1. Foundations and Finite Geometries¶

Axiomatic Systems¶

Consider a proof of a theorem in mathematics, say, the Pythagorean Theorem. Mathematicians cannot prove this theorem is true everywhere, all the time. Mathematicians can only operate within a given set of ground rules. Primitive ideas must be assumed to be true, not proven. More advanced theorems are built upon the basics, but different sets of primitive ideas are possible.

Formal axiomatics take a subject and analyze the best set of primitive ideas and convert them into undefined terms, defined terms and axioms. Upon this foundation, every single result, theorem and lemma must rely. Each must be proven in turn starting from the most basic to the most advanced. Geometry is typically studied as an axiomatic system, but we must be very careful how we choose. Tiny changes in one axiom can completely alter the structure of the system and what theorems can be proven within it.

Formal Axiomatics¶

An axiomatic system must have four distinct components.

Undefined Terms: Set of technical terms that are deliberately chosen as undefined terms and are subject to the interpretation of the reader.
Defined Terms: All other technical terms of the system are defined terms the definitions of which rely upon the undefined terms.
Axioms: The axioms of the system are a set of statements dealing with the defined and undefined terms and will remain unproven.

Postulate

A synonym for axiom that is often used in its place. Euclid, for example, stated postulates.

Theorems: All other statements must be logical consequences of the axioms. These derived statements are called the theorems of the axiomatic system.

An axiomatic system is consistent provided any theorem which can be proven using the system can never logically contradict any of the axioms or previously proved theorems. An idividual axiom is independent if it cannot be proven by use of the other axioms. An axiomatric system is independent provided each of its axioms is independent. For more than two thousand years, mathematicians wondered if Euclid’s Parallel Postulate was actually a theorem we could prove using his other postulates. It’s not, but the search for the proof of it’s dependence or independence led to the discovery of many important theorems in Euclidean geometry and, eventually, to the creation of non-Euclidean geometries.

Mathematicians generally prefer independent sets of axioms although independent systems can be tedious to work with because every result – many of which seem to be trivially true – must be proven individually. This labor-intensive process can be shortened by taking a few essential theorems as axioms. Though the system will be dependent, students of mathematics will be able to advance quickly to the proof of non-trivial results.

An axiom set must also be complete. We must have enough axioms that every theorem that can possibly be stated using the terms and axioms of the system can be proven either true or false. Proving the completeness of an axiomatic system is quite difficult and beyond the scope of this course. However, you can see the importance of this idea. We would like for our notions of geometry to all be proveable, so our set of axioms must be robust enough to establish results that match the physcial properties we need for 2D and 3D geometry. Physics relies upon geometry. Einstein said his theory of relativity would have not have possible were he not familiar with hyperbolic geometry.