John Chrysostom famously thought that there were not really ‘Paul’ and ‘Apollos’ factions in the Corinthian church, but that these names were applied by Paul to factions that were actually associated with local would-be leaders. The reason for using false names at first (before revealing his play in chapter 4) was for the sake of gaining a good hearing. Chrysostom bases this particularly on the fact that the verb metaschematizo in 4:6 normally means ‘transform’ or ‘disguise,’ so: ‘I have disguised these things so as to be about Paul and Apollos, for your sake…’
This viewpoint has been defended recently by David R. Hall and others. Hall says:
The meaning is that Paul has disguised his argument, so that what really applies to other people has been applied to himself and Apollos.
Thomas Aquinas (who, like most, was influenced by John Chrysostom), follows this same line:
it should be noted that above when the Apostle tried to repress the rivalry about ministers among the Corinthians, he had used the names of good ministers of Christ, as when he said, ‘Each of you says, “I belong to Paul,” or “I belong to Apollos,” or “I belong to Cephas”‘ (1 Cor 1:12) and again: ‘Whether Paul or Apollos or Cephas’ (1 Cor 3:22). But in fact they were not glorying in Christ’s good ministers or disagreeing over them but over the false apostles, whom he chose not to name, lest it seem that he was speaking against them from hatred or envy. Rather he had employed his own name and the names of other good preachers. (paragraph 199, commentary on 1 Cor)
Erasmus also follows this view, in his paraphrases of Paul’s letter:
It is not that anything of this sort has arisen from my name or Apollos’, but I wanted to use these names to that you might sense more clearly the impropriety of the matter. (Collected Works 43, p52)
But don’t misunderstand: in this discourse so far I have used Apollos and myself metaphorically, not because we are the occasion for any such sects (for neither do we claim anything for ourselves, nor does anyone among you boast of being Pauline or Apollonian); rather, knowing that there are among you various partisan attachments, and fearing that I might provoke someone to anger, I preferred to introduce the matter with names disguised so that when the letter is read in public, each one may quietly recognise himself. (Collected Works 43, p62)
John Calvin writes:
Hence we may infer, that it was not those who were attached to Paul that gave rise to parties, as they, assuredly, had not been so instructed, but those who had through ambition given themselves up to vain teachers. But as he could more freely and less invidiously bring forward his own name, and that of his brethren, he preferred to point out in his own person the fault that existed in others. (Commentary, p157)
So this perspective has some weighty history – Chrysostom, Aquinas, Erasmus, Calvin…! But I don’t think we have to say either that there was rivalry between those who attached to Paul and those who attached to Apollos or that there was rivalry between those who attached to certain local leaders. Why can’t it be both? That is, in the charismatic leadership vacuum that followed Apollos’ departure, some would-be orators in the Corinthian church wanted to continue his style of ministry, thus saying ‘I belong to Apollos,’ – and others then joined their sphere of influence. In response, perhaps, others reacted negatively and associated themselves with Paul. In other words, it is feasible that there were attachments to local leaders, who represented the ministry styles of the external figureheads. But Paul chooses to speak using only the names of those external figureheads.