What is an Axiom?

The definition of an axiom according to the dictionary is, “a statement or proposition that is regarded as being established, accepted, or self-evidently true.”  In mathematics, we add the stipulation that is the basis for an argument.  This means that we can build any argument we want as long as we start with the most basic axioms.

This is what Euclid did when he discovered Euclidean geometry.  He started with five axioms, and built an entire geometry dependent on them.  These axioms are so fragile that if one was shown to be inherently false, then the entire geometry would fall apart and work spanning over a hundred years would be meaningless.  This is because they all rely on the same basic “self-evident” truths.

Axioms are also seen in other parts of mathematics.  They are the building blocks of Algebra, Analysis, and Topology.  Without a basis for each of these mathematics, we as humans could not further these fields.  Without the axiomatic definitions of algebraic structures, we could not classify different sets into types of groups or rings.  These are universally given simply because we have decided that they are true.

Overall, axioms are little more than a way for humans to have a launching point to further our understanding of mathematical topics.  We have established them to be true simply to have almost a safety net of ideas that we can always fall back on.

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