Are "proofs" that are contingent upon physical reality valid?

Question

Consider the following statement:

Let $P$ be any polygon and let $A$ be a point inside of $P$. Then there exists at least one side of $P$ such that the perpendicular from $A$ to said side touches the side within $P$.

Now consider the following attempt at a "proof":

We can construct a physical object with the shape of $P$ and with center-of-mass at $A$. If we lay this object on one of its sides (say, $S$), it will "roll" onto its next side if the perpendicular from $A$ to $S$ touches $S$ outside of $P$. If the above statement does not hold, this object will continue to roll indefinitely, which is impossible. Therefore, the above statement must be true.

My question is this: Is this a valid proof, even though it is based on physical reality as opposed to purely mathematics?

Please let me know if what I've written above is unclear in any way. If it is, I can try to make some diagrams to convey what I mean.

Answer 1

The proof strategy is valid, but it needs some formalization if we want to be confident that it proves exactly what it claims. In this case, the statement is false as written; you need to assume that the polygon is convex!

Here's one way to formalize your proof. In physics, gravity tends to minimize the distance from the center of mass to the floor. Mathematically, then, we want to argue that there exists a point $M$ on $P$ with minimal distance to $A$, and that $M$ lies in the interior of a side $S$, rather than on a vertex. (Here we need $P$ to be convex. Otherwise, we'll get stuck.) Then, we can argue that $MA\perp S$.

Looking back, our physical intuition helps us to decide which mathematical tools to employ, and the mathematical deductive process forces us to clarify the statement before we claim a full proof.

Answer 2

Unfortunately not. Physical explanations like that are a great way to develop intuition that might lead to a proof, but it's not a mathematical proof. A mathematical proof is a set of logical statements that lead from axioms or theorems to the desired conclusion.

In fact, physical intuition can lead to some major errors in math. A great example is the axiom of choice. The axiom of choice says something like: if there's a set of stuff than one can choose one element of the stuff. This seems reasonable. But if one accepts this axiom, then one can split a ball into two balls of exactly the same volume. And this, of course, is impossible physically.