Purpose of the Law? (Gal 3:19-20)
Things are designed for certain purposes. When they are not used for their intended purposes, it may be harmful to the users. God gave us His Law for a purpose. However, the false teachers in the Galatian church wrongly preached that believing in Jesus was not the only way to be saved; and that the Law was an alternative means of salvation. Thus, Paul had to address the purpose of the Law.
Before God gave the Law, the children of God were not aware that the sins they committed against their fellowmen were also sins against God. For example, when someone sinned by stealing from his neighbour, he only needed to repay his neighbour to be reconciled with him. However, when the Ten Commandments were given, it taught the people that a transgression against their fellowmen is also a transgression against God; and there is nothing one can do on his own to be forgiven of his sin committed against God.
After the children of God left Egypt, the Law was given to them so that they can be a national witness for God. When the neighbouring nations observed Israel, they saw how different Israel was from them. Today, we too must live for Christ so that those around us can see the light of Christ in our lives. This is the reason why God leaves us on earth after we are born-again. We must not be afraid of disrupting the false peace and tranquility in our homes, workplaces, etc. When we dare not stand up for Christ, people around us will not be able to see Christ.
During the giving of the Ten Commandments at Mount Sinai, God wrote the Commandments and the angels gave them to Moses (Gal 3:19c), Moses being the mediator between two parties: God and the people of Israel. Priests too are mediators. They accept animal sacrifices on behalf of the people, and present the sacrifices to God. While Moses was the mediator between God and Israel only, Jesus is the Mediator between God and the whole world. The Law given to Moses was never meant to be an alternative way to salvation. As promised by God, salvation is by faith in Jesus. We must believe and receive Him as our Lord and Saviour in order to be saved. The Law cannot save.