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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.

arshajii
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  • It's perfectly clear and a really excellent proof. Those physical properties are provable in physics, which is just math applied to our universe, so I don't see why it wouldn't be valid. – genisage Sep 24 '14 at 23:01
  • You have to prove every single postulate you take as granted in the physical-reality proof. Why will it roll? Can you prove it mathematically?(You'll have to use physics, but constructivicists may say it's not enough for a proof). Why is it not possible to roll indefinitely? And the real question; Will you be able to prove these postulates on ANY proof you'll have to do in the future? – UserX Sep 24 '14 at 23:01
  • Why would you want to use a mostly-coherent pile of experimental observations and well-reasoned assumptions in the foundations of a mathematical proof? Don't get me wrong, I don't have anything against physics here, I'm just saying that the assumptions required for this would be complicated and numerous. – rschwieb Sep 24 '14 at 23:11
  • Sorry, but what does "touches the side within P" mean? – Zimul8r Sep 25 '14 at 01:09
  • @Zimul8r from the "inside", in a Jordan curve theorem sense. Hence the statement failing (as pointed out in the answers) for non-convex bodies. – Harry Wilson Sep 30 '14 at 09:52

2 Answers2

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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.

Chris Culter
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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.

NicNic8
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    Perhaps more to the point, there are a lot of unfortunate examples in mathematical history of times when people depended on their physical intuition to come to some conclusion which then turned out to be false. One of my favorite examples is: Let $S$ be a family of intervals whose union covers every point of $\Bbb Q\cap[0,1]$. Then the total length of the intervals must be at least 1. This is physically obvious, but false. – MJD Sep 29 '14 at 23:51
  • And there's some horrible story about a group of Italian algebraic geometers in the 1950s(?) who relied overmuch on physical intuition and intuited a bunch of theorems that turned out to be false. There's a post about that somewhere on this web site, but I couldn't find it. – MJD Sep 29 '14 at 23:54
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    Here's the Wikipedia article about it, anyway. "By about 1950 it had become too difficult to tell which of the results claimed were correct..." – MJD Sep 30 '14 at 05:11