
The property of rigour (Latin rigor, â€˜unbendingnessâ€™) is considered by mathematicians to be the greatest virtue. It is applied to arguments and proofs. It has been the main goal of mathematicians since the early 19th century, when the haphazard results and arguments of earlier mathematicians began to be systematized and questioned, following the uncertainty caused by the discovery and acceptance of nonEuclidean geometry. Rigour means arguing with the strictest application of the principles of logic, and not using any assumptions which are not explicitly stated at the outset. The reason that mathematicians sought rigour is that the results it gives cannot be questioned, whereas results which rely on intuition, your ideas about what should happen, can often be mistaken.
The search for rigour has led to pure mathematics becoming increasingly dependent on logic and set theory; some have even viewed mathematics as only a part of logic. The reason these two areas were chosen to be the basis of mathematics is that logic is supposed to be universal and to be the arbiter of a correct argument (though both those statements are open to question), and that set theory is sufficiently general and powerful for every mathematical concept to be defined in terms of sets. SMcL 
