> I really love the idea that pure mathematics and nature are the same thing
This might not be what you mean, but there is a philosophical position that holds that all mathematical structures exist, and that our universe is simply one of these structures:
If you assume some notion of an object's description "really existing", it's easy to then assume the Universe must have a zero information description, since otherwise the description would be "outside" the Universe. And then you've backed yourself into a corner: what could zero information describe that also contains our observed universe? All mathematical structures it is.
This might not be what you mean, but there is a philosophical position that holds that all mathematical structures exist, and that our universe is simply one of these structures:
https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...