The Analyst

For other uses, see The Analyst (disambiguation).

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 (Burton 1997, 477). 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 calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal 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 nx^n-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, “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct”(Grabiner 1997).

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 (Sherry 1987).

Influence

Two years after this publication, Thomas Bayes 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 (Grabiner 1997).

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 (Burton 1997). The concept of limits had already appeared in the work of Newton (Pourciau 2001), but was not stated with sufficient clarity to hold up to the criticism of Berkeley.(Edwards 1994)

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.

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 (Boyer 1991), 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 (Edwards 1994). 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 (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).

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., ed., Landmark Writings in Western Mathematics. Elsevier: 121–30.

References

• Arkeryd, Leif (Dec. 2005), "Nonstandard Analysis", The American Mathematical Monthly 112 (10): 926–928 
• Robert, Alain (1988), Nonstandard analysis, New York: Wiley, ISBN 0-471-91703-6 
• Burton, David (1997), The History of Mathematics: An Introduction, McGraw-Hill, pp. 374 
• Boyer, C; Merzbach, U (1991), A History of Mathematics (2 ed.) 
• Edwards, C. H. (1994), The Historical Development of the Calculus, Springer 
• Grabiner, Judith (1997), "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions", The American Mathematical Monthly (Mathematical Association of America) 104 (5): 393–410, doi:10.2307/2974733, JSTOR 2974733 
• Grabiner, Judith V. (Dec 2004), "Newton, Maclaurin, and the Authority of Mathematics", The American Mathematical Monthly 111 (10): 841–852 
• Kleiner, I.; Movshovitz-Hadar, N. (Dec. 1994), "The Role of Paradoxes in the Evolution of Mathematics", The American Mathematical Monthly 101 (10): 963–974 
• Leader, Solomon (May 1986), The American Mathematical Monthly, 93, pp. 348–356 
• Pourciau, Bruce (2001), "Newtion and the notion of limit", Historia Math. 28 (1): 393–30 
• Sherry, D. (1987), "The wake of Berkeley’s Analyst: Rigor mathematicae?", Studies in Historical Philosophy and Science 18 (4): 455–480 
• Wren, F. L.; Garrett, J. A. (May 1933), "The Development of the Fundamental Concepts of Infinitesimal Analysis", The American Mathematical Monthly 40 (5): 269–281