Mathematical subjects encompass topics like numbers (arithmetic and number theory), formulae and associated structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (from the Ancient Greek “knowing, study, learning”) (calculus and analysis). The majority of mathematical activity involves using pure reason to determine or support the characteristics of abstract objects, which can be either natural abstractions or, in modern mathematics, things that are predetermined to have specific attributes, or axioms. A mathematical proof is a series of applications of certain deductive rules to already established facts, such as theorems that have already been established, axioms, and (in the case of abstraction from nature) some fundamental properties that are regarded as the real foundations of the theory in question.