The Analyst - Content

Content

The Analyst was a direct attack on the foundations and principles of the infinitesimal calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change. In section 16, Berkeley criticizes

the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn-1. But, notwithstanding all this address to cover it, the fallacy is still the same.

Its most frequently quoted passage:

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

To quote Judith Grabiner, “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct” (Grabiner 1997).

Sherry argues that Berkeley's criticism of infinitesimal calculus consists of a logical criticism and a metaphysical criticism. The logical criticism is that of a fallacia suppositionis, which means gaining points in an argument by means of one assumption and, while keeping those points, concluding the argument with a contradictory assumption. The metaphysical criticism is a challenge to the existence itself of concepts such as fluxions, moments, and infinitesimals, and is rooted in Berkeley's empiricist philosophy which tolerates no expression without a referent (Sherry 1987). Katz et al. argue that the logical criticism is based in Berkeley's misunderstanding of Leibniz's procedures for manipulating infinitesimals;(Błaszczyk, Katz & Sherry 2012) and that the force of Berkeley's criticisms has been overestimated; that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof; and that Leibniz's system for differential calculus was free of logical contradictions.(Katz & Sherry 2012)