In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. Euclid, detail from The School of Athens by Raphael. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ...

Every mathematical theory is based on a set of axioms. Usually these axioms are not mentioned when a mathematical equation is presented. Mathematicians know from their education on which axioms mathematical theories are based. Indeed, mathematical theories usually are based on very few axioms. Some of them are mentioned in the example below. An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ...

Example: The axiomatization of natural numbers

The mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system that was first written down by the mathematician Peano in 1901. He defined the axioms (see Peano axioms) for the set N of natural numbers as being: A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... 1901 (MCMI) was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ...

• There is a natural number 0.
• Every natural number a has a successor, denoted by a + 1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if a ≠ b, then a + 1 ≠ b + 1.
• If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but the truth is that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that is merely a limitation on the purposes that deductive logic serves.its crazy right.