It is probable that this quote was popularized by the writings of Morris Kline, e.g. in Mathematics for Liberal Arts (1967):

[Michel Rolle] taught that the calculus was a collection of ingenious fallacies.

The earliest instance I can find of this quote is in Rouse Ball, A Short Account of the History of Mathematics (1888):

He taught that the differential calculus was nothing but a collection of ingenious fallacies.

This quote has since been repeated many times elsewhere. For example, Stewart, Calculus (2016):

He was a vocal critic of the methods of his day and attacked calculus as being a "collection of ingenious fallacies."

Bomans & Rogers, How We Got From There to Here: A Story of Real Analysis (2014):

In fact, Rolle was disdainful of both Newton and Leibniz's versions of calculus, once deriding them as a collection of "ingenious fallacies."

Bradley & Sandifer, Cauchy’s Cours d’analyse: An Annotated Translation (2010):

Rolle, for example, said that calculus was "a collection of ingenious fallacies"

Did Rolle ever say/write any such thing?

  • Rouse Ball is his surname. Don't ask why it doesn't have a hyphen, because I don't know. – Peter Taylor Nov 26 at 8:43
up vote 12 down vote accepted

"Did Rolle ever say/write any such thing (as that the calculus was 'a collection of ingenious fallacies')?"

Michel Rolle (France, 1652-1719) certainly did attack the mathematical basis of the infinitesimal calculus. I haven't found the exact phrase attributed to him by the authors quoted in the question, but there are plenty of broadly similar attacking phrases from Rolle about the calculus in Rolle's tract of 1703 (Remarques De M. Rolle De L'Académie Royale Des Sciences Touchant Le Problesme General Des Tangentes) -- such as 'absurd' {'absurde' - p.36}, 'entirely inconceivable' {'tout à fait inconcevable[s]' - p.41}, 'impossibilities and contradictions arise in crowds' {les impossibilitez & les contradictions se presenteroient en foule - p.41}, &c. This tract formed part of a disputatious exchange about the foundations of the new calculus, between Rolle and Joseph Saurin (1659-1737). [Also see edit/postscript below for a second and very relevant paper by Rolle found just now.] (It has been reported that later on Rolle changed his views: Michel Blay, 'La Naissance de la mécanique analytique', Paris, 1992, p.49.)

Rolle's criticisms had much to do with the seemingly ambiguous character (zero or non-zero?) of an infinitesimal, and he was not alone as a critic of the subject: he lived at a period when the logical basis of newer mathematical methods related to calculus was for a long time in widespread doubt and dispute. Rolle expressed respect for the work of Descartes, Fermat and Hudde, earlier contributors to what might be called 'pre-calculus', but in 1703 he aimed his violent expressions of criticism especially at the well-known 1696 exposition of Leibniz's infinitesimal calculus, Guillaume (Marquis) de L'Hôpital's 'Analyse des infiniment petits' (see https://www.maa.org/press/periodicals/convergence/mathematical-treasure-l-hospital-s-analyse-des-infiniment-petits). Rolle's criticisms had much in common with other and better-known critiques. An earlier stage in these mathematical developments was the 'method of indivisibles' of Bonaventura Cavalieri (c.1598-1647): this had been given poor mathematical justification at best, and it soon attracted fundamental and destructive criticisms, notably from Paul Guldin (Switzerland, 1577-1643). A much later and still-better-known critique of similar kind, aimed especially at the idea of infinitesimals, was given by George Berkeley in 'The Analyst' (1734) (see https://en.wikipedia.org/wiki/The_Analyst).

Edit/postscript: There is another paper by Rolle stating his attacks, which perhaps comes closer (than the paper linked above) to the content of the phrase quoted in the current question -- because its final sentence, following all of the attacks, acknowledges that the infinitesimal analysis is 'very ingenious'. It also gives another set of Rolle's condemnatory conclusions about the calculus: 'fort défectueux' (highly defective), 'ces Infinis fourmilleroient de contradictions' (these infinite[simal]s teem with contradictions), 'insoûtenable' (unsustainable), and the calculus involves 'petition de principe' (a petitio principii -- one of the well-known forms of fallacy), &c: see Rolle's Du nouveau systeme de l'infini, Mém. acad. roy. des sci., année 1703, 312-336. This combination of contents arguably suggests that the short phrase quoted in the question could have arisen as a brief paraphrase of this paper by a later commentator or possibly by Rolle himself. Blay (1986), cited below, shows that not all of Rolle's attacking papers were printed.

Among mathematical historians, Carl B Boyer has given a historical account of the long-lasting disputes about the foundations of calculus, in chapter 6 of his 'History of the Calculus and its Conceptual Development' (also published as 'Concepts of the Calculus') (editions of 1949 and 1959). This says rather little about the history of Rolle's attacks, though, and more detail on these is given by Michel Blay, in Deux moments de la critique du calcul infinitésimal : Michel Rolle et George Berkeley; Revue d'histoire des sciences, 39 (1986) 223-253). Blay reports that Rolle's earlier attacks on the Leibnitzian calculus were defended by Pierre Varignon, who made several justificatory appeals to Newton's first section from Book 1 of the Principia. (The material thus used by Varignon offers justification of Newton's geometrical form of calculus in terms of 'first and last ratios', i.e. limits.) In respect of the arguments relied on by Varignon against Rolle, a recent study argues that Boyer's and some other modern commentaries give unjustly poor assessments of the cogency of these earlier foundations offered for Newton's work in the calculus (Bruce Pourciau (2001), Newton and the notion of limit, Historia Mathematica 28, 18-30). Also, Jean-Étienne Montucla's well-known Histoire des mathématiques, vol.3 (1802), especially at pp. 110-119, gives more background on the history and merits/demerits of Rolle's attacks, and indicates that in the late 18th-c., it was considered that such attacks had by then been rigorously rebutted.

  • Could you quote those sentences/passages where he makes remarks that come close? – Kenny LJ Nov 26 at 10:13
  • @KennyLJ Thanks for your comment: Yes, I've now added the original French phrases in an amendment to the answer, and indicated their locations in the original paper to give the context. It can be seen that Rolle's technique of attack was to develop examples that when treated by calculus methods would lead (so he claimed) to something false, absurd, contradictory or inconceivable. His opponents accused him of not understanding the subject. ../.. (ctd next comment) – terry-s Dec 1 at 3:01
  • @KennyLJ ... (ctd) .. Also cited in the amended answer is a second paper by Rolle, found just now, with a content that arguably comes a little closer yet to the phrase that you quoted in the question -- because it combines the condemnations with a final acknowledgement that the infinitesimal analysis was 'very ingenious'. – terry-s Dec 1 at 3:02
  • So, my attempt at a tl;dr one-sentence answer: "We can't find him having said/written this exact phrase, but the spirit of his attacks is such that this is something he could very well have said." – Kenny LJ Dec 1 at 3:21

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