Abstract
This paper consists of three main sections. In the first section, we consider how early attempts at understanding the relationship between mathematics and philosophy in Leibniz’s thought were often made within the framework of grand reconstructions guided by intellectual trends such as the search for “the ideal of system”. In the second section, we proceed to recount Leibniz’s first encounter with contemporary mathematics during his four years of study in Paris presenting some of the earliest mathematical successes which he made there. In particular, we argue that recently published letters and papers reveal how his youthful mathematical reflexions were deeply intertwined with important philosophical insights that, in turn, acted as guiding ideas for his mathematical research. Finally, in the third section, we situate the central themes of the essays of the present volume within the new understanding of the interrelations between philosophy and mathematics in Leibniz’s thought briefly indicated in the opening section.
Les Mathematiciens ont autant besoin d’estre philosophes, que les philosophes d’estre Mathematiciens.
Leibniz to Malebranche, 13/23 March 1699 (A II, 3, 539)
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- 1.
In an exchange with Basnage de Bauval, Leibniz revealed his intention to publish his correspondance with Arnauld and advanced what was to be expected from the content of his letters in these terms: “Il y aura un melange curieux de pensées philosophiques et Mathematiques qui auront peut-estre quelque fois la grace de la nouveauté”; Leibniz to Basnage de Bauval, 3/13 January 1696 (A II, 3, 121).
- 2.
Leibniz to L’Hospital, 27 December 1694 (A III, 6, 253): “Ma metaphysique est toute Mathematique pour dire ainsi, ou la pourroit devenir”.
- 3.
See Russell (1903).
- 4.
Brunschvicg (1912, 204). Unless otherwise stated, all the translations are ours.
- 5.
Concerning Leibniz scholarship in the twentieth Century, see Albert Heinekamp (1989) who distinguished three main lines of study: first, the view that focuses on the ideal of system (“à la recherche du vrai système leibnizien”); second, the defense of the “structuralist” reading (“les interprétations structuralistes”); third, the view that denies any systematic structure in Leibniz’s philosophy (“refus du caractère systématique de la philosophie leibnizienne”) which, according to Heinekamp, begins to be present only in the 80’s. The first line of reading may be regarded as the most widely represented amongst scholars interested in studying Leibniz from the perspective of the interrelations between mathematics and philosophy. Amongst French scholars, Serres (1968) and Belaval (1960) may be mentioned as cases where the indirect impact of mid-twentieth Century foundational philosophy of mathematics and logic can be detected. One could also mention the work of G.-G. Granger (1981), who emphasizes the epistemic value of Leibniz’s guiding ideas at the basis of his mathematical contributions (vis-à-vis the work of other great seventeenth Century contributions to mathematical analysis) but also sees Leibniz’ mathematical work as a possible anticipation of modern non-standard analysis. For a contextual study of the development of formal logic in the late nineteenth and early twentieth Century and the exact role played by Leibniz’s work as a possible anticipation of modern approaches in logic and mathematics, see Peckhaus (1997). Despite revealing historical studies, the ‘logicist’ trend is still represented explicitly in recent times, for instance, by Sasaki (2004, 405), who goes so far as to speak of “Leibniz’s ‘logicist-formalist’ philosophy of mathematics”.
- 6.
Couturat (1903), Preface.
- 7.
See Russell’s Preface to the second edition of his book on Leibniz, Russell (1937): in composing his original book, Russell conceded that he ignored all material relevant to Leibniz’s mathematical studies and contributions, but still insisted that his “interpretation of Leibniz’s philosophy is still the same” as in 1900.
- 8.
- 9.
However, even Mahnke tried to rescue the idea of system by proposing a view which was conceived as a synthesis of both leading interpretations at his time in his book Leibnizens Synthese von Universalmathematik und Individualmetaphysik (Mahnke 1925).
- 10.
See Couturat (1901, vii): “Les philosophes, séduits à bon droit par sa métaphysique, n’ont accordé que peu d’attention à ses doctrines purement logiques, et n’ont guère étudié son projet d’une Caractéristique universelle, sans doute à cause de la forme mathématique qu’il revêtait. D’autre part, les mathématiciens ont surtout vu dans Leibniz l’inventeur du Calcul différentiel et intégral, et ne se sont pas occupés de ses théories générales sur la valeur et la portée de la méthode mathématique, ni de ses essais d’application de l’Algèbre à la Logique, qu’ils considéraient dédaigneusement comme de la métaphysique. Il en est résulté que ni les uns ni les autres n’ont pleinement compris les principes du système, et n´ont pu remonter jusqu´à la source d´où découlent à la fois le Calcul infinitésimal et la Monadologie”.
- 11.
Leibniz to Nicolas Malebranche, 13/23 March 1699 (A II, 3, 539): “Les Mathematiciens ont autant besoin d’estre philosophes, que les philosophes d´estre Mathematiciens”.
- 12.
Leibniz to Duke Johann Friedrich, autumn 1679 (A II, 1 (2006), 761); Leibniz to Fabri, beginning of 1677(A II, 1 (2006), 442); Leibniz to Conring, 24 August 1677 (A II, 1 (2006), 563).
- 13.
Leibniz to the Pfalzgräfin Elisabeth, November 1678 (A II, 1 (2006), 66).
- 14.
See for example Leibniz, De solutionibus problematic catenarii vel funicularis in Actis Junii A. 1691. aliisque a Dn. I. B. propositis (GM V, 255); Historia et origo calculi differentialis (GM V, 398); Leibniz to Huygens, first half of October 1690 (A III, 4, 598); Leibniz to Remond, 10 January 1714 (GP III, 606): “Il est vray que je n’entray dans les plus profondes [sc. mathematiques] qu’apres avois conversé avec M. Hugens à Paris”.
- 15.
See Antognazza (2009, 140–141).
- 16.
Leibniz to Duke Johann Friedrich, 21 January 1675 (A I, 1, 491−492): “Paris est un lieu, ou il est difficile de se distinguer: on y trouve les plus habiles hommes du temps, en toutes sortes des sciences, et il faut beaucoup de travail, et un peu de solidité, pour y establir sa reputation”. See also Leibniz to Gallois, first half of December 1677 (A III, 2, 293−294); Leibniz to Bignon, 9/19 October 1693 (A I, 10, 590) .
- 17.
Boineburg to Conring, 22 April 1670, Gruber (1745, II, 1286−1287): “Leibnizio literae tuae maximo sunt solatio. Est iuvenis 24 annorum, Lipsiensis, Juris Doctor: imo doctus supra quam vel dici potest, vel credi, Philosophiam omnem percallet, veteris et novae felix ratiocinator. Scribendi facultate apprime armatus. Mathematicus, rei naturalis, medicinae, mechanicae omnis sciens et percupidus; assiduus et ardens”.
- 18.
Oldenburg to Huygens, 28 March 1671, Hall and Hall (1965−1986, VII, 537−538/538−539): “Il ne semble pas un Esprit du commun, mais qui ait esplusché ce que les grands hommes, anciens et modernes ont commenté sur la Nature, et trouvant bien de difficultez qui restent, travaillé d’y satisfaire. Je ne vous scaurois pas dire comment il y ait reussi; j’oseray pourtant affirmer que ses pensees meritent d’estre considerées.” See also Oldenburg to Huygens, 8 November 1670, Hall and Hall (1965−1986, VII, 239−240/241−242).
- 19.
Leibniz to Malebranche, end of January 1693 (A II, 2, 659): “Au commencement de mes etudes mathematiques je me fis une theorie du movement absolu, où supposant qu’il n’y avoit rien dans le corps que l’étendüe et l’impenetrabilité, je fis des regles du mouvement absolu que je croyois veritables, et j’esperois de les pouvoir concilier avec les phenomenes par le moyen du systeme des choses.”
- 20.
Leibniz to Pellisson-Fontanier, 7 May 1691 (A I, 6, 195−196): “L’envie de me rendre digne de l’opinion favorable qu’on avoit de eue de moy, m’avoit fait faire quelques decouvertes dans les Mathematiques, quoyque je n’eusse gueres songé à cette science, avant que j’estois venu en France, la philosophie et la jurisprudence ayant esté auparavant l’objet de mes études dont j’avois donné quelques essais.” See also Leibniz to Duke Johann Friedrich, 29 March 1679 (A I, 2, 155); Leibniz to Duke Ernst August, early 1680? (A I, 3, 32); Leibniz to Foucher, 1675 (A II, 1 (2006), 389); De numeris characteristicis ad linguam universalem constituendam (A VI, 4, 266).
- 21.
See Leibniz, Historia et origo calculi differentialis (GM V, 395).
- 22.
I, 8, § 25; Hobbes (1651, 72).
- 23.
See Hofmann (1974, 15).
- 24.
Leibniz to Oldenburg, 26 April 1673 (A III, 1, 83−89, 88): “At ego totius seriei in infinitum continuatae summam invenio methodo mea: 1/3 1/6 1/10 1/15 1/21 1/28 etc. in infinitum; quod jam publice propositum esse, vel ideo non credidi, quia a Nobilissimo Hugenio mihi primum propositum est hoc problema in numeris triangularibus; ego vero id non in triangularibus tantum, sed et pyramidalibus etc. et in universum in omnibus ejus generis numeris solvi ipso Hugenio mirante”.
- 25.
See Bos (1978, 61).
- 26.
Leibniz for Gallois, end of 1672 (A II, 1 (2006), 342): “Quis enim sensu duce persuaderet sibi, nullam dari posse lineam tantae brevitatis, quin in ea sint non tantum infinita puncta, sed et infinitae lineae (ac proinde partes a se invicem separatae actu infinitae) rationem habentes finitam ad datam; nisi demonstrationes cogerent.”
- 27.
Ibid, 349: “at Axioma illud fallere impossibile est, seu quod idem est, Axioma illud nunquam, ac non nisi in Nullo seu Nihilo fallit, Ergo Numerus infinitus est impossibilis, non unum, non totum, sed Nihil.”
- 28.
Leibniz, Mathematica (A VII, 1, 657): “Nam 0 + 0 = 0. Et 0–0 = 0. Infinitum ergo ex omnibus unitatibus conflatum, seu summa omnium esr nihil, de quo scilicet nihil potest cogitari aut demonstrari, et nulla sunt attributa.” See also De bipartitionibus numerorum eorumque geometricis interpretationibus (A VII, 1, 227).
- 29.
See Beeley (2009).
- 30.
Leibniz to Malebranche, end of January 1693 (A II, 2, 661): “La marque d’une connoissance imparfaite chez moy, est, quand le sujet a des proprietiés, dont on ne peut encor donner la demonstration.”
- 31.
Leibniz to the Pfalzgräfin Elisabeth?, November 1678 (A II, 1 (2006), 662): “Mais pour moy je ne cherissois les Mathematiques, que par ce que j’y trouvois les traces de l’art d’inventer en general […].” See also Leibniz to Duke Johann Friedrich, February 1679 (A II, 1 (2006), 684).
- 32.
See Leibniz, De arte characteristica inventoriaque analytica (A VI, 4, 321): “Duobus maxime modis homines inventores fieri deprehendo, per Synthesin scilicet sive Combinationem et per analysin; utrumque autem vel facultati natura usuve comparatae, vel methodo debere.” See also ibid. (A VI, 4, 329).
- 33.
Leibniz, Zur Ars signorum von George Dalgarno (A VI, 3, 170): “sed vera Characteristica Realis, qualis a me concipitur, inter [ap]tissima humanae Mentis instrumenta censeri deberet, [invin]cibilem scilicet vim habitura et ad inveniendum, et ad retinendum et ad dijudicandum. Illud enim efficient in omni material, quod characteres Arithmetici et Algebraici in Mathematica.” See also Antognazza (2009, 162).
- 34.
See Leibniz, De alphabeto cogitationum humanarum (A VI, 4, 271−272).
- 35.
Leibniz to Gallois, 19 December 1678 (A III, 2, 570): “Je suis confirmé de plus en plus de l’utilité et de la realité de cette science generale, et je voy que peu de gens en ont compris l’étendue. Mais pour la rendre plus facile et pour ainsi dire sensible; je pretends de me servir de la Characteristique, dont je vous ay parlé quelques fois, et dont l’Algebre et l’Arithmetique ne sont que des échantillons. Cette Characteristique consiste dans une certaine ecriture ou langue, (car qui a l’une peut avoir l’autre) qui rapporte parfaitement les relations des nos pensées. Ce charactere seroit tout autre que tout ce qu’on a projetté jusqu’icy. Car on a oublié le principal qui est que les caracteres de cette écriture doivvent servir à l’invention et au jugement, comme dans l’algebre et dans l’arithmetique”.
- 36.
Leibniz to Mariotte, July 1676 (A II, 1 (2006), 424): “ce seroit pour ainsi dire une algebre universelle, et il seroit aussi aisé d’inventer en morale, physique ou mechanique, qu’en Geometrie”.
- 37.
Leibniz to Oldenburg, [18]/28 December 1675 (A III, 1, 331): “Haec algebra, quam tanti facimus merito, generalis illius artificii non nisi pars est. Id tamen praestat, ut errare ne possimus quidem, si velimus, et ut veritas quasi picta velut machinae ope in charta expressa deprehendatur. Ego vero agnosco, quicquid in hoc genere praebet algebra, non nisi superioris scientiae beneficium esse, quam nunc combinatoriam, nunc characteristicam appellare soleo, longe diversam ab illis, quae auditis his vocibus statim alicui in mentem venire possent”.
- 38.
Leibniz, De synthesi et analysi universali seu Arte inveniendi et judicandi (A VI, 4, 540).
- 39.
Concerning this issue, see the material gathered in Jesseph and Goldenbaum (2008) .
- 40.
For a discussion of the different “levels of abstraction” in Leibniz´s mathematical practice, see Breger (2008b) and, in particular, in connection with the present idea, see Breger (2008a, 193): “[…] it was only by proving many theorems and gaining experience with the new material that Leibniz arrived at the higher level of abstraction from which he was able to recognize and explicitly formulate the rules of calculus”.
- 41.
Leibniz, Analyseos tetragonisticae pars tertia (A VII, 5, 313): “Pleraque theoremata Geometriae indivisibilium quae apud Cavalerium, Vincentium, Wallisium, Gregorium, Barrovium extant statim ex calculo patent”.
- 42.
See Leibniz to Schmidt, 3 August 1694 (A I, 10, 499): “Novum Calculi Analytici genus a me in Geometriam introductam […] Usum inprimis habet ad ea analysi subjicienda, in quibus quantitates finitae determinantur interveniente aliqua consideratione infiniti, quemadmodum saepe praesertim cum Geometria applicatur ad naturam. Ubique enim infinitum Naturae operationibus involvitur. See also Leibniz to Kochański, 10/20 August 1694 (A I, 10, 513−514); Leibniz to the Electress Sophie for the Duchess Elisabeth Charlotte of Orléans, 28 October 1696 (A I, 13, 85): “Et c’est une chose estrange, qu’on peut calculer avec l’infini comme avec des jettons, et que cependant nos Philosophes et Mathematiciens ont si peu reconnu combien l’infini est mêlé en tout”.
- 43.
Leibniz to Foucher, end of June 1693 (A II, 2, 713): “Je suis tellement pour l’infini actuel, que au lieu d’admettre que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte partout, pour mieux marquer les perfections de son auteur. Ainsi je crois qu’il n’y a aucune partie de la matiere, qui ne soit, je ne dis pas divisible mais actuellement divisée, et par consequent la moindre particelle doit estre considerée comme un monde plein d’une infinité de creatures differentes.”
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Goethe, N., Beeley, P., Rabouin, D. (2015). The Interrelations Between Mathematics and Philosophy in Leibniz’s Thought. In: Goethe, N., Beeley, P., Rabouin, D. (eds) G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes, vol 41. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9664-4_1
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