William M. Ramsay on cites Dio Chrysostom to the effect that the custom of women going veiled in Tarsus was an oriental and non-Greek custom, Paul is merely reflecting his own (Jewish) background by requiring women wear head coverings (The Cities of St. Paul, 201-5). Because of the popularity of Ramsey’s works on Paul, this theory is often repeated in modern commentaries, but it seems odd that Paul would impose this one Jewish custom on congregations when he frees them from so many other Jewish customs.
The application of this rather obscure command is usually some vague platitude that women should be dressed modestly. If the culture includes head coverings in this then the woman ought to not offend the culture. No one ever points out that if this is the true application, then a woman visiting a culture which is comfortable with public nudity is free to “fit right in” when they visit the beach!
I seriously doubt that modesty is the issue Paul is trying to get at in 1 Corinthians 11. There is clear evidence in the Greco-Roman world of prostitutes wearing head coverings. There are several artistic representations of groups of women with or without head coverings. There is simply no evidence that head coverings were universal in the Greco-Roman world!
Based on his study of Roman statues, D. W. J. Gill has argued that it was a Roman convention to cover the head while praying or offering a libation. There are two well know statues from Corinth, one of Nero and one of Augustus with their heads veiled. It was the leader of a prayer or sacrifice that would cover their heads, the congregation (if any) would not necessarily do so. Gill argues that the social elite in Corinth also practiced head covering while praying or participating in a sacrifice. Since the passage seems to cover the whole congregation, though, perhaps it is only the prophets who are speaking in the congregation that are covering their heads while prophesying (in 14:29 only two or three ought prophesy.)
The problem in Corinth is that the Christians are (continuing) to take their cues for worship from the pagan world. They are worshiping in the same way that they would have in a pagan rite, Paul is rejecting this mixing of the world with the Church.
If the problem that is at the heart of the veiling of men / unveiling of women is taking worship cues from the pagan world, then there is a most serious application possible. How far we want to take this application is quite controversial, from the mega-church movement to modern praise and worship services, it is possible that the American church has taken its cues from the pagan world rather than from the Bible. The modern American church seems to be following MTV rather than the NIV.
There is always a tension between cultural relevancy for the sake of evangelism and participating in the world because we enjoy it. It is possible that is what was happening in Corinth. The members of the church of Corinth were routinely acting like the world without taking into consideration how their new Christian world view speaks to a practice (sexual mores, lawsuits, feasts and banquets at temples, etc.)
The veiling of women / men may seem like a minor problem to use (“it’s just cultural”) but that misses the whole point. If these people were indistinguishable from the world in their worship, how were they going to effectively evangelize their culture?
D. W. J. Gill, “The Importance of Roman Portraiture for Head-coverings in 1 Corinthians 11:2-16,” Tyndale Bulletin 41 (1990): 246-60.
C. T. Thompson, “Hairstyles, Headcoverings, and St. Paul: Portraits from Roman Corinth” BA (1988): 99-115.