We often hear the words conjecture, theorem, postulate, axiom, corollary, and lemma in the field of Mathematics. And not one of them is synonymous to the other – at least to me.
Mathematicians are usually keen to patterns. Mathematical results, just like in any other hard science, are derived from observations. If, for example, you have spotted a pattern that is seemingly present in all the things you are observing, you will CONJECTURE its being true. Through a rigorous logical and mathematical argumentation from your set of AXIOMS (propositions regarded as self-evidently true), you will prove that your conjecture is true. And when proven so, it will be a THEOREM. When another idea is an immediate consequence of your theorem, then it is COROLLARY. A corollary, hence, is also a theorem which is a result of some other theorems.
Let us have a very simple and relevant illustration.
Consider on one hand Jose Luis Miranda claiming that June 30, 2012 is the end of the world. On the other hand, Eliseo Soriano saying that June 30 is not yet the end of the world. Also take as an axiom the biblical words:
“When a prophet speaks in the name of the LORD, if the word does not come to pass or come true, that is a word that the LORD has not spoken; the prophet has spoken it presumptuously. You need not be afraid of him” (Deut. 18:22)
Both Miranda and Soriano made their claims sufficient days before the 30th of June. So we may really view the situation as a prophet’s versus another prophet’s pronouncement. (Click this for more background on the issue.)
But TODAY is July 10, 2012 and my world is still mathematical! Just to say, Miranda is wrong and from our axiom, WE SHOULD NOT BE AFRAID OF HIM.
Our theorem could now be stated as: “June 30, 2012 is not end of the world and Miranda has just spoken presumptuously and that he is false!”
Corollary to our theorem is that SORIANO IS RIGHT IN HIS CLAIMS.
To GOD be the glory!