Mathematics is a system for building abstract statements that can be mapped to reality. The axioms of a mathematical (or other axiomatic) model define the conditions that a system (such as a pair of apples in the real universe) must satisfy in order for the abstract model to be applicable as well as providing a schema for mapping the abstract model to the concrete system.

There are other kinds of abstractions we could meaningfully talk about and they need not be defined as precisely as an axiomatic model like mathematics. An abstract model could be defined as a relationship between abstract ideas that can be mapped to a concrete system by pinning down... (read 458 more words →)

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