Q. Due to advances in techno, why do we have to see the moon with the naked eye? Why not follow calculation and do something better with our time than go and look for the moon?
A. First off, to look for the moon is sunnah. Can you find a better thing to do than follow a sunnah? Second of all, Muslims knew mathematics and thus knew how to count as well. Despite this knowledge,
Rasulullah (Sallallaahu alayhi wa sallam) said:
Fast when you see the moon (Ramadhan) and break your fast when you see the moon (Shawwal).
Thus, Muslims have a duty to go and look the moon of Ramadaan (and, indeed, all other months). Should the moon not be visible, they must begin Ramadaan after the completion of the 30th day of shabaan. Doing so utilizing other methods would be a violation of shariah, thus causing for a negation of the first fast, as well as Taraweeh.
Allah certainly knows best.