OP said "draw an arbitrary triangle in chalk on the ground", and from there gave a set of instructions that showed you how to check that the interior angles of that triangle added up to 180 degrees. You're right that that concrete set of instructions only proves it for that one triangle.
But that's not the whole proof. If I followed these instructions for a right triangle with angles 45,45,90, I'd only have proved that that triangle's angles added up to 180 degrees.
The full proof is that if you imagine doing this, for any triangle, it's clear that it will always work. When I picture this in my head, it's clear that I'm going to end up having turned 180 degrees, regardless of the measurements of the triangle.
This leap: starting with a set of concrete instructions that you can do on a particular object, but then verifying that they would work no matter what object you started with, is common in proofs.
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.
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?
(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.
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.
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.
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.
Interesting. For me, there is no clear gap between intuitive proofs and formal proofs. Sure, intuition can lead you astray, but as you do mathematics, you develop your intuition so that you stop being intuitively certain of false things. Contrariwise, formal proofs are more likely to contain dumb algebraic errors, but as you do mathematics you learn to be exceedingly careful in your calculations.
But the wider point I want to make is that there's no gap between the two. _Real proofs aren't fully expanded._ If you've worked with a theorem prover like CoQ, it becomes painfully clear how many steps even the simplest proof skips. For example, the proof that the sqrt of 2 is irrational is really easy:
But look at how many steps this skips, if you want to get close to actual axioms and definitions:
- You squared both sides of an equation. In this case, that's fine because you're only doing forward reasoning, but if you wanted to reason backwards you'd also have to check that both sides had the same sign to start.
- You multiplied by q^2. That's only valid if q^2 is nonzero. Now intuitively we know that q^2 is nonzero, since q is nonzero. But it needs a proof.
- You deduced that p was even from the fact that p^2 was even. How do you prove that? My first thought is to use the fundamental theorem of arithmetic. I don't know about you, but my intuition completely glossed over the fact that the proof that sqrt(2) is irrational made use of the fundamental theorem of arithmetic when I first read it. Either that, or there's another way to prove this; what is it?
Now, you could expand this proof to include all the steps. But we don't bother, because it's painstaking and not actually that likely to catch flaws in the proof, because we intuitively know these things. Likewise, I feel like the triangle proof is the same way: it skips over some things, but we intuitively know it's okay and you could expand it (to talk about commutativity of translation and rotation) but there's usually not much reason to bother.
Although it's a lot more obvious how to go about expanding the proof that sqrt(2) is irrational, I'll give you that.
There is an implicit, widely-held sense that the detail in proofs should scale with the expected training of the people who will be reading them. If your intended audience has internalized their field so well that the expansion and checking happens completely automatically and subconsciously, more power to them - but on the other end of the scale you have the proof that the square root of two is irrational, or this one about interior angles. Other than machine-aided proofs the most thoroughly expanded you ever see anything is in highschool geometry!
So, what justifies this scheme? I would say that once you have seen a technique used in full detail, you don't need to see the detail elsewhere because there's a meta-theorem in your head that applies to every case where things line up in a pattern where the technique works. Slowly this replaces your natural intuition.
