While these “false brothers” are not named, Cephas (Simon Peter) and Barnabas are specifically mentioned for their conduct that weakened the message of the gospel and in turn would mean that person’s faith would be built on a wrong foundation.
One thing we learn from Paul in this is that he used “grace and peace from God our Father and the Lord Jesus Christ” (1:3) in addressing such a difficult issue. If we come to difficult issues like this from the resources of God’s grace and are “eager to maintain the unity of the Spirit in the bond of peace” (Ephesians 4:3), confronting will not be a tearing experience, but rather one of building up. Paul also knew when issues were not those that required his insistence so as not to “stumble a brother” but “pursue what makes for peace and for mutual upbuilding” (Romans 14:13-23).
Be prepared to confront issues that affect the pure gospel. The glory of God in Christ is at stake!