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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| The Analyst | 1/2 | https://en.wikipedia.org/wiki/The_Analyst | reference | science, encyclopedia | 2026-05-05T08:43:13.533929+00:00 | kb-cron |
The Analyst: A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious Mysteries and Points of Faith, is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated that Isaac Newton was intended. The book contains a direct attack on the foundations of calculus, specifically on Isaac Newton's notion of fluxions and on Leibniz's notion of infinitesimal change.
== Background and purpose ==
From his earliest days as a writer, Berkeley had taken up his satirical pen to attack what were then called 'free-thinkers' (secularists, sceptics, agnostics, atheists, etc.—in short, anyone who doubted the truths of received Christian religion or called for a diminution of religion in public life). In 1732, in the latest installment in this effort, Berkeley published his Alciphron, a series of dialogues directed at different types of 'free-thinkers'. One of the archetypes Berkeley addressed was the secular scientist, who discarded Christian mysteries as unnecessary superstitions, and declared his confidence in the certainty of human reason and science. Against his arguments, Berkeley mounted a subtle defense of the validity and usefulness of these elements of the Christian faith. Alciphron was widely read and caused a bit of a stir. But it was an offhand comment mocking Berkeley's arguments by the 'free-thinking' royal astronomer Sir Edmund Halley that prompted Berkeley to pick up his pen again and try a new tack. The result was The Analyst, conceived as a satire attacking the foundations of mathematics with the same vigour and style as 'free-thinkers' routinely attacked religious truths. Berkeley sought to take apart the then foundations of calculus, claimed to uncover numerous gaps in proof, attacked the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not so much to mock mathematics or mathematicians, but rather to show that mathematicians, like the Christians they criticized, relied upon unknowable mysteries in the foundations of their reasoning. Moreover, the existence of these "superstitions" was not fatal to mathematical reasoning; indeed, it was an aid. So too with the Christian faithful and their mysteries. Berkeley concluded that the certainty of mathematics is no greater than the certainty of religion.
== Content == The Analyst was a direct attack on the foundations of calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change. In section 16, Berkeley criticises
...the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn-1. But, notwithstanding all this address to cover it, the fallacy is still the same. It is a frequently quoted passage, particularly when he wrote:
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? Berkeley did not dispute the results of calculus; he acknowledged the results were true. The thrust of his criticism was that Calculus was not more logically rigorous than religion. He instead questioned whether mathematicians "submit to Authority, take things upon Trust" just as followers of religious tenets did. According to Burton, Berkeley introduced an ingenious theory of compensating errors that were meant to explain the correctness of the results of calculus. Berkeley contended that the practitioners of calculus introduced several errors which cancelled, leaving the correct answer. In his own words, "by virtue of a twofold mistake you arrive, though not at science, yet truth."
== Analysis ==
The idea that Newton was the intended recipient of the discourse is put into doubt by a passage that appears toward the end of the book:
"Query 58: Whether it be really an effect of Thinking, that the same Men admire the great author for his Fluxions, and deride him for his Religion?"
Here Berkeley ridicules those who celebrate Newton (the inventor of "fluxions", roughly equivalent to the differentials of later versions of the differential calculus) as a genius while deriding his well-known religiosity. Since Berkeley is here explicitly calling attention to Newton's religious faith, that seems to indicate he did not mean his readers to identify the "infidel (i.e., lacking faith) mathematician" with Newton.
Mathematics historian Judith Grabiner comments, "Berkeley's criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct". While his critiques of the mathematical practices were sound, his essay has been criticised on logical and philosophical grounds.
For example, David Sherry argues that Berkeley's criticism of infinitesimal calculus consists of a logical criticism and a metaphysical criticism. The logical criticism is that of a fallacia suppositionis, which means gaining points in an argument by means of one assumption and, while keeping those points, concluding the argument with a contradictory assumption. The metaphysical criticism is a challenge to the existence itself of concepts such as fluxions, moments, and infinitesimals, and is rooted in Berkeley's empiricist philosophy which tolerates no expression without a referent. Andersen (2011) showed that Berkeley's doctrine of the compensation of errors contains a logical circularity. Namely, Berkeley's determination of the derivative of the quadratic function relies on Apollonius's determination of the tangent of the parabola.