In a Nutshell
Paul writes to stress their departure from Christ's gospel, explaining the great significance of Christ's revelation to them..
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There are the various accounts of Paul's confrontation with Jesus. They are in Acts in Chapters 9 (by Luke), 22 (by Paul to a Jerusalem Crowd) and 26 (by Paul in discussion with King Agrippa). Why are they different?
Let me re-affirm to you, brothers [and sisters ref 3:28], that what I preach to you is no human gospel. I did not receive it from anyone, nor was I taught it, but it came to me [into my hands] by Jesus Christ's revelation [of Himself]. For you have heard about my way of life in Judaism, how I relentlessly persecuted the church of God seeking its destruction and I thereby advanced in Judaism far beyond others of my age, so completely a zealot was I for my ancestral traditions. But when the One who had set me apart, from [the time I was in] the womb of my mother, having called me by His grace, wanted to reveal His Son in me, in order that I might proclaim Him among the nations [Gentiles], my immediate concern was not to confer with flesh and blood, nor was it go up to Jerusalem to those who were apostles ahead of me, but [instead], I went into Arabia and then finally returned to Damascus. It was after three years that I went up to Jerusalem to visit Cephas [i.e. Peter], and stayed with him for fifteen days. Except for James, the brother of the Lord, I saw no other apostles.
Take note: what I am writing here, before God, is no lie. Then I went into the regions of Syria and Cilicia. And I was still unknown, unsighted by the churches of Judaea; what they heard was only [the report that]: "the one who used to persecute us is now preaching the faith he had previously been destroying." And so they praised God because of me.
Why then does Paul give such emphasis to his own "path"? What is he trying to say? He is insistent. Whatever it is that has been going on among the Galatian churches, and between him and them, it has a direct and immediate impact upon his call, upon his own standing as a servant of Christ. He insists that the call he received to proclaim Christ to the Gentiles was from Christ and not handed to him by the apostles in Jerusalem. They may have had to confirm it later on, but now, with the Galatian churches on the precipice of deserting him, he looks back on his own call, on how it came about, and affirms that for him to "give an inch" and accept these [Galatian] developments would be an outright denial of the call to proclaim the Gospel among the Gentiles which came to him from the Lord Himself. This would be a denial of the Lord. He cannot do that. He will not.
He appeals to the fact that his own reputation, as one who proclaimed the Gospel, was well established before he met Peter and James and before he became known to the Judaean churches. His work does not take its cue from the Judaean churches as its source, as if now the Galatian churches, if they are to be true to Jesus Christ, will have to take the role of "daughter churches" of those in Judaea. No. And the work that had begun in Galatia is also a genuine work of the Holy Spirit Himself - no church community has "apostolic precedence" over the others.
It is as if Paul is asking them: Why should I be so concerned about those unsettling you? In effect, his answer goes something like this:
"This move of yours not only shifts where you stand; it implies a different interpretation of the work Jesus has called me to do! And I should know. The ones trying to unsettle you are, whether they intend to or not, seeking to add to the Gospel, are trying to wreck God's work in your midst, just as I had been trying to do in my former life in Judaea. You now need to be stopped in your tracks like the Lord Jesus stopped me in mine!"
Paul is unsparing in his determination to tell these Christians - congregations of Jewish and Gentile believers - that this curséd intrusion not only departs from the Gospel, it is an attack upon the Gospel. It's no mere "add-on".
The rest of the letter, then, explains why. He is not excluding Jews from the people Jesus Christ gathers together. He could hardly exclude himself. No, it is to emphasize that all peoples are now called to receive the King of all Kings, the Lord of all Lords. The Messiah is of the Jews and His Revelation has now been given and from them it is to be proclaimed to all peoples. He is now to be revealed in the midst of those who believe, whether Jew or Gentile, without any further qualification. Paul has already alluded to this when he refers to James, the brother of the Lord Himself (v. 19). He is insistent. He can't have received his commission from others - he is an apostle by "revelation of Jesus Christ". Likewise, the purpose of this letter is to tell the members of the churches in Galatia, that they also now receive membership in God's family from Jesus Christ Himself. It is not so much a matter of becoming a fully-fledged relative in Jesus' earthly family, tribe and nation. The Kingdom which the resurrected Messiah of the Jews now rules, includes all the families and tribes and nations - all those who believe full stop.
There are, of course, many, many questions raised by Paul's account of His encounter with Jesus. And there is more to what he has to say, much more.