I am, by nature, a timid, centrist soul. I listen to Coldplay, drink lattes, and always begin calculus with a unit on limits. I trot out the Squeeze Theorem and the Intermediate Value Theorem and the definition of continuity like a diligent textbook follower.
But writing a book about calculus has radicalized me.
I’ve now fallen in with those hooligans and scofflaws who denigrate limits. Not the mathematical concept, of course; I just question the idea that, before meeting derivatives and integrals, the student must undergo a thorough, decontextualized study of the local behavior of abstract functions.
Where did the limit framework come from? In short, Augustin-Louis Cauchy and sundry 19th-century pals. Shaken by the tricky questions of Fourier analysis, which cast the whole concept of “convergence” into doubt, they sought to rebuild calculus from the ground up, at a level of technicality and arithmetical precision that no prior generation had felt any need for.
But calculus students haven’t seen any Fourier analysis! They don’t feel the intellectual need that Cauchy felt. The IVT to them is a piece of blinding obviousness, dressed up in ornate technicality.
Next time I teach calculus, I intend to join the rebels by diving right into differentiation, circling back to notions of convergence and continuity only when they arise naturally in context (e.g., discussing what “nondifferentiable” behavior might mean). If it was good enough for the Bernoullis, it’s good enough for me.