P.S. One "human" remark. Your statement about the
ancient Greek straightjacket does not apply
to ancient Greeks. As Archimedes' papers show,
he used heuristic methods and algorithms to
discover his results and only then did he
prove them formally. This was also probably
true of all the other great ancient
mathematicians. In fact, Euclid probably
did not discover what he wrote in his Elements,
so is not a good example. Basically, judging
Ancient Greek mathematics from Euclid is like
judging 20th century math from Bourbaki, or
from a college Calculus and Analytic Geometry
textbook.