ABSTRACT
The shortest distance between two points is a straight line. Two parallel lines will tend to stay parallel. The internal angles of a triangle add up to 180 degrees. Such statements belong to the apparently secure domain of high school mathematics. They have as their basis the axioms of Euclidean geometry. The more useful, but complex theorems of Euclidean geometry are typically deductions from the basic definitions, axioms, and postulates of the system. 1 Why do we accept these basic ideas: because they are self-evidently true, because they are inherently rational, or because they are intuitively correct? All these reasons have indeed been advanced. But, instead of committing ourselves so wholeheartedly to the building-blocks of Euclidean geometry as somehow necessarily true, we might just accept them because they provide us with an internally consistent system which seems to underwrite so much that we say about the physical world. Then, if we are faced with a choice between geometries, we can make decisions without feeling the need to justify our choice from first principles.
