Moral improvement through mathematics: Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie


This paper examines the ethical and religious dimensions of mathematical practice in the early modern era by offering an interpretation of Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie (1667). According to these important figures of seventeenth-century French philosophy and theology, mathematics could achieve extra-mathematical or non-mathematical goals; that is, mathematics could foster practices of moral self-improvement, deepen the mathematician’s piety and cultivate epistemic virtues. The Nouveaux éléments de géométrie, which I contend offers the most robust account of the virtues cultivated by mathematics in the period, was envisaged by its authors to cultivate moral, Christian and epistemic virtues that could serve in the fulfilment of moral and Christian obligations. In this paper, I set out the goals of mathematical inquiry for the Port-Royalists and describe the specific virtues they believed a revised edition of the Elements of Euclid could foster. I show that Arnauld and Nicole believed that an acquaintance with mathematics could render a student of Euclid more just, truth-loving, attentive and humble, and better able to discern truth from falsity.

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  1. 1.

    For a discussion, see Jones (2006, p. 1).

  2. 2.

    For representative passages of this view, see Plato (2006, p. 527b).

  3. 3.

    As Angela Axworthy argues in her discussion of the propaedeutic value of Renaissance mathematics, “thanks to the rediscovery of the works of Plato and the commentary of Proclus on the first book of Euclid's Elements, the reassertion of the propaedeutic status of mathematics went hand in hand with the restauratio mathematicarum, which was undertaken especially in Italy in the mid-sixteenth century” (2009, p. 33).

  4. 4.

    In The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue, Jones argued that the idea that mathematics has propaedeutic value is found in the works of early modern mathematical thinkers such as Descartes, Pascal and Leibniz. Briefly, Jones’ view is that Descartes, Pascal, and Leibniz’s “efforts to improve techniques for living and thinking well helped shape the kind of mathematics and experimentation they did, what they thought made good mathematics and method, and what they saw as the lessons illustrated by their innovations” (2006, p. 2). In addition to this being a contentious interpretation of Descartes, Pascal and Leibniz, a claim I defend in another paper, I argue in this paper that the view ascribed to these figures is more accurately attributed to Arnauld and Nicole. This issue, as it emerges in Descartes’ philosophy, has more recently been treated by Nelson (2019).

  5. 5.

    I use the following standard abbreviations to refer to works:

    Nouveaux éléments: Nouveaux éléments de géométrie [Edited by D. Descotes]

    Logique: Logique, ou l’art de penser [Edited by J. V. Buroker]

    OA: Oeuvres de Messire Antoine Arnauld. 1–43 vols

    CSM: The philosophical writings of Descartes.

    CSMK: The philosophical writings of Descartes: The correspondence.

  6. 6.

    According to correspondence with Vatier, Descartes was not convinced that the ideas on method articulated in the Discours were made use of in the Géométrie. He writes, “I couldn't show the usage of the method in the three treatises which I gave because [the method] requires an order for investigating things that is very different from that which I thought necessary to use to explain them” (CSMK, 1991, p. 85). I develop this line of inquiry in full elsewhere. See also Gaukroger (1995) and Garber (2002, chapter 2) for a complimentary assessment.

  7. 7.

    Translation of “C'est une ignorance très blâmable que de ne pas savoir, que toutes ces spéculations stériles ne contribuent rien à nous rendre heureux; qu'elles ne soulagent point nos misères; qu'elles ne guérissent point nos maux; qu'elles ne nous peuvent donner aucun contentement réel et solide.”

  8. 8.

    Compare the two positions Nicole adopts in the Preface about the disutility of mathematical practice: “et je ne sais si l’on ne peut point dire qu’elles sont toutes inutiles en elles-mêmes” (Nouveaux éléments, p. 94) and “puisqu’il est si vain et si vide de vrai bien, qu’il est capable de s’occuper tout entier à des choses si vaines et si inutiles” (Nouveaux éléments, p. 95).

  9. 9.

    Translation of “Je serais plutôt porté à diminuer l'idée trop haute que quelques personnes en pourraient avoir, étant très persuadé qu'il est beaucoup plus dangereux d'estimer trop ces sortes de choses, que de ne les pas estimer assez.”

  10. 10.

    As we will see, Nicole’s critical remarks are echoed throughout the Nouveaux éléments. Yet Arnauld also praises the purely intellectual achievements of contemporary mathematicians and natural philosophers. Arnauld’s remarks to Du Bois in the “Réflexions sur l’éloquence des prédicateurs” and his praise of Descartes in the “Examen d'un écrit qui a pour titre: Traité de l'essence du corps, et de l'union de l'âme avec le corps, contre la philosophie de M. Descartes” show not only that he believes that the mathematical and natural sciences are not entirely useless but also that he believes the discoveries of contemporary natural philosophers and mathematicians were impressive. See particularly Antoine Arnauld, the “Examen d'un écrit qui a pour titre: Traité de l'essence du corps, et de l'union de l'âme avec le corps, contre la philosophie de M. Descartes” (OA 38 1775–1783, pp. 96–97) and available in partial translation as “Eulogy on Descartes’s Philosophy,” in Arnauld 1899, pp. 311–314).

  11. 11.

    Translation of “ils sont obligés d'être justes, équitables, judicieux dans tous leurs discours, dans toutes leurs actions, et dans toutes les affaires qu'ils manient; et c'est à quoi ils doivent particulièrement s'exercer et se former” (Arnauld and Nicole 1683, p. 2). Buroker translates “s'exercer et se former” as “train and educate.” In my view, “practice and train” may be a more accurate translation.

  12. 12.

    For a more recent discussion of Arnauld’s theology which touches on questions of God’s will and grace see Nadler (2008).

  13. 13.

    Translation of “Il est vrai qu’il n’y a que la grâce et les exercices de piété qui puissent la guérir véritablement: mais entre les exercices humains qui peuvent le plus servir à la diminuer, et à disposer meme l’esprit à recevoir les vérités chrétiennes avec moins d’opposition et de dégoût, il semble qu’il n’y en ait guère de plus propre que l’étude de la géométrie.”

  14. 14.

    Translation of “Néanmoins comme il est impossible de se passer absolument d'une science qui sert de fondement à tant d'arts nécessaires à la vie humaine, il peut y avoir quelque utilité à montrer aux hommes de quelle sorte ils en doivent user, et de leur rendre cette étude la plus avantageuse qu'il est possible.”

  15. 15.

    This discussion is of interest to contemporary virtue epistemologists, such as Baehr (2011), whose attempt to ground the distinction between moral and intellectual virtues informs the account given here. See Baehr (2011, particularly Appendix).

  16. 16.

    This is just as true of Malebranche who writes in The Search After Truth that as spiritual beings our duties are to use our freedom to avoid vanity and falsity. More than this, we are obliged to cultivate our understanding, urge it towards new knowledge, and knowledge of truths based on meditations on worthy subjects. According to Malebranche, we have a moral duty to perfect our minds (1997, p. 11). Of course, what distinguishes this general attitude from the account developed by Arnauld and Nicole is the extent to which Malebranche’s mathematical work constituted the context for the cultivation of these virtues. Unlike Arnauld and Nicole, Malebranche did not author a work of mathematics to facilitate the cultivation of virtue.

  17. 17.

    The more general question of whether epistemic virtues are to be understood as reducible to moral virtues, a subset of moral virtues, or distinct from moral virtues is beyond the scope of this paper. I have bracketed this discussion since it requires a lengthier treatment of Port-Royalist virtue epistemology, and because the epistemic virtues to be treated in this paper are among the intellectual virtues that are reducible to moral virtues.

  18. 18.

    The terms of reference for this discussion are Pritchard, “Truth as the Fundamental Epistemic Good” (2014, pp. 112–115).

  19. 19.

    Elsewhere, Arnauld and Nicole write that “[w]e should also govern ourselves in such a way that we can watch them stray without going astray ourselves, and without wandering from the goal we ought to set for ourselves, which is to be enlightened by the truth we are investigating.” Emphasis added. Logique (p. 211).

  20. 20.

    Mathematics was not the only route by which an individual could counter materialism. For Descartes, language showed us the existence of a soul since only humans can arrange words or other signs in ways that allow them to declare their thoughts to others (CSM I 1985, p. 141).

  21. 21.

    For a more detailed account, see Brown (2007).

  22. 22.

    Thank you to an anonymous Synthese reviewer for drawing my attention to Clauberg’s Logica vetus et nova.

  23. 23.

    Translation of “telle répétition est si désagréable que nous prenons bien vite l'habitude d'une plus grande attention.” See also Clauberg (1654, p. 12).

  24. 24.

    Translation of “Or l’étude de la géométrie est encore un remède à ce defaut; car en appliquant l’esprit à des vérités abstraites et difficiles, elle lui rend faciles toutes celles qui demandent moins d’application; comme en accoutumant le corps à porter des fardeaux pesants, on fait qu’il ne sent presque plus le poids de ceux qui sont plus légers.”

  25. 25.

    The Port-Royalists also recommended non-cognitive routes for the cultivation of moral and epistemic virtues. Working hard through manual labour was itself beneficial, according to the Port-Royalists since they believed slovenliness was a great source of sin. See Barnard (1913, p. 95ff).

  26. 26.

    Elsewhere, Arnauld and Nicole put this in slightly different terms. The knowledge that we gain by ourselves depends, they write, on reason. The truths of geometry are ordinarily of this kind since it is we that undergo this process of learning through proofs by demonstration. The truths concerning infinity are unlike this in cases where our reason cannot penetrate these truths. There is another kind of path to knowledge which comes from “the authority of persons worthy of credence who assure us that a certain thing exists, although by ourselves we know nothing about it” which “is called faith or belief.” The truths of infinity—like the truths of religion - demand faith since in the case of the person for whom infinity is not amenable to comprehension by reason, authority demands that we believe the proofs to be true. See Logique, p. 260ff for Arnauld and Nicole’s discussion of those things we know by faith and reason.

  27. 27.

    For a more detailed account, see Lennon (1996). A thorough treatment of scepticism in the French context, which includes a discussion of the attitudes of Saint-Cyran, an early Abbot of Port-Royal, is offered by Popkin (1979, particularly chapters 5 and 6). It is worth noting that Saint-Cyran and Arnauld’s views differ, though a discussion of their respective attitudes to sceptisim is beyond the scope of the current paper.

  28. 28.

    See Nouveaux éléments (p. 99) for this example. What Nicole prescribes in this context is using reductio proofs to show how absurd is the alternate case. Though in the Nouveaux éléments and Logique Arnauld and Nicole argue that direct proofs are to be preferred to indirect (including reductio) proofs, they nevertheless endorse indirect proofs in cases where positive proofs are impossible. As I have just shown, Arnauld and Nicole believed that matters in morality and religion are of this kind. See Logique (p. 255) for Arnauld and Nicole’s discussion of demonstrations by impossibility. For Arnauld and Nicole’s pragmatic account of how best to direct our reason in matters that concern faith, see the Logique (pp. 262–265).

  29. 29.

    A sol, later sou, was a unit of money.

  30. 30.

    For a complementary assessment of this example see Descotes’ editorial note in the Nouveaux éléments (2009, p. 148): “The concern for moral edification is just as present in the New elements of geometry as it is in the logic.”


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I thank John Carriero, Millicent Churcher, Daniel Garber, Stephen Gaukroger, Timothy Smartt, Hannah Tierney and two anonymous Synthese reviewers for helpful comments on this paper. I also thank audiences at Vrije Universiteit Brussel, Oxford University, The University of Nottingham, Macquarie University and the University of Sydney for feedback on earlier versions of this paper.

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Kotevska, L. Moral improvement through mathematics: Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie. Synthese (2020).

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  • Early modern mathematics
  • Early modern moral philosophy
  • Elements
  • Euclid
  • Arnauld
  • Moral virtue
  • Epistemic virtue