The Analyst, subtitled
A DISCOURSE Addressed to an Infidel Mathematician, is a book published by
George BerkeleyGeorge Berkeley , also known as Bishop Berkeley, was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" . This theory contends that individuals can only know directly sensations and ideas of objects, not abstractions such as "matter"...
in 1734. The "infidel mathematician" is believed to have been
Edmond HalleyEdmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist, who is best known for computing the orbit of Halley's comet, which is named for him.- Biography and career :...
or Sir
Isaac NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
. 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
infinitesimal calculusInfinitesimal calculus was independently invented by both Leibniz and Newton in the 1660s, drawing on the work of such mathematicians as Barrow and Descartes...
, specifically on Newton's notion of
fluxionsMethod of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
and on
LeibnizGottfried Wilhelm Leibniz was a German philosopher, polymath and mathematician who wrote primarily in Latin and French....
's notion of
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
change.
The Analyst, subtitled
A DISCOURSE Addressed to an Infidel Mathematician, is a book published by
George BerkeleyGeorge Berkeley , also known as Bishop Berkeley, was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism" . This theory contends that individuals can only know directly sensations and ideas of objects, not abstractions such as "matter"...
in 1734. The "infidel mathematician" is believed to have been
Edmond HalleyEdmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist, who is best known for computing the orbit of Halley's comet, which is named for him.- Biography and career :...
or Sir
Isaac NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
. 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
infinitesimal calculusInfinitesimal calculus was independently invented by both Leibniz and Newton in the 1660s, drawing on the work of such mathematicians as Barrow and Descartes...
, specifically on Newton's notion of
fluxionsMethod of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
and on
LeibnizGottfried Wilhelm Leibniz was a German philosopher, polymath and mathematician who wrote primarily in Latin and French....
's notion of
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
change. According to historian of science Judith Grabiner, “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct”. Berkeley sought to defend
religionA religion is a system of human thought which usually includes a set of narratives, symbols, beliefs and practices that give meaning to the practitioner's experiences of life through reference to a higher power, deity or deities, or ultimate truth...
by showing that the calculus, which grounded religion's new rival,
natural philosophyNatural philosophy or the philosophy of nature , is a term applied to the study of nature and the physical universe that was dominant before the development of modern science...
(the predecessor of today's
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
), 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 quantitiesThe expression ghosts of departed quantities, familiar to many students of infinitesimal calculus, was coined by Bishop Berkeley in his work The Analyst...
?
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 BayesThomas Bayes , was a British mathematician and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name: Bayes' theorem, which was published posthumously.- Biography :...
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 MaclaurinColin Maclaurin was a British mathematician. Due to changes in orthography since that time , his surname is alternatively written MacLaurin. In Gaelic the name is "Cailean MacLabhruinn", which is literally 'Colin, the son of Laurence.'-Life and work:Maclaurin was born in...
'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 .
But only beginning around 1830, first in the hands of Augustin Cauchy, later in those of
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
and
Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :...
, were the
derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
and
integralIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
redefined using a rigorously defined new concept, that of
limit. And finally in 1966, with the publication of
Abraham RobinsonAbraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
's book
Non-standard AnalysisNon-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson...
, 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
- 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-Guinness, I.
Ivor Grattan-Guinness is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
, ed., Landmark Writings in Western Mathematics. Elsevier: 121–30.