We’re told that Moses ‘sat down’ to deliver the judgments interpreting the law (Exodus 18:13) and Jesus refers to this tradition when he says the Scribes and Pharisees occupy the ‘Seat of Moses’ (Mt.23.2)
So from these two Scriptural details we can infer several things: the Jews had not only the written Scriptures, but an agreed, recognized, living, successive interpretive authority alongside their Scriptures. That authority was not individualistic, but communal. That authority had a particular function and a there was a way to know who occupied the seat of Moses and who did not.
Jesus and his followers would have taken it as given that this was part of their religion. In fact, in the Matthew passage Jesus actually tells his followers to obey that interpretive authority. This Jewish understanding of the relationship between Scripture and Tradition, and the need for a dynamic, successive interpretive authority would have come into the early Christian Church naturally.
Thus we see Peter as the second Moses, and the Chair of Peter becomes the symbol of apostolic authority. Similarly, the cathedra for each bishop is a symbol of the teaching authority which he shares with the Prince of the Apostles.
Is that the bell? Apologetics 101 is now over.
PS: Owl of the Remove has some good quotes from the Apostolic Fathers for today’s feast here.
PPS: Shawn Tribe has this picture of another papal chair: Pius IX’s faldstool