Simplicity in physics and mathematics
TL;DR: Mathematics and physics are biased towards simple laws and structures
In both physics and mathematics, simplicity is often seen not only as a desirable feature, but also as a hallmark of a good theory. There is an implicit sense that the right theories ought to be simple… for some idea of right and simple that is not so simple to pin down. Nevertheless, there is the idea that this simplicity is really a feature of the object that is being described: it is the mathematical structures themselves that are simple, it is the laws of nature themselves that are simple. In contrast, economic systems, biological systems, sociological systems are inherently complicated and we cannot hope to find simple fundamental laws for them. My suspicion is that math and physics are simple because if a topic becomes too complicated it stops being math or physics. It becomes something else.
When presented with the infinite complexity of the real world, one is forced to narrow his focus. Most disclipines focus on a particular subject domain: substances and how they can change into each other (i.e. chemistry), the human body and its proper functioning (i.e. medicine), expression using sounds (i.e. music) and so on. Obviously, in each of these subjects there will be simple instances and impossibly complex ones. For example, in ancient times people already knew that honey was helpful to dress a wound, that copper and tin could be combined into bronze or that strings could be used to produce music. Oxidation, germ theory and tonality, instead, are much more recent notions. We have no expectation that any of these subjects, or parts of these subjects, will be finished in the sense that nothing else can be discovered.
Physics, however, does not work like this. In fact, it is a bit difficult to even define what physics is. For example, one may say physics studies the basic constituents of matter and how they interact, but this does not seem to work. We study orbits of planets in physics, and planets are not at all basic constituents of matter. Moreover, studying how atoms interact with each other to form molecules would qualify in that definition as well, and in fact in physics we do study the electron orbitals for hydrogen atoms. Yet, things are moved to chemistry pretty quickly. Similarly, in physics we study simple heat engines, simple electric circuits, simple mechanical structures, but the more advanced cases are not studied in physics anymore: they become engineering, electronics, and so on. Therefore, we can consider concluded the part of physics that studies those topics. Some advanced cases do stay in the physics department, but it’s not just physics anymore: it is biophysics, astrophysics, geophysics and so on.
It seems, then, that physics studies simple systems that can be replicated and studied reliably under experimentation and obey simple rules: things that fall, heat transfer between two objects, flow of current in simple circuits and so on. The phenomenon being studied is less important than the ability to carve it out from the rest of the world and find strong predictable relationships. If we imagine going through nature at different scales, we will find some scales in which certain objects are very complicated and others that are simple. Physics, then, will stop at the latter ones, and study those setups in isolation. That is, physics looks for simple objects with simple rules, assumes all other objects are complex arrangements of those simple objects and simple rules, and leaves the study of the complex ones to other disciplines. Physics is biased towards simpler systems.
The more interesting question, then, would be: is there something particular about nature that allows for these simple levels of description at specific scales? Or is it something that is inevitable, due to results like the zero-one law? Personally, I do not know.
But what about mathematics? The above argument does not work for it: more complex mathematics is still mathematics. So, how does mathematics create its own simplicity? It does so by throwing out everything that does not fit into a precise formal system, so that, technically, mathematics becomes the manipulation of abstract symbols that in principle do not actually represent tangible objects. Geometry is about points with no size, lines with no thickness, perfect circles: none of these literally exist, they are simplified idealizations. The starting point becomes simple axioms and simple definitions, and all that mathematics does is prove or disprove propositions within its boundaries. It is the real world that has to be made to fit into the mathematical world through assumptions and idealizations. The practice of physics, in a sense, is finding those physical systems that can be reasonably described by such mathematical idealizations.
In this view, simplicity in math and physics may not be mainly due to some intrinsic simplicity of the real world, but rather to the obstinacy and cleverness of our minds.