Symmetries in laws

Theists have often noticed that theism provides a nice aesthetically-based explanation for why we have simple laws, namely that such laws are beautiful and this gave God reason to enact them. (One can run this in two ways: (1) such laws are objectively beautiful, and God made them because of their objective beauty; (2) such laws are beautiful to us, and God created a world where the laws are beautiful to the intelligent creatures therein.)

Another interesting question about the fundamental laws is why they exhibit such nice symmetries. This question on its face seems independent of the question of why the laws are simple. You can have simple but asymmetric laws, and complex but symmetric ones. Again, an aesthetic theistic explanation seems to work well here (and again, it comes in two forms: either the symmetries are objectively beautiful or God made a world where the aesthetic properties of the laws fit with the aesthetic sensibilities of the intelligent creatures).

One might hope that symmetry considerations would thus allow one to run a teleological argument for the existence of God that escapes from the difficulty of making the notion of simplicity precise. However, while I think there is hope of a symmetry-based theistic argument, I don't think it escapes from the difficulties of theoretical simplicity. Any set of laws of nature that has an infinite space of solutions has an infinite number of symmetries: any bijection of the space of solutions onto itself is a symmetry. When we are excited by a potential symmetry like charge-parity-time invariance, we are excited by the fact that the symmetry can be specified in a simple way with respect to physically natural quantities. And if we can make sense of these twin notions (simplicity and physical naturalness), then we can likewise make sense of the notion of the simplicity of laws. So while a symmetry-based argument may provide additional evidence for the existence of God, it is subject to the same main difficulty as the simplicity of laws argument. (That said, I think this difficulty is not fatal.)