What Did Mathematics Do to Physics?

Aristotle , Physics II.2 194a 8–9.

Laird W. R. , “Patronage of mechanics and theories of impact in sixteenth-century Italy”, in Moran Bruce T. (ed.), Patronage and institutions: Science, technology, and medicine at the European court 1500–1750 (Rochester, 1991), 51–66.

There is a considerably large literature on the mixed sciences, but, to my knowledge, there is still no general account of its extension to fields other than the canonical ones (Astronomy, Optics, Harmonics) as suggested here. For the case of Galileo, see Machamer Peter , “Galileo and the causes”, in Butts R. E. Pitts J. C. (eds), New perspectives on Galileo (Dordrecht, 1978), 161–80; Lennox James G. , “Aristotle, Galileo, and ‘mixed sciences’”, in Wallace W. (ed.), Reinterpreting Galileo (Washington, D.C., 1986), 29–51; Laird W. R. , “Galileo and the mixed sciences”, in Di Liscia Daniel A. Kessler E. Methuen C. (eds), Method and order in the Renaissance philosophy of nature (Aldershot, 1997), 253–70. More generally, see Rose Paul Lawrence , The Italian renaissance of mathematics (Geneva, 1975); Dear Peter , Discipline and experience: The mathematical way in the Scientific Revolution (Chicago, 1995); Mandosio Jean-Marc , “Entre mathématiques et physique: Note sur les ‘sciences intermédiaires’ à la Renaissance”, in Comprendre et maitriser la nature au Moyen Age: Mélanges d'histoire des sciences offerts à Guy Beaujouan (Geneva, 1994), 115–38; McKirahan Richard D. , “Aristotle's subordinate sciences”, The British journal for the history of science, xi (1978), 197–220; Cleary John J. , Aristotle and mathematics (Leiden, 1995); Brown Gary I. , “The evolution of the term ‘mixed mathematics’”, Journal of the history of ideas, lii (1991), 81–102; Garrison James W. , “Newton and the relation of mathematics to natural philosophy”, Journal of the history of ideas, xlviii (1987), 609–27.

le Bovier de Fontenelle B. , “Sur l'utilité des mathématiques et de la physique et sur les travaux de l'Académie des Sciences”, Oeuvres complètes, vi (Paris, 1994), 37–50; Arbuthnot John , “An essay on the usefulness of mathematical learning”, in Aitken George A. (ed.), The life and works of John Arbuthnot (Oxford, 1892), 409–35; Boyle's 1671 essay “On the usefulness of mathematics to natural philosophy” is in vol. iii of The works of the Honourable Robert Boyle, ed. by Birch Thomas (London, 1744), 425–34. For Bacon's views on mathematics, see Rees Graham , “Mathematics and Francis Bacon's natural philosophy”, Revue internationale de philosophie, xl (1986), 399–426.

For Boyle's views on mathematics see Shapin Steven , “Robert Boyle and mathematics: Reality, representation and experimental practice”, Science in context, ii (1988), 23–58.

Cited by Malet Antoni , “Isaac Barrow on the mathematization of nature: Theological voluntarism and the rise of geometrical optics”, Journal of the history of ideas, lvii (1997), 265–87, pp. 280–1. Barrow's Essay was first published in Latin in 1683. On the tradition of the mixed sciences in England, see Bennett J. A. , “Christopher Wren: Astronomy, architecture, and the mathematical sciences”, Journal for the history of astronomy, vi (1975), 149–84.

Wallis to Oldenburg, 5 December 1668, in Hall A. R. Hall Marie B. (eds), The correspondence of Henry Oldenburg (Madison and London, 1965–86), v, 221.

Huygens to Marquis de l'Hôpital, 29 December 1692, in Huygens C. , Oeuvres complètes (The Hague, 1888–1950), x, 354; the quotation from Seneca was “calculis ludimus, in supervacuis subtilitas tertur”. On Huygens, see Yoder Joella G. , Unrolling time: Christiaan Huygens and the mathematization of nature (Cambridge, 1988).

Shapiro Alan S. (ed.), The optical papers of Isaac Newton (Cambridge, 1984), i, 86; cited by Gabbey Allan , “Newton's mathematical principles of natural philosophy: A treatise on ‘mechanics’?”, in Harman P. M. Shapiro Alan E. (eds), The investigation of difficult things (Cambridge, 1992), 312–13.

I distinguish ‘quantification’ from ‘mathematization’. The first refers to the production of numbers for measuring phenomena through the construction of a metric (a graduated thermometer for example) whereas the second refers to the writing of abstract geometric or algebraic formulations (like the law of free fall or the law of refraction). The first can exist without the second and the latter can be formulated before the former.

For a useful survey of the traditional discussion of mathematization in physics, see Cohen H. Floris , The Scientific Revolution: A historiographical inquiry (Chicago, 1994). Koyré's view is presented in his classic paper “Galileo and Plato”, published in 1943 and reprinted as chap. 2 of his Metaphysics and measurement (Cambridge, 1968). For a synthetic presention see Jorland Gérard , La science dans la philosophie: Les recherches épistémologiques de Alexandre Koyré (Paris, 1981).

See for example Bennett J. A. , “The mechanics' philosophy and the mechanical philosophy”, History of science, xxiv (1986), 1–28; Dear , op. cit. (ref. 3); Biagioli Mario , Galileo courtier (Chicago, 1993).

For a general discussion of mathematization, see Bochner Salomon , The role of mathematics in the rise of science (Princeton, 1966); Bellone Enrico , A world on paper (Cambridge, Mass., 1982). A recent historical analysis of the role of mathematics in physics is provided by Elizabeth Garber, The language of physics: The calculus and the development of theoretical physics in Europe, 1750–1914 (Boston, 1999). See also Heilbron J. L. , Weighing imponderables and other quantitative science around 1800, Historical studies in the physical sciences, Supplement to vol. xxiv/1 (1993). For case studies of mathematization, see Buchwald Jed Z. , “William Thomson and the mathematization of electrostatics”, Historical studies in the physical sciences, viii (1977), 101–36; Wise Norton , “William Thomson's mathematical route to energy conservation: A case study of the role of mathematics in concept formation”, Historical studies in the physical sciences, x (1979), 49–83. None of these works, however, emphasizes the resistances to mathematization.

Henry John , The Scientific Revolution and the origins of modern science (New York, 1997), 21.

Stigler Stephen M. , “Apollo mathematicus: A story of resistance to quantification in the seventeenth century”, Proceedings of the American Philosophical Society, cxxxvi (1992), 93–126. Debus Allen G. , “Mathematics and nature in the chemical texts of the Renaisssance”, Ambix, xv (1968), 1–28.

Gingras Yves , “La substance évanescente de la physique”, Actes du XX^e Congrès International d'Histoire des Sciences (Liège, in press). It should be noted that by concentrating on what mathematics did to physics, I do not wish to suggest that the development of new instruments and new methods of experimentation did not also affect access to the practice of physics; that would be absurd. Although I cannot here develop this aspect of the transformation of the discipline, I do allude to it. As one reviewer suggested, one could also analyse what physics did to mathematics; but that of course would require writing a different, though complementary, paper.

Duhem Pierre , The aim and structure of physical theory (Princeton, 1954), 38–39.

For such an approach, see Cunningham Andrew , “How the Principia got its name: Or, taking natural philosophy seriously”, History of science, xix (1991), 377–92; see also his “Getting the game right: Some plain words on the identity and invention of science”, Studies in the history and philosophy of science, xix (1988), 365–89. The author seems to suggest that it is historiographically ‘illegitimate’ to use words in ways that differ from how the authors themselves used them, as if words could have but a single meaning at any given time.

See for example Boudon Raymond , The unintended consequences of social action (New York, 1982).

On this see Cunningham , “How the Principia got its name” (ref. 18).

For a discussion of the changing content of ‘physics’, see Cannon Susan Faye , “The invention of physics”, chap. 4 of her book Science in culture: The early Victorian period (New York, 1978), 111–36; Kuhn Thomas , The essential tension (Chicago, 1977), 60–64; Heilbron John L. , “Experimental natural philosophy”, in Rousseau G. S. Porter Roy (eds), The ferment of knowledge: Studies in the historiography of eighteenth-century science (Cambridge, 1980), 357–87.

I think, for example, that much of the discussion about the ‘non-existence’ of the Scientific Revolution is based on a confusion of levels and categories of analysis.

For a recent discussion of that question see Jardine Nick , “Uses and abuses of anachronism in the history of the sciences”, History of science, xxxviii (2000), 251–70. Though I essentially agree with the author, I would not use the term ‘anachronism’ to refer to the application of sociological categories to past events (p. 261). Since analytical categories are defined by analysts they need not have been part of the actor's repertoire. It is thus a category mistake to call them ‘anachronistic’ since they are not used as if they were actors' categories. Otherwise we would have to call anachronistic any explanations of past diseases invoking ‘virus’ or ‘microbes’ before the ‘discovery’ of these entities. I know of no historian ready to pay that price. And it should be clear that these categories do not prevent us from looking at how the actors themselves explained these diseases in the absence of the categories of ‘virus’ or ‘microbes’. They are simply different levels of analysis.

Fatio to Huygens, 24 June 1687, in Huygens , Oeuvres (ref. 8), ix, 167–8.

On the concept of force, see Westfall Richard S. , Force in Newton's physics: The science of dynamics in the seventeenth century (London, 1971); Jammer Max , The concept of force: A study in the foundations of dynamics (Cambridge Mass., 1957).

Mouy Paul , Le développement de la physique cartésienne, 1646–1712 (Paris, 1934), 144.

Journal de Trévoux, x (1710), 356.

On Varignon, see Blay Michel , La naissance de la mécanique analytique: La science du mouvement au tournant des XVIIe et XVIIIe siècles (Paris, 1992).

de la Harpe Jacqueline , Le Journal des Savants et l'Angleterre, 1702–1789 (University of California Publications in Modern Philology, xx, no 6; Berkeley, 1941); Desautels Alfred R. , Les Mémoires de Trévoux et le mouvement des idées au XVIIIe siècle, 1701–1734 (Rome, 1956); Pappas John N. , “Berthier's Journal de Trévoux and the philosophes”, Studies on Voltaire and the eighteenth century, iii (1957), 13–63; Ehrard Jean Roger Jacques , “Deux périodiques français du 18^e siècle: Le Journal des Savants et les Mémoires de Trévoux”, in Bolleme G. (eds), Livre et société dans la France du XVIII^e siècle (Paris, 1965), 33–59; Carlson C. Lennart , A history of the Gentleman's Magazine (Providence, 1938); Pailler Albert , Edward Cave et le Gentleman's Magazine (1731–1754) (Lille, 1975).

In his classic book La formation de l'esprit scientifique (Paris, 1938), Gaston Bachelard studied typical works of 18th-century science but only to contrast them with modern 19th- and 20th-century science. Completing this approach, I wish to follow the process of exclusion that transformed scientific practice and led to the situation described by Bachelard as “scientific” as opposed to what he called the “pre-scientific” spirit of the 18th century. It is obvious that I do not have to use these normative categories in order to describe that process. For a more detailed discussion of Bachelard's views in relation to our project, see Gingras Yves , “Mathématisation et exclusion: Socio-analyse de la formation des cités savantes”, in Wunenburger Jean-Jacques (ed.), Gaston Bachelard et l'épistémologie française (Paris, in press).

For a detailed analysis of Leibniz's reaction to Newton's mathematization of natural philosophy, see Meli Domenico Bertoloni , Equivalence and priority: Newton versus Leibniz (Oxford, 1993). As he explains, Leibniz stressed “the insufficiency of purely mathematical laws [and] the need for physical explanations …” (p. 24); see also Gingras Yves , “La dynamique de Leibniz: Métaphysique et substantialisme”, Philosophiques, xxii (1995), 395–405. In the 1730s Jean Bernoulli also devoted two prize-winning essays to the question of the physical cause of gravitation, trying to reconcile vortex motions with the mathematical laws of Kepler and Newton. For him, Newton's vacuum and attraction were “incomprehensible for a physicist” who had to “search the causes of the facts”; “Essai d'une nouvelle physique céleste” in Opera omnia (Geneva, 1742), iii, 266–7. For details, see Shea William , “The unfinished revolution: Johann Bernoulli (1667–1748) and the debate between the Cartesians and the Newtonians”, Shea William (ed.), Revolutions in science: Their meaning and relevance (Canton, 1988), 70–92.

Mémoires de l'Académie Royale des Sciences, 1733, 311.

Histoire de l'Académie Royale des Sciences, 1733, 94.

Aristotle , Metaphysics, 995a, 15–18.

Cited by Home R. W. , “The notion of experimental physics in the early eighteenth-century”, in Pitt J. C. (ed.), Change and progress in modern science (Dordrecht, 1985), 107–31, p. 124.

Newton Isaac , Principia, translated by Cohen I. Bernard Whitman Anne (Berkeley, 1999), 381.

Collinson P. to Colden C. , 27 March 1747, in “The letters and papers of Codwallader Colden, iii: 1743–1747”, Collections of the New York Historical Society for the year 1919 (New York, 1920), 368.

Colden Cadwallader , Principles of action in matter, the gravitation of bodies and the notion of the planets, explained of those principles (London, 1751), preface. Also mentioned in C. Colden to Dr Betts, 25 April 1750, in “The letters and papers of Codwallader Colden, iv: 1748–1754”, Collections of the New York Historical Society for the year 1920 (New York, 1921), 204.

Colden Cadwallader , Explication des premières causes de l'action de la matière et de la cause de la gravitation (Paris, 1751). Since Colden was, through his relation with Franklin, in indirect contact with Abbé Nollet, the latter may have been at the origin of the translation.

Gentleman's magazine, December 1752, 499–500, 570–1, 589–90; and January 1753, 65–66.

Colden , Principles of action in matter (ref. 38), 3.

C. Colden to Collinson P. , 20 June 1745, in “The letters and papers of Codwallader Colden, iii” (ref. 37), 119. The Anglo-American fleet was then attacking the French settlement of the LouisBourg fortress which capitulated on 26 June.

Colden , Principles of action in matter (ref. 38), 2. A similar statement is also found in the 1745 edition, p. v.

Ibid., 3.

Keys Alice M. , Cadwallader Colden: A representative eighteenth century official (New York, 1906), 13–14.

Euler to Wetstein, 21 November 1752, vera copia in the letter from P. Collinson to Colden, 7 March 1753, in “The letters and papers of Codwallader Colden, iv” (ref. 38), 356.

Euler to Le Sage, 16 April 1763, in Notice de la vie et des écrits de George-Louis Le Sage. Redigée d'après ses notes par Pierre Prévost. Suivie d'un opuscle de Le Sage sur les causes finales: D'extraits de sa correspondance avec divers savants et personnes illustres (Geneva, 1805), 386.

On Euler's views on mechanical explanation, see Wilson Curtis , “Euler on action-at-a-distance and fundamental equations in continuum mechanics”, in Harman Shapiro (eds), The investigation of difficult things (ref. 9), 399–420.

Euler to Wetstein , op. cit. (ref. 38), 356.

On these exchanges, see “The letters and papers of Codwallader Colden, iv” (ref. 38), 378 (letter from Collinson, March 10, 1754,), 395–396 (Colden to Collinson, July 7 1753); the citation is on p. 406 (Collinson to Colden Sept. 1, 1753). He seems not to have succeeded for I could find nothing on gravitation by Colden in the Magazine after the extracts from his book appeared in January 1753. In his letter to Colden on 10 March 1754, Collinson noted that “your answer to pro: Euler is not yett publis'd…”, ibid., 378.

Shapin , op. cit. (ref. 5), 42.

Varignon P. , Nouvelles conjectures sur la pesanteur (Paris, 1690). It is interesting to note that though historians have studied in detail Varignon's contribution to analytical mechanics, they pass over in silence this essay totally devoted to a mechanical explanation of gravity. Somehow in the 1690s, Varignon seems to have had a conversion to the mathematical approach and to have completely abandoned this project.

Journal de Trévoux, xlii (1742), 1093; Healy George R. , “Mechanistic science and the French Jesuits: A study of the response of the Journal de Trévoux (1701–1762) to Descartes and Newton”, Ph.D. dissertation, University of Minnesota, 1956, p. 198, attributes, plausibly, the text to Castel, though it is not signed.

Castel Père Louis , Vrai système de physique générale de M. Isaac Newton. A la portée du commun des physiciens (Paris, 1743).

Cited by Bachelard , op. cit. (ref. 30), 230.

Cited by Schier Donald S. , Louis-Bertrand Castel, anti-Newtonian scientist (Iowa, 1941), 113.

Golinski J. , “Precision instruments and the demonstrative order of proofs in Lavoisier's chemistry”, Osiris, n.s., ix (1994), 30–47 and idem, Science and public culture: Chemistry and enlightenment in Britain, 1760–1820 (Cambridge, 1992), 138.

By contrast, the relatively wide distribution of X-ray apparatus at the end of the nineteenth century made that phenomena accessible to non-professional physicists, see Gingras Yves , “La réception des rayons X au Québec: Radiographie des pratiques scientifiques”, in Fournier Marcel Gingras Yves Keel Othmar (eds), Sciences et médecine au Québec: Perspectives sociohistoriques (Sainte-Foy, 1987), 69–86.

Castel exchanged letters with Diderot on his Lettre sur les aveugles; see Schier , op. cit. (ref. 56), 48.

See Pappas John , “Lesprit de finesse contre l'esprit de géométrie: Un débat entre Diderot et d'Alembert”, Sudies on Voltaire and the eighteenth century, lxxxix (1972), 1229–53; and more generally Hankins T. L. , Jean d'Alembert: Science and the Enlightenment (Oxford, 1970); Paty Michel , D'Alembert (Paris, 1998).

Diderot D. , “De l'interprétation de la nature”, Oeuvres philosophiques (Paris, 1961), 177–244, p. 216.

See for example ibid., 214 where he writes: “nos faiseurs de cours d'expérience ressemblent un peu à celui qui penserait avoir donné un grand repas parce qu'il aurait eu beaucoup de monde à sa table.”.

Histoire de l'Académie des Sciences, 1745, 28; cited by Brunet Pierre , Les physiciens hollandais et la méthode expérimentale en France au XVIIIe siècle (Paris, 1926), 132.

Journal encyclopédique, February 1769, 131.

D'Alembert J. R. , “Introduction aux recherches sur la précession des équinoxes et sur la nutation de l'axe de la Terre dans le système newtonien”, in Oeuvres complètes (Geneva, 1967), i, 437–50, p. 437. On Buffon's critiques of mathematics, also published in 1749, see Roger Jacques , Buffon (Paris, 1989), 263–5.

D'Alembert , “Introduction”, 450.

See for example ibid., 344, 353.

Ibid., 438.

D'Alembert , “Discours préliminaire ou analyse des recherches sur différents points importants du système du monde”, in Oeuvres complètes (ref. 65), i, 349–91, p. 355.

See for example, ibid., 352, 356, 358, 438; see also Paty Michel , “Rapports des mathématiques et de la physique chez D'Alembert”, Dix-huitième siècle, no. 16 (1984), 69–79.

On the diffusion of Newtonianism among the public, see Stewart Larry , The rise of public science: Rhetoric, technology, and natural philosophy in Newtonian Britain, 1660–1750 (Cambridge, 1992).

Clairault to Euler, 19 June 1749, in Juskevic Adolf P. Taton René (eds), Correspondance de Leonhard Euler avec A. C. Clairault, J. D'Alembert et J. L. Lagrange (Basel, 1980), 186. On the debate over the shape of the Earth, see Greenberg John L. , The problem of the Earth's shape from Newton to Clairault (Cambridge, 1995). For a fascinating cultural history of this period see Badinter Elizabeth , Les passions intellectuelles I: Désirs de gloire (1735–1751) (Paris, 1999).

Halley to Newton, 29 June 1686, in Turnbull H. W. (ed.), The correspondance of Isaac Newton, ii: 1676–1687 (Cambridge, 1960), 443. Mordechai Feingold analysed the debate on the place of mathematics in the Royal Society in “Mathematicians and naturalists: Sir Isaac Newton and the Royal Society”, in Buchwald Jed Z. Cohen I. Bernard (eds), Isaac Newton's natural philosophy (Cambridge, Mass, 2000), 77–102.

Cited by Brunet , Les physiciens hollandais (ref. 63), 123.

Massière M. , Réflexions critiques sur le système de l'attraction (Nice, 1759), p. v.

Ibid., p. viii.

Ibid., p. x.

Ibid., p. xvii.

Ibid., 402.

de La Ville Cte de Lacépède Bernard-Germain Etienne , Théorie des comètes pour servir au système de l'électricité universelle, suivie d'une lettre critique sur l'attraction (London and Paris, 1784), 66.

Mangin J. , Le tombeau de l'attraction universelle ou démonstrations incontestables de la fausseté du système de l'attraction newtonienne (Verdun, 1826), 13.

Demonville Antoine-Louis Guénard , Vrai système du Monde, Part 2 (Paris, 1837), Avis de l'auteur.

Viel Charles-François , De l'impuissance des mathématiques pour assurer la solidité des batimens et Recherches sur la construction des ponts (Paris, 1805). For a general discussion of the debates accompanying the use of mathematics in the design of bridges, see Kranakis Eda , Constructing a bridge: An exploration of engineering culture, design, and research in nineteenth-century France and America (Cambridge, 1997); and Picon Antoine , L'invention de l'ingénieur moderne: L'École des Ponts et Chaussées 1747–1851 (Paris, 1992), 76.

Heilbron notes, for example, that the Coffee House Physical Society “banned everything that smelt of mathematics”, Heilbron , “Experimental natural philosophy” (ref. 21), 364. In Weighing imponderables (ref. 13), 31, he notes that at the end of the 18th century, the German textbook-writer Gren F. A. C. , “who had been brought up in the older qualitative, inclusive natural science … no doubt felt menaced by the calculators”. Other reactions are noted on pp. 147–9. For another example of an exchange over the appropriate use of mathematics in physics, see the discussion between Franz Ernst Newman and Ludwig Moser in Olesko Kathryn M. , Physics as a calling: Discipline and practice in the Königsberg seminar for physics (Ithaca, 1991), 93–95.

Bourdieu Pierre , “The scientific field and the social conditions for the progress of reason”, Social science information, xiv/6 (1975), 19–47. The formation of a field necessarily implies a form of “boundary work” concerning the proper attribution of domains and methods to ‘physics’, ‘mathematics’ or ‘applied mathematics’: Including or excluding mathematical techniques in the study of physical phenomena was a way of imposing the legitimate definition of a field as well as the legitimate methods of inquiry. On boundary work, see Gieryn Thomas F. , Cultural boundaries of science (Chicago, 1999).

Biot Jean-Baptiste , Traité de physique expérimentale et mathématique (Paris 1816), p. xi. On Biot, see Frankel Eugene , “J. B. Biot and the mathematization of experimental physics in Napoleonic France”, Historical studies in the physical sciences, viii (1977), 33–72.

Heilbron , “Experimental natural philosophy” (ref. 21), 367–75.

J. D. Forbes to Whewell W. , 29 May 1831, cited by Smith Crosbie , “Mechanical philosophy and the emergence of physics in Britain”, Annals of science, xxxiii (1976), 3–29, p. 25.

Forbes J. D. to Whewell W. , 8 August 1833, cited by Smith , op. cit. (ref. 88), 27. On Whewell's attitude towards the relation between physics and mathematics, see Becher Harvey W. , “William Whewell and Cambridge mathematics”, Historical studies in the physical sciences, xi (1980), 1–48; Fish Menachem , “A philosopher's coming of age: A study of erotetic intellectual history”, in Fish Menachem Schaffer Simon (eds), William Whewell: A composite portrait (Cambridge, 1991), 31–66.

Faraday to Maxwell, 25 March 1857; Harman P. M. (ed.), The scientific letters and papers of James Clerk Maxwell (Cambridge, 1990), 548.

Faraday to Maxwell, 13 Nov. 1857; ibid., 552, note 13.

Morus Iwan Rhys , “Different experimental lives: Michael Faraday and William Sturgeon”, History of science, xxx (1992), 1–28.

Faraday M. , “On the conservation of force”, Philosophical magazine, 4th ser., xiii, no. 86, April 1857, 225–39, p. 238.

Brücke E. , “On gravitation and the conservation of force”, Philosophical magazine, 4th ser., xv, no. 86, February 1858, 81–90, p. 82.

Harman , op. cit. (ref. 90), 429, 671.

On the physics discipline, see Caneva Kenneth L. , “From galvanism to electrodynamics: The transformation of German physics and its social context”, Historical studies in the physical sciences, ix (1978), 63–169; Wilson David B. , “Experimentalists among the mathematicians: Physics in the Cambridge natural sciences tripos, 1851–1900”, Historical studies in the physical sciences, xii (1982), 325–71; Olesko , op. cit. (ref. 84); Jungnickel Christa McKormach Russell , Intellectual mastery of nature: Theoretical physics from Ohm to Einstein (Chicago, 1986); Silliman Robert , “Fresnel and the emergence of physics as a discipline”, Historical studies in the physical sciences, iv (1974), 137–62; Sviedrys R. , “The rise of physics laboratories in Britain”, Historical studies in the physical sciences, vii (1976), 405–36; Kevles D. , The physicists: The history of a scientific community in modern America (New York, 1978); Gingras Yves , Physics and the rise of scientific research in Canada (Montreal and Kingston, 1991).

Bachelard Gaston , L'activité rationaliste de la physique contemporaine (Paris, 1951), 42.

McLellan James E. III , Science reorganized: Scientific societies in the eighteenth century (New York, 1985), 360 (his chap. 7 deals with tighter control of access to membership in scientific academies); on specialized scientific journals, see ibid., 257–9, and McLellan J. E. , “The scientific press in transition: Rozier's Journal and the scientific societies in the 1770s”, Annals of science, xxxvi (1979), 425–49.

Brunet Pierre , L'introduction des théories de Newton en France au XVIIIe siècle: Avant 1738 (Paris, 1931); Aiton A. J. , The vortex theory of planetary motion (London, 1972); Guerlac Henry , Newton on the Continent (Ithaca, 1981), chap. 3; Koyré Alexandre , Newtonian studies (London, 1965).

For an excellent discussion of changing notions of explanation in the history of physics, see Gaukroger Stephen , Explanatory structures: Concepts and explanation in early physics and philosophy (Hassocks, 1978). For the case of mathematics, see Mahoney Michael S. , “Changing canons of mathematical and physical intelligibility in the later 17th century”, Historia mathematica, ii (1984), 417–23.

For the different interpretations given by historians to the term ‘mechanical philosophy’, see Cohen Floris , op. cit. (ref. 11), 142–5. Here I mean by ‘mechanical explanation’ one that provides an efficient cause based on contact forces; two roads were open: If void was admitted, that force could be obtained through corpuscular interactions (as in Lesage's theory of gravitation); if void was excluded it could be through the action of a fluid (as in Euler's theory) or through the movement of corpuscules of different size filling all space as in Descartes's system. From this point of view, Newton's mathematization of gravitation was a demechanization of the world picture and not a mechanization as suggested by Dijksterhuis E. J. , The mechanization of the world picture (Princeton, 1986).

Shea William R. , “Descartes as critic of Galileo”, in Butts Pitt (eds), New perspectives on Galileo (ref. 3), 139–59.

Westman R. S. , “The astronomer's role in the sixteenth century: A preliminary study”, History of science, xviii (1980), 105–47; for a detailed analysis of the sixteenth-century debate on the relation between natural philosophy and astronomy and on the nature of astronomical explanation, see Jardine N. , The birth of history and philosophy of science: Kepler's “A defence of Tycho against Ursus” with essays on its provenance and significance, rev. edn (Cambridge, 1988).

Koyré A. , La révolution astronomique (Paris, 1961), 364.

On Newton's style of mathematical physics, see Cohen I. Bernard , The Newtonian revolution (Cambridge, 1980); Kroes P. A. , “Newton's mathematization of physics in retrospect”, in Scheurer P. B. Debrock G. (eds), Newton's scientific and philosophical legacy (Dordrecht, 1988), 253–67.

This is a ‘category mistake’ from the point of view of the actors. For us of course, it is no longer a category mistake since we have accepted Newton's view of what physics is. The same applies to Kepler's “celestial physics”.

Journal des sçavans , 2 August 1688, 154; on the first reviews of the Principia, see Cohen I. B. , “The review of the first edition of Newton's Principia in the Acta eruditorium, with notes on the other reviews”, in Harman Shapiro (eds), The investigation of difficult things (ref. 9), 323–53.

Cited by Schier , op. cit. (ref. 56), 199.

Note that the actors used the categories of “Cartesians” and “Newtonians” in a variety of ways. A common feature to all is, I think, that the first must involve a mechanical explanation of all phenomena in the sense defined above (ref. 101) while the other can content itself with a mathematical formulation and leave unanswered the question of the mechanical action.

On habitus, see Bourdieu Pierre , The logic of practice (Stanford, 1990).

Schier , op. cit. (ref. 56), 58.

Ibid., 89.

Castel , op. cit. (ref. 54), 13.

Ibid. , 37. On Huygens's reaction to Newton's Principia, see De A. Martins Roberto , “Huygens's reaction to Newton's gravitational theory”, in Field J. V. James Frank A. J. L. (eds), Renaissance and revolution: Humanists, scholars, craftmen and natural philosophers in early modern Europe (Cambridge, 1993), 203–13.

Castel , op. cit. (ref. 54), 52.

Ibid., 94.

Ibid., 95, italics in the original.

Ibid., 97.

Ibid., 98–99.

Ibid., 121.

Ibid., 121.

Ibid., 253.

Ibid., 304.

Ibid., 302.

Ibid., 348.

D'Alembert , “Introduction aux recherches sur la précession des équinoxes et sur la nutation de l'axe de la Terre dans le système newtonien”, in Oeuvres complètes (ref. 65), i, 437–50, p. 450.

D'Alembert , Essai sur les élémens de philosophie , in Oeuvres complètes (ref. 65), i, 115–348, p. 345.

Ibid., 341.

le Bovier de Fontenelle B. , Oeuvres (Paris, 1996), vii, “Préface de l'éditeur”, 377–82, p. 377.

Ibid., 378.

de Marivetz Baron et Goussier , Physique du monde (Paris, 1780), v, 57, cited by Bachelard , La formation de l'esprit scientifique (ref. 30), 231.

D'Alembert , op. cit. (ref. 127), i, 341. This views recalls Max Planck's when he wrote: “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it”; Planck's own scientific autobiography and other papers, transl. by Gaynor F. (New York, 1949), 33–34.

Ibid., 346.

Diderot , op. cit. (ref. 61), 216.

Biot Jean-Baptiste , Traité de physique expérimentale et mathématique (Paris 1816), p. xii. On Biot, see Frankel , “J. B. Biot and the mathematization of experimental physics in Napoleonic France” (ref. 86).

Biot , Traité (ref. 135), p. xiv.

Ibid., p. xxiii.

Ibid., p. xv.

Harman (ed.), op. cit. (ref. 90), i, 671.

Ibid., 671.

Ibid., 672.

Fish suggests that, for Whewell , “the formal system employed should fully ‘cash out’ empirically”, op. cit. (ref. 89), 45.

University College London, Fleming Coll. Ms Add 122/37. My thanks to Sungook Hong for giving me access to his notes taken from this manuscript.

Harman P. M. , Metaphysics and natural philosophy: The problem of substance in classical physics (Brighton, 1982), 145–6.

Cited by Harman , ibid., 146.

Cited in ibid., 132. On the meaning of the use of Lagrangian formulations, see Klein Martin J. , “Mechanical explanations at the end of the nineteenth century”, Centaurus, xvii (1972), 58–82; Bunge Mario , “Lagrangian formulation and mechanical explanation”, American journal of physics, xxv (1957), 211–18.

Cassirer E. , Substance and function (New York, 1953), provides a philosophical analysis of this process.

Comte A. , Cours de philosophie positive (Paris, 1968), ii, 340.

Lorentz H. A. , “Considerations on gravity”, Amsterdam Koninklijke Akademie Physica, ii (1900), 559–74. On Le Sage, see Aronson Samuel , “The gravitational theory of Georges-Louis Le Sage”, The natural philosopher, iii (1964), 53–74.

For the period from 1700 to 1900, very few proposed explanations of gravity were published in journals controlled by the physics community. I will later publish a more detailed history of the various attempts at a mechanical explanation of gravity. For a brief survey, see van Lunteren F. H. , “Gravitation and nineteenth-century physical worldviews”, in Scheurer P. B. Debrock G. (eds), Newton's scientific and philosophical legacy (Dordrecht, 1988), 161–73. Outside the field of physics, attempts at providing a mechanical explanation still goes on; for relatively recent examples, see Larson Dewey B. , Beyond Newton: An explanation of gravity (Portland, 1964); de Puymorin René , L'origine de la gravitation (Paris, 1975).

Lorentz H. A. , The theory of the electron (New York, 1953), 43; cited by Jammer M. , Concepts of mass in contemporary physics and philosophy (Princeton, 2000), 36.

For the conceptual evolution of the concept of mass, see Jammer Max , Concepts of mass in classical and modern physics (Cambridge, Mass., 1961) and his more recent Concepts of mass in contemporary physics and philosophy (ref. 151).

For a survey of these developments, see Cantor G. N. Hodge M. J. S. (eds), Conceptions of ether: Studies in the history of ether theories, 1740–1900 (Cambridge, 1981).

Wigner E. P. , “The unreasonable effectiveness of mathematics in the natural sciences”, Communications on pure and applied mathematics, xiii (1960), 1–14. For more examples of such preoccupations by scientists and for a philosophical approach to the question, see Steiner Mark , The application of mathematics as a philosophical problem (Cambridge, Mass., 1998). Husserl also provided a philosophical analysis of the meaning of the mathematization of nature in Edmund Husserl, The crisis of European sciences and transcendental phenomenology (Evanston, 1970), 21–59.

Miller Arthur I. , “Redefining anschaulichkeit”, in Shimony Abner Feshbach Herman (eds), Physics and natural philosophy (Cambridge, Mass., 1982), 376–411; Serwer Daniel , “Unmechanischer zwang: Pauli, Heisenberg, and the rejection of the mechanical atom, 1923–1925”, Historical studies in the physical sciences, viii (1977), 189–256.

Bohm David Peat F. David , Science, order, and creativity (New York, 1987), 7, 9.

Bohm's view are now undergoing a revival; see Holland Peter R. , The quantum theory of motion (Cambridge, 1993), and Olwell Russell , “Physical isolation and marginalization in physics: David Bohm's cold war exile”, Isis, xc (1999), 738–56.

For a recent critique of the lack of physical explanations in the modern mathematical approach to physics, see Athearn Daniel , Scientific nihilism: On the loss and recovery of physical explanation (Albany, 1994).

For very recent examples, see Nature, cccciv, issue of 2 March 2000, 28–29; Science, cclxxxvii, issue of 7 January 2000, 49–50.

See, for example, Parshall Karen Hunger , “Chemistry through invariant theory? James Joseph Sylvester's mathematization of the atomic theory”, in Therman Paul H. Parshall Karen Hunger (eds), Experiencing nature (Dordrecht, 1997), 81–111; Simoes Ana Gavroglu Kostas , “Quantum chemistry qua applied mathematics: The contributions of Charles Alfred Coulson (1910–1974)”, Historical studies in the physical and biological sciences, xxix (1999), 363–406, and idem, “Quantum chemistry in Great Britain: Developing a mathematical framework for quantum chemistry”, Studies in history and philosophy of modern physics, xxxi (2000), 511–48; Israel Giorgio , “The emergence of biomathematics and the case of population dynamics: A revival of mechanical reductionism and darwinism”, Science in context, vi (1993), 469–509.

Gingras Yves , “La substance évanescente de la physique” (ref. 16).