Define 'Axiom Postulate Theorem Corollary Hypothesis'
When explaining to a friend of mine some of the fundamentals of mathematics, I found myself giving definitions of these base terms. A quick search with Google returned this great excerpt that efficiently and eloquently defines the words and relations, and merits posting here.
The words axiom and postulate are synonymous in mathematics. They are statements that are accepted as true in order to study the consequences that follow from them. Suppose that you were studying a deductive system called Jabbermetry and the words toves and mome appeared in the statement For every two toves P and Q there is a unique mome PQ that contains both P and Q. Given no other information, you would be unable to prove that this statement is true. In fact, this statement defines an important relationship between toves and momes; it is an axiom or postulate of Jabbermetry. Notice that if you are informed that Jabbermetry actually means Geometry, toves means points, and mome means line, the statement becomes more familiar: axiom 1: For every two points P and Q there is a unique line PQ that contains both P and Q. Do not be misled by your familiarity with points and lines, however. It remains that this statement cannot be proven since you are given no other information and is thus an axiom or postulate.
Three other terms that occur in the hierarchy of deductive reasoning are theorem, corollary, and hypothesis. A theorem is a statement that is a logical consequence of axioms and other theorems. A theorem, unlike an axiom or postulate, exists only if a proof can be given for the statement. For example, if you allow the additional axioms axiom 2: There exist three points not all in one line, and axiom 3: Two lines L and M are parallel if they do not intersect or if L = M, then you can show that the statement Two distinct lines intersect in at most one point is a consequence of axioms 13; the proven statement is a theorem. A corollary is a trivial theorem, that is, a theorem that so closely follows another axiom or theorem that it practically does not require a proof. For example, a corollary of axiom 3 above is If L is a line, then L is parallel to itself. The proof of this corollary is the definition of equals: L = L. Finally, a hypothesis is a statement that has not been proven but is expected to be capable of proof. For example, you may hypothesize If point B is between points A and C, then point C is not between points A and B. This is equivalent to a conjecture. If this statement does not follow from your present system of axioms, you may wish to include an additional axiom or make the hypothesis an axiom itself and see where it leads you. For the hypothesis given above, you may want to introduce an axiom that establishes a coordinate system.
Thus, axioms and postulates form the roots of a particular deductive system; theorems and corollaries are the logical consequences that fill out the deductive system; hypotheses drive theoretical development forward.
From: The American Heritage Book of English Usage