"If you will only ask yourself, "Is what I am being told really true?," it is amazing how much you can find is, or borders on, being false, even in a… - Richard Hamming

When you yourself are responsible for some new application in mathematics... then your reputation... and possibly even human lives, may depend on the results you predict. It is then the need for mathematical rigor will become painfully obvious to you. ...Mathematical rigor is the clarification of the reasoning used in mathematics. ...a closer examination of the numerous "hidden assumptions" is made. ...Over the years there has been a gradually rising standard of rigor; proofs that satisfied the best mathematicians of one generation have been found inadequate by the next generation. Rigor is not a yes-no property of a proof... it is a vague standard of careful treatment that is currently acceptable to a particular group.

There is no unique, correct answer in most cases. It is a matter of taste, depending on the circumstances... and the particular age you live in. ...Gradually, you will develop your own taste, and along the way you may occasionally recognize that your taste may be the best one! It is the same as an art course.

Vicarious learning from the experiences of others saves making errors yourself, but I regard the study of successes as being basically more important than the study of failures. There are so many ways of being wrong and so few of being right, studying successes is more efficient.