It is difficult to find a brief layman's explanation of the "New Perspective on Paul" (NPP). To rectify the problem I humbly offer "The Idiot's Guide to the New Perspective on Paul" (myself being the idiot). Comments welcome! I posted something similar to this a while back on an Arminian group discussion, so this may be a repeat for a few.
Imagine a church that has the following rule: EVERYONE WHO ATTENDS THIS CHURCH MUST WEAR A SUIT. The church has the rule to keep out the bums, reasoning that bums can't afford suits.
The people in the church know very well that wearing a suit in and of itself doesn't make one a Christian. They simply have the requirement for the purpose of excluding others who are not like them. They want the church for themselves, and want to keeps the bums out.
In this analogy the root problem is that the Christians want to exclude others. There is nothing inherently wrong with someone wanting to wear a suit. The problem is that they are using their preference for suits as an arbitrary rule to exclude others.
New Perspective Explained:
Among Protestant circles, Paul's writings have generally been interpreted as a criticism of legalism (the suit in the analogy). NPP argues that Paul's writings are instead a criticism of exclusivism (keeping the bums out).
NPP points out that during Paul's ministry the Jewish Christians were reluctant to accept Gentiles as fellow believers. The Jews often acted in ways to prevent the inclusion of others. One way that this manifested itself was by the demand that the Gentiles follow detailed adherence to the OT law. The law was used as a way of excluding others.
NPP argues that the Jewish Christians knew that salvation came by grace through faith. Or put another way, they did not believe that following the law would save them in and of itself. NPP argues that root problem was that the Jewish Christians wanted to keep Christianity exclusive to themselves. Thus, Paul's criticism of works should be interpreted in this light.
In short, NPP proponents argue that:
1) The Jewish Christians knew that salvation was by grace through faith, and that following the law didn't merit salvation.
2) The Jewish Christians were using the law as grounds to exclude Gentiles from fellowship.
3) Paul's beef was with the exclusion, not with those who wanted to obey the law.
Why does this matter today?
A key component of Protestant theology is that we are saved by grace through faith, and not by works (Eph 2:8-9). During the Reformation, Luther and others appealed to Paul's criticism of works in order to point out some of the severe abuses of the Roman Catholic church.
If Paul's primary concern was exclusivism rather than works, some think that this could undermine the scriptural justification for the Reformation. Exclusivism is a different problem than works based salvation.
I personally believe that NPP has some merits. But I also believe that the traditional Protestant view does as well. I think that both exclusivism and works salvation were addressed in the scope of Paul's writing. I also believe that both of these views can co-exists along side each other without conflict, and in fact compliment each other.
How does this relate to the Arminian / Calvinist debate?
It is often the case that Arminians are more open to NPP. Calvinists tend to be more critical of it, though this is not always the case. John Piper is a strong critic of NPP, while NT Wright is a proponent. Both are Calvinists. I personally see no reason why Arminians or Calvinists must reject NPP, as it has little to say about the definition of election or the scope of the atonement. As mentioned earlier, it does touch on certain aspects of Reformational thought, however, this is not in and of itself a good reason to reject the theory.
Further reading on NPP
8 hours ago