characteristics of formal methods in discrete mathematics or mathematics?

Answer 1

For a carefully structured, coherent, and comprehensive course of discrete mathematics, the approach is traditional, deductive, and straightforward, with no unnecessary abstraction. It is self-contained including all the fundamental ideas in the field. It can be approached by anyone with basic competence in arithmetic and experience of simple algebraic manipulations. Students of computer science whose curriculum may not allow the study of many ancillary mathematics courses will find it particularly useful. Mathematics students seeking a first approach to courses such as graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra will also enjoy a clear introduction to these more specialized fields. You will want to present descriptions of numerous algorithms on a form close to that of a real programming language. To enable students to develop practical programs from the design of algorithms. Students of mathematics and computer science seeking an eloquent introduction to discrete mathematics should see the the source for further information.

Answer 2

I would say the Axiomatic Method, the concept of "proof", adherence to logic.

However, I really don't know if any of these methods is really what mathematicians "do" when doing mathematics, or just how they present the results.
I suggest books dealing with Mathematical Logic or Foundations.

I find it just as difficult to describe doing math as an artist might describe doing artwork. For me math is a human social activity, with certain rules, some explicit and many implicit. Even though I have learned much math from books, those books were still written by mathematicians, and the product of a culture with certain shared values, I don't know if these can ever be totally described.