Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἁξίωμα, meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

In different areas of study, the exact definition of an axiom can change. In classical philosophy, an axiom is something so evident or well-known that people accept it without argument. In modern logic, an axiom is simply a starting point for reasoning.

In mathematics, an axiom can be a "logical axiom" or a "non-logical axiom". Logical axioms are always true within the rules of the logic system they use. Non-logical axioms are special statements about the topics of a mathematical theory, like saying a + 0 = a when talking about whole numbers. These can also be called "postulates," "assumptions," or "proper axioms." Creating a system of knowledge using axioms means showing that all its ideas come from a small, clear group of starting points.

Etymology

The word axiom comes from the Ancient Greek word ἀξίωμα (axíōma), which means "to deem worthy" or "to require." Ancient Greek philosophers and mathematicians used axioms as statements that were obviously true and did not need proof. They were like the basic ideas that many areas of study shared.

The word postulate means to "demand." For example, the mathematician Euclid asked people to accept certain ideas, like the idea that any two points can be connected by a straight line. Some ancient mathematicians, like Proclus, thought there was a difference between axioms and postulates, but others, like Boethius, used the words in different ways.

Historical development

The ancient Greeks developed a method where conclusions follow logically from basic assumptions, which is the foundation of modern mathematics. They believed that certain basic ideas, called axioms and postulates, were self-evident truths that didn’t need proof. For example, they thought it was obvious that equal things subtracted from equals would leave equals.

Euclid, a Greek mathematician, listed several postulates and common notions in his work Elements. These included ideas like being able to draw a straight line between any two points, and that all right angles are equal. Over time, mathematicians began to treat these basic ideas more abstractly, separating them from specific examples. This allowed math to become more general and useful in many different areas. Today, axioms are seen as rules that define a system, and mathematicians study what follows from these rules without needing to know what the rules actually mean.

Mathematical logic

In mathematical logic, axioms are statements that are accepted as true without proof. They serve as the foundation for building more complex ideas and theories. There are two main types of axioms: logical and non-logical.

Logical axioms are rules that apply to all areas of mathematics. For example, in propositional logic, one logical axiom states that if a statement implies another statement, and that second statement implies the first, then both statements must be true. These axioms help build the rules for how we can reason and prove things in mathematics.

Non-logical axioms are specific to particular areas of mathematics. For example, the Peano axioms describe the basic properties of numbers, and Euclid’s postulates describe the rules of geometry. These axioms help define the unique characteristics of different mathematical systems.