I think the threshold was literacy. There must have been a big divide between the illiterate rural poor (known originally as pagans) and the city dwelling literate citizens. Even if you were a slave you could advance in the Roman world once you had gained your freedom and could read and write and learn the wisdom from earlier generations of writers.
Within the privileged group of the well educated, there were the few who excelled and had extraordinary insights into the physical universe and its workings not the least of these was the genius Archimedes. There is a small museum in modern Olympia devoted to his inventions--his achievements are astonishing.
The Antikithera device is really all the evidence needed to tell us that the ancients were highly accomplished astronomers, geometers and mathematicians and precision metal workers to boot.
As a detail of the scientific approach for achieving accuracy, the designer built into this astronomical calculating device a mechanism to allow for the eccentricity of the orbit of the moon! This tiny fact alone places their understanding at an early modern level of consciousness yet must have been known for a minimum of twenty one centuries. Yet the Antikithera device looks to me (I saw it in Athens this year) like it was already a highly refined and miniaturised machine which had been developed from much older mechanisms, analogous for example to the development of a chronometer from the cathedral clock. This would then put back the understanding of astronomy, geometry and gear making to an even more distant past.