Hilbert's axioms

Hilbert's axioms are a set of 20 basic ideas that help us understand Euclidean geometry, the kind of geometry most people learn in school. They were proposed by a famous mathematician named David Hilbert in 1899. Hilbert wrote about these ideas in his book called The Foundations of Geometry.

These axioms give a clear and modern way to build up all the rules of geometry from simple starting points. Before Hilbert, people used the ideas of the ancient Greek mathematician Euclid, but his work had some gaps that Hilbert wanted to fix.

Other mathematicians, like Alfred Tarski and George Birkhoff, also created their own sets of axioms for Euclidean geometry later on. Hilbert's work is still very important for learning geometry and for advanced mathematical studies.

The axioms

Hilbert's axioms are a set of rules that help us understand basic geometry. David Hilbert created these rules in 1899 to make geometry clearer and easier to study. They use simple ideas like points, lines, and planes to explain how shapes work.

The axioms start with six basic ideas:

• Point: A spot in space, like a dot you might draw on paper.
• Line: An imaginary straight path that goes on forever in both directions.
• Plane: A flat surface that stretches out in all directions.

These ideas connect in three important ways:

• Incidence: This tells us when a point is on a line or a plane.
• Betweenness: This helps us understand the order of points on a line — which points come between others.
• Congruence: This lets us compare lengths and angles to see if they are the same.

These rules help build the foundation for studying shapes, sizes, and spaces in geometry. Other famous sets of rules for geometry were also created by mathematicians like Alfred Tarski and George Birkhoff.

Hilbert's discarded axiom

David Hilbert had an extra rule in his geometry work called Pasch's theorem. It helped explain how points line up. Later, mathematicians E. H. Moore and R. L. Moore found that this rule wasn't needed. The other rules already covered the same ideas. So, Pasch's axiom was moved to a different spot in the list.

Main article: Pasch's theorem

Editions and translations of Grundlagen der Geometrie

David Hilbert wrote Grundlagen der Geometrie in 1899 for a special talk. The book was quickly translated into French and English. The French version added a rule called the Completeness Axiom. In 1902, an English version by E.J. Townsend included these changes and more.

Over the years, many new editions of the book were published in German. The seventh edition was the last one Hilbert himself saw. Later editions had updates mostly in extra sections at the end. A new English version was made in 1971 by Leo Unger. This version was based on a later German edition and included more changes suggested by Paul Bernays. These updates renamed some rules and changed how the axioms were organized.

Application

Hilbert's axioms help us describe Euclidean geometry, the study of shapes and spaces. By changing some rules, we can look only at flat, two-dimensional geometry.

These axioms changed modern mathematics. They showed new ways to think about rules and proofs. They inspired other mathematicians to create their own systems for understanding geometry. Later, when computers were used to check Hilbert's work, some of his proofs needed clearer explanations.

Main articles: Euclidean plane geometry, Tarski's axioms, first-order logic, metamathematical, formal systems