For over two millennia, the Elements of Euclid served as the template for mathematics. The ideas of Euclidean geometry were considered to be more than mere axioms and logical constructions from the mind of man, in fact they were taken to be reflections of truth and perfection; Ideals that exist beyond the imperfections of space and time.

During the nineteenth century, when one of the five axioms, the "parallel lines" axiom of Euclidean geometry was replaced by another contradictory axiom, many believed that the resulting logical system must be inconsistent, but instead of nonsense, mathematicians watched as a new noneuclidean geometry emerged! With this noneuclidean set of axioms, the geometry of Relativity was born. After the shock of this event, mathematicians began to consider all sorts of new mathematical structures. I call this the game of mathematics.

For the Third Abstraction, Vector Spaces, we put together some definitions and axioms and then look for mathematical sets that satisfy these conditions. On this foundation we continue to build our Linear Algebra.