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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 calculus, specifically on Newton's notion of fluxion 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.
Nothing much came of Berkeley's criticisms in the 18th century, if only because Berkeley was neither mathematician nor natural philosopher. But 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 rigourously defined new concept, that of limit. But only 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 Analyst is available online at David R. Wilkins' website, which 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 Uni. Press.
Ewald believes that Berkeley's objections to the calculus of his day were, by and large, well taken.
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