An axiom is defined as "a proposition that is assumed without proof for the sake of studying the consequences that follow from it." Axioms are self-evident truths or universally accepted principles or rules that require no proof. Axioms are used as a starting point by mathematicians and other scholars for proving propositions.
What is the role of axioms in mathematics?
"Mathematics is the science of proving theorems, and a theorem is a statement that, given the premises laid down by the axioms and certain agreed-upon rules of inference, is apodictically true." The mathematical community has agreed upon a set of axioms used for doing mathematics. Axioms cannot be proven absolutely true but in turn, are used as a premise or starting point in mathematics. Most proofs are built on axioms because mathematicians need some basic knowledge when proving conjectures. It would be extremely tedious and time consuming to prove propositions if mathematicians had to prove everything from the basics upward every time.
As can be seen, axioms play a huge role in mathematics. Axioms allow mathematicians to be able to progress in the mathematical community without having to prove the basics every time a conjecture is to be proven.
References
http://www.mrc.uidaho.edu/~rwells/Critical%20Philosophy%20and%20Mind/Chapter%2023.pdf
http://scienceblogs.com/goodmath/2007/03/07/basics-axioms/
clear, coherent: +
ReplyDeletecomplete, content: good start, just need more depth. Examples, your experience, historical development... your pick.
consolidated: once you've written more, figure out a way to tie it up. One framework to use to summarize is answer one or more of: what? (important bits) so what? (why important) or now what? (what's next).