In my experience, the two greatest difficulties in mathematics are:
The obvious is not always true.
The truth is not always obvious.
Rigour is the essence of mathematics. A rigorous proof provides an explanation of why a particular mathematical statement is true, and, at the same time, takes care of all the "Yes, but what if"s.
Rigour and proof provide the guarantee of correctness and reliability.
Rigour and proof refine our mathematical insights and instincts so that the superficially "obvious" misleads us less frequently.
When I pose the problem "1, 2, 3, x Find x." the initial response is usually a derisory laugh, of disbelief that I am serious, because "the answer is obviously 4". It is easy to demonstrate using practical examples that this statement is, as it stands, nonsense. A rigorous analysis is required.
