What are the “elementary principles of the world” in Galatians 4:3?

First, you need to put the statement “elementary principles of the world” in its context. The general context is the transition from the Old Testament order to the new order in Christ, which is the new covenant. Paul is looking back to his past life as a Jewish person and thinking about how he was under the law of God. In his case, he was under the law of God but not resting in the fulfillment of the promise of God. Because that was so, the principles of the law of God brought condemnation into his life.

There is a sense in which one could view each of the commandments as a basic principle of life. Adam and Eve were created to follow these principles. We as Christians are created to follow these principles. However, in each of these areas, instead of the commandments being energized by faith in the promise, or for us as Christians being energized by faith in the living Christ, all we’ve got is these commandments about the basic elements of life as it should be lived. Instead of living in the free Spirit of the power of Jesus Christ that enables us to live in accordance with God’s commandments, we are actually in bondage to them.

That’s by no means a universal view, but I think it makes sense of the fact that Paul is speaking within the context of the flow of redemptive history. He’s speaking about things that are absolutely basic to life, and he understands that those are the very issues that the law addresses.

When Paul becomes a Christian, instead of being in bondage with respect to these aspects that the law addresses, he’s actually empowered by the Spirit of Christ to live in the freedom of them. Without Christ, those duties actually create bondage, whereas in Christ those duties become our delight. In living this way, empowered by the Holy Spirit, we’re living in a way that a) we know is how God created us to live, and b) is pleasing to our heavenly Father, and that is delight and joy.