rigor
Imagine assembling flat-pack furniture: ‘it looks about right’ is not enough — every joint must actually lock, or the shelf collapses under a real book. Rigor is the analyst's insistence that every joint in an argument actually locks, with no step justified only by a picture or by intuition.
Rigor is the standard of fully justified, gap-free reasoning: each assertion follows from the axioms, definitions and previously proved results by valid logical steps, and every quantity invoked is precisely defined. It is what separates analysis from the informal calculus of the 1700s, where powerful but unjustified manipulations of infinitesimals sometimes produced correct answers and sometimes produced nonsense.
Rigor is not pedantry for its own sake. The historical drive to make calculus rigorous (Cauchy, Bolzano, Weierstrass) was forced by genuine paradoxes — divergent series summed to absurd values, ‘obvious’ claims about continuity and convergence that turned out false. The honest caveat: rigor guarantees correctness given the axioms, but it cannot by itself tell you which definitions are the fruitful ones; that still takes mathematical taste.
The series 1 - 1 + 1 - 1 + ... was once ‘summed’ to 1/2 by formal manipulation. A rigorous treatment first defines what a sum is (the limit of partial sums) and then observes those partial sums are 1, 0, 1, 0, ... which have no limit, so the series simply diverges.
Define the object first; only then ask whether it exists.