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.
Example: The axiomatization of natural numbers
The mathematical system of natural numbers 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:
- 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.