Now, let's just think about it: We have state functions and we can prove that some property is a state function or a path function by showing that it is an exact or inexact differential. Showing exact or inexact differential is purely mathematics. And by intuition we can show that a property is state/path function.
Now, with this perspective, we can represent every intuitive idea mathematically. Now comes my confusion:
How we can mathematically show that adiabatic processes can only occur in thermally isolated systems and isothermal process can only be done in thermally non-isolated systems?