Clement writes to a real congregation in Corinth about the removal and restoration of ministers. Ignatius names bishops, presbyters, and deacons and ties Eucharistic unity to the bishop. Irenaeus points readers to publicly known churches, their teaching, and their successions. Cyprian’s correspondence shows discipline and reconciliation operating through a network of bishops and clergy.
The spiritual nature of the Church is not in question. The mistake is turning ‘spiritual’ into ‘structureless.’ The early Church appears in the sources as a worshipping body with concrete assemblies, offices, sacraments, boundaries, and discipline. Its ministers could fail and its communities could divide, but it was not an invisible abstraction.
Primary sources
1 Clement 42–44: https://www.newadvent.org/fathers/1010.htm
Ignatius, Smyrnaeans 8: https://www.newadvent.org/fathers/0109.htm
Irenaeus, Against Heresies 3.3: https://www.newadvent.org/fathers/0103303.htm