Axiom Concept – In logic, in philosophy, in mathematics and examples

We explain what an axiom is in logical thinking, philosophy and mathematics. Also, examples of famous axioms.

The axioms serve as the foundation for a body of deductive theories and reasoning.

What is an axiom?

In the world of knowledge and knowledge, it is known as the axiom to any proposition or premise that is considered self-evident, that is to say, obvious, easily demonstrable, and that serves as the foundation for a body of deductive theories and reasoning.

Axioms they are general and basic rules of logical thinking, which exist in a myriad of disciplines, scientific or not, and which are distinguished from the postulates by the fact that they do not require a demonstration (being evident) and therefore must be simply accepted. It is possible to compare them with a seed: in them is condensed what is necessary for a whole theoretical tree of thought to sprout.

The word axiom comes from the Greek axiom (“Authority“), in turn derived from the noun axios (“Meritorious” or “adequate”), as it was already used by the classical philosopher Aristotle (384-322 BC): “everything that is assumed as the basis of a demonstration, a principle in itself evident”.

In fact, it was the great Greek mathematicians who bequeathed history a very small set of simple axioms, obtained after the logical reduction of various mathematical theorems and problems.

Axioms are very common in formal disciplines, such as logic or mathematicsBut it is also possible to find them in very different disciplines, although the term is often used in a metaphorical sense, to say that an idea is fundamental or indispensable.

Examples of axiom

Some examples of axioms are as follows:

  • The elements of Euclid, formulated by this Greek mathematician and geometrist (ca. 325-265 BC) in the 4th century BC. C., are composed of a set of “common notions”, which we can perfectly qualify as axioms.
  • The axiom of choice, formulated in 1904 by the German mathematician Ernst Zermelo (1871-1953), establishes that every set can be well ordered, that is, that for each family of non-empty sets that exists, there is also another set that contains one element of each of yours.
  • The axioms of human communication, formulated by the Austrian theorist Paul Watzlawick (1921-2007), establish the five fundamental and self-evident principles according to which all forms of communication between human beings occur.