Rigorous Foundations of the Calculus

Source: Grabiner, Judith V. (1981). The Origins of Cauchy’s Rigorous Calculus. Cambridge, MA: MIT Press.

The basic problem can be illustrated by a simple example. Basically everyone who “did” calculus in the 18th century did something like the following: Starting with y= x2 , let x take the value x + h for some small value h, and then yx= 2+2xh+h2. If the change in x is h, then the change in y is x22+ 2xh +−h x2=2xh +h2and the ratio of changes is 2xh + h2 =2x +h . So far, all is well. The problem begins in getting from 2x+ hto 2x. h The differential quotient (or fluxion, or in our terms, derivative) is 2x, not 2x + h. What happened to the h?

Eighteenth Century Answers

Patch Proponents Problems Infinitesimals: h is infinitesimally Leibniz, Newton: Infinitesimals do not small and so can be neglected with L’Hospital, obey the Archimedean Principle respect to the finite quantity 2x. Johann and thus have no legitimate Thus 2x + h and 2x are essentially Bernoulli mathematical standing equal. Newton, Berkeley: Unless h = 0, neglecting it is an error, and “errours, tho’ never so small, are not to be neglected in Mathematicks.”

Fluxions: x and y, as quantities that Newton, Berkeley, d’Alembert, Lagrange: could change, were called flowing Maclaurin, There is no definition of quantities or fluents. Their everyone else “velocity” clear enough to serve velocities were finite quantities in Britain. as a foundation for the calculus. called fluxions. Over small time Besides, that is a notion from intervals, increments in x and y were physics, not from mathematics. assumed to be proportional to the velocities of change. This ratio is “Indefinitely little” sounds still 2x+ h. Then, in Newton’s suspiciously like an infinitesimal. words, h is “indefinitely little” so it Later, Newton preferred to talk in will be “nothing in respect to the terms of limits or “ultimate rest.” ratios”

Early Limit Ideas, Version 1: A Newton, Berkeley, Lagrange: First, by this limit is a quantity which the ratio can d’Alembert, very definition of limit, 2x + h can never surpass, but can approach to Lacroix, never really become 2x, and the within any given difference. Then Maclaurin secant line can never become the 2x becomes the limit of 2x + h as h tangent line. Moreover, the goes to zero. whole process is unintelligible: Either h is zero or it isn’t. If it is Early Limit Ideas, Version 2: The 2xh+ h2 zero, the ratio is limit is the value of the ratio at the h last instant of time before h – an undefined. If it isn’t 0, 2x + h is “evanescent quantity” -- has never really 2x. Summing up, h vanished. At that moment, 2x+ h ”cannot be got rid of.” And becomes 2x. “errours, tho’ never so small, are not to be neglected in Mathematicks.” (Berkeley used this line, from Newton’s critique of Leibniz, to attack all versions of calculus.)

Compensation of Errors: The Berkeley, Lagrange: Although the offsetting methods of the calculus get correct Carnot errors could actually be verified results even though the foundations for some functions, it was not at didn’t make sense, because the error all clear that it could be verified in made in replacing 2x + h by 2x is general. offset by the error made in treating the curve at the given point as though it coincided with the tangent at two different points, which it does not.

Greek Exhaustion: The process Maclaurin This might work as a proof could be made rigorous by proving method, but still the only method that, if the rate of change of y is R, R of finding the rates of change cannot be less than 2x and cannot be involved the suspect methods of more than 2x, so R must be exactly infinitesimals, fluxions, etc. In 2x. This method is similar to that addition, the proofs in general used by Archimedes prove that the cases were quite difficult. area of the circle couldn’t be less than 2π, and could not be greater than 2π.

Orders of Zeros: The quantity h can Euler; Lagrange: This is essentially be made less than any given quantity Laplace equivalent to the limit argument and so when added to a finite above, so the critiques of quantity like 2x it is actually zero. Berkeley and Lagrange still hold. But when considered in ratios, it is not zero, so the expression 2xh+ h2 is not meaningless. h Algebraic Methods: The equation Lagrange; It is not clear how the series is ()x +=hx22+2xh+h2is just a John Landen, related to other approaches special case of the more general L.F.A. tangents or rates of change. How series representation of a function: Abrogast, do we know there is a series for J.-P. Gruson each function? Two functions 2 can have the same Taylor series. fx( +=h) f()x+hp()x+hq()x+....

We then define the rate of change of y with respect to x as p(x). There is no need to describe what happened to h. In the case of y= x2 , the derivative is exactly 2x.

These were the attempts, such as they were, to address the problems with the foundations of the calculus in the 18th century. Of all the 18th century mathematicians, Lagrange probably moved farthest in the direction of recognizing the need for a more rigorous footing for calculus, and he helped lay the foundation for that work.The real solution to these issues didn’t really surface until the work of Cauchy and his colleagues in the 19th century. We look at that next.