What Are Proofs, and Why Do We Write Them?

Abstract

A statement in Mathematics is just a sentence which could be designated as true or false. The sentences “$1 + 1 = 2$” and “x = 4 implies x ² = 16” are true statements while “All rational numbers are positive” and “There is a real number x such that $x^{2} + 5 = 2$” are false statements. Some sentences like “Green is nice” or “Authenticity runs hot” are too ambiguous, a matter of opinion, or are just plain nonsense and cannot be said to be true or false, so mathematicians would not consider them to be statements. Mathematicians have a lot of words for kinds of statements including many that you have heard: definition, axiom, postulate, principle, conjecture, lemma, proposition, law, theorem, contradiction, and others.