An example: The statement that it does not matter whether I do my (thought) experiment today or tomorrow, can be used mathematically deduce the conservation of

An example: The statement that it does not matter whether I do my (thought) experiment today or tomorrow, can be used mathematically deduce the conservation of some quantity which we can intuit is the mechanical energy. More technically it's a differentiable symmetry: we can do the experiment a day, a week, a year in the future - this time can be varied smoothly. If you know calculus read the first few chapters of Landau and Lifshitz book on classical mechanics.