I've heard often that the number 7, when seen in scripture, mean completeness or fullness in some way.
I'm aware that the number 7 exists many times in scripture, but nowhere does it state or necessarily imply that it's meant to represent completeness. Where did this idea of completeness or fullness come from?
It comes from the Shabbat, the seventh day of creation. The creation of the universe was completed by the creation of the Shabbat. The day that Hashem "stopped" the creation. That is also why the eighth day (the day of bris milah) symbolizes going over and above the universe of space-time.
Rav Hirsch among other meforshim says that this is the basic connection to the concept of the completion of the natural world. The reference to completion mean that hashem explicitly stopped creation, but that everything that happened including the stopping for Shabbat was part of creation. Also see the comments on the actions of man eating the Aitz Hadaas and how this too was "part of creation" and helped create the nature of the world.
As an example The importance of the eighth day
The world was created in seven days. Beyond the physical world - beyond seven days - is the eighth day, which represents the metaphysical, the spiritual. Since bris symbolizes taking the physical and elevatiing it to serve G-d, bris is done on the eighth day, a day that represents a level beyond physicality. Similarly, another explanation is that the number eight is composed of seven and one. Seven represents the world, which was created in seven days. One represents the one G-d who created this world. Eight, therefore symbolizes G-d's absolute sovereignty over His world, and our service to Him. It is befitting then that the bris milah a commandment which reminds us that life's purpose is to use the physical gifts that G-d gave us to serve Him, is performed on the eighth day.
