Science: the Quest for Symmetry

by Yohan J. John

Attitudes toward science in the public sphere occupy an interesting spectrum. At one extreme there are the cheerleaders — those who seem to think that science is the disembodied spirit of progress itself, and will usher us into a brave new world of technological transcendence, in which we will merge with machines and upload our minds to the cloud. At the other extreme there is decidedly less exuberance. Science in its destructive avatar is often called scientism, and is seen as a hegemonic threat to religions and to the humanities, an imperial colonizer of the mind itself.

The successes of science give the impression that it has no limitations, either in outer space or inner space. But this attitude attributes to science somewhat magical powers. The discourse surrounding science might benefit from an awareness that its successes are closely tied to its limitations. The relationship between scientists and the rest of society needs mutual understanding and constructive criticism, rather than a volatile mix of reverence, fear, and mistrust. The veil of the temple of knowledge must be torn in two — or at least lifted up from time-to-time.

To this end, it might be illuminating to see scientific ideas as tools forged in workshops, rather than spells divined by wizards in ivory towers. The tool metaphor also reminds us that science is not merely an outgrowth of western philosophy — it is also the result of the painstaking work of “miners, midwives and low mechanicks” whose names rarely feature in the annals of Great Men. [1]

So what sort of toolbox is science? I'd like to argue that it's a set of lenses. These lenses allow us to magnify and clarify our perceptions of natural phenomena, setting the stage for deeper understanding. The lenses of science reveal the symmetries of nature, so we might call them the Symmetry Spectacles. The Symmetry Spectacles are normally worn by mathematicians and theoretical physicists, but I think that even laypeople interested in science might find that the world looks quite interesting when viewed through them.


So what is symmetry? Most people have an intuitive sense of what symmetry is. Symmetry connotes evenness, or balanced sameness. An asymmetrical object is skewed, off-balance, and uneven. It is easy to grasp that a scalene triangle is asymmetrical, whereas an isosceles triangle is symmetrical. To many people an equilateral triangle seems even more symmetrical than an isosceles triangle. The goal of a formal approach is to make explicit our intuitions about symmetry. Mathematicians and theoretical physicists define symmetry as follows:

Symmetry is immunity to a possible change.

How can we apply this notion of symmetry to the scalene, isosceles and equilateral triangles?

Imagine that each triangle is a uniformly colored plastic block, placed flat on a table. We can rotate and flip each of them in infinitely many ways. Consider the equilateral triangle. If we rotate the equilateral triangle by 120 or 240 degrees around its center, it will end up looking just as it did initially. Flipping the equilateral triangle as shown in the diagram has the same effect. Each of these changes that we can apply to a triangle has the potential to reveal a corresponding type of symmetry. The isosceles triangle possesses a “flip” symmetry but no rotational symmetries, and the scalene triangle possesses neither type of symmetry. From the formal perspective it doesn't make sense to just say that an object is symmetrical — we also need to specify the specific actions or changes that reveal the object's symmetries.

The two essential elements of symmetry are (1) The possibility of a physical change, and (2) immunity to that change, which is also known as invariance. In the triangle example, the rotations and reflections were the 2 possible changes. The threefold rotational symmetry of an equilateral triangle is an immunity to rotations in steps of 120 degrees.

An object has a symmetry if some aspect of it stays the same even after we apply a change to it. The possibility of change is crucial to symmetry, and so we need some way of telling that a change has in fact taken place. In other words, we need a reference frame, or a measuring device. The key feature of a measuring device is that it is not the same when we apply the change to the object being measured. It must be asymmetrical with respect to that change. This leads us to a very interesting observation: every symmetry implies an underlying asymmetry. In the case of the equilateral triangle, we need a device that tells us that we have successfully rotated the triangle by 120 degrees. This device is not immune to the rotation — it is asymmetric with respect to the rotations of the triangle. Before we rotate the triangle, the device reads “0 degrees”, and after the rotation, it reads “120 degrees”. Since the measuring device does not look the same as before, we can tell that a change has taken place, and we can then discern that the equilateral triangle looks the same after the rotation.

The Symmetry Spectacles really demonstrate their power when we moves out of the domain of geometry, where our informal intuitions linger, and into the realm of concepts and relations. We can show that an analogy is a key type of symmetry. Here is a simple analogy of the sort found in standardized tests: “Puppy is to Dog as Kitten is to Cat.” To see why an analogy is a type of symmetry, we need to describe it in terms of a possible change, and an immunity to that change. To help us do this, let’s rewrite our example as “X is a young Y”. If X is “Puppy” then Y is “Dog”, and if X is “Kitten” then Y is “Cat”. The possible change in this analogy is the replacement of the pair of words {X,Y} with another pair. The validity of the relation is immune to certain changes. It holds for some pairs of words. X and Y could also be “Cub” and “Bear” or “Gosling” and “Goose”, but not “Piano” and “Guitar”. A classic analogy from the history of science was the discovery that the Moon is to Planet Earth as the Planet Earth is to the Sun. The analogy can be expressed as “X revolves around Y”, where the pair {X ,Y} can be {Moon, Earth} , or {Earth,Sun}, or {Europa, Jupiter}. The humble analogy can spark a revolution.

A mathematical equation is a concise way of expressing an analogy among sets of numbers. In the equation for Newton's Law of Gravitation, the validity of the dependence of the gravitational force between two bodies on their masses and on the distance between them is immune to changes in the values of the masses and the distance. Classification is a type of analogy and is therefore also a type of symmetry. The validity of the statement “X is a subspecies of Canis lupus” holds if X is “Dog”, or “Wolf” or “Dingo”. Geology and taxonomy are two fields that show that classification is no mere stamp-collecting exercise, but can serve as a launchpad for further scientific discovery.

We can now make explicit the link between symmetry and science. The laws of science are the symmetries of nature — the aspects of nature that do not change when other aspects are changed. Scientific statements capture the invariant phenomena that show up against our changing, asymmetrical reference frames. But because of the requirement of an underlying asymmetry, no phenomenon can be shown to be invariant across all possible reference frames. The history of science is the story of how we discover new symmetries in nature, and also of how we discover where symmetries break down. Science, in short, is the quest for invariance. [3]

The symmetry perspective might provide a framework within which to view seemingly opposed philosophies of science. When we empirically confirm that a particular symmetry holds, we are in the domain of verification. But when a symmetry seems to break down, we might be tempted to invoke Karl Popper's notion of falsification. But we don’t always discard old symmetries when we find new ones. Newtonian classical mechanics is still used for most terrestrial problems, where Einstein’s general relativity is too cumbersome. The process of investigating symmetries and their violations operates at the level of individual scientific statements or experiments, but it also mirrors how society assesses the worth of an entire theory, research programme or paradigm, as Thomas Kuhn or Imre Lakatos might have pointed out. A paradigm shift might be described as symmetry-breaking at the level of theories or programmes. But such shifts are relatively rare, and are separated by Kuhn's “normal science” periods, during which puzzle-solving scientists refine existing techniques and apply them in new areas, validating the symmetries implicit in mainstream scientific paradigms. Physicists in the 18th and 19th centuries were not necessarily looking to poke holes in Newtonian physics. Many of the greatest triumphs of physics before the 20th century involved “rotating” unexplained phenomena in conceptual space — much as we did with our triangles — until they appeared analogous to the phenomena that were already in the “explained” category. As the physicist Willard Gibbs wrote, “One of the principal objects of theoretical research is to find the point of view from which the subject appears in the greatest simplicity.”

Science clearly involves the quest for symmetry, but we can also argue that science itself is a form of symmetry. To do this we need to show that three central pillars of science — reproducibility, predictability, and reduction — are also forms of symmetry. If a phenomenon is reproducible, that means that it can be obtained reliably in the same laboratory as well as in other laboratories. (Ideally it will show up outside laboratories too!) A phenomenon is reproducible if it is immune to changes in (A) the location of the lab in space and time and, (B) the identities of the experimentalists studying the phenomenon. This also gives us a symmetry-based conception of scientific objectivity, or at least “intersubjectivity”. Objective reality consists of those aspect of the universe that are immune to a change of observer.

Predictability means that the relationship that a theory proposes to exist between experimental inputs and experimental outputs is immune to changes in inputs. In other words, a good scientific theory doesn’t have to be tinkered with each time new data comes to light. When you plug new data into a reliable theory, you get an accurate prediction without having to reformulate the theory. The most celebrated scientific theories can predict the outcomes of experiments that have never been conducted before. A recent example of this sort of prediction was the discovery of the Higgs boson.

Reduction may well be the most crucial aspect of modern science, and one that is frequently overlooked, misunderstood, or even vilified. Reduction is how scientists confront the integral whole that is our experience of nature. To make headway in science, we reduce experience by dividing it into parts that we hope are disconnected, even though we know full well that ultimately they are not completely disconnected. Diametrically opposed to reduction is holism — the idea that since everything is related to everything else, every phenomenon must be understood in terms of the whole of which it forms a part. As Carl Sagan memorably put it, 'If you wish to make an apple pie from scratch, you must first invent the universe.' Holism may be an honest way to think about nature, but it is not always the most productive way to interact with nature. Reduction is the attempt to slice up the world into manageable parts that can be studied and manipulated in isolation. Reduction draws a frame around a phenomenon. The hope when drawing such a frame is that everything outside the frame will interact only minimally with what is inside the frame. Two key scientific reductions are (1) Observer versus Observed, and (2) System versus Environment. Let’s look at these reductions in more detail, because they will reveal not only why science is useful, but why its methods are of limited power in many situations of importance to us.

(1) The Observer versus Observed reduction: When this symmetry holds, an observed phenomenon is immune to changes of observer. The movements of heavenly bodies are indifferent to the identity of the astronomer looking through the telescope. The same sort of invariance holds for chemical reactions (indifferent to the chemist) and for cellular structures (indifferent to the microscopist). But this symmetry may not hold when studying phenomena involving people or animals. Do you behave the same irrespective of who is watching? Does your dog or cat? Perhaps, in some situations, if the observer is well hidden. But in other situations of obvious interest — small-scale social interactions among humans, for instance — the observer/observed reduction is either unethical or impractical, and usually both. [4]

(2) The System versus Environment reduction: When this symmetry holds, the behavior of the system under consideration is immune to certain environmental changes. Simple experiments that can be performed in a high school science lab are immune to changes in the weather, for instance. Maintaining environmental independence can be an elaborate and expensive process, however. The Boston University Photonics Center, for instance, was built on a reinforced concrete foundation to isolate the sensitive instruments in the building from the vibrations of the nearby Massachusetts Turnpike. But no matter how hard we try to isolate a system from its environment, there are always cracks in our defenses — a fact that we are periodically and brutally reminded of when a disaster like the Fukushima Daiichi nuclear reactor meltdown occurs. A system can only ever be quasi-isolated: it can never be completely cut-off from its surroundings. A system such an a country’s economy, which is always strongly interlinked with the economies of other countries, may never be amenable to the kind of system/environment reduction that empowers physics, chemistry and molecular biology. Many psychological and cultural phenomena may be similarly irreducible — we may find that they are too strongly dependent on forces external to the apparent physical boundaries of the system.

A successful scientific theory reduces a phenomenon in order to discover the aspects of it that are predictable and reproducible. All technology is dependent on phenomena that can be reproduced, predicted, and easily disentangled from observers and environments. A car, for instance, would be unusable if it responded in unique and unpredictable ways to the attitude of the driver or to the vagaries of the weather. The aspects of science that makes it so successful also reveal where its limitations lie. Scientists isolate systems from environments and attempt to study them as disinterested observers, looking for the symmetries that render phenomena reproducible and predictable. But there is no guarantee that all natural phenomena are reproducible, predictable, or reducible. Perhaps more controversially, in order to discover the symmetries of nature, there must be underlying asymmetries.

Metaphysical speculations about the “fundamental” nature of the universe often posit the existence of a basic substance: something that exists at all times and in all places, uniform and unchanging. The symmetry perspective seems to preclude the scientific discovery of any such universal invariant. In order to discover a symmetry, we need an underlying asymmetry — a measuring stick or reference. If we take seriously the idea that science is symmetry, we can ask ourselves what the underlying asymmetry might be. What is the ultimate reference frame — the messy backdrop against which the neat symmetries of science show up in stark relief? Can we conceive of anything that is fundamentally asymmetrical?

I can only offer some wild speculation. There seem to be three grand metaphysical concepts that no one can really agree on: existence, time, and mind. The symmetry perspective does not really shed much light on these imponderables, but it might allow us to link them to each other. Symmetry is immunity to a possible change. What exists — what is visible through the Symmetry Spectacles — must be susceptible to change. Change and time are thoroughly entangled. To measure a change is to compare two points in time. But to compare two points in time, something must be measurably changed. And it seems to me that in order to measure anything at all there must be a mind — a subjectivity that is restlessly aware of the passage of time and the accretion of changes. Perhaps the mind is the fundamental asymmetry: the process that is never quite the same from one moment to the next.


Notes and References

[1] There is no doubt that science owes a lot to philosophy, but not everything. Many people think of science as a “top-down” activity in which a rare genius discovers knowledge and bequeaths it to lesser mortals. But historically, science has also depended on the “bottom-up” process by which practical, hands-on knowledge is collected by workers and craftspeople. Clifford D. Conner, in his book A People's History of Science: Miners, Midwives and Low Mechanicks, documents a variety of such neglected contributions to human knowledge.

[2] For this treatment of symmetry I am largely indebted to the book Symmetry Rules: How Science and Nature Are Founded on Symmetry by Joe Rosen.

[3] It would be remiss to talk about symmetry without mentioning Noether's Theorem, a major discovery in theoretical physics made by the celebrated German mathematician Emmy Noether. Simply put, Noether's Theorem established that every physical symmetry of a certain class is associated with a conservation law. Invariance with respect to changes in time is associated with the conservation of total energy. Similarly, invariance with respect to changes in position is associated with the conservation of momentum, and invariance with respect to rotation is associated with the conservation of angular momentum. Armed with Noether's theorem, much of physics was reformulated in terms of the quest for symmetry and, increasingly, for symmetry-breaking. The most recent success of this approach was the discovery of the Higgs boson.

[4] The contribution of the observer/observed distinction to the success of science may be one reason quantum physics causes so much consternation. According to some interpretations of quantum mechanics, the specific observations that an observer chooses to make can alter the system under observation in irreversible, non-deterministic ways. The idea that sub-microscopic particles capriciously change their behavior depending on what the observer does is so absurd to some people that they conjure up the (oxymoronic?) image of infinite universes as an ontological escape route.