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Fragments of Other Letters?
The material covered in 2 Corinthians is diverse, so much so that various scholars have suggested it is actually portions of up to five different letters that have been spliced together. For example 6:14-7:1 seems so out of place in the flow of the letter that some have suggested it might be an excerpt from Paul's lost first letter to the Corinthians on sexual immorality. These verses say not to "hook up" with unbelievers, and we know the lost letter had some material on not associating with sexually immoral people.  It is an interesting thought, but not one we could really conclude with any certainly one way or another.
Some scholars also suggest that 1 Corinthians 8 and 9 might come from two different letters as well. It is true that they deal with a completely different topic than 2 Corinthians 1-7, namely, the collection Paul was collecting to take to Jerusalem. But 1 Corinthians also covers numerous different topics as well. 1 Corinthians 8 flows very nicely on from the earlier chapters. Paul seems to be in Macedonia as he writes, north of Greece (8:1), which is where he wrote the earlier chapters from as well (e.g., 7:5). 1 Corinthians 9:2 also fits well with Paul writing from Macedonia.
The key to understanding 2 Corinthians 8-9 is to recognize that Paul is about to send Titus back to Corinth again (8:17), even though he has just arrived in Macedonia from there (7:6). Two other "brothers" were also going with Titus (8:18, 22), individuals the Corinthians apparently knew, but whom Paul strangely does not mention by name. One explanation is that one or more of these individuals stood in some way at the center of some controversy between Paul and the Corinthians.
The first is possibly one of a list of people mentioned in Acts 19:22 and 20:4, individuals who were with Paul in Macedonia: Timothy, Erastus, Sopater, Aristarchus, Secundus, Gaius, Tychicus, Trophimus, and--maybe--Luke, assuming he is part of the "we" that pops up occasionally in Acts. Since the trip is to keep the Corinthians from being embarrassed in relation to the collection (9:3-4), we can possibly eliminate those of these that seem to represent various regions. That would leave Erastus and Timothy as good candidates for these two brothers.
Probably the best alternative for the second (8:22)) is Timothy. Paul had apparently sent Timothy to the Corinthians before (1 Cor. 4:17), and it is at least possible that as Paul's representative he had been directly involved in some of the conflict. Indeed, perhaps the reason Paul sent Titus with the harsh letter was because Timothy had become, to some extent, "scorched earth" with them. Acts 20:4 also mentions Timothy as part of the company that accompanied Paul to Jerusalem.  It thus seems very likely that Timothy was the one who accompanied Titus to Corinth to try to secure their portion of the Jerusalem collection.
This collection seems to have been a sore spot with the Corinthians. They had apparently expressed their desire to contribute a year earlier (2 Cor. 8:10; 9:5), but the intervening conflict with Paul had apparently derailed that process. We would like to think that after the Corinthians submitted, especially given the tone of 1 Corinthians 1-7, everything was back on track between Paul and the Corinthians. Unfortunately, it was apparently not the case.
Once we get to 2 Corinthians 10, we immediately sense a significant change in tone from the earlier chapters. While 2 Corinthians 1-7 include some of the most uplifting material in the Bible, Paul suddenly goes on the defensive in chapter 10, so much so that some have even suggested 10-13 are an excerpt from the earlier, lost, harsh letter. But 12:18 speaks of Titus having already returned from a visit, which places this material after the harsh letter, after Titus' initial visit to Corinth on Paul's behalf.
Some thus suggest that Paul is now addressing a different segment of the congregation, a part that has not submitted to him, unlike his audience in the first seven chapters. The problem with this suggestion is that Paul gives us no clue at all that he has switched audiences. We have the same word "you" used continuously throughout. There is simply no basis in the text of 2 Corinthians to justify a switch in who Paul is addressing in the audience.
Indeed, these last four chapters do not operate on the same assumptions as the first 9 chapters. In the first part of 2 Corinthians, the insubordinate individual in the congregation has repented and submitted (e.g., 2:5-11). As we have seen, the first half breathes the relief Paul feels at his reconciliation with the community. But in 12:20-21, Paul expresses deep concerns that when he comes he will not find them as they ought to be. He is afraid he will find that many who had sinned earlier will not have repented. He fears that he will find the same problems he tried to address way back when he wrote 1 Corinthians.
Paul is not mentally unstable, so we seem to face two basic options. First, perhaps Paul received new information before he sent the first part of the letter with Titus and the other two brothers. If so, however, we can wonder why he did not go back and revise the first part of the letter. What is more likely, however, is that 2 Corinthians 10-13 is an excerpt from a subsequent letter. 12:18 mentions that Paul has not only sent Titus before, but also another brother.
It thus would make a good deal of sense if Paul sent 2 Corinthians 1-9 with Titus and perhaps Erastus and Timothy. But the solicitation of the collection apparently did not go well--and apparently they found that the Corinthians had not submitted nearly as much to Paul as he had thought. Paul writes a follow up letter, which included chapters 10-13. The tone is not like Galatians. The tone is more one of discouragement and sarcasm.
So Paul defends himself once more, giving us another great autobiographical passage to add to Galatians 1-2 and Philippians 3. The Corinthians are apparently putting down Paul in comparison to other "super-apostles" (11:5). Perhaps our first thought might be that they are comparing Paul to Apollos again, especially since they do mention Paul's lack of rhetorical skill (10:10; 11:6). But the general tenor sounds also like some of Paul's criticisms of Judaizers elsewhere.
Whoever they are, Paul calls them "false apostles" (11:13). Like Satan, they disguise themselves as angels of light. Paul sarcastically says he was too weak to push them around and slap them in the face (11:20), like these super-apostles. They are so wise that they take pride in fools (11:19).
Paul goes on to catalog the kinds of things he has endured for the sake of the gospel, things these dainty super-apostles have not had to face. He speaks of how many times he has been flogged and beaten by both Jews and non-Jews (11:23-25). He speaks of the kinds of revelations God has granted him (12:1-4). Although he speaks as if he is talking about a different person, he goes on to talk about God keeping him humble in 12:7 in a way that reveals he has been talking about himself. God allowed him to live with a physical problem, perhaps difficulty with eye sight (cf. Gal. 4:15), so that Christ's strength would be shown in Paul's weakness (2 Cor. 12:9).
Paul's last letter to the Corinthians thus ends on a somber note. He has had trouble with the Galatians. He has perhaps been imprisoned and banished from Ephesus. The Corinthians remain in rebellion to his authority, unmentioned among those who accompanied Paul to Jerusalem (Acts 20:4). Presumably they do not contribute to the collection. It is thus with great pathos that we read Paul's polite greeting from Gaius in Romans 16:23. And it is with great pathos that we read Paul's words in Romans 15:23: there is "no further place for me in these regions." So he looks toward Rome and Spain beyond.
 Of course Paul clarifies in 1 Corinthians 5:9-13 that he primarily had immoral believers in mind, not unbelievers. 2 Corinthians 6:14-7:1 has often been used in popular teaching to argue against dating or marrying unbelievers, although the topic of sex and marriage is not clearly what is under discussion there.
 Although, interestingly, Acts never mentions the collection Paul raised for Jerusalem, a curiousity that has given rise to its own share of speculation.