Axiomatization

A mathematical axiom is a "first principle", a basic assumption, which a mathematical theory is based on.

Each and every mathematical theory is based on a set of axioms. Usually these axioms are not mentioned when a mathematical equation is presented. Mathematician know from their education on which axioms mathematical theories are based on. Indeed, mathematical theory is based on very few axioms. Some of them are mentioned in the example below.

Example: The mathematical system of natural numbers 1, 2, 3, 4, ... is based on an axiomatic set that was first written down by the mathematician Peano. He defined the axioms for the set N of natural numbers as being:

1. There is a element in N we will call '1' (pronounced 'one'). 2. There is an operation we will call '+' (pronounced 'plus').

  This operation associates to any two elements x,y in N a new element z in N. In such a case we write z=x+y for short.

3. For each element x in N there is an element y in N such that x+1=y. 4. For any elements x,y in N we have x+y=y+x. 5. For any elements x,y,z in N we have (x+y)+z=x+(y+z), where the content of the brackets are processed first.

Note that these axioms do not mention the word 'number' at all. The set of natural numbers with the '+' operation only comes into existence by applying the underlying axioms.

It is perfectly alright if you invent your own axioms and base your own mathematical theory on them. Be aware, however, that such an arbitrary axiomatic set will not necessarily by free of contradicions. You might deduct two statements from your set of axioms which contradict each other! Such contradictory axiomatic systems are usually not dealt with in mathematics, because the logic system we base all consideration on is usually one that says "either a statement is true or it is false, but not both at the same time".

Some scientists, however, investigate in other types of logic which allow such contradictions.

General Background:

In order to understand the importance of axioms it is important to understand that Mathematics is not immediately related to real things. Mathematics is a game where you set up arbitrary rules on an arbitrary playing board and combine them in a 'mindless' and playful way (any computer can do this, at least in priciple).

Only when mathematical results are applied to reality (or 'mapped' onto the physical reality, as a mathematician might say) is the time when Mathematics (then called 'Applied Mathematics') comes in contact with the 'real' world. This mapping is always an approximation in many ways. From this it is clear that Mathematics needs a foundation that is abstract. This is where the idea of axioms comes in.