Fascinating use of physics to prove a math theorem

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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?

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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?

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  • #2
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Agreed it is a looser use of the term "proof" and indeed even if you were to allow such empirical proof you cannot physically carry out all possible configurations of the three points.

Rather you would have to break out the theoretical physics to prove all cases. That would utilize first principles and deductions which parallel and indeed invoke the same mathematics you would (or might) use to prove the result in a pure mathematical setting.

You might however find it an interesting exercise to reconstruct and define geometric quantities analogous to the physical ones used in the theoretical argument, i.e. potential energy and force. Note that the physical argument is implicitly using vector nature of the forces and resolution into components when it makes the "bisection" symmetry argument. You'd want to do something similar on the pure mathematical side... say by defining a gradient of each string segments length as a function of vertex position? That's what the physical construct calls a force when you make length equate to energy.
  • #3
I always think math is more fundamental than physics.
I'm going to have to disagree with that premise...

If I put an apple on a table, then put another apple beside it, then there are 2 apples. Hence 1+1=2.

I can empirically observe that there are not 3 or 4 or more apples, nor are there 0 or 1 apples unless I get hungry. Based on these observations, I can formulate a principle of conservation of apples. At its core, mathematics was invented to describe the real world, so if nature says that a thing is true, the math better agree or the mathematician has made a mistake.

I do understand that it's possible to invent self consistent, mathematical structures that do not dirrectly correspond to anything physical. Imaginary numbers being an obvious example. All this demonstrated though is that a mathematical proof does not constitute a proof of physical reality. I'd still maintain that physical reality does constitute mathematical proof.

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