What is really fundamental in science?

What do we mean when we say something in science is fundamental? When is an entity or a theory more fundamental or less fundamental than something else? For example, are quarks and leptons more fundamental than atoms? Is statistical mechanics more fundamental than thermodynamics? Is physics more fundamental than chemistry or biology? In a fractional quantum Hall state, are electrons or the fractionally charged quasiparticles more fundamental?

Answers depend on who you ask. Physicists such as Phil Anderson, Steven Weinberg, Bob Laughlin, Richard Feynman, Frank Wilczek, and Albert Einstein have different views.

In 2017-8, the Foundational Questions Institute (FQXi) held an essay contest to address the question, “What is Fundamental?” Of the 200 entries, 15 prize-winning essays have been published in a single volume. The editors give a nice overview in the Introduction.

This post is mostly about the essay, Fundamental? of the first prize winner, Emily Adlam, a philosopher of physics. She contrasts two provocative statements.

Fundamental means we have won. The job is done and we can all go home.

Fundamental means we have lost. Fundamental is an admission of defeat.

This raises the question of whether being fundamental is objective or subjective.

Examples are given from scientific history to argue that what is considered to be fundamental has changed with time. The reductionism has led to the drive to explain everything in terms of smaller and smaller entities, that are deemed 'more fundamental". But we find that smaller does not always mean simpler.

Perhaps we should ask what needs explaining and what constitutes a scientific explanation. For example, Adlam asks whether explaining the fact that the initial state of the universe had a low entropy [the "past hypothesis"] is really possible or should be an important goal.

She draws on the issue of the distinction between objective and subjective probabilities. Probabilities in statistical mechanics are subjective: they are a statement about our own ignorance about the details of the motion of individual atoms and not any underlying randomness in nature. In contrast, probabilities in quantum theory reflect objective chance.

as realists about science we must surely maintain that there is a need for science to explain the existence of the sorts of regularities that allow us to make reliable predictions... but there is no similarly pressing need to explain why these regularities take some particular form rather than another. Yet our paradigmatic mechanical explanations do not seem to be capable of explaining the regularity without also explaining the form, and so increasingly in modern physics we find ourselves unable to explain either. 

It is in this context that we naturally turn to objective chance. The claim that quantum particles just have some sort of fundamental inbuilt tendency to turn out to be spin up on some proportion of measurements and spin down on some proportion of measurements does indeed look like an attempt to explain a regularity (the fact that measurements on quantum particles exhibit predictable statistics) without explaining the specific form (the particular sequence of results obtained in any given set of experiments). But given the problematic status of objective chance, this sort of nonexplanation is not really much better than simply refraining from explanation at all. 

Why is it that objective chances seem to be the only thing we have in our arsenal when it comes to explaining regularities without explaining their specific form? It seems likely that part of the problem is the reductionism that still dominates the thinking of most of those who consider themselves realists about science

In summary, (according to the Editors) Adlam argues that "science should be able to explain the existence of the sorts of regularities that allow us to make reliable predictions. But this does not necessarily mean that it must also explain why these regularities take some particular form." 

we are in dire need of another paradigm shift. And this time, instead of simply changing our attitudes about what sorts of things require explanation, we may have to change our attitudes about what counts as an explanation in the first place. 

Here, she is arguing that what is fundamental is subjective, being a matter of values and taste.

In our standard scientific thinking the fundamental is elided with ultimate truth: getting to grips with the fundamental is the promised land, the endgame of science. 

She then raises questions about the vision and hopes of scientific reductionists. 

In this spirit, the original hope of the reductionists was that things would get simpler as we got further down, and eventually we would be left with an ontology so simple that it would seem reasonable to regard this ontology as truly fundamental and to demand no further explanation. 

But the reductionist vision seems increasingly to have failed. 

When we theorise beyond the standard model [BSM] we usually find it necessary to expand the ontology still more: witness the extra dimensions required to make string theory mathematically consistent.

It is not just strings. Peter Woit has emphasised how BSM theories, such as supersymmetry, introduce many more particles and parameters.

... the messiness deep down is a sign that the universe works not ‘bottom-up’ but rather ‘top-down,’ ... in many cases, things get simpler as we go further up.

Our best current theories are renormalisable, meaning that many different possible variants on the underlying microscopic physics all give rise to the same macroscopic physical theory, known as an infrared fixed point. This is usually glossed as providing an explanation of why it is that we can do sensible macroscopic physics even without having detailed knowledge of the underlying microscopic theories. 

For example, elasticity theory, thermodynamics and fluid dynamics all work without knowing anything about atoms, statistical mechanics, and quantum theory.

But one might argue that this is getting things the wrong way round: the laws of nature don’t start with little pieces and build the universe from the bottom up, rather they apply simple macroscopic constraints to the universe as a whole and work out what needs to happen on a more fine-grained level in order to satisfy these constraints.

This is rather reminiscent of Laughlin's views about what is fundamental.

Finally, I mention two other essays that I look forward to reading as I think they make particularly pertinent points.

Marc Séguin (Chap. 6) distinguishes "between epistemological fundamentality (the fundamentality of our scientific theories) and ontological fundamentality (the fundamentality of the world itself, irrespective of our description of it)."

"In Chap. 12, Gregory Derry argues that a fundamental explanatory structure should have four key attributes: irreducibility, generality, commensurability, and fertility."

[Quotes are from the Introduction by the Editors].

Some would argue that the Standard Model is fundamental, at least on some level. But it involves 19 parameters that have to be fixed from experiment. Related questions about the Fundamental Constants, have been explored in a 2007 paper by Frank Wilczek.

Again, I thank Peter Evans for bringing this volume to my attention.

Why do deep learning algorithms work so well?

I am interested in analogues between cognitive science and artificial intelligence. Emergent phenomena occur in both, there have been some fruitful cross-fertilisation of ideas, and the extent of the analogues is relevant to debates on fundamental questions concerning human consciousness.

Given my general ignorance and confusion on some of the basics of neural networks, AI, and deep learning, I am looking for useful and understandable resources.

Related questions are explored in a nice informative article from 2017 in Quanta magazine, New Theory Cracks Open the Black Box of Deep Learning by Natalie Wolchover.

Like a brain, a deep neural network has layers of neurons — artificial ones that are figments of computer memory. When a neuron fires, it sends signals to connected neurons in the layer above. During deep learning, connections in the network are strengthened or weakened as needed to make the system better at sending signals from input data — the pixels of a photo of a dog, for instance — up through the layers to neurons associated with the right high-level concepts, such as “dog.” 

After a deep neural network has “learned” from thousands of sample dog photos, it can identify dogs in new photos as accurately as people can. The magic leap from special cases to general concepts during learning gives deep neural networks their power, just as it underlies human reasoning, creativity and the other faculties collectively termed “intelligence.” 

Experts wonder what it is about deep learning that enables generalization — and to what extent brains apprehend reality in the same way.

The article describes work by Naftali Tishby and collaborators that provides some insight into why deep learning methods work so well. This was first described in purely theoretical terms in a 2000 preprint

The information bottleneck method, Naftali Tishby, Fernando C. Pereira, William Bialek 

The idea is that a network rids noisy input data of extraneous details as if by squeezing the information through a bottleneck, retaining only the features most relevant to general concepts.

Tishby was stimulated in new directions in

2014 after reading a surprising paper by the physicists David Schwab and Pankaj Mehta

 An exact mapping between the Variational Renormalization Group and Deep Learning 

[They] discovered that a deep-learning algorithm invented by Geoffrey Hinton called the “deep belief net” works, in a particular case, exactly like renormalization [group methods in statistical physics... When they]. applied the deep belief net to a model of a magnet at its “critical point,” where the system is fractal, or self-similar at every scale, they found that the network automatically used the renormalization-like procedure to discover the model’s state. 

Although this connection was a valuable new insight, the specific case of a scale-free system, is not relevant to many deep learning situations.

Tishby and Ravid Shwartz-Ziv discovered that 

Over the course of training, common patterns in the training data become reflected in the strengths of the connections, and the network becomes expert at correctly labeling the data, such as by recognizing a dog, a word, or a 1.

...layer by layer, the networks converged to the information bottleneck theoretical bound: a theoretical limit derived in Tishby, Pereira and Bialek’s original paper that represents the absolute best the system can do at extracting relevant information. At the bound, the network has compressed the input as much as possible without sacrificing the ability to accurately predict its label...

...deep learning proceeds in two phases: a short “fitting” phase, during which the network learns to label its training data, and a much longer “compression” phase, during which it becomes good at generalization, as measured by its performance at labeling new test data.

What these new discoveries teach us about the relationship between learning in humans and in machines is contentious and explored briefly in the article. Although neural nets were inspired by the structure of the human brain the connection with the neural nets used today is tenuous.

The mystery of how brains sift signals from our senses and elevate them to the level of our conscious awareness drove much of the early interest in deep neural networks among AI pioneers, who hoped to reverse-engineer the brain’s learning rules. AI practitioners have since largely abandoned that path in the mad dash for technological progress, instead slapping on bells and whistles that boost performance with little regard for biological plausibility.

Demonstrating polymer entanglement

From Steve Spangler I learnt this "party trick" demonstration of how the polymer molecules (polyethylene) in a plastic bag are entangled with one another. 

I was not sure that it would work as easily as it did for him. But it did!

Condensed Matter Physics: A Very Short Introduction out now!

Hard copies of my book can now be purchased directly from Oxford University Press. 

After a long wait and a lot of work, it was great to finally see it in print. I am very happy with the quality of the typesetting and the figures.

I look forward to getting feedback from readers.

Spontaneous symmetry breaking and the thermodynamic limit

 Spontaneous symmetry breaking is a fundamental concept in condensed matter and quantum field theory. Amongst philosophers of science the concept is receiving increasing attention, particularly in the context of discussions about emergence.

How do we understand the following two observations about a system at zero temperature?

At zero temperature for a finite-sized system there is no symmetry breaking. The ground state transforms as the trivial representation of the symmetry group of the Hamiltonian. It is non-degenerate.

In the thermodynamic limit, there is a family of degenerate ground states. They are related to one another by a transformation of the symmetry group. This concept is captured in picture below of the Mexican hat potential.

Motion around the trough is associated with the Goldstone mode. Motion perpendicular to the trough is associated with the "Higgs boson".

How does this picture connect with a finite system?

An intuitive picture is that the ball in the trough has a finite mass and so motion in the trough is like the quantum mechanics of a rotor with finite moment of inertia. Then there is a non-generate ground state with equal probability to be located at any angle. What might the moment of inertia be? For reasons described below it turns out to be related to the superfluid stiffness.

For the case of a Heisenberg antiferromagnet, the physics was worked out by Phil Anderson in 1952 where he introduced the concept of a "tower of states" that become degenerate in the thermodynamic limit. 

They are described by the following effective Hamiltonian

c is the speed of magnons (Goldstone bosons). vec(S) is the total spin, V is the volume of the system, and rho_s is the spin stiffness associated with the broken-symmetry. As the thermodynamic limit is approached the energy of these states scale with L^-d where d is the dimension of the system. In contrast, the magnon states scale with L^-1. Thus, for exact diagonalisation of sufficiently large systems, the "tower of states" should be clearly be below the magnons states.

 In 1992 all of the above was confirmed for the triangular lattice in numerical work by Bernu, Hluillier, and Pierre.  

In the figure below, the top panel shows the low-lying eigenstates. The lowest energy states do scale with S^2. The middle panel shows how these states do separate from the magnon states. 

The figure below shows how the moment of inertia [proportional to the denominator in the tower of states equation above] does scale with the system size.

 

More recently there has been some interesting work that explores how the tower of states appears in the entanglement entropy.

Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry

Max A. Metlitski, Tarun Grover

But for now, discussing that is above my pay grade 😀

I thank Gerard Milburn for asking me questions that led to me finally getting a better physical picture of the issues discussed here.

The incomplete veil: from macroscopic to the microscopic

 It is natural to assume that scientists need to probe a system at the microscopic scale to learn about what is happening at that scale. If we take this view we will necessarily be pessimistic about the "bottom-up" research strategy for quantum gravity advocated by Bei Lok Hu. It goes from macro- to micro-, the opposite to the more popular approaches of string theory and loop quantum gravity. However, the history of science shows that we can learn a lot about microscopics from probing systems at much greater length scales. Here are some examples.

Following Perrin's experiments and Einstein's theory of Brownian motion, almost all scientists believed that atoms were not just a mathematical convenience but did exist and were the basic constituents of liquids and solids. All this was before X-ray diffraction allowed the more direct study of crystals at the atomic scale.

Crystallography was pretty much settled as a field before there was any direct evidence of the atomic constituents and their spatial arrangement. Cleavage of crystals, facets observed in minerals, and group theory provided a complete classification of all possible crystal structures. Observations of crystal facets and different modes of sound can be sufficient to determine (or at least constrain options for) the crystal class. 

Figure from Traité de minéralogie (1801) by Rene Hauy See also this.

In 1935, Linus Pauling proposed the crystal structure of common ice without any information from X-ray crystallography. He only used the measured value of the residual entropy, simple models of hydrogen bonding, and the Bernal-Fowler ice rules.

In 1961, the biochemist Peter Mitchell deduced the mechanism of the synthesis of ATP, the molecule responsible for energy transport in cells, without knowing any details of the molecular structure of cell membranes. He reasoned from thermodynamics and the fact that there was an electric potential across the cell membranes. His work led to the discovery of the enzyme ATP synthase, a molecular motor. The underlying physics is beautifully described by Phil Nelson in his text, Biological Physics.

I see two important and related lessons for today from these historical examples.  

1. We have access to amazing computational power and microscopic probes. However, before rushing off to use them, ponder what constraints on the microscopic might be deduced from macroscopic observations.

2. Given that a quantum theory of gravity seems so elusive more resources might be invested in the macro- to micro- strategy.

Aside: Overall, I think this post is going against the strong claims that Bob Laughlin makes in "The Dark Side of Protection", chapter 12 in A Different Universe.

Condensed Matter Physics: A Very Short Introduction now available on Kindle

The good news is that if you read books on Kindle, my book Condensed Matter Physics: A Very Short Introduction can now be purchased as an e-book on Amazon for US$7.50.  It seems to only be available on the USA site but I used my USA Amazon account and downloaded it.

I am no fan of Amazon and minimise my purchases from them. They are my shop of last resort. I understand that some readers will not want to go this route.

The bad news is that the production of print copies continues to progress slowly. Depending on the country, different sites advertise it being available at various dates over the next three months...  

I look forward to readers feedback.