matter Physics: A Very Short Introduction 

There are many more states of matter than just solid, liquid, and gas. Examples include liquid crystal, ferromagnet, glass, superfluid, and superconductor. New states are continually, and unexpectedly, being discovered. A superconductor can be like Schrodinger’s cat and in the macroscopic world exhibit the weirdness associated with the microscopic world of atoms, photons, and electrons, that is described by quantum theory. Condensed matter physics investigates how states of matter, and their distinct physical properties emerge from the atoms that a material is composed of. Such a system composed of many interacting parts can have properties that the parts do not have. Water is wet, but a single water molecule is not. Your brain is conscious, but a single neuron is not. Such emergent phenomena are central to condensed matter physics and occur in many fields, from biology to computer science to sociology, leading to rich intellectual connections. When do quantitative differences become qualitative differences? Can simple models describe rich and complex behaviour? What is the relationship between the particular and the universal? How is the abstract related to the concrete? Condensed Matter Physics: A Very Short Introduction is concerned with such big questions. The materials in silicon chips, liquid crystal displays, and magnetic computer memories, may have transformed society. But, understanding them has transformed how we think about complex systems. Key concepts explored include phase diagrams, phase transitions, symmetry, types of order, spontaneous symmetry breaking, spatial dimensionality, scaling, universality, macroscopic quantum states, topology, metrology, and emergence.

Keywords: condensed matter physics, states of matter, phase transition, broken symmetry, emergence, solid-state physics, superconductivity, Physics Nobel Prize, theoretical physics, metrology

Chapter 6. The critical point

“The critical point” in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature is the same. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close the critical point. The successful theory developed in the 1970s exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics can be made tractable by consider the abstract idea of a system with variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.  

Keywords: Critical point, phase diagram, phase transition, critical exponent, universality, Ken Wilson, renormalisation group, scaling, fractals

7. Quantum matter

“Quantum matter” describes how some states of matter, such as superconductors and superfluids, have properties like those found in the strange quantum world of atoms, electrons, and photons. In quantum theory, properties such as energy cannot have any possible value, but are quantised. That is they come in discrete lumps. Also, particles can act like waves and so interfere with themselves. The weirdness of quantum theory is captured in the paradox of Schrödinger’s cat, where quantum effects occur on the macroscopic scale. This is realised in a superconductor, where the magnetic flux is quantised. Josephson proposed an electrical circuit that is the basis for a Superconducting Quantum Interference Device (SQUID). These devices are of technological significance as they are used in quantum computing, and to make precise measurements of fundamental physical constants and of magnetic field strengths. SQUIDs are now used in metrology, the science of precise measurements, being the basis for the international standard for the Volt, the unit of electrical voltage, and are used in quantum computing.  

Keywords: Schrödinger’s cat, quantum theory, Josephson effect, SQUID, superconductor, superfluid, metrology, magnetic flux, macroscopic quantum effect, quantum computing

8. Topology matters

“Topology matters” in condensed matter physics because topology provides a means to describe the unusual types of order found in some states of matter in which there is no symmetry breaking. Topology is the field of mathematics covering the properties of geometric objects that do not change when smoothly deformed. The Hall effect is the existence of a new type of electrical resistance in a conductor caused by an electric current moving transverse to a magnetic field. In a two-dimensional metal (Flatland) in a large magnetic field, the Hall resistance is quantised in units defined by fundamental constants. This is a macroscopic quantum effect and provides a means to make highly precise measurements of electrical resistance. This quantum Hall effect is now used in metrology, being the basis for the international standard for the ohm, the unit of electrical resistance. Topology helps explain why the Hall resistance is so precisely quantised and is independent of so many details such as the chemical composition of the material in which the electrons move. Haldane showed that topology is also key to understanding the unusual properties of chains of magnetic atoms. They exhibit new states of matter, quite distinct from those found in three-dimensional magnets. Haldane also laid the foundation for proposals of topological insulators, a state of matter which is an electrical conductor on its surfaces, but an insulator in its interior.

Keywords: Topology, quantum Hall effect, mathematics, topological invariant, metrology, Duncan Haldane, topological insulator, Flatland

9. Emergence: more is different

“Emergence: more is different” discusses how the concept of emergence is central to condensed matter physics. An emergent property of a system composed of many interacting parts is a property that the individual parts do not have. The whole is greater than the sum of the parts. Other characteristics of emergent properties such as irreducibility, universality, and unpredictability are discussed. Emergent properties are illustrated with an example involving language, grammar, and literature. States of matter are emergent, as are the quasiparticles present in many systems. Phil Anderson argued that a hierarchy of scales illuminates the relationship between different scientific disciplines and shows the limitations of reductionism. Emergence explains why condensed matter physics works as a unified discipline. Due to universality, there are concepts and theories that describe phenomena in a wide range of materials. An emergence perspective highlights how discerning the relevant scale is central to effective scientific strategies aiming to understand complex systems, whether in physics, biology, or sociology.

Keywords: Emergence, reductionism, condensed matter physics, universality, philosophy of science, Phil Anderson, biology, stratification, quasiparticles, complex systems

10. An endless frontier

“An endless frontier” discusses how is difficult to predict the future of condensed matter physics, but it is likely to be an exciting one as new discoveries of emergent phenomena, such as novel states of matter, are often not anticipated. Grand challenges are identified including understanding glasses, Schrödinger’s cat, and exploring new materials, and new extremes of temperature, pressure, and magnetic field. Condensed matter physics has a rich history of contributing concepts, techniques, and personnel to other fields, including chemistry, biology, computer science, and materials engineering. This interdisciplinarity is likely to continue with investigations of complex systems, soft matter, and the social sciences. New technologies will aid new discoveries in condensed matter physics and vice versa. Threats to the future vitality of the field are like those for other intellectual enterprises imbedded in institutions increasingly controlled by financial values: a preoccupation with short-term commercial outcomes, hype, metrics, and managerialism. Advances have historically come from individuals and groups working within contexts and institutions that valued intellectual freedom, creativity, patience, curiousity, and serendipity.

Keywords: condensed matter physics, glass, complexity, emergence, science funding, hype, scientific discovery, materials science, interdisciplinarity

Any suggestions for improvement?

Chapter abstracts for A Very Short Introduction (part 1)

I am pleased to announce that Condensed Matter Physics: A Very Short Introduction is scheduled to be published on December 29.

The manuscript is currently with copy editors and in production. 

For each chapter, I have been asked to provide abstracts and keywords for the online version of the book. This turns out to be somewhat interesting as it is an issue of marketing, using internet searches to sell books. There are no chapter abstracts for the hard copy. Here is some of the background provided by Oxford University Press.

High quality A&K (Abstracts and Keywords) are those that help readers get to the content they are looking for, first by making the relevant content appear high in search results, and then by accurately describing the work so that they can decide whether it will be relevant to their needs. High quality A&K become even more important when readers are able to choose and purchase the relevant content from the results. 

The availability of A&K is now an industry standard. Referrals from Google represent a higher percentage of total visits for sites that have free A&K (up to 40-70%), compared to those that do not (around 13-30%).

An example of an abstract is that for the first chapter of Globalisation: A Very Short Introduction which is one of the best-selling titles in the series.

Chapter abstracts should be 100–250 words.

Here are my current versions for the first five chapters. I welcome suggestions for improvement.

1. What is condensed matter physics?

What is condensed matter physics?  It is the science concerned with characterising and understanding all the possible states of matter that can exist. Solid, liquid, and gas are not the only states of matter. There is also liquid crystal, superconductor, superfluid, crystal, glass, ferromagnet, and antiferromagnet. The central question of condensed matter physics is “how do the physical properties of a state of matter emerge from the interactions between the atoms of which the material is composed?” This is illustrated with the distinct properties of graphite and diamond, two distinct solid states of carbon. The recent discovery of graphene, a material composed of single layer of carbon atoms, and its unique electrical properties is an example of how the field continues to produce exciting surprises. Condensed matter physics is one of the largest and most vibrant subfields of physics. As it is concerned with materials and with emergent phenomena there is significant cross-fertilisation of concepts and techniques with other sub-fields of physics, science, and engineering.

5–10 keywords that can be used for describing the content of the chapter

Condensed matter physics, states of matter, physical properties, graphite, diamond, graphene, materials science, emergence, Kamerlingh Onnes

2. A multitude of states of matter

There are “a multitude of states of matter”. Materials composed of just one or two types of atoms can form many different states of matter.  Each has qualitatively different physical properties. Transitions between distinct states are defined by abrupt changes in properties. Dramatic examples include superconductivity and superfluidity. Phase diagrams encode which state of a material is stable under specific conditions defined by variables such as temperature and pressure. The phase diagrams of water, carbon dioxide, and carbon are discussed. At the critical point there is no distinction between liquid and gas. Topics discussed are relevant to making artificial diamonds, freeze-dried food, decaffeinated coffee, and dry ice.

Keywords: tipping point, phase transition, phase diagram, critical points, temperature, superconductivity, superfluidity, magnetism, sublimation, supercritical fluid

3. Symmetry matters

“Symmetry matters” in condensed matter physics because the mathematics of symmetry is key to characterising the qualitative difference between distinct states of matter. The concept of symmetry is illustrated by considering how the appearance of specific objects do not change when they undergo transformations such as rotations, reflections, and translations. Crystals are composed of repeat units of atoms. Symmetry constrains the number of types of different repeat units that are possible. The spatial arrangement of the atoms in each repeat unit can be determined by diffraction of a beam of x-rays by the crystal. The symmetry of a state of matter constrains what physical properties it can have. Symmetry aids a connection between the macroscopic and microscopic properties of a state of matter, such as explaining why snowflakes have six-fold rotational symmetry. The unexpected discovery of a quasicrystal, a distinct state of matter, revised the definition of a crystal.

Keywords: symmetry, Bravais lattice, crystal, crystallography, x-ray diffraction, structure, quasicrystal, chemistry, molecular biology, Bragg

4. The order of things

“The order of things” describes how distinct states of matter are associated with distinct types of ordering, such as the regular pattern of atoms in the material. The change in symmetry between different states of matter can be quantified in terms of an “order parameter”. Determining the relevant symmetry and order parameter for a state of matter often takes decades as it requires significant scientific insight. Lev Landau proposed a general theory to describe the amount of ordering present in any material that undergoes a phase transition to a distinct state of matter. The associated concept of spontaneous symmetry breaking is central to condensed matter physics, and to the theory of elementary particles and fundamental forces and led to the prediction of the existence of the Higgs boson. There is a rigidity associated with the type of order in a state of matter, whether a crystal or a superconductor. This rigidity determines the type of deviations from perfect order that are possible. Examples of defects include dislocations in crystals and vortices in superconductors and superfluids.

Keywords: States of matter, rigidity, symmetry breaking, magnetism, Lev Landau, Higgs boson, order parameter, liquid crystal, vortex, topological defect

5. Adventures of Flatland

“Adventures of Flatland” describes how condensed matter physics is different in a two-dimensional world, than in our three-dimensional one. This Flatland can be accessed in a laboratory because it is possible to fabricate materials, such as graphene, that are two-dimensional or close to it. Theory can also be used to investigate this different world and led to predictions of new states of matter and new types of phase transitions. A simple theoretical model for magnetic phase transtions is the Ising model. The two-dimensional version of the model illustrates concepts such as spontaneous symmetry breaking, long-range order, critical points, emergence, and more. A class of (almost) two-dimensional materials of great interest are crystals consisting of layers of copper and oxygen atoms. These materials superconduct at higher temperatures than any other, have a rich phase diagram, exhibit unusual metallic states, and remain a theoretical puzzle.  

Keywords: Flatland,  spatial dimension, phase transition, Ising model, two-dimensional material, high-temperature superconductor, Kosterlitz-Thouless transition

What do you think? I welcome suggestions for improvement.

A guide through hype about computational chemistry on quantum computer

One of the many problems with hype in science is that it glosses over problems that means they do not get addressed which ultimately hinders real scientific progress. 

There is a lot of hype about how quantum computers will be able to solve problems in materials science that are of industrial significance and thus "herald a new era of chemical research". Such claims are carefully examined and deconstructed in the following preprint. Most of the authors are at Schrodinger, Inc.

How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?

V.E. Elfving, B.W. Broer, M. Webber, J. Gavartin, M.D. Halls, K. P. Lorton, A. Bochevarov

The article is also a useful guide to current state-of-the-art computational chemistry on classical computers.

I reproduce most of the paper abstract below as it is helpful summary.

Numerous reports claim that quantum advantage, which should emerge as a direct consequence of the advent of quantum computers, will herald a new era of chemical research because it will enable scientists to perform the kinds of quantum chemical simulations that have not been possible before. Such simulations on quantum computers, promising a significantly greater accuracy and speed, are projected to exert a great impact on the way we can probe reality, predict the outcomes of chemical experiments, and even drive design of drugs, catalysts, and materials. 

In this work we review the current status of quantum hardware and algorithm theory and examine whether such popular claims about quantum advantage are really going to be transformative. We go over subtle complications of quantum chemical research that tend to be overlooked in discussions involving quantum computers. 

We estimate quantum computer resources that will be required for performing calculations on quantum computers with chemical accuracy for several types of molecules. In particular, we directly compare the resources and timings associated with classical and quantum computers for the molecules H2 for increasing basis set sizes, and Cr2 for a variety of complete active spaces (CAS) within the scope of the CASCI and CASSCF methods. The results obtained for the chromium dimer enable us to estimate the size of the active space at which computations of non-dynamic correlation on a quantum computer should take less time than analogous computations on a classical computer. Using this result, we speculate on the types of chemical applications for which the use of quantum computers would be both beneficial and relevant to industrial applications in the short term.

The authors present a useful typology of claims of quantum advantage that are irrelevant.

1. Irrelevance due to availability of accurate experimental results. 

2. Irrelevance due to availability of conventional computational results. 

3. Irrelevance due to real world complexity:

When simulated chemical processes are very complicated and involve potentially hundreds of intermediates, conformations, or reaction paths, as in catalytic and metabolic pathways, the real research bottleneck lies in a combinatorial explosion of possibilities to probe with simulation.

4. Irrelevance to industrial applications

Many of the issues discussed in the preprint are not unrelated to those associated with hype about using machine learning in computational materials science, and are beautifully critiqued by Roald Hoffmann and Jean-Paul Malrieu.

de Gennes enthuses about Chemistry and skewers Comte

Pierre-Gilles de Gennes (1932-2007) was arguably the founder of soft matter as a research field, as recognized by the Nobel Prize in Physics in 1991. After this de Gennes gave many lectures in French high schools, which were then published as a book, Fragile Objects: Soft Matter, Hard Science, and the Thrill of Discovery. Previously, I mentioned the book with regard to whether condensed matter physics is too abstract.

 One of many fascinating sections of the book is a chapter entitled, The Imperialism of Mathematics. de Gennes sings the praises of chemistry, and rants about the weaknesses of the French system, laying the blame at the feet of his compatriot Auguste Comte (1798-1857). Comte was one of the first philosophers of modern science and a founder of sociology and of positivism.

Below I reproduce some of the relevant text. When reading it bear in mind that de Gennes was a theoretical physicist and did work that often involved quite abstract mathematics and concepts.

        THE "AUGUSTE COMTE" PREJUDICE

I now come to a prejudice typical of French culture, inherited from the positivism of Auguste Comte. This nineteenth-century philosopher achieved some degree of fame by inventing a classification system of the sciences. 

At the top of his hierarchy was mathematics; at the bottom was chemistry, which according to him "barely deserved the name of science"; in the middle were astronomy and physics. This classification dismissed out of hand geography and mineralogy, sciences which were declared concrete and descriptive, retaining only those that were theoretical, abstract, and general. The tone was set! It is ironic that this philosophical concept came from an individual who had once written in a letter "The only absolute truth is that everything is relative," and who claimed to be steeped quasi-religiously in factually observable laws, in other words, laws verifiable by experiments. 

The "Auguste Comte" prejudice corrupts to this day the teaching of the sciences, the scientific disciplines, and even the scientists themselves. It also contains the seed of contempt for manual labor, which has interfered for years by curbing every attempt at reform to revalue the manual trades and their apprenticeship... 

An example comes to mind, of some graduates of the Polytechnic School of Paris attending an advanced program at Orsay to learn solid-state physics. They would often show up convinced that they knew everything on the basis of calculations. 

... But the typical Polytechnic graduate I inherited at the time would remain stumped in front of his bare blackboard. One of them finally blurted out (I will never forget his comment): "But, sir, what Hamiltonian should I diagonalize?" He was trying to hang on to theoretical ideas which had no connection whatsoever with this practical problem. This kind of answer explains, in large part, the weakness of French industrial research.

Among all the catastrophes brought about by the positivist prejudice, none is worse than the widespread contempt for chemistry. I have already pointed out the importance of this discipline for our industrial future, the importance of chemists, these marvelously inventive sculptors of molecules, to whom the French teaching establishment does not do nearly enough justice. An undergraduate math major once told me about a teacher who, on opening day, announced: "I personally dislike chemistry, but I have to talk about it. So, I will start by giving you two hours of chemical nomenclature: what the name of an obscure and com- plex molecule is, and the like." At the conclusion of the two hours, the entire class was turned off chemistry for life!

When Lucien Monnerie, the director of studies, and I took over re- sponsibility for courses at the Institute of Physics and Chemistry, we had to wage a determined battle to overcome the antichemistry prejudice. Just before our arrival, the students had organized a strike: they all wanted to become physicists. Slowly, we climbed back up the slope with a series of measures: changing labels, opening up several new channels, turning the entire curriculum upside down, and launching a verbal propaganda campaign. It was rather easy for me to sound persuasive; being a theoretical physicist, nobody could accuse me of protecting my own turf. But it took us 10 years to restore the proper balance. 

To anyone who wants to form a more precise idea of chemistry, of the life of a typical chemical engineer, I would advise reading the magnificent collection of essays by Primo Levi, The Periodic Table. They recount real-life stories. They possess an authenticity and a vitality which give a universal impact to the account of an ordinary fact, the description of minute events. It is an excellent antidote to the poison spread by Auguste Comte's classification scheme.

Can emergent properties be explained?

An important question about emergent properties is whether they can be explained solely in terms of the properties of the components of the system. Here I explore the question from the point of view of Hempel's covering law of scientific explanation, discussed in my last post.

According to Hempel, a scientific explanation E of a specific phenomena P is a logical argument that starts with some premises, at least one of which is a scientific law L, and which logically implies P.

I now give a version of this that describes a microscopic scientific explanation of some emergent property.

Suppose that a macroscopic system S has property X. S is composed of many interacting microscopic components whose properties, including their interactions, have a finite enumeration x1, x2, x3,...xn. None of these properties is X. Hence, in the sense of novelty, X is an emergent property of S. Let l1, l2, l3,.., lm be a finite number of microscopic laws. Then X has a microscopic scientific explanation if it can be deduced from the x's and l's.

A possible problem with most microscopic "explanations" of emergent properties may be whether they at some point implicitly assume some "emergent" scientific law, such as spontaneous symmetry breaking, or the existence of X. Let me illustrate this possible problem with some examples.

Irreversibility. Microscopic laws are invariant under time-reversal. But macroscopic systems exhibit irreversible behaviour such as the mixing of two distinct fluids. This is encoded in the second law of thermodynamics. This problem of the "arrow of time" is nicely discussed by Tony Leggett in The Problems of Physics, in a chapter entitled "Skeletons in the Cupboard." An alternative perspective is that of Joel Lebowitz, who claims Boltzmann solved the problem.

Superconductivity. One could claim that BCS theory provides a microscopic explanation of superconductivity. We start with the properties of electrons, ions, Coulomb's law, quantum mechanics, and statistical mechanics. These properties and microscopic laws can be used to show that there is an effective attractive interaction between electrons. One then considers the BCS variational wavefunction and calculates the properties of the macroscopic system. They are consistent with experimental observations of superconductivity. It is explained!

However, there are several problems on the way, which all in some sense involve assuming that superconductivity does occur. First, investigating the variational wave function only shows that the superconducting state has lower energy than the normal metallic state. This does not prove it is the true ground state. In fact, in one dimension it is not.

But potentially more fatal to the claimed microscopic explanation is that it assumes that spontaneous symmetry breaking is allowed, including (in some subtle sense that people still argue about) the breaking of the gauge symmetry of electromagnetism. One of the major points that Phil Anderson was trying to make in More is Different is that spontaneous symmetry breaking is a law of nature that should be viewed as of similar status to microscopic laws such as Schrodinger's equation. 

Mean-field theory of antiferromagnetism. One might claim that one can start with a classical Heisenberg or Ising model, and classical statistical mechanics, crank the mathematical handle and get antiferromagnetic. If one does mean-field theory, then one is not really doing statistical mechanics as one is considering a weird ensemble and a Hamiltonian that is no longer microscopic. Suppose instead one does the exact solution of the Ising model. That can give the magnetic state and all the critical exponents. But, it is not clear to me that when one takes the thermodynamic limit, one assumes that the broken symmetry state is allowed. Similar questions arise for me if one does a computer simulation on large lattices and uses clever finite-size scaling techniques to deduce physical properties of the emergent state. Does the assumption of the validity of these techniques amount to some extra (macroscopic) law of nature?  

I wonder whether some of these issues would be clarified (or just muddied) by considering the Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics by David Ruelle. In particular, does he make clear how an equilibrium broken symmetry magnetic state is fundamentally different from the microscopic equilibrium state associated with a finite number of spins.

I welcome ideas on how to clarify these issues.

What is a scientific explanation?

An important issue concerning emergent phenomena is whether part of their definition should be that they "cannot be explained/predicted" solely in terms of the properties of the components of the system.  As I have discussed before, there are several alternatives such as difficult to explain, extremely difficult to explain, impossible to explain, and possible to explain in principle, but not in practice...

To further consider this issue, it is helpful to back up and consider the general question, "What is a scientific explanation of a specific natural phenomena?" This has received a lot of attention over the past century from philosophers of science. A nice accessible introduction is found in Philosophy of Science: A Very Short Introduction by Samir Okasha. In chapter 3, he discusses at length the Covering Law model of explanation, developed by Carl Hempel.

Here are the basic elements of the model. Hempel observed that "scientific explanations typically have the logical structure of an argument, i.e., a set of premises followed by a conclusion." More specifically, 

1. The argument should be a deductive one. The conclusion is the phenomena to be explained.

2. All the premises must be true.

3. One of the premises should be a scientific law.

Here is an example:

Premise a. Astronomical data about planetary motion.

Premise b. Kepler's laws of planetary motion can describe the data.

Premise c. Newton's theory of gravity.

Conclusion. The motion of planets can be described by Newton's theory of gravity.

Hempel considered that explanation and prediction were two sides of the same coin. 

Okasha also discusses some weaknesses of the covering law, namely that it is too broad allowing "explanations" that are not really scientific. Specifically, there is the problem of symmetry and the problem of irrelevance.

In the next post, I will explore how this model for scientific explanation might help frame discussions about whether emergent phenomena can be predicted.