One thing that proves Mesopotamian society was advanced was they made the first system of writing, which was called cuneiform. This language was used to make the first laws that were written in stone (Hammurabi's code). Written records were important to them. They wanted men to learn to read and write. They also made the first story, which was called the Epic Tale of Gilgamesh.
Another aspect of Mesopotamian life that proved they were advanced was the invention of the wheel. The wheel was used to make plows, chariots, wagons, clocks, and spinning wheels.
The third thing that proves Mesopotamia was advanced is they made irrigation systems. Irrigation systems make sure your crops and livestock don't die in a flood.