- The Analyst
"The Analyst", subtitled "A DISCOURSE Addressed to an Infidel Mathematician", is a book published by
George Berkeley in1734 . The "infidel mathematician" is believed to have beenEdmond Halley or SirIsaac Newton . In the latter case, no reply would have been possible, as Newton died in1727 ."The Analyst" was a direct attack on the foundations and principles of the
calculus , specifically on Newton's notion of fluxions and on Leibniz's notion ofinfinitesimal change. Berkeley sought to defendreligion by showing that the calculus, which grounded religion's new rival,natural philosophy (the predecessor of today'sphysics ), 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 ofBernhard Riemann , andKarl Weierstrass , thederivative andintegral were redefined using a rigorously defined new concept, that of limit. And finally in1966 , with the publication ofAbraham 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
*" [http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/ 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-Guiness, I., ed., "Landmark Writings in Western Mathematics". Elsevier: 121-30.
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