On Truth, Beauty & Simplicity: Laws of Nature

 

The relationship between beauty and truth has always been fascinating to me, especially when we add in the elegance of algorithmic simplicity. Simplicity in scientific laws often appeals to us for its clarity and efficiency, offering the most straightforward explanation with minimal assumptions. However, beauty in these laws can reveal deeper truths, providing a more profound understanding that transcends mere simplicity. While simplicity seeks the shortest path to an answer, beauty often uncovers hidden connections and symmetries that weave disparate phenomena into a cohesive and elegant framework, offering insights that simplicity alone may overlook.

Aesthetic qualities like symmetry and unity are part of the substrate of the laws of physics - like, say, how Maxwell’s equations elegantly unify electricity and magnetism, or how Einstein’s field equations in general relativity reveal the curvature of spacetime in a remarkably concise and harmonious form. Beauty in scientific theories often manifests in the form of deep connections between seemingly disparate concepts, symmetries, and unifications – like in the aforementioned example of how electricity and magnetism were once considered separate forces, until they were unified by Maxwell’s equations.

Simplicity can push us towards an algorithmic or computationally efficient law, whereas beauty might hint at a more abstract or elegant formulation that unifies multiple phenomena under a single framework that simplicity alone may not capture. For example, the inverse-square law governing both gravitational and electromagnetic forces can aptly be seen as beautiful, especially for its ability to unify distinct forces while reflecting the geometric structure of the universe in a coherent and elegant way. But the beauty might lie more in its generality and symmetry extended across diverse contexts, rather than in mere simplicity. That is to say, a simple law would be algorithmic, and beauty in laws of nature can be apprehended with a kind of elegance that transcends mere simplicity.

Or a conservative force (such as gravitational or electrostatic forces), where the work undertaken is path-independent, can be derived from a potential function, where the relationship between a conservative force and a potential might be seen as expressing beauty (to the beholder) in nature when it ties together concepts in a mathematically elegant way. However, this elegance doesn’t necessarily equate to simplicity. The mathematical elegance of a relationship between force and potential - such as deriving forces from the gradient of a potential - can still involve complex expressions for the potential itself, especially in more intricate or realistic systems. Simplicity might hold for idealised cases (like point masses or charges), but when extended to real-world scenarios, the potential functions often become more elaborate and intractable, even though they remain conceptually beautiful.

There are even cases in nature where the potential function is more difficult to express than the force law itself, such as in the aforementioned gravitational potentials in general relativity (like dealing with the curvature of spacetime), extended mass distributions (like how the Earth’s gravity is distributed across a volume), and continuous charge distributions (like the electric fields in capacitors). 

Maxwell's equations and the conservation laws provide a simpler way to derive and understand complex behaviours in various macroscopic systems – they can perhaps be thought of as beauty uncovering a simpler order, with equations that reveal clear, well-structured laws that govern a wide range of phenomena in everyday life, and an elegant and unified description of electromagnetism, revealing the interconnectedness of electric and magnetic fields in a way that feels intuitive and ordered (although at this level I’d say it’s more an apparent simplicity that masks deeper complexity beyond the intention of this post).

Whereas relativity and quantum mechanics are beautiful in a different way, and can perhaps be thought of as beauty uncovering greater complexity, as they reveal deeper and often counterintuitive layers of reality. Although, of course, on the other hand, constancy of the speed of light and spacetime as a geometric entity also offer elegant simplicities that may beautify our conceptions too. While mathematically complex, Einstein's relativity simplifies the understanding of gravity as curvature in spacetime rather than as a force, which to me is conceptually beautiful. And although quantum mechanics takes us to deep and mysterious places, Schrödinger's equation, wave functions and the uncertainty principle are based on relatively simple mathematical rules of quantum mechanics, which present a kind of beautiful succinctness to the picture. And physics itself is also a spectrum of complexity, which I think is beautiful in its own way too.

Many scientific laws possess a kind of beauty in that they offer the most elegant, coherent, and far-reaching explanations, provoking an aesthetic appreciation as well as an epistemic one. But whereas I think simplicity is attractive in a different way – especially parsimonious models that explain complex phenomena with fewer assumptions or parameters, or even Occam's Razor, which suggests that the simplest explanation is usually the correct one - we often need a sense of elegant rigour to capture more profound truths that only become apparent when we consider the deeper structures and connections in nature, which are often revealed through the beauty of the mathematical relationships governing physical laws.

Perhaps the most enchanting thing about beauty, then, is that it can simultaneously point to a richer, more unified understanding of reality, while at the same time hint at a deeper truth that simplicity alone would fail to capture in its full gravitas.  

Part II tomorrow.