It is not outrageous. For example, conservation of energy tells you that if you observe the surface of a box, you can know things inside the box. If you observe how much energy enters or exits the box, you know the change of energy inside the box. And that's true independently of what is inside the box.

Inside the box, there may be one rat, a galaxy, a nuclear bomb, or whatever thing you can imagine, and still you know exactly the change in stored energy. It only depends on what happened on the surface of the box.

Also, when you solve a differential equation, you need the contour conditions. The contour completely determines what happens inside.