The Axiomatic System: Definition & Examples
The Axiomatic system (Definition, Properties, & Examples)
Though geometry was discovered and created around the globe by different civilizations, the Greek mathematician Euclid is credited with developing a system of basic truths, or axioms, from which all other Greek geometry (most our modern geometry) springs.
What is an axiom?
An axiom is a basic statement assumed to be true and requiring no proof of its truthfulness. It is a fundamental underpinning for a set of logical statements. Not everything counts as an axiom. It must be simple, make a useful statement about an undefined term, evidently true with a minimum of thought, and contribute to an axiomatic system (not be a random construct).
The axiomatic system
An axiomatic system is a collection of axioms, or statements about undefined terms. You can build proofs and theorems from axioms. Logical arguments are built from with axioms.
You can create your own artificial axiomatic system, such as this one:
Every robot has at least two paths
Every path has at least two robots
A minimum of one robot exists
This might describe a routine for a computer to control activity in a warehouse, but it is also a set of axioms. We have two undefined terms, "robot" and "path." We have not defined "robot" or "path," but we can build on those undefined terms to construct various proofs. Let's prove a path exists:
By the third axiom, a robot exists.
By the first axiom, the existing robot must have at least one path.
Therefore, at least one path for a robot exists.
Such an axiomatic system is limited, but it would be enough to build a network of robots to work in a warehouse. Euclid, the ancient Greek mathematician, created an axiomatic system with five axioms. From that basic foundation we derive most of our geometry (and all Euclidean geometry).
Euclid's five Axioms
Euclid (his name means "renowned," or "glorious") was born circa (around) 325 BCE and died 265 BCE. He is the Father of Geometry for formulating these five axioms that, together, form an axiomatic system of geometry:
A straight line may be drawn between any two points.
Any terminated straight line may be extended indefinitely.
A circle may be drawn with any given point as center and any given radius.
All right angles are equal.
If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.
Mathematicians have, for centuries, accepted the first four axioms and built great achievements on them. The fifth axiom has provoked a lot of controversy over those same centuries.
A different translation or wording produced this alternative:
For any given point not on a given line, there is exactly one line through the point that does not meet the given line.
That is the "parallel postulate," but it is also a recasting of the fifth axiom. The reason for the controversy about the fifth axiom is that axiomatic systems usually fulfill three conditions, or have three properties.
Three properties of axiomatic systems
For an axiomatic system to be valid, from our robot paths to Euclid, the system must have only one property: consistency.
An axiomatic system is stronger for also having independence and completeness. Let's look at each quality in turn.
Consistency
An axiomatic system is consistent if the axioms cannot be used to prove a particular proposition and its opposite, or negation. It cannot contradict itself. In our simple example, the three axioms could not be used to prove that some paths have no robots while also proving that all paths have some robots.
Independence
An axiomatic system must have consistency (an internal logic that is not self-contradictory). It is better if it also has independence, in which axioms are independent of each other; you cannot get one axiom from another.
All axioms are fundamental truths that do not rely on each other for their existence. They may refer to undefined terms, but they do not stem one from the other.
Completeness
The third important quality, but not a requirement of an axiomatic system, is completeness. Whatever we attempt to test with the system will either be proven or its negative will be proven.
Mathematicians have argued for centuries that Euclid's fifth axiom is really a theorem, but others counter that the other four axioms cannot be used to prove it. Without the fifth axiom, Euclid's axiomatic system lacks completeness.
Your world
Axioms may seem a little removed from your everyday life. Rather than pointing to some commonplace object and saying, "That shows an axiom," consider that the shaping of your mental processes - the way you think - depends on axioms. To do well in geometry, you learn to think logically, building proofs from axioms.
When you branch out into other mathematics, like non-Euclidean geometry, different axioms produce different results, like allowing parallel lines to meet. Axiomatic systems like those are useful for ideas like geosynchronous orbits for satellites, radio communications, and land surveying.