Disagreements, misunderstandings, variations of viewpoints, inconsistencies, etc. are abound. You are right in that mathematics has unusually broad applications in which it is remarkably successful. The breadth of applicability comes from a high degree of abstraction. But abstraction does not mean lack of intuition. I can't explain this in a short reply.
Concrete knowledge comes from schemata with the addition of specific genes.
Alexander Bogomolny 2789868 .