The Analyst

It has been suggested that Ghosts of departed quantities be merged into this article or section. (Discuss)

The Analyst, subtitled A DISCOURSE Addressed to an Infidel Mathematician, is a book published by George Berkeley in 1734. The "infidel mathematician" is believed to have been Edmond Halley or Sir Isaac Newton. In the latter case, no reply would have been possible, as Newton died in 1727.

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. Berkeley sought to defend religion by showing that the calculus, which grounded religion's new rival, natural philosophy (the predecessor of today's physics), led to paradox and absurdity.

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?

A more modern paraphrase:

What are these "instantaneous" rates of change? The ratios of vanishing increments? And what are these "vanishing" Increments? They are neither finite quantities nor "infinitesimal" quantities, nor yet nothing. May we not call them the ghosts of departed quantities?

Two years after this publication, Thomas Bayes published anonymously "An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst" (1736), which he defended the logical foundation of Isaac Newton's calculus against the criticism outlined in The Analyst.

But only beginning around 1830, first in the hands of Augustin Cauchy, later in those of Bernhard Riemann, and Karl Weierstrass, the derivative and integral were redefined using a rigorously defined new concept, that of limit. And finally in 1966, with the publication of Abraham Robinson's book Non-standard Analysis, was the object of Berkeley's strongest ridicule, Leibniz's intuitive notion of the infinitesimal, made fully rigorous, thus showing another way of overcoming the difficulties which Berkeley pointed out in Newton's approach.

[edit] The text

• The Analyst at David R. Wilkins' website. Includes links to some responses by Berkeley's contemporaries.

The Analyst is also reproduced, with commentary, in:

• Ewald, William, ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 1. Oxford Univ. Press.

Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken.

[edit] Commentary

• Jesseph, D.M., 2005, "The analyst" in Grattan-Guiness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 121–30.