Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built. One might attempt to distinguish mathematics from other disciplines by arguing that, ultimately, we know that its results express truth in a way no other subject can boast; however, philosophical arguments of the early 1900s call even this claim into question.

While the story of calculus features plenty of intrigue and debate, readers should rest assured that controversy is often a hallmark of mathematical discovery. As with every pursuit of the mind, mathematics advances when its explorers resist attempts to settle matters. Think of cubism challenging the dominance of linear perspective in Western art, jazz releasing music from traditional notions about rhythm and harmony, or free verse perturbing the boundaries of poetic meter. Every intellectual pursuit needs people who respect the rules yet stretch them.

Calculus shifted in its focus from geometry and puzzling claims about the infinite to functions, limits, and picture-free algebra. During the transition, each proposed definition or rule weathered a barrage of exceptions. The goal of the debate was certainty.