The Analyst
Encyclopedia
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 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.
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 vigor and style as 'free-thinkers' routinely attacked religious truths.
Berkeley took mathematics apart, uncovering numerous gaps in proof, attacking the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not to mock mathematics or mathematicians (Berkeley himself was an accomplished mathematician in his youth), but rather to show that mathematicians, like Christians, relied upon incomprehensible '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.
, specifically on Newton's notion of fluxions
and on Leibniz
's notion of infinitesimal
change. In section 16, Berkeley criticizes
Its most frequently quoted passage:
To quote 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”.
According to Sherry, advances in science and mathematics have attested to the validity of Berkeley's insights. He explains that The Analyst 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 of a similar type as the logical criticism and amounts to showing that concepts like moments and fluxions have contradictory properties .
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 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 .
Despite these attempts calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann
and Karl Weierstrass
, redefined the derivative
and integral
using a rigorous definition of the concept of limit
. The concept of using limits as a foundation for calculus had been suggested by d'Alembert
, but d'Alembert's definition was not rigorous by modern standards . The concept of limits had already appeared in the work of Newton , but was not stated with sufficient clarity to hold up to the criticism of Berkeley.
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 that was in a similar spirit to the way calculus was done before the (ε, δ)-definition of limit had been fully developed.
Edwards describes this as the most memorable point of the book . Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals , but it is also used when discussing differentials , and adequality
.
The Analyst is also reproduced, with commentary, in:
Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken.
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...
in 1734. The "infidel mathematician" is believed to have been Edmond Halley
Edmond Halley
Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career:Halley...
or Sir Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
. In the latter case, no reply would have been possible, as Newton died in 1727.
Purpose
From his earliest days as a writer, Berkeley had taken up his satirical pen to attack what were then called 'free-thinkers' (secularists, skeptics, agnostics, atheists, etc. - in short, anyone who doubted the truths of received Christian religion and/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 spiritualism and 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 vigor and style as 'free-thinkers' routinely attacked religious truths.
Berkeley took mathematics apart, uncovering numerous gaps in proof, attacking the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not to mock mathematics or mathematicians (Berkeley himself was an accomplished mathematician in his youth), but rather to show that mathematicians, like Christians, relied upon incomprehensible '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 and principles of the infinitesimal calculusInfinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
, specifically on Newton's notion of fluxions
Method of Fluxions
Method 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 Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
's notion of infinitesimal
Infinitesimal
Infinitesimals 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. In section 16, Berkeley criticizes
- 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.
Its 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?
To quote Judith Grabiner
Judith Grabiner
Judith Victor Grabiner is an American mathematician who is among the best-known historians of mathematics. She earned her B.S. in mathematics at the University of Chicago in 1960 and her M.A. and Ph.D. in the history of science at Harvard University, and is currently Flora Sanborn Pitzer...
, “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct”.
According to Sherry, advances in science and mathematics have attested to the validity of Berkeley's insights. He explains that The Analyst 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 of a similar type as the logical criticism and amounts to showing that concepts like moments and fluxions have contradictory properties .
Influence
Two years after this publication, Thomas BayesThomas Bayes
Thomas Bayes was an English mathematician and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name: Bayes' theorem...
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 Maclaurin
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
'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 .
Despite these attempts calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
and Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
, redefined the derivative
Derivative
In calculus, a branch of mathematics, 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
and integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
using a rigorous definition of the concept of limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
. The concept of using limits as a foundation for calculus had been suggested by d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...
, but d'Alembert's definition was not rigorous by modern standards . The concept of limits had already appeared in the work of Newton , but was not stated with sufficient clarity to hold up to the criticism of Berkeley.
In 1966, Abraham Robinson
Abraham Robinson
Abraham 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....
introduced Non-standard Analysis
Non-standard analysis
Non-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. He wrote:...
, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation that was in a similar spirit to the way calculus was done before the (ε, δ)-definition of limit had been fully developed.
Ghosts of departed quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities , Berkeley wrote:
It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. 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?
Edwards describes this as the most memorable point of the book . Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals , but it is also used when discussing differentials , and adequality
Adequality
In the history of infinitesimal calculus, adequality is a technique developed by Pierre de Fermat. Fermat said he borrowed the term from Diophantus. Adequality was a technique first used to find maxima for functions and then adapted to find tangent lines to curves...
.
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-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, 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.