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Fermat’s Dilemma: Why Did He Keep Mum on Infinitesimals? And the European Theological Context

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Abstract

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function.

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Notes

  1. See further in note 35.

  2. “Following the verdict of the Tribunal of the Holy Office, the split–which would prove to be irreversible–between the truth of science and the truth of faith appeared in broad daylight.”

  3. Cf. Katz et al. (2013), Bascelli et al. (2014, Section 3), Błaszczyk et al. (2017, Section 4).

  4. These two themes were explored more fully in recent articles in Erkenntnis (Katz and Sherry 2013), HOPOS (Bascelli et al. 2016), Journal for General Philosophy of Science (Bair et al. 2017) and elsewhere.

  5. To a modern reader familiar with the calculus the procedure is reminiscent of quotients of increments occurring in the definition of the derivative but the period under discussion precedes the calculus of Newton and Leibniz.

  6. “[Let there be] adequated, to speak like Diophantus” (here we use adequate as a verb–rather than as an adjective–in a neologism aiming to convey the meaning of the Latin). See further in Sect. 1.11.

  7. Herbert Breger attempts to account for Fermat’s treatment of the cycloid without mentioning any notion of smallness but ends up falling back on the condition “if the point E is not too far away of the point D, then...” (Breger 1994, p. 206). Breger’s biased criticism of Paul Tannery’s Fermat scholarship is dealt with in Sect. 2.3. Breger’s flawed rendering of Fermat’s method is dealt with in Sect. 2.6.

  8. Though apparently straighforward, Weil’s thesis has been resisted by a small number of Fermat scholars recently; see Sect. 2.1.

  9. We therefore reject Breger’s claim that “[t]he idea that the mathematically conservative Fermat ... is supposed to have calculated using infinitesimals is a bold hypothesis for which there are no textual proofs” (Breger 2013, p. 23).

  10. Without referring to them explicitly as infinitely small, of course. We should mention that Mahoney’s interpretation of Fermat’s method of adequality is incompatible with ours.

  11. The pair may be familiar to the reader as the leaders of opposing schools in the debate over geometric algebra; see e.g., [Weil 1978/79].

  12. Jean d’Espagnet was interested in the work of Robert Fludd (1574–1637), whom Kepler had attacked. That alone might have brought d’Espagnet to Kepler’s books. While it is plausible that d’Espagnet’s library at Bordeaux should have included Kepler’s works, the question of Kepler’s possible influence on Fermat’s method is a subject of long-standing scholarly controversy exhaustively covered in Cifoletti (1990, pp. 40–60). For a summary of Kepler’s contribution to infinitesimal techniques of barrel measuring see Jongmans (2008).

  13. A possible exception is the work of Mengoli, cf. (Massa 1997), though Mengoli carefully avoided the language of indivisibles.

  14. For each pair of complementary infinite subsets of \({\mathbb {N}}\), such a measure \(\xi \) “decides” in a coherent way which one is “negligible” (i.e., of measure 0) and which is “dominant” (measure 1).

  15. Leibniz actually used a symbol that looks more like \(\sqcap \) [see Katz and Sherry (2013)] but the latter is commonly used to denote a product.

  16. In Ad eamdem methodum [item III in Fermat (1891)], Fermat refers to A as incognitam (unknown) (Fermat 1891, p. 140). A text starting with the words “Je veux par ma méthode” is a translation (of the above) dating from 1638 in old French first published in Fermat (1922). This text refers to A as an inconnue (Fermat 1922, p. 74, line 7) and similarly at Fermat (1922, p. 80, line 4). Fermat similarly describes A as inconnue in a 22 october 1638 letter to Mersenne (Fermat 1894, pp. 170, 175).

  17. Thus Breger’s claim to the effect that “in the first text on the method sent to Descartes, adaequare means equality” (Breger 2013, p. 21) has no basis; see Sect. 2.1.

  18. Mahoney is using adequate as an English verb; cf. note 5.

  19. This is contrary to Breger’s claim at Breger (2013, p. 28, line 3). Breger reproduces only a misleadingly truncated fragment of Fermat’s Latin phrase, in footnote 50 there to support his dubious claim. Breger’s fragment is “duo homogenea maximae aut minimae aequalia.” However, Breger’s fragment does not make sense grammatically, because the words “maximae/minimae” only make sense in Fermat’s full phrase as Datives construed with “aequalia”, namely “two homogeneous terms equal to the maximum or the minimum”. The two homogeneous expressions are not equated but rather adequated by Fermat. In the 1638 text mentioned in note 16 Fermat writes (in old French) explicitly: “comme s’ils estoient esgaux, bien qu’en effect ils ne le soient pas” (Fermat 1922, p. 74, lines 15–16).

  20. Latin original: “Consideramus nempe in plano cujuslibet curvae rectas duas positione datas, quarum altera diameter, si libeat, altera applicata nuncupetur. Deinde, jam inventam tangentem supponentes ad datum in curva punctum, proprietatem specificam curvae, non in curva amplius, sed in invenienda tangente, per adaequalitatem consideramus et, elisis (quae monet doctrina de maxima et minima) homogeneis, fit demum aequalitas quae punctum concursûs tangentis cum diametro determinat, ideoque ipsam tangentem” (Fermat 1891, p. 159).

  21. For the sake of completeness we reproduce lines 14, 15, 16, and 17 exactly as they appear in Fermat (1891, p. 159):

    • (14) in invenienda tangente, per adaequalitatem consideramus et, elisis

    • (15) (quae monet doctrina de maxima et minima) homogeneis, fit demum

    • (16) aequalitas quae punctum concursûs tangentis cum diametro determinat,

    • (17) ideoque ipsam tangentem.

  22. Some of the extant texts have égalité instead.

  23. Latin original: “et demum (quod operae pretium est) portiones tangentium jam inventarum pro portionibus curvae ipsis subjacentis sumantur, et procedat adaequalitas ut supra monuimus: proposito nullo negotio satisfiet.” (Fermat 1891, p. 162)

  24. More precisely, the term occurs once on each of the pages 159, 160, 161, three times on 162, twice on 163, and twice on 164.

  25. From an editorial comment on page 145 in the same volume it emerges that Aubry developed his remarks (in Note XXV) especially for the 1912 volume.

  26. Of 43,500 livres (Mahoney 1994, p. 16) in Fermat’s case. At the time all public functions were sold under certain conditions (Mousnier 1971).

  27. Fermat’s probable mother, Claire née de Long, came from a Huguenot family from Montauban, a Huguenot stronghold (Gairin 2001). His wife (and distant cousin) Louyse was also a de Long ((Mouranche 2017) prefers the spelling “Louise” while Codicille au testament de Pierre de Fermat uses the spelling “Louyse” [Chabbert 1967, p. 347)]. In addition, Fermat had close protestant friends in the Chambre d’Édit at Castres, namely Pierre Saporta and Jacques de Ranchin (Chabbert 1967). The claim that Fermat “was baptized (and most probably born) on 20 August 1601 to Dominique Fermat ... and his wife Claire, née de Long” (Mahoney 1994, p. 15) is incorrect, since we know that in 1603 Dominique was still married to Françoise Cazeneuve (Spiesser 2008, p. 172). Claire de Long’s grandfather Jean de l’Hospital was a Huguenot expelled from the Parliament for religious reasons; see Sect. 5.1.

  28. This refers to the Edit de Nantes (1598) or the Edict of Nantes, where a compromise was reached between catholics and protestants that held until its abolition in 1685 by the l’Etat, c’est Moi (The State, it is Myself) king who apparently found no room within either his catholic self or his State for protestantism.

  29. Lalouvère specifically complained about this choice of terminology, as noted in Descotes (2015, p. 269).

  30. This means that the dimension of the entity is one less than the dimension of the ambient figure.

  31. Being contrary to Aristotle was viewed as a serious offense by the catholic hierarchy. Thus, part of jesuit Biancani’s book was censored on the grounds that “The addition to Father Biancani’s book about bodies moving in water should not be published since it is an attack on Aristotle and not an explanation of him (as the title indicates). Neither the conclusion nor the arguments to prove it are due to the author, but to Galileo. And it is enough that they can be read in Galileo’s writings. It does not seem to be either proper or useful for the books of our members to contain the ideas of Galileo, especially when they are contrary to Aristotle.” [quoted in (De Ceglia 2003, p. 162)]

  32. This refers to a scholastic dispute as to whether qualities that differ in intensity differ in number: is there a numerical difference between the anger of a person who is very angry and one who is only slightly so?

  33. This passage of the ban alludes to the Aristotelian doctrine of hylomorphism; see note 31.

  34. See note 30 on the role of aristotelianism.

  35. Jean Bertet (1622–1692), jesuit, quit the Order in 1681. In 1689 Bertet conspired with Leibniz and Antonio Baldigiani in Rome to have the ban on Copernicanism lifted (Wallis 2012).

  36. “panis et vini una cum corpore et sanguine” in the original Latin. This is the key term here—this is the doctrine favored by Luther—consubstantiation. It was favored also by Scotus and Ockham, who rejected it only because it was inconsistent with the 1215 Lateran council. Artigas et al. note that “the concept of substance [in 13.2] was borrowed from Aristotelian philosophy” (Artigas et al. 2005). The substance/form dichotomy is the content of the Aristotelian doctrine of hylomorphism.

  37. Grassi’s objections [see (Festa 1991, pp. 100–101)] closely parallel those contained in G3, leading Redondi to conjecture that Grassi was in fact the author of G3. Many Galileo scholars, however, feel that there is insufficient evidence for this, and specifically reject Redondi’s claim of similarity of handwriting. See Sect. 4.5 for further details.

  38. See note 27.

  39. See also http://www.etymonline.com/index.php?term=hocus-pocus.

  40. “The accused were right to flee. Toulouse would have certainly sent them to the stake.” One of the 18 Marranos, named Roque de Leon, was an ancestor of Jacques Blamont; see Blamont (2000, p. 17).

  41. We find the following period detail: “...l’intendant du Languedoc, Claude Bazin de Bezons, a émis en 1663 un jugement aussi lapidaire que péjoratif sur son [Fermat’s] activité de magistrat. Dans une note adressée au ministre Colbert sur les membres du parlement de Toulouse, il déclare que Fermat est ‘homme de beaucoup d’érudition, a commerce de tous costés avec des sçavans, mais assez intéressé, n’est pas trop bon rapporteur et est confus’” (Mouranche 2017, p. 52). Bazin added that Fermat “n’est pas des amys du premier président” (ibid.), namely Gaspard de Fieubet. President Fieubet overruled Fermat in the matter of the priest Raymond Delpoy and had Delpoy hanged. “Fermat, qui n’était pas convaincu de sa culpabilité, en fut choqué” (ibid.).

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Acknowledgements

We are grateful to Catherine Goldstein, Israel Kleiner, Eberhard Knobloch, David Schaps, and Maryvonne Spiesser for helpful comments. We thank Thomas Willard for the information on Fludd and Kepler given in note 12. M. Katz was partially supported by the Israel Science Foundation Grant No. 1517/12.

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Bair, J., Katz, M.G. & Sherry, D. Fermat’s Dilemma: Why Did He Keep Mum on Infinitesimals? And the European Theological Context. Found Sci 23, 559–595 (2018). https://doi.org/10.1007/s10699-017-9542-y

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