Main Question or Discussion Point
The math theorem to be proven
We want to join three given points using any number of straight lines of any length while minimising the total length of the straight lines. Show that this is achieved by using three lines that are 120##^\circ## apart as shown above.
The following is the answer to the question. It uses physics and no calculus. I always think math is more fundamental than physics. How is it possible that physics can be used to prove a math theorem? Does the following constitute a proof?
The answer uses concepts of potential energy and/or gravitational field. It seems strange that these concepts form part of the proof. My concern is that this answer may be using a principle that depends on empirical verification. If so, then it cannot constitute a proof. For instance, suppose it is found empirically that the equilibrium position is not one with the lowest potential energy, then this answer is invalid. This makes it seems like the truth of a math theorem depends on how nature behaves. But this cannot be the case. Hence, an answer that uses a principle that depends on how nature behaves cannot constitute a proof. But I think there is nothing empirical about this answer, because we can perform this thought experiment in our mind and dictate that the masses obey Newtonian mechanics, even though nature may not. Then, can I say that Newtonian mechanics (together with concepts of forces and energy) is a mathematical tool that can be used to proof math theorems? And whether this mathematical tool actually describes the real world is irrelevant to the validity of using it for proving math theorems.
On the other hand, if we were to use a more "mathematical" approach, I guess we need to use calculus. But how?