Does anyone else find it ironic that PizzaPlanet's cited definition of 'curvature' uses the curvature of the Earth as a demonstration example?
That aside, I'm not really sure your proof is convincing from a FE point of view, momentia. It relies on the assumption that two points designated at different latitudes and longitudes are actually points on a sphere in the first place, rather than say radial points on a disc. The mathematics are rigorous enough to be sure, but the haversine formula is derived by considering points on a sphere -- not points on an arbitrary surface for which we wish to determine the curvature. It simply doesn't apply if the surface is not a sphere.
Similarly, other methods for determining the Gaussian curvature of the Earth fail for this sort of proof. Although the Gaussian curvature is an intrinsic property of a surface, we still must know something about the geometry of that surface in order to make any progress towards finding its curvature. If we assume a totally arbitrary surface, there is no way to determine whether the Gaussian curvature is positive, negative or zero. You could always assume the surface geometry then check predictions against reality, but without that data a purely mathematical proof isn't going to get anywhere.
Of course, if you accept that practical use of the haversine formula has shown that it yields accurate results in real life, I think you'll have a hard time arguing with that proof. If it could be shown that it is accurate, then the whole issue would be resolved. Demonstration of its accuracy would provide sufficient information about the shape of the Earth to follow through with the rest of the proof you provided and determine that the Earth is indeed spherical (or approximately so).
Unfortunately, I don't really count myself much in the navigation, cartography or surveying departments so I can't really vouch for the formula's results. Now, if only there were some profession dedicated to the study and practice of making accurate measurements over long distances... (what ever happened to Theodolite?)
Alternately, it would be a pretty simple matter to try and travel in a triangle shape on the Earth's surface. Travel some arbitrarily long distance, turn left by 120 degrees, travel the same distance again, repeat the turn, then repeat the distance once more. If the Earth is in fact flat you should end up exactly where you started; if it is round you will be off by a distance proportional to the length of your travel leg.
Heck, you wouldn't even need to worry about trying to drive or walk in a straight line with this one -- you could do it with a protractor, some mirrors and a laser, mounted up on tall towers so you don't get things like trees or hills in the way. But then, we've tried the whole "shine a laser over a long distance" proof before, and FE always rejects the notion either through semantic dissembling or bendy light. I tried to make a similar point in a thread about the sun's path over a flat Earth vs. its apparent location, and the whole thing got lost in the notion of what constitutes travel in a straight line.
PizzaPlanet,
You might want to be careful about rejecting mathematics as entirely abstract. While technically correct, your position neglects the fact that this abstraction is derived from real concepts and from ideas, shapes and properties that are grounded in reality. From these things new truths about reality can be deduced. As a FE supporter you should be especially sensitive to things like this, since the zetetic method is predicated upon the idea that a real concept can be analyzed using deductive logic in order to derive new truths about reality.
For example, it is easy enough to find an entirely abstract proof of the Pythagorean theorem. It's all geometry and algebraic relations. But if I show you a rectangular object (say, a picture frame) and ask you to find the distance from one corner to the other, I'm pretty sure you would find that your measurement matches exactly with predictions made using that abstract mathematical proof.
Likewise, if it can be demonstrated that the distance between any two points on Earth can be calculated accurately using a particular method, then the mathematical properties of that method could be exploited to determine the shape of the Earth. In this case, if the haversine formula accurately reflects reality then it can be conclusively demonstrated that the Earth is a globe, as momentia has shown.