[whatshisface]

Why is it clear that it will always work? It would be a proof if that part was written down. Since we all already know that the angles add up to 180, it might just be the intuition taught to us by another proof of the statement leaking backwards into the implicit step in this attempt at proving the statement. The part that's left off (the proof that it always works) is actually the bigger part of the contraption.

[justinpombrio]

Because you end up back on the line you started at, facing in the opposite direction. That's a 180 degree turn.

[whatshisface]

Suppose I found a triangle whose internal angles added up to 190 degrees. If I did the experiment on it, I would end up 10 degrees away from where I was predicted to be. How can this scenario be ruled out?

[justinpombrio]

Both of these things are true:

(i) Like you said, if you found a triangle whose internal angles added up to 190 degrees, and you followed the procedure, you would have turned 190 degrees, rather than 180 degrees. This is true because during the procedure, you turned three times, each by one of the angles of the triangle, so the total amount you turned was the sum of the angles.

(ii) You would end up back at the line you started with, facing in exactly the opposite direction. This is true because the last step of the instructions is to turn until you're facing in the direction of this line. Thus you have turned 180 degrees.

Now of course this is nonsense: you can't have turned 180 degrees, but also turned 190 degrees. How did we arrive at nonsense like this? The logic is sound, so it must have been one of the assumptions. Which assumption is questionable? Oh, right, the triangle whose angles added up to 190 degrees.

This is a proof by contradiction, that shows that a triangle whose angles add up to 190 degrees cannot exist.

[whatshisface]

Inside of that is the assumption that the sum of the inner angles that I rotate by as I walk around a closed loop is equal to the angle between my initial and final directions at the starting point, so that having turned 180 means that my feet have shuffled by a total of 180. That isn't true outside of Euclidean geometry, which indicates that its proof might not be as trivial as it seems.

[justinpombrio]

Bah, I almost talked about what would happen if you did this on a sphere. Yes, there is an assumption that rotations and translations are commutative and associative. We're so used to this that our intuitions sensibly hide it.

The fact that there are hidden assumptions doesn't invalidate the proof, though. There are always hidden assumptions. Even if I give you formal axioms to reason with, you need a system in which to interpret those axioms.

[whatshisface]

Although I see the reasoning, I'm still not comfortable with the proof. It sounds to me like "plug something in to the Zeta function, observe that it isn't a nontrivial zero, conclude that there are no nontrivial zeros." Even if it seems completely intuitve to me I still wouldn't consider it proven.