ax·i·om
noun
1.
a self-evident truth that requires no proof.
2.
a universally accepted principle or rule.
3.
Logic, Mathematics . a proposition that is assumed without proof for the sake of studying the consequences that follow from it.
I find it interesting that we as humans even has a word for "a self-evident truth that requires no proof."
The best teachers, for instance Jed McKenna, stress that we cannot be certain of anything. Everything is based on belief, and a belief that you can't change is a trap.
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3 comments:
Eolake said...
"I find it interesting that we as humans even has a word for "a self-evident truth that requires no proof.""
Funny: that's exactly what I thought when I was reading it! :-D Such an arrogant race we humans are! ;-)
Yes, that about says it.
Usually, a mathematical axiom is described as such because, while the proof exists, it would probably add fifty pages to even the most rudimentary paper just to go through the basics of, say, square numbers or complex numbers.
An axiom presented at degree or above level is one where the author is basically saying "Assume that all the crap you learned up to degree level is true, and for proof just go back through all the boring lecture notes you took as a kid, or go and look up the relevant pre-university book you once used as a text book."
Proofs you can accept as axioms include such simple things as "anything times zero equals zero" and "one plus one equals two." They have been proved solid since the dawn of mathematics in written history, and so no need exists to go over Euclid or Pythagoras when making the founding assumptions of your mathematical paper.
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