Getting There Without Arriving

For about a hundred and fifty years, calculus worked and nobody could say why it was allowed.

The procedure, in Newton's hands and Leibniz's, involved a quantity that was not zero (you divided by it) and then, a line or two later, was zero, since you discarded every term still containing it. Both steps were necessary. Together they are a contradiction.

Bishop Berkeley said so, with some relish, in 1734. He was arguing that mathematicians who mocked theology for demanding faith were themselves reasoning about quantities they could not define. 'And what are these fluxions?' he wrote. 'The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?'

The criticism was unanswerable and the mathematics kept working. Euler produced a century of correct and often astonishing results with foundations that could not survive an afternoon's scrutiny, because he had an instinct for when the manipulations were legitimate. Instinct is not transferable, and eventually the subject met functions that instinct got wrong.

Fourier's work on heat in 1807 forced the issue. He represented functions as infinite sums of sines and cosines, including functions with corners and jumps, and the results were right, and they raised a question nobody had a good answer to: what is a function? If it is a formula, Fourier's series are baffling. If it is any rule at all pairing inputs with outputs, then all the comfortable assumptions about smooth curves become claims requiring proof.

Then the counterexamples arrived. In 1872 Weierstrass exhibited a function continuous everywhere and differentiable nowhere: a curve with no breaks that has a corner at every point. Hermite wrote that he turned away with fear and horror from this lamentable plague. Standard proofs broke, because they had assumed pictures.

The repair was to replace intuition with inequalities. Cauchy started it and Weierstrass finished it, and the definition of a limit is the result.

It runs like this. A sequence approaches a limit L if, for any tolerance you care to name (any positive number, however small) there is a point in the sequence beyond which every term is within that tolerance of L.

Nothing is ever reached. Reaching is not mentioned, and no infinitely small quantity appears anywhere. The definition is a challenge and a response: you name how close, I name how far along. Every notion in calculus (continuity, derivative, integral, convergence) is then rebuilt on it, and the rebuilding is what makes the subject teachable to people without Euler's instincts.

This is where the standard schoolroom argument gets settled. Does 0.999… equal 1?

It does, exactly. The notation denotes the limit of the sequence 0.9, 0.99, 0.999, and that limit is 1. Not approaching 1, not infinitesimally short of it: equal to it, because the limit is defined as the value the terms get and stay arbitrarily close to. If the two differed there would be a number strictly between them, and there is none. The stubborn feeling that something remains comes from reading the notation as a process that stops somewhere, which is precisely the reading the definition removes.

The definition also sorts out which infinite sums have values. Terms shrinking to zero turns out to be necessary and not sufficient: 1 + 1/2 + 1/3 + 1/4 + … has terms tending to zero and exceeds any bound you like, though it needs around ten to the power of forty-three terms to pass a hundred. Meanwhile 1/2 + 1/4 + 1/8 + … converges to exactly 1, which is Zeno's dichotomy answered: the infinitely many stages of the journey have a finite total, and the total is where you arrive.

There is a coda. In the 1960s Abraham Robinson showed that infinitesimals can be made rigorous after all, by extending the real numbers with quantities smaller than every positive real and proving the extension consistent. In non-standard analysis, Leibniz's original manipulations are valid as written, and the ghosts turn out to have been respectable all along.

It has not replaced the limit in teaching, partly because the logical machinery required is heavier, and mostly because by 1960 everything had already been rebuilt without it. The vindication arrived three centuries late, to a subject that had stopped needing it.