I don't know if the Philomathean Society did anything to recognize existance of the civil war (like going inactive/dormant). I had always recieved the impression that it existed continuously at the school until sororities were disbanded at Wesleyan College. You might find more answers by checking out Wesleyan's history...for example: the society would have gone dormant if the school had closed for the duration of the war, etc.
It's a really good question though. I'm a history teacher and I've always wondered (being from the north) about how universities, especially for females, survived the war.
History of Wesleyan College There isn't anything about the Civil War included, but it is interesting all the same.