The Judaizers of Paul's day wanted the Gentiles (the "outsiders") to strip themselves of their cultural and ethnic backgrounds and fully adopt the customs and practices of the "insiders" in order to be fully received in the fold of God.
The Judaizers wanted the lines to be distinct. Gentile ways of living, thinking, speaking were to be abandoned and replaced with the "right way" of living, thinking, and speaking. Taking the yoke of the Law of Moses and its customs involved "insider" and "outsider" language.
This is no different than the tactics of theological policing that goes on in conservative Reformed and Evangelical circles today, where the "insiders" - the "Truly Reformed," the "orthodox," the "conservatives" - only use their accepted terms, expressions, and slogans.
The Judaizing tendency lives on in the church, if we have eyes to see it. Like in Paul's day, the gospel - and walking in step with it - tears it down.
