- Every concept, such as a theorem, is matter in Platonic realm for being able to exist.

Only in math can you 100% prove something. => The concept has to materialize itself in our world, because the theorem is perfect not only in the Platonic realm, but also in our world. This means that it is either a perfect copy of the concept in Platonic realm, or it brings the concept from that realm. => If latter, the concept may still have a tendency to stay in the Platonic realm.

Then, there must be some conditions when a concept materializes itself in our world. It could be that it only materializes itself when somebody is thinking it clearly enough, aka. casting a spell by going the theorem in mind or in word. People think a bit differently when they speak (it even has a difference when sitting or standing). That might explain why the mages speak when casting.