"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why? ...Proofs have gaps and are... inherently incomplete and sometimes wrong. ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification. ...Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored... Social pressure often hides mistakes in proofs. In a seminar lecture... most mathematicians sit quietly... understanding very little... and applauding politely... One of the joys of Gel'fand's seminar... he would constantly interrupt... to ask questions and give elementary examples... [T]he audience would actually learn some mathematics. There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's '. Mathematics can be done scrupulously."