The Analyst (subtitled 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.
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.
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.
According to Burton, Berkeley introduced an ingenious theory of compensating errors that were meant to explain the correctness of the results of calculus.
"[8] 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?"
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.
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.
[13] The concept of limits had already appeared in the work of Newton,[14] but was not stated with sufficient clarity to hold up to the criticism of Berkeley.
[15] 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.
[17] Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus.
[20] The full text of The Analyst can be read on Wikisource, as well as on David R. Wilkins' website,[21] which includes some commentary and links to responses by Berkeley's contemporaries.
The Analyst is also reproduced, with commentary, in recent works: Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken at the time.