The History of Plane Axioms

The history of plane axioms dates back to ancient Greek mathematics and philosophy. Particularly, Euclidean geometry, named after the ancient Greek mathematician Euclid (circa 300 BCE), is a discipline that studies plane and solid figures based on axioms and theorems. Euclid’s work, “Elements,” laid the foundation for the axiom system of classical geometry.

For over two thousand years, Euclidean geometry was the only known geometric system and was believed to describe the actual state of the physical world. This belief was based on the axiom system proposed by Euclid, which started from a series of undefined terms like “point” and “line,” along with five axioms that seemed self-evident. These axioms were considered the cornerstone from which all other properties of geometry could be formally deduced using logic.

The first four axioms are direct and simple, including statements such as a straight line can be drawn between any two points, and all right angles are equal. However, the fifth axiom, the so-called parallel postulate, is much more complex. It states that if a line crosses two lines so that the sum of the interior angles on one side is less than two right angles, then these two lines, if extended indefinitely, will meet on that side. Due to its complexity, this axiom seems more like a theorem and is not as intuitively obvious as the others.

For centuries, mathematicians attempted to derive the fifth axiom from the first four but were never successful. It was not until the 19th century that it was proven that the fifth axiom is not derived from the first four, leading to the development of non-Euclidean geometry.

The importance of Euclid’s axiomatic method lies in its elimination of the need to remember a long list of geometric facts. Instead, one only needs to understand a small set of axioms and apply reasoning rules to deduce the entire set of geometric truths.

Although early geometry attempted to describe the real world, it was also considered a kind of idealized abstraction that could only be fully realized in thought. Despite terms like “point” and “line” being undefined, the ancient Greeks believed these to be real objects and trusted that their developed system was an increasingly accurate description of the real world.

Euclid and the Greeks’ use of the axiomatic method not only laid the foundation for geometry but also for the development of logic and deductive reasoning in mathematics. This historical journey demonstrates the evolution of mathematical thought and the pursuit of understanding the basic structure of space and shapes that make up our world.