Saturday, May 10, 2014

The Nature of Mathematics

What is an axiom? 

 According to the Merriam-Webster online dictionary, an axiom is,
In mathematics terms, an axiom is typically a proposition or rule that does not require a proof for supporting evidence of its truth. Axioms can take different forms and can also be used in other areas of study other than mathematics such as law, the sciences, etc..

What is the role of axioms in mathematics?

Axioms in mathematics play a large role since mathematics focuses on proving conjectures. These conjectures are proved in a similar way of building blocks: you must lay a foundation and build upon those blocks. These foundations typically contain axioms. When considering axioms as building blocks, these do not need to be proven every use but simply recalled when used. It reminds me of the idea of infinity and how if we had to prove each axiom with its use would also then call for a proof of the axiom's axiom, and this could go on for an unfortunate long period of time. This causes too much unnecessary work. Thus, axioms are very convenient for mathematicians.

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