At the time of Paul's writings, I believe the NT was not yet written, incomplete. The "impefect" I beleive is the miraclious spiritual gifts which ceased. What do you think the "perfect" is in 1 Cor. 13:10?
"The perfect" in this passage is the coming completion of the revelation of God to man in what we have today as the New Testament. It is the "perfect law of liberty" according to James 1:25. Ephesians 4 also speaks about the coming of the unity of "the faith." So it was not the incomplete revelation, but the coming completed revelation. That is "that which is perfect/complete."
This makes sense in the context too. What was incomplete? Prophecies, tongues, knowledge. What are those things? They are ways of revealing the truth. Well, when you have all of the truth, then you no longer need those things that reveal the truth to you. Jesus promised that the apostles would be guided into "all truth" (John 16:13). So that is what we must believe that happened.
Paul was speaking at a time when that was not yet accomplished. So, they still needed those prophecies, tongues, and knowledge. However, when that which is perfect/complete is come, then that which is incomplete would be done away.