The Pythagorean theorem isn't proved or or even checked by measuring right triangles and noticing that a^2 + b^2 = c^2.
I am having trouble with this as a statement of historical fact. Isn't that how they did it?
I'm not sure that's how it was motivated historically. Note that Euclid's proof (Edit: not Euler) doesn't require measuring anything at all.
To use a different example, how would one go about measuring whether there are more real numbers than integers? The proof is pretty easier, but it doesn't require any empirical facts as far as I can tell.
Here's the new thread for posting quotes, with the usual rules: