Emmy Noether and the symmetry aesthetic

Note: I originally wrote this for the QRI website. It can be thought of both as context for the intuitions which generated the Symmetry Theory of Valence (original; condensed primer), and offering Noether’s work as a generally promising research thread for formalizing phenomenology.

Emmy Noether is perhaps the world’s most famous female mathematician, and worked in the Göttingen cluster in the 1910s and 20s. She’s most well-known for Noether’s theorem, which states that “every differentiable symmetry of the action of a physical system has a corresponding conservation law.” In other words, when there’s an invariance (in the form of a symmetry) in the equations describing a system, there always exists a corresponding invariance (in the form of a conservation law) in the behavior of that system, and vice-versa. As Max Tegmark puts it,

German mathematician Emmy Noether proved in 1915 that each continuous symmetry of our mathematical structure leads to a so-called conservation law of physics, whereby some quantity is guaranteed to stay constant … All the conserved quantities that we discussed in Chapter 7 correspond to such symmetries: for example, energy corresponds to time-translation symmetry (that our laws of physics stay the same for all time), momentum corresponds to space-translation symmetry (that the laws are the same everywhere), angular momentum corresponds to rotation symmetry (that empty space has no special “up” direction) and electric charge corresponds to a certain symmetry of quantum mechanics. (Our Mathematical Universe, 2014)

Noether’s theorem sounds simple, but it’s arguably one of the most significant pillars of mathematical physics, since it provided a focal point for aligning centuries of mathematical research into symmetry (e.g. Pythagoras, Hamilton, Hilbert, Lorentz, Klein & Lie) with centuries of physics research into conservation laws (e.g. Newton, Huygens, Galileo, Einstein, Weyl), offering inspiration for both fields and providing the mathematical basis for gauge theory, the framework modern physics uses to characterize the Strong, Weak, and Electromagnetic forces. Furthermore, Noether’s work helps illuminate the elegance aesthetic at the heart of physics, the idea that the laws of reality embody a remarkable conceptual beauty, and this beauty is ultimately derives from symmetry. Here’s Nobel laureate Frank Wilczek:

… the idea that there is symmetry at the root of Nature has come to dominate our understanding of physical reality. We are led to a small number of special structures from purely mathematical considerations–considerations of symmetry–and put them forward to Nature, as candidate elements for her design. … In modern physics we have taken this lesson to heart. We have learned to work from symmetry toward truth. Instead of using experiments to infer equations, and then finding (to our delight and astonishment) that the equations have a lot of symmetry, we propose equations with enormous symmetry and then check to see whether Nature uses them. It has been an amazingly successful strategy. (A Beautiful Question, 2015)

Another Nobel laureate, Philip Warren Anderson, goes even further: “It is only slightly overstating the case to say that physics is the study of symmetry.”

Relevance to formalizing phenomenology:

It’s possible that Noether’s theorem may apply simply and literally to phenomenology: if we can represent a conscious system in the same way we represent physical systems (as a Hamiltonian or Lagrangian), then symmetries in these equations should naturally correspond to invariances in phenomenology. This would be an enormous breakthrough. It’s also possible that the aesthetic inherent in Noether’s work can provide inspiration about how to approach formalization in general. Some components of this aesthetic:

• Systematizing a phenomenon often involves pointing to what people are already doing implicitly and describing it explicitly;
• Making a formal connection between two things (in Noether’s case, mathematical invariances and conserved quantities) helps make progress on both;
• The invariances of a system are a key entry point for understanding the nature of that system;
• Symmetry is much more than just a ‘neat quirk of geometry’; it’s one of the few properties that are well-defined on all mathematical objects and is functionally and foundationally important to modern physics, and by implication it will be important in any field that inherits a theoretical aesthetic from physics or math.

Further reading:

• Wikipedia: Noether’s theorem.
• Conover, E. In her short life, mathematician Emmy Noether changed the face of physics. Sciencenews.org, 2018.
• Quigg, Chris. Colloquium: A Century of Noether’s Theorem. Arxiv.org, 2019.
• Wilczek, Frank. A Beautiful Question: in Pursuit of the Hidden Logic of the Universe. Allen Lane, 2015.
• Brading, Katherine, and Elena Castellani. Symmetries in Physics: Philosophical Reflections. Cambridge University Press, 2003.
• Gross, David J. The role of symmetry in fundamental physics. PNAS December 10, 1996 93 (25) 14256-14259.