Modeling: A physical problem is usually reduced to a simplfied problem, which can then be addressed through mathematics. For example, a field is replaced by a rectangle or a trapezoid. The areas may not be exactly the same but may be close enough to answer a question.

Abstraction: Physical quantities are replaced by variables. A particular problem is presented as an instance of a more general problem.

Conjectures: On the basis of specific results, a more general result is postulated. This is often tied to inductive reasoning.

Hypotheses: In the event that a particular result can not be logically deduced, the problem is constrained (or restricted) by adding further qualifying assumptions, called hypotheses.

Proofs: Using deductive logic (If P then Q, and the like) the Hypotheses are shown to imply the desired result. Note, finding a counter-example nullifies the proof, and implies different or additional hypotheses.

Modern proofs rely heavily on typical Hypotheses, Lemma, Proof, Corrollary structure.

Generalizations: Once one has made a rigorous proof of a statement, one can try to extend the statement to be of greater applicability.