The Analyst

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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. According to historian of science 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). 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^{[citation needed]}.

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?

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), in which he defended the logical foundation of Isaac Newton's calculus against the criticism outlined in The Analyst. Colin Maclaurin's two-volume Treatise of Fluxions published in 1742 also began as a response to Berkeley attacks, intended to show that Newton's calculus was rigorous by reducing it to the methods of Greek geometry (Grabiner 1997).

But only beginning around 1830, first in the hands of Augustin Cauchy, later in those of Bernhard Riemann and Karl Weierstrass, were the derivative and integral 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.

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.

Commentary

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

References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (March 2009)

• Grabiner, Judith (1997). "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions". The American Mathematical Monthly (Mathematical Association of America) 104 (5): 393–410. http://www.jstor.org/stable/2974733. Retrieved 2008-12-22. 
• Robert, Alain: Nonstandard analysis, Wiley, New York 1988. ISBN 0-471-91703-6