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Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.

Category Theory is at the highest level of abstraction; then comes Set Theory which can be used to talk about the system in little more detail?

Could you please demonstrate Category Theory and Set Theory with an example?

My attempt to know the System in Math that lets me communicate with other Mathematicians.

Example: There is a book. I can study the information that it has. To describe what the book is about I will use Category Theory and to see what words are on what pages I would use Set Theory?

In short, I want to know the System in Mathematics that lets me communicate my ideas with other mathematicians.

Thanks.