kb/data/en.wikipedia.org/wiki/The_Analyst-1.md

title	chunk	source	category	tags	date_saved	instance
The Analyst	2/2	https://en.wikipedia.org/wiki/The_Analyst	reference	science, encyclopedia	2026-05-05T08:43:13.533929+00:00	kb-cron

== Influence == 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. Despite these attempts, calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann and Karl Weierstrass, redefined the derivative and integral using a rigorous definition of the concept of limit. The idea of using limits as a foundation for calculus had been suggested by d'Alembert, but d'Alembert's definition was not rigorous by modern standards. The concept of limits had already appeared in the work of Newton, but was not stated with sufficient clarity to hold up to the criticism of Berkeley. In 1966, Abraham Robinson introduced Non-standard Analysis, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation, the way it was done before the (ε, δ)-definition of limit had been fully developed.

=== Ghosts of departed quantities === Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities, Berkeley wrote:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. 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?

Edwards describes this as the most memorable point of the book. Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions. Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals, but it is also used when discussing differentials, and adequality.

== Text and commentary == The full text of The Analyst can be read on Wikisource, as well as on David R. Wilkins' website, which includes some commentary and links to responses by Berkeley's contemporaries. The Analyst is also reproduced, with commentary, in recent works:

William Ewald's From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken at the time.

D. M. Jesseph's overview in the 2005 "Landmark Writings in Western Mathematics".

== References ==

== Sources == Kirsti, Andersen (2011), "One of Berkeley's arguments on compensating errors in the calculus.", Historia Mathematica, 38 (2): 219–318, doi:10.1016/j.hm.2010.07.001 Arkeryd, Leif (Dec 2005), "Nonstandard Analysis", The American Mathematical Monthly, 112 (10): 926–928, doi:10.2307/30037635, JSTOR 30037635 Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18: 43–74, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8, S2CID 254508527 Boyer, C; Merzbach, U (1991), A History of Mathematics (2 ed.) Burton, David (1997), The History of Mathematics: An Introduction, McGraw-Hill Edwards, C. H. (1994), The Historical Development of the Calculus, Springer Grabiner, Judith (May 1997), "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions", The American Mathematical Monthly, 104 (5): 393–410, doi:10.2307/2974733, JSTOR 2974733 Grabiner, Judith V. (Dec 2004), "Newton, Maclaurin, and the Authority of Mathematics", The American Mathematical Monthly, 111 (10): 841–852, doi:10.2307/4145093, JSTOR 4145093 Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, 78 (3): 571–625, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y, S2CID 254471766 Kleiner, I.; Movshovitz-Hadar, N. (Dec 1994), "The Role of Paradoxes in the Evolution of Mathematics", The American Mathematical Monthly, 101 (10): 963–974, doi:10.2307/2975163, JSTOR 2975163 Leader, Solomon (May 1986), "What is a Differential? A New Answer from the Generalized Riemann Integral", The American Mathematical Monthly, 93 (5): 348–356, doi:10.2307/2323591, JSTOR 2323591 Pourciau, Bruce (2001), "Newton and the notion of limit", Historia Math., 28 (1): 393–30, doi:10.1006/hmat.2000.2301 Robert, Alain (1988), Nonstandard analysis, New York: Wiley, ISBN 978-0-471-91703-8 Sherry, D. (1987), "The wake of Berkeley's Analyst: Rigor mathematicae?", Studies in Historical Philosophy and Science, 18 (4): 455–480, Bibcode:1987SHPSA..18..455S, doi:10.1016/0039-3681(87)90003-3 Wren, F. L.; Garrett, J. A. (May 1933), "The Development of the Fundamental Concepts of Infinitesimal Analysis", The American Mathematical Monthly, 40 (5): 269–281, doi:10.2307/2302202, JSTOR 2302202

== External links == Works related to The Analyst: a Discourse addressed to an Infidel Mathematician at Wikisource