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The controversy Paul addressed in his letter to the Galatians was not a debate over whether or not true Christians obey God's commandments. Rather, it was a debate over whether or not Gentiles must be circumcised and keep the Law of Moses in order to be saved. The clear answer is that they don't, and for a number of good reasons, many of which Paul mentions in this chapter.
Keep in mind that God has given three major "sets of rules" in human history. The first is the "law of the conscience," given to every person since Adam. That set of rules existed thousands of years before the Law of Moses, and God has always expected everyone to obey those laws, which is why He has wired every human in history with a conscience (see Rom. 2:14-16).
Second, God gave the Law of Moses to the descendants of Israel. It was given only to them, and it included many laws that, unlike the law of conscience, were not moral in nature, such as laws about feast days, a priesthood, animal sacrifices and so on. The Law of Moses was only intended to be temporary, lasting until the time of the new covenant (see Jer. 31:31-34; Heb. 7:12; 8:6-13). It included many moral laws that were also part of the "law of conscience."
Third, God gave the Church the "law of Christ," which includes everything Christ commanded. The law of Christ includes all the moral laws that were also part of the "law of conscience." All Jewish and Gentile believers in Jesus are obligated to follow the law of Christ, and they do strive to follow it, by virtue of the fact that they are born again. None, including Jewish believers, are obligated to follow the Law of Moses any longer (see 1 Cor. 9:19-21). It took quite a while, however, for early Jewish Christians to accept this fact. As we've seen, some even thought that Gentiles should be obeying the Law of Moses.
In today's reading, Paul succinctly lists many reasons why Gentiles don't need to be circumcised and keep the Law of Moses to be saved. The foremost is, of course, that God forgives and regenerates them when they believe in Jesus. God gives His Spirit to those who believe, not first requiring their circumcision and Mosaic law-keeping (3:2-5).
Salvation has always been received through faith, as proven in the Old Testament (see 3:11), and by Scripture's testimony regarding Abraham, who "believed God, and it was reckoned to him as righteousness" (3:6). Abraham's true descendants follow his example of faith (3:7), and they include believing Gentiles, as God had originally promised Abraham (3:8-9).
In fact, when God promised Abraham that all the nations would be blessed through his seed, that is, through Christ, it was a promise made 430 years before the Law of Moses was given. Paul argues that, even in a covenant made between men, the conditions can't be altered after the agreement is made. Surely when God gave the Law of Moses, He wasn't adding conditions to the covenant He made with Abraham 430 years earlier to bless Gentiles through his seed! So the Law of Moses has absolutely nothing to do with Gentiles receiving the blessing that God promised them through Jesus, Abraham's singular seed.
Another reason salvation can't be gained by keeping the Law of Moses is because that Law promises a curse on any who don't keep it perfectly, something no one has ever done (with the exception of Jesus).
So what purpose did the Law of Moses ultimately serve? It helped Jews, who because of God's great dealings with them might have been tempted to think they had salvation "in the bag," to realize their need to be saved, and saved by some means other than the Law. So the Law became, for Jews, a tutor to lead them to Christ (3:24). Paul clearly states that believing Jews are no longer under that tutor (3:26).
So Jews and Gentiles are in the same boat. All can be saved only through faith. When they do believe, they are born again to be sons of God, "clothed in Christ" (3:27), now acting like Him!