I started out as an engineering undergrad at Purdue. As a freshman I realized that I am primarily interested in mathematics of all my subjects, and I had an intensifying craving for a kind of precision I felt was missing from the engineering math courses, so I switched my major to applied math. Now I know that I was sensing the lack of rigour - or precision of thinking and derivation in mathematics. In complicated proofs and theories, even the greatest mathematicians have made small mistakes (usually natural-sounding assumptions), and the challenge of rigour is to avoid them at all costs. This is a fundamental requirement for the integrity of a mathematician. The slightest mistake in a theorem, can lead to entirely erroneous theories built upon it.

Today I have come across a quote from Rolle stated in 1703 on the fledgling theory of Calculus:

Currently Analysis is considered entirely rigorous, made so by Dedekind's Axiom (alternatively the Archimedean Axiom), as well as the Zermelo-Fraenkel Axioms in Set Theory. From these, all the theorems of Analysis can be rigorously derived, primarily built from the epsilon-delta bricks of Cauchy.

Today I have come across a quote from Rolle stated in 1703 on the fledgling theory of Calculus:

"Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. Among its principles one could only find true axioms and all the theorems and problems posed were either soundly demonstrated or capable of sound demonstration. And if any false or uncertain propositions were slipped into it they would immediately be banned from this science. But it seems that this feature of exactness does not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes."Surprisingly this skeptical quote questioning the rigour of the early Calculus was highly unique in its day, as no-one really understood what Rolle meant, virtually everyone opposed him. The point of reference for his criticism was Euclidean Geometry, which was then and still is considered entirely rigorous. By the 19th Century however, it became clear to the majority of mathematicians, that Analysis (Calculus) lacks fundamental rigour, and there came a burning need to make it so. In that unrigorous form, it could not be applied to prove for instance Fourier's theorems in the developing Thermodynamics. The quest was thus undertaken by a number of mathematicians, such as Cauchy (epsilon-delta definitions and proofs), Weierstrass, Dedekind and others.

Currently Analysis is considered entirely rigorous, made so by Dedekind's Axiom (alternatively the Archimedean Axiom), as well as the Zermelo-Fraenkel Axioms in Set Theory. From these, all the theorems of Analysis can be rigorously derived, primarily built from the epsilon-delta bricks of Cauchy.