Unlike the false teachers mentioned in 6:12–13, Paul wanted to increase God's glory. He did not want to increase his own reputation. Boasting about following the law would take away from recognizing the cross of our Lord Jesus Christ.
In the Greco-Roman world, the cross seemed foolish to non-believers (5:11; 1 Corinthians 1:23; compare John 6:53–61). But it is the foundation of a Christian's faith and hope (1 Corinthians 1:17–18; 15:3; Philippians 2:8–11; Colossians 1:20–22; 2:14–15).