Postulates and theorems are the building blocks for proof and deduction in any mathematical system, such as geometry, algebra, or trigonometry. By using postulates to prove theorems, which can then prove further theorems, mathematicians have built entire systems of mathematics.

Postulates, or axioms, are the most basic assumptions with which a reasonable person would agree. An example of an axiom is "parallel lines do not intersect." Postulates must be consistent, meaning that one may not contradict another. They are also independent, meaning not one of them can be proved by some combination of the others. There may also be a few undefined terms and definitions.

Postulates or axioms can then be used to prove propositions or statements, known as theorems. In doing so, mathematicians must strictly follow agreed-upon rules of argument known as the "logic" of the system. A theorem is not...