Abstract
This chapter focuses on three episodes in the history of observatories between 1793 and 1846, to examine the epistemic and social foundations of the conception of mathematics as a tool. This is an interesting case because in the observatory culture sophisticated uses of mathematics have always been deployed side-by-side with tools and instruments taken in a material sense. We look at the Paris Observatory under the French Revolution to introduce crucial distinctions between tools and instruments. We then turn to the Royal Observatory Greenwich to show how specific social techniques were developed in order to improve the precision of the mathematical tool. We finally consider the case of Bessel functions developed at the Könisberg Observatory to show that astronomers also tinkered with the mathematical instrument, like they did with telescopes, to increase its precision. This chapter is intended as a contribution to the epistemological debate regarding the “unreasonable effectiveness,” or applicability of mathematics to the natural sciences. I argue that if we take seriously the analogy between mathematics and tools and instruments, this effectiveness is rather similar to that of tools and instruments; that is, the effectiveness of mathematics is the product of specific practices performed by socially diversified communities precisely to this effect.
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Notes
- 1.
William Herschel to Caroline Herschel, June 5, 1782 (Lubbock 1933, 115).
- 2.
“Remarks on the neglect, by the Junior Assistants, of the course of education and scientific preparation recommended to them.” Airy Papers, Cambridge University Library, RGO 6/43, 235. About this memo, see Aubin (2009), 273 and 276–277.
- 3.
Jedes Instrument wird auf diese Art zweimal gemacht, einmal in der Werkstatt des Künstlers von Messing und Stahl; zum zweitenmale aber von dem Astronomen aud seinem Papiere, durch die Register der nöthingen Verbesserungen, welche er durch seine Untersuchung erlangt.
- 4.
In this article, I shall not refer to the “mathematical instruments” tradition, which obviously played a role in shaping the conception of instruments in the observatory culture. By the late eighteenth century, astronomers were relying on highly skilled makers, such as Jesse Ramsey and Edward Troughton, fellows of the Royal Society whose names were, as we shall see, routinely attached to the high-precision instruments they produced (Bennett 2011; Chapman 1995).
- 5.
As expressed by Gottlob Frege in 1884, there is a simple solution to the applicability problem: “The laws of numbers, therefore, are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgments holding good of things in the external world: they are laws of the laws of nature” (Frege 1980), quoted by (Withold 2006, 74). For another visions of mathematics as a language in this context, see Sarukkai (2005).
- 6.
- 7.
See “Pièce justificative N ∘ X: Inventaire des instrumens de l’Observatoire national de Paris en 1793,” Archives nationales F17/1219; Archives de l’Observatoire D.5.38; repr. (Cassini 1810), 208–217). The date of this report is uncertain, but September 19, 1793 is a reasonable estimate (Wolf 1902, 349). Dated “19 of the first month of Year II” [which should be 19 vendémiaire, that is, October 10], the report was said to be completed when discussed by the Commission temporaire des arts on September 26. According to his biographer, Cassini left the Observatory never to come back, on October 3rd (S.-Devic 1851, 205).
- 8.
- 9.
Under the Terror, Cassini was jailed in the English Benedictine Convent on February 14, 1794. He was freed after Robespierre’s downfall, on August 5, 1794, but never returned to the Observatory. For more information on the history of the Paris Observatory during the French Revolution, I refer to Chapin (1990) and Aubin (2013); more complete accounts in French can be found in Cassini (1810), S.-Devic (1851), Wolf (1902).
- 10.
N ∘ 1. Lunette achromatique de Dollond, objectif à trois verres de 42 lignes d’ouverture, 3 pieds et demi de foyer ; elle a trois oculaires, un terrestre et deux célestes ; elle est montée sur un pied d’acajou à colonne de cuivre avec tous ses mouvemens; à cette lunette s’adapte un héliomètre de Bouger, objectif simple, plus un micromètre filaire de Hautpois.
- 11.
About instruments in state of disrepair, see Schaffer (2011).
- 12.
Outil, (…) instrument dont les ouvriers & artisans se servent pour travailler aux différens ouvrages de leur profession, art & métier ; tels sont les marteaux, les compas, les rabots, les équerres, les villebrequins, &c. (…) Nous ajoutons seulement que les ouvriers mettent quelque différence entre les outils & les instrumens ; tout outil étant instrument, & tout instrument n’étant point outil.
- 13.
Father Cotte, a cleric, is known for his work on meteorology and on the popularization of natural history, physics, and astronomy (Pueyo 1994).
- 14.
On entend par Machine une combinaison de plusieurs machines simples, telles que le levier, le treuil, la poulie, etc. dont le résultat est de suppléer aux forces de l’homme et de produire de grands effets en peu de tems et avec peu de dépense dans toutes les Opérations mécaniques où elles sont employées (…). L’Instrument est bien aussi une espèce de machine, mais susceptible d’une très-grande précision, pour pouvoir être employée dans les Opérations scientifiques qui demandent de l’exactitude, comme l’astronomie, la géométrie pratique, la chirurgie, etc. L’Appareil est une combinaison de différens instrumens dont la réunion concourt à démontrer les vérités physiques, mathématiques, chimiques, etc. L’Outil est un instrument simple, le plus souvent de l’espèce du coin, qui sert dans les Opérations manuelles et habituelles des arts et des métiers.
- 15.
It is interesting to note that, in the definition of the word instrument, the Oxford English Dictionary today explains, similarly, that the distinction between tools, instruments, and machines, is based on social, rather than lexicographical, grounds: “Now usually distinguished from a tool, as being used for more delicate work or for artistic or scientific purposes; a workman or artisan has his tools, a draughtsman, surgeon, dentist, astronomical observer, his instruments. Distinguished from a machine, as being simpler, having less mechanism, and doing less work of itself; but the terms overlap.” Note added to the definition of “Instrument” (www.oed.com).
- 16.
On the history of the Dudley Observatory, see Wise (2004).
- 17.
Airy, “Remarks on the neglect, by the Junior Assistants, of the course of education and scientific preparation recommended to them” (Dec. 4, 1861). RGO 6/43, 235.
- 18.
Several slightly different copies of this memo are extent in Airy’s papers at the Cambridge University Library. A first draft was written on November 20, 1856 and a slightly revised version was adopted on May 10, 1857. In the following I quote from RGO 6/43, 170–175. For a more detailed analysis of this memo, see Aubin (2009, 277).
- 19.
On the division of computing labor at Greenwich, see Grier (2005).
- 20.
RGO 6/24/33: Airy’s Diary
- 21.
Airy to Bowman, January 8, 1839. RGO 6/526, 86.
- 22.
Thomas to Airy, January 21, 1839. RGO 6/525, 29, orig. emphasis; (Dunkin 1999, 72).
- 23.
All quotations in the above paragraph from Thomas to Airy, January 21, 1839. RGO 6/524 File 10bis, 352, orig. emphasis. “Units of the same kind but of different magnitudes, as pounds, shillings, pence, and farthings, which are units of value …are called units of different denominations” (De Morgan 1836, 25).
- 24.
- 25.
All quotes are from Thomas to Airy, January 21, 1839. RGO 6/525, 29.
- 26.
Thomas to Airy, undated [1838]. RGO 6/525, 15, orig. emphasis. Esq. Mathematical Master at the Royal Naval School, Camberwell, Foley was later elected a Fellow of the Royal Astronomical Society; see Monthly Notices of the Royal Astronomical Society 6 (1844), 52.
- 27.
Stratford’s opinions about computers are in Thomas to Airy, January 21, 1839. RGO 6/525, 29.
- 28.
Thomas to Airy, January 21, 1839. RGO 6/525, 29.
- 29.
- 30.
Bessel to Airy, October 5, 1845. RGO 6/530, 75–76. I quote from one of the two (nearly identical) English translations in Airy’s papers (ibid., 76–78 and 79–80).
- 31.
(Bessel 1819), 19; trans. (Hoffmann 2007), 346, my emphasis: “die eine enthält die eigentlichen Beobachtungsfehler, die von unzähligen zufälligen Ursachen abhängen und deshalb den allgemeinen Sätzen der Wahrscheinlichkeitsrechnung folgend angesehen werden können; die andere begreift die von beständig einwirkenden Ursachen herrührenden, der Abweichung der Instrumente von ihrer mathematischen Idee, oder ihrer Behandlungsart zuzuschreibenden.”
- 32.
- 33.
- 34.
- 35.
Bessel (1824); repr. Bessel (1875–1876), 1:86: “Ob aber die astronomischen Theorien allenthalben in so grosser übereinstimmung mit den Beobachtungen sind, dass dadurch jeder Zweifel an der Wahrheit der Newton’schen Annahme zurückgewiesen wird, dieses ist eine Frage, welche wohl Niemand bejahen wird, deren genaue Erörterung jedoch sehr wichtig ist und die grössten Fortschritte der Wissenschaft verheisst.” For a study of Bessel’s cosmological understanding of Newton’s law, see Merleau-Ponty (1983), 119–122.
- 36.
Excerpt from Le Verrier inaugural lecture at the Sorbonne; quoted in Revue scientifique et industrielle 28 (1847), 131.
- 37.
Humboldt to Jacobi, December 22, 1846 (Humboldt and Jacobi 1987, 103).
- 38.
Jacobi to Humboldt, December 26, 1846; (Humboldt and Jacobi 1987, 104): “Du lieber Gott! Hier heißt es nicht, Gedanken tief, sondern Hand flink.” Below, Jacobi added: “dieser Sachen kriegen erst durch die wirkliche numerische Ausfürung Werth, und es ist langweilich, jeder dünnen Gedanken sogleich mit 10,000 Logaritmen escortiren so müssen” (Humboldt and Jacobi 1987, 105–106). Humboldt himself called Jacobi’s a “donnerdend Brief” (Humboldt and Jacobi 1987, 109).
- 39.
Jacobi to Humboldt, December 21, 1846 (Humboldt and Jacobi 1987, 100): “wenn Leverrier mit dem Auge der Mathematik einen neuen Planeten gesehen hat, habe ich damals der Mathematik ein neues Auge eingesetzt.” Jacobi was referring to his theory of elliptic functions (1829) which was used by Le Verrier in his work.
- 40.
Jacobi to Legendre, July 2, 1830 (Jacobi 1881–1884, 1:453): “le but unique de la science, c’est l’honneur de l’esprit humain, et (…) sous ce titre, une question de nombres vaut autant qu’une question de système du monde.” In a complementary approach, one might also consider Bessel acting as a research-technologist in the sense put forward in (Shinn 2008).
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Aubin, D. (2017). On the Epistemic and Social Foundations of Mathematics as Tool and Instrument in Observatories, 1793–1846. In: Lenhard, J., Carrier, M. (eds) Mathematics as a Tool. Boston Studies in the Philosophy and History of Science, vol 327. Springer, Cham. https://doi.org/10.1007/978-3-319-54469-4_10
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