The question
Paul opposed proselyte circumcision for his gentiles-in-Christ, and did so fiercely. The chapter asks a question the literature has treated as already answered: who was promoting it?
The standard answer is Jewish people, whether Christ-believing or not. Novenson argues that the answer is not an inference from the letters but an inheritance from 1830s Tübingen, where Baur read early Christianity as a Hegelian conflict between Petrine Jewish Christianity and Pauline gentile Christianity. That scheme is a two-hundred-year-old grand unified theory which the field still grants the benefit of the doubt, and the step from "Paul opposed circumcision" to "Jews were urging it" is a non sequitur it licenses.
The evidence
Walked through the letters, the case against the standard answer is cumulative.
The Jerusalem pillars, James, Cephas and John, expressly agreed with Paul that gentiles-in-Christ should not undergo proselyte circumcision. At Antioch the charge against Cephas is hypocrisy, believing the right thing and acting otherwise, and the dispute is about table-separation; the people from James urge circumcision on nobody. The phrase οἱ ἐκ περιτομῆς means those from among the Jews, and the common rendering "the circumcision party" imports a faction that is not in the Greek. Paul's one secure clash with Jewish apostles is 2 Corinthians 10 to 13, "are they Hebrews? so am I", and those opponents show no interest in circumcising anyone; Watson describes them as Jewish Christians uninterested in imposing circumcision on gentiles.
The positive case, which Novenson takes from Johannes Munck's coolly received 1954 argument, runs the other way. Galatians 6:13 says that those getting circumcised do not themselves keep the law, and the participle is present: these are people currently undergoing the procedure, not people born to it. In Philippians 3 Paul claims not equality with his rivals, as at Corinth, but superiority, boasting that he was circumcised on the eighth day, of the race of Israel, of the tribe of Benjamin; the boast only works if his rivals have not-eighth-day circumcision, which is to say proselyte circumcision. The abuse in the same passage, the dogs and the κατατομή, a coarse play on περιτομή, points the same way, as does the man of Romans 2:17 who calls himself a Jew.
The conclusion is that Paul's circumcision-preaching opponents were recently circumcised gentile proselytes, foreskins by nature rather than Jews by nature; the book's phrase is gentile poseurs. The overlap between Paul's Jewish opponents and his circumcision-preaching opponents is nil.
The second century corroborates it. Ignatius warns the Philadelphians that it is better to hear Χριστιανισμός from a man who has circumcision than Ἰουδαϊσμός from a foreskin, which makes the imagined preacher of Judaism an uncircumcised gentile. The Epistle of Barnabas frets that its own gentile audience might become proselytes. Meanwhile the texts that plausibly come from Jewish Christ-believers, Matthew, James, the Didache and Revelation, show no interest in circumcising gentile converts, which agrees with the verdict of non-Christian Jewish sources: Jews do not make proselytes, proselytes present themselves.
The objections
Sanders would praise the diagnosis and withhold the verdict. He spent a career removing one anachronism from the field, and would recognise that Novenson has found an older one beneath it; Baur's scheme was always more Hegel than history and was taught anyway for a century and a half. He would enter two disciplines. Somebody at Jerusalem did argue otherwise, since Paul mentions false brothers, and Jerusalem is a Jewish city; the chapter itself calls this a good chance but by no means certain, and the concession should stay visible. And, echoing Barclay's old objection to Munck, insults tell you what a man was angry about and are poor evidence for who his enemies were. He would find the thesis probable and decline to find it proved.
Fredriksen would find it overdue and supply a motive. Recent converts are the fiercest, and a man who has paid the knife's price for belonging will not watch the next man take the seat free. On her reading of ancient religion these proselytes are deserters of their ancestral gods, and the fullest available regimen of Torah would recommend itself as protection against powers they had just abandoned. Paul's anger is then the anger of a man who finds his converts paying tribute to beings the End has dethroned.
Hart would take the argument as a demonstration of how bad reception history can get. Two centuries of criticism deduced a conflict between Jewish and gentile Christianity from a scheme of German idealism, while the letters were using obscene language about men freshly under the bandage. He rendered those epithets at full strength in translation and was criticised for coarseness; the coarseness turns out to be load-bearing. He would add that the finding makes the tragedy worse rather than better, since a polemic of gentile against gentile was read for two millennia as a case against the Jews.
Paul's own account
The letters are unusually specific here, and the specificity is the argument.
Paul reports that the pillars gave him the right hand of fellowship, and that the dispute at Antioch concerned a table rather than a knife. He describes the men from James as having frightened Cephas into eating apart, which he calls hypocrisy, not a circumcising mission.
Where he curses, he identifies the hands. "Those who are getting circumcised do not even keep the law themselves": the tense is his, and it names converts fresh from the procedure who are urging on others what they have just undergone. The extremity of the language is in the text and has usually been softened in translation; Paul writes that he wishes the men unsettling the Galatians would go the whole way and castrate themselves.
What the letters do not contain is a curse against Israel. Where Paul raises the question directly he answers it with μὴ γένοιτο. The anger is aimed at recent proselytes, and the chapter's contribution is to notice that the grammar has been saying so all along.
The remaining disagreement
Nobody defends Baur. The disagreement is about what standard of proof a reconstruction of this kind should meet.
Sanders would hold that a thesis resting substantially on the tone of insults has reached probability and not proof, and that the false brothers remain unexplained. Fredriksen would hold that the reconstruction is confirmed by everything known about how ancient people changed gods. Hart would hold that the philology settles it and that the resistance is theological rather than evidential.
The chapter states its own confidence carefully, which is why Sanders's objection can be entered without damaging it.
The same chapter as a dialogue sets these positions in the participants' own voices, which is how this was first written.