More Is Different

On the limitations of reductionism

Photo by NASA on Unsplash

“The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe”.
P.W.Anderson

What is the most real thing in the world? Is it the number pi? Is it the quantum fields merrily vibrating us into existence? Or is it the white and black on the screen on which you are reading this text?

In describing reality, the natural sciences go through a transition from the abstract universal to the idiosyncrasies of our lived reality: from quantum physics and relativity to molecules, cells, organisms, individuals, from solving Schrödinger’s equation in a Coulomb potential to the contingency of history and the uniqueness of its evolution.

The farther we move up the ladder, the more specific things get: from the mathematical simplicity of the gauge groups of the standard model, we get to the huge assortment of molecules in chemistry, whose expressivity, in turn, gives rise to the language of DNA, encoding a mind-boggling number of organisms, from bacteria to insects to reptiles to human beings. And when we think about humanity, there is an immense individual variety yet again: all the details of your existence are insanely specific, from our gender, nationality, and interests, to our eye color, hair color, personality, etc.

We like to think that a fundamental theory of the universe should be as universal as possible, and include as little specificity as we can get away with.

In searching for a deep connection between certain seemingly disparate phenomena, we are trying to understand how their specificity is dissolved in universal principles: Darwin found in evolution an explanatory principle behind all living beings, and with the discovery of DNA, this organizing principle was further reduced to the four letters of the genetic code.

Photo by Hulki Okan Tabak on Unsplash

But some questions remain: how much can we actually explain by reducing, and can we keep on reducing until we reach the bottom? Is strident reductionism actually enough, and is the most fundamental theory really the best theory? In other words: does a theory of everything really end up explaining everything?

“Everything should be made as simple as possible, but no simpler.”
Albert Einstein

In every scientific theory, we like to hunt for simplicity. Finding timeless and simple universal gives us a deep sense of understanding. Therefore, when scientists encounter a theory that doesn’t appear as simple as it could be, they are on the look-out for deeper explanations.

To paint a simple caricature, the ultimate aim of reductionism is to find a simple, unified, and concise theory, with perhaps the most popular (and controversial) contender to date being string theory. These theories, roughly speaking, aim at finding mathematical structures that reflect reality in such a profound way that from them everything else follows the rigorous logic of mathematical deduction.

Particle physics is the reductionist discipline par excellence. It is concerned with finding the most basic entities constituting reality. Particle physics discovered that what we once called atoms are not truly indivisible, but composed of protons and neutrons, which are in turn made up of quarks, and electrons, and that these indivisible particles interact by 4 irreducible forces, summed up in this beautiful figure:

The Standard Model of Particle Physics. Cush, Public domain, via Wikimedia Commons

But consequently, not having the simplest possible theory is especially frustrating to particle physicists. As pretty as the figure is, and as orderly as it looks, the 18 free parameters of the standard model of particle physics cannot be nicely determined by any theory but have to be plugged in by hand into the model from measurements (for those more deeply involved in physics, these are the fine-structure constant, α, The Weinberg angle, the coupling constant g3 of the strong interaction, the Higgs potential vacuum expectation value v, the Higgs mass mH, the three mixing angles and the CP-violating phase of the CKM matrix, and the nine Yukawa coupling constants that determine the masses of the nine charged fermions (six quarks, three charged leptons)).

They are called free parameters because they are not constrained by theory. We don’t understand if there are any connections between them or where they come from. This is one of the reasons why the standard model of particle physics, despite its enormous success, continues to be unsatisfying from a theoretical point of view, and physicists have been looking for models beyond the standard model that could show deeper relationships between the seemingly disparate free parameters for many years.

In my previous article on symmetries, I went into what crucial role they play in modern physics, and how they relate to us to identify deep connections in our theories. As to Anderson, “It is only slightly overstating the case to say that physics is the study of symmetry”.Physics, Symmetries, and BeautyThe most beautiful of all links is that which makes, of itself and of the things it connects, the greatest unity…medium.com

A symmetry is a property of something that remains the same under a specific set of changes, which is called a symmetry transformation. A simple example is a circle being invariant under rotations. You can rotate it around its center, and it will still look the same. You can also flip space around one axis once, and it will seem unchanged (which is equivalent to looking at the circle in the mirror).

In modern particle physics, symmetries play a more subtle role but are nevertheless formulated in a mathematically equivalent framework: fundamental particles are described in terms of representations of the symmetry transformations of the gauge groups of the standard model, and symmetries also regularly pop up in all other areas of physics.

Our most fundamental equations, the Hamiltonians and Lagrangians that tell us how quantum states evolve and particles interact, are built on these symmetries. Symmetries unite disparate phenomena and show us that what seemed different was only a different facet of the same thing.

And so it may not come as a surprise that the 18 parameters of the standard model could actually be reduced by finding an overarching gauge symmetry, and this symmetry could explain how they just 18 facets of one larger, underlying thing, whatever it was (first attempts were made with larger groups such as SU(5) in the Georgia-Glashow model, and string theory also provides some solutions, but as of yet to no avail).

Symmetry breaking and emergence

Simple laws, rules, and mechanisms can, when applied to very large assemblages, lead to qualitatively new consequences.
– P.W.Anderson

Emergent phenomena have become increasingly important in the study of physics, especially when it comes to many-body phenomena that play such a defining role in solid-state and condensed matter physics.

They provide a counterargument to the reductionist quest for the simplest possible theory. But the word emergence “is not an idea coming without a certain measure of obscurity and even ambiguity as to what it ultimately amounts to explanatorily”, and it tends to be thrown around in the context of many things both difficult and controversial (such as consciousness and free will) without a precise definition.

W.Anderson, one of the 20th century’s most important physicists, gave emergence a more formal underpinning by connecting it to the idea of spontaneous symmetry breaking (SSB).

To understand how this works, let’s look at our circle again: if you introduce a detail like another small circle into the circle, you can’t rotate the picture without it looking differently, and so its rotational symmetry is broken.

Rotational invariance is lost by adding a symmetry-breaking factor to the circle. This figure was expertly crafted by myself, using only Microsoft Paint.

A more abstract, but rather similar step occurs in the now world-famous Higgs mechanism.

The Higgs field is responsible for what is called “electroweak symmetry breaking” in the standard model. It provided physicists with a way to model how the electromagnetic force and the weak force are actually two sides of the same coin, and only appear distinct because when the universe cooled down, the Higgs field developed a vacuum expectation value, breaking the symmetry of the electroweak force. Roughly, you can think of the Higgs field as the little dot in the circle above: once it’s there, the rotational symmetry of the circle no longer holds, and the universe appears strangely specific (why is the dot/vacuum expectation value at this precise location and not somewhere else?).

… it is not enough to know the ‘fundamental’ laws at a given level. It is the solutions to equations, not the equations themselves, that provide a mathematical description of the physical phenomena. ‘Emergence’ refers to properties of the solutions in particular, the properties that are not readily apparent from the equations.
P.W.Anderson

A magnet floating over a high-temperature superconductor. Peter Nussbaumer, CC BY-SA 3.0

But the Higgs mechanism isn’t the only place where symmetries are broken. If you move up the ladder to many-body-physics, symmetry breaking becomes much more common-place.

Perhaps most prominently, it plays a crucial role in understanding surprising many-body-phenomena such as the BCS theory of superconductivity (a technology that could become very relevant to our societies with loss-free energy transport allowed by high-temperature superconductors).

A material is superconductive once it starts conducting electricity without any resistance by forming so-called Cooper pairs. At normal temperatures, the material is not superconductive and reflects the symmetries present in the Hamiltonian (you can think of this as the mathematical description containing all the fundamental physics of the material).

But once the material is cooled and the superconductive state is reached, it departs from this gauge symmetry, developing completely novel, counterintuitive properties.

The overall point here is that the states of big systems tend to not reflect the symmetry properties of the underlying equations. When we move into the limit of many particles interacting, they seem to care increasingly less about the symmetry laws that dictate their behavior: as just one example, we think that space is homogeneous, and the electromagnetic force is symmetric under parity, but still we observe in nature that all sugar molecules break parity (all sugar molecules in nature are right-handed molecules called D-glucose)

Structure of D-glucose (left) and L-glucose (right). Chem Sim 2001, Public domain, via Wikimedia Commons

When we move to a different level of description of reality, it is as if the perfect symmetries contained in that level of reality suddenly disappear, and make way to something less symmetric and more unique.

The central task of theoretical physics in our time is no longer to write down the ultimate equations but rather to catalogue and understand emergent behavior in its many guises…
Laughlin and Pines (2000)

In the latter half of the 20th century, these different approaches to physics lead to a clash between what has been called “strong reductionists” vs. “emergentists”. While particle physicists were successfully reducing and unifying, condensed matter physicists were following a different route.

More is different.
W.Anderson

Anderson gave perhaps the most condensed summary (pun intended) of the emergentist viewpoint in his famous paper titled “more is different”, from which many of the quotes here are derived.

He was experiencing what he considered a denigrating arrogance by the school of reductionists and was defending an approach to science investigating all of those strange effects arising at the frontier of complexity that can not be satisfyingly explained from a description of the fundamentals alone.

It is tricky to get from the Schrödinger equation to molecules because already these molecules stray away from the symmetries dictated by the Hamiltonian that solves the Schrödinger equation. But it seems impossible to get from the Schrödinger equation to human beings.

The overarching point of Anderson’s paper was that while in principle, all physics would arise from the Lagrangians and Hamiltonians of fundamental physics, just looking at the Hamiltonians isn’t enough most of the time because on every new level of analysis and interactions entirely new properties can arise.

He was not so much arguing against reductionism in general, but more for a holistic approach to understanding and modeling our universe.

I think the key take-away message is that we need to be aware of the potential limitations of our theories. Even if we would prove that string theory was actually the correct description of particle physics and gravity, it still wouldn’t necessarily help us build better computer chips or rocket engines or help us deal with our psychological issues on a daily basis.

The study of cellular automata, like Wolfram’s rule 110 or Conway’s game of life (see here for a cool animation), shows us that what looks really complex to us can arise from very simple rules and building blocks, but that it can be difficult or actually impossible to actually predict what will happen from simply looking at rules without actually running it. I discussed this in more detail from a computational perspective in my article on Chaos, Computational Irreducibility, and the Meaning of Life.

And so, as Anderson says, chemistry is not only applied physics, and biology not only applied chemistry, and we will always have to think about reality from many levels of analysis. I actually think it makes the scientific enterprise all that much richer: to look at the elegance and simplicity at the bottom of it, and then to raise our eyes and look at the world with all its whirling maddening richness of diversity.

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