Nina E. Livesey, in the Westar Forum series (Fall 2013: 'Rethinking the Book of Acts', page 156) argues against the historical reliability of Luke’s account in Acts of the Apostles, after having compared it with Paul’s views on circumcision in his extant letters. She says, in a following essay ('Circumcision as a Means of Testing the Historicity of Acts 16:1–5', page 217) that "Acts 16:1–5 reverses Paul’s position concerning circumcision. Rather than representing any of Paul’s various understandings of circumcision, Luke creates his own Jew-pleasing image of Paul to advance his own theological position."
Dennis E. Smith and Joseph B. Tyson say, in Acts and Christian Beginnings, page 182, the circumcision of Timothy "would appear to be a violation of Paul's deepest principles." They point out that in Galatians 2:3, Titus was not required to be circumcised, then go on to ask why Timothy was circumcised now if Paul had been so opposed to circumcision and Timothy had until then been an uncircumcised believer. Their answer is that Luke's Timothy is not the historical Timothy of the epistles, but was created to appear in this scene and serve a narrative role in Acts. They say, "The Timothy whom Luke creates is a hybrid, part Greek and part Jewish, making him in a sense a narrative stand-in for the hybridity of the church in Luke's story."
It is true that Paul was a proud Jew, but Acts of the Apostles consistently portrays him as seeking to please the Jews, even as, time and again, they rebuff his approaches. That Paul held strong views against circumcision is most clearly demonstrated by Galatians 5:12:
Galatians 5:12 (NIV): As for those agitators, I wish they would go the whole way and emasculate themselves!