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Mendelssohn, Wolff, and Bernoulli on Probability

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Moses Mendelssohn's Metaphysics and Aesthetics

Part of the book series: Studies in German Idealism ((SIGI,volume 13))

Abstract

In his “On Probability,” Moses Mendelssohn proposed to use a mathematical ­formulation of the definition of probability, found in the work of Christian Wolff to support the validity of induction (against Hume) and the view that all our actions, even including those supposed to be the result of free will, are predetermined (in agreement with Leibniz). Mendelssohn went into few of the details of mathematical probability as they had been developed by mathematicians like Jacob Bernoulli in the century before he wrote.

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Notes

  1. 1.

    See Edith Sylla, “Introduction,” in The Art of Conjecturing together with Letter to a Friend on Sets in Court Tennis by Jacob Bernoulli (Baltimore: The John Hopkins University Press, 2006).

  2. 2.

    Christian Wolff, Philosophia Rationalis sive Logica (1740), part 2, 443: “Quemadmodum Mathematici artem inveniendi veritatem in Mathesi adeoque artis huius partem sibi propriam in Algebra tradunt; ita Logicae probabilium specimina dedere in sorte ludorum determinanda. Pertinent huc praeter Hugenii tractatum de ludo aleae Jacobi Bernoulli Ars conjectandi, Remondi de Monmort Analysis ludorum a fortuna pendentium, cuius altera editio priori auctior cum epistolis Joannis & Nicolai Bernoullii prodiit, atque Abrahami de Moivre doctrina sortis, seu Methodus computandi probabilitatem in ludis. Principia, quae de his insunt, generalia suo trademus loco, quando de Logica probabilium ex instituto agemus.”

  3. 3.

    See Alexander Altmann, Moses Mendelssohn: A Biographical Study (London: Routledge and Kegan Paul, 1973), 76. More detail of alternate versions of the story in Altmann, Moses Mendelssohns Frühschriften zur Metaphysik (Tübingen: Mohr/Siebeck, 1969), 209–11.

  4. 4.

    Altmann, Mendelssohn’s Frühschriften, 212. The 1771 version is translated by Daniel Dahlstrom in Mendelssohn, Philosophical Writings (Cambridge: Cambridge University Press, 1997). The paragraph starting “But from what source does God take these reasons...” on page 248, and the ending of the essay beginning with “This case should not be confused with the one previously mentioned...” on page 249 and continuing to the end, are new in comparison to the 1756 edition, which has a different two paragraphs at the end (Mendelssohn, Gesammelte Schriften. Jubiläumsausgabe, vol. 1, 164, hereafter cited as JubA and volume number, followed by a colon and page number.) I have relied on Dahlstrom’s translation except for some slight emendations where I thought they might make the sense clearer.

  5. 5.

    Mendelssohn, Philosophical Writings, 234. Here, I have revised the translation of the last sentence.

  6. 6.

    Wolff, Philosophia Rationalis, 443. See quotation in note 2 above.

  7. 7.

    Luigi Cataldi Madonna, “Wahrscheinlichkeit und wahrscheinliches Wissen in der Philosophie von Christian Wolff,” Studia Leibnitiana 19/1 (1987): 8, identifies the 1709 review of Montmort’s work and the 1714 review of Bernoulli’s Ars Conjectandi as having been written by Wolff.

  8. 8.

    Altmann, Mendelssohns Frühschriften, 214.

  9. 9.

    Mendelssohn, Philosophical Writings, 240; Nicholas Bernoulli, “Specimina artis conjectandi ad quaestiones juris adplicatae,” Acta Eruditorum, Supplementum, vol. 4 (1711), sec. 4: 159.

  10. 10.

    Altmann, Mendelssohns Frühschriften, 225, says: “Mendelssohn, dem die algebraische Demonstration des Bernoullischen Theorems in der ‘Ars Conjectandi’ sicherlich vertraut war, vermeidet es, auf sie einzugehen und gibt statt dessen eine logische Erklärung des von ’s Gravesande mitgeteilten Verfahrens.” If Mendelssohn had studied Bernoulli’s proof of the weak law of large numbers, he might understandably have avoided getting into it in a talk to the club group, but he would surely have said something more nuanced about the relation of the proportions of black and white balls drawn from the urn in relation to the numbers inside at the start.

  11. 11.

    Mendelssohn, Philosophical Writings, 234.

  12. 12.

    In abstracting for the Zentralblatt Database Cataldi Madonna, “Wahrscheinlichkeit und wahrscheinliches Wissen,” Ivo Schneider writes, that “Wolff sehr wohl von den Ansätzen der Glücksspiel- und Wahrscheinlichkeitsrechnung seiner Zeit beeinflusst war, dass aber seine Versuche einer Analyse der umgangssprachlich und in der philosophischen Tradition vor allem von Leibniz benutzten Wahrscheinlichkeitsbegriffe mit Hilfe des Prinzips des unzureichenden Grundes schon an den einfachsten Aufgaben der Glückspielrechnung scheiterten. Von daher erklärt sich das Fehlen jeder Wirkung von Wolff nicht nur auf die Entwicklung der mathematischen Wahrscheinlichkeitstheorie, sondern auch der induktiven Logik.” As Cataldi Madonna, in the article in question, points out, Leibniz himself made errors in such basic matters as counting the numbers of different ways of throwing given sums with two dice (Cataldi Madonna, “Wahrscheinlichkeit und wahrscheinliches Wissen”, 24–25).

  13. 13.

    Jacob Bernoulli, introduction to Art of Conjecturing, 58–60.

  14. 14.

    Mendelssohn, Philosophical Writings, 235.

  15. 15.

    Mendelssohn, Philosophical Writings, 235. The Latin of Wolff’s Philosophia Rationalis, section 578, states, “Si praedicatum subjecto tribuitur ob rationem insufficientem, propositio dicitur probabilis. Patet adeo, in probabili propositione praedicatum subjecto tribui ob quaedam requisita ad veritatem.”

  16. 16.

    Wolff, Philosophia Rationalis, sections 573, 578.

  17. 17.

    Wolff, Philosophia Rationalis, section 579, “Probabilior igitur est propositio, si subjecto predicatum tribuitur ob plura requisita ad veritatem, quam si tribuitur ob pauciora.”

  18. 18.

    Bernoulli, Art of Conjecturing, 315.

  19. 19.

    Bernoulli, Art of Conjecturing, 318.

  20. 20.

    See Cataldi Madonna, “Wahrscheinlichkeit und wahrscheinliches Wissen,” 8.

  21. 21.

    Editors’ notes to JubA 1:636–37. Wilhelm Jacob ’s Gravesande, Einleitung in die Weltweisheit (1755), § 587, “Wenn wir eine Erkenntnis von den Dingen ausser uns erlangen sollen, so muss meistenteils verschiedenes zusammen kommen. Sind nun einige dieser Dinge da, indem andere fehlen, so werden wir nicht völlig überzeugt, dass unser Begrif mit der Sache überein komme, welche er vorstellen soll. Es gibt also verschiedene Stufen, durch welche wir endlich zu dieser Überzeugung gelangen; und es liegt uns ob, zu zeigen, wie diese bestimmen sind. Wir werden sie Grade der Wahrscheinlichkeit oder Probabilität nennen.”

  22. 22.

    For Bestimmungsgrund, see “On Evidence in the Metaphysical Sciences,” in Mendelssohn, Philosophical Writings, 288, at the bottom of the page, where Dahlstrom translates “compelling reason.”

  23. 23.

    See Bernoulli, Art of Conjecturing, 318.

  24. 24.

    Bernoulli, Art of Conjecturing, 319. He lists as a rule of reasoning “It is not sufficient to weigh one or another argument. Instead we must bring together all arguments that we can come to know and that seem in any way to work toward a proof of the thing.” In urging that more arguments be used, Bernoulli is acknowledging that there is no obvious set of all the relevant arguments.

  25. 25.

    Mendelssohn, Philosophical Writings, 236. Titius, Caius, Sempronius, and Maevius are all names very typically used in examples about law cases.

  26. 26.

    Wolff’s definition of probability as cited by Mendelssohn is only the first part of Wolff’s treatment of the logic of probability. In a later section (Wolff, Philosophia Rationalis, section 595), Wolff says that special principles are needed for estimating probability, principles which depend upon ontology and philosophy, and he goes on to discuss how principles of a scientific demonstration might be gained from experience a posteriori.

  27. 27.

    Altmann, Mendelssohns Frühschriften, 221, 239, and elsewhere, refers to Mendelsssohn’s use of “Wolff’s Begriff des zureichenden Grundes.” Whether or not sufficient (or conclusive?) grounds are enough to support Mendelssohn’s argument that in the case of God the denominator of the probability fraction will be infinite is still unclear to me. If so, then my discussion, which was written before I saw Altmann’s Mendelssohns Frühschriften, might be reframed to use this terminology.

  28. 28.

    Mendelssohn, Philosophical Writings, 4.

  29. 29.

    Mendelssohn, Philosophical Writings, 249–50. This passage was not in the 1756 edition, but a similar statement was there: “Es ist also klar, dass man entweder Gott sogar die wahrscheinliche Präscience in Ansehung unsrer freyen Handlungen absprechen, oder den freyen Handlungen eine determinirte Wahrheite zuschreiben muss, dadurch sie vorher gewusst werden können” JubA 1:164.

  30. 30.

    Bernoulli, Art of Conjecturing, 327–28.

  31. 31.

    Mendelssohn, Philosophical Writings, 238.

  32. 32.

    Mendelssohn, Philosophical Writings, 239.

  33. 33.

    ’s Gravesande, Einleitung in die Weltweisheit, 86–87, §§ 611–12; §§ 616–18.

  34. 34.

    ’s Gravesande, Einleitung in die Weltweisheit, 83–84, §§ 589–91.

  35. 35.

    Mendelssohn, Philosophical Writings, 240.

  36. 36.

    Mendelssohn, Philosophical Writings, 242.

  37. 37.

    Mendelssohn, Philosophical Writings, 242.

  38. 38.

    See Hans Lausch, “Moses Mendelssohn: ‘Wir müssen uns auf Wahrscheinlichkeiten stützen,’” Acta historica Leopoldina 27 (1997): 201–13; and Lausch, “‘The Ignorant Hold Back their Judgment and Await the Conclusions of the Knowing’: Moses Mendelssohn and Other Mathematicians,” Aleph 2 (2002): 107–9. Altmann, Mendelssohns Frühschriften, 235, also mentions the relation of Mendelssohn’s formula to that of Thomas Bayes.

  39. 39.

    Cf. Mendelssohn, Philosophical Writings, 246.

  40. 40.

    Mendelssohn, Philosophical Writings, 249.

  41. 41.

    Abraham De Moivre, Doctrine of Chances (1718), 1, quoted in Sylla “Introduction”, in Art of Conjecturing by Bernouilli, 112.

  42. 42.

    Mendelssohn, Philosophical Writings, 245.

  43. 43.

    Mendelssohn, Philosophical Writings, 243.

  44. 44.

    See Reinier Munk, “Moses Mendelssohn’s Conception of Judaism,” in Studies in Hebrew Language and Jewish Culture, eds. Martin F. J. Baasten and Reinier Munk (Dordrecht: Springer, 2007).

  45. 45.

    Nachum L. Rabinovitch, “Studies in the History of Probability and Statistics XXII. Probability in the Talmud,” Biometrika 56/2 (1969): 438. In a slightly different case, if the meat were found in the street without reference to entering a particular store, then the chances would be 9 to 1 that it was ritually slaughtered.

  46. 46.

    Rabinovitch, “Probability in the Talmud,” 440. Most likely this means it has a probability greater than half.

  47. 47.

    Mendelssohn, Philosophical Writings, 247.

  48. 48.

    Altmann, Mendelssohns Frühschriften, 243, states that in this case Mendelssohn is supporting Leibniz’s determinism, and so he looks for opponents of Leibniz that might be Mendelssohn’s target here, mentioning Crusius, Formey, Merian, and Prémontval. I have not looked for contemporaries of Mendelssohn who might have held the views that Mendelssohn opposes.

  49. 49.

    Robert Sleigh, Jr., Vere Chappell, and Michael Della Rocca, “Determinism and Human Freedom” in The Cambridge History of Seventeenth-Century Philosophy, eds. Daniel Garber and Michael Ayers (Cambridge University Press, 1998), 1196.

  50. 50.

    Sleigh, “Determinism and Human Freedom,” 1195–96.

  51. 51.

    Sleigh, “Determinism and Human Freedom,” 1258.

  52. 52.

    Sleigh, “Determinism and Human Freedom,” 1258. The quotation comes from Leibniz’s Conversatio cum Domino Stenonio de libertate (1677).

  53. 53.

    Sleigh, “Determinism and Human Freedom,” 1259.

  54. 54.

    Bernoulli, Art of Conjecturing, 315.

  55. 55.

    Mendelssohn, Philosophical Writings, 238.

  56. 56.

    Mendelssohn, Philosophical Writings, 248. Mendelssohn, JubA 1:163. As usual I have modified Dahlstrom’s translation of “Bewegungsgründe.”

  57. 57.

    Mendelssohn, Philosophical Writings, 248–49. I have changed Dahlstrom’s translation of “Bewegungsgründe” from “compelling reasons” to “motives,” to avoid the possible misinterpretation that these rational grounds for a conclusion compel assent.

  58. 58.

    Mendelssohn, Philosophical Writings, 249. This may be the point at which the original reader of Mendelssohn’s essay pronounced zero as “oh,” leading to Mendelssohn’s spontaneous correction, revealing his authorship. See Altmann, Moses Mendelssohn, 76.

  59. 59.

    The 1756 version of “On Probability” ends with the paragraph-long sentence (replaced by something else in the 1771 edition and possibly also in the 1761 edition, which I have not seen): “Da nun vermöge eben dieser Schlüsse erhellet, dass gar keine moralische Wahrscheinlichkeit vorhanden seyn könnte, wenn unser Will nicht zureichend durch die Bewegungsgründe determiniret werden sollte, weil sich der Grad der Wahrscheinlichkeit zur Gewissheit verhalten würde, wie eine endliche, zu einer unendlichen Grösse, so gebe ich denen Weltweisen, die der gleichgültigen Freyheit zugethan sind, zu bedenken, ob sie auch diese Folge annehmen können, ohne gewissermassen der Erfahrung zu widersprechen?” JubA 1:164.

  60. 60.

    Bernoulli, Art of Conjecturing, 329.

  61. 61.

    Mendelssohn, Philosophical Writings, 248.

  62. 62.

    Bernoulli, Art of Conjecturing, 318–20.

  63. 63.

    Mendelssohn, Philosophical Writings, 285–86. The German comes from Moses Mendelssohn, Abhandlung über die Evidenz in metaphysischen Wissenschaften, neue Auflage (Berlin, 1786), 87.

  64. 64.

    Mendelssohn, Philosophical Writings, 250. Cf. Leibniz’s claim that Molina’s argument involved a chimera.

  65. 65.

    JubA 1:515: “Wir wollen setzen, Cassius wäre im Stande gewesen, alle Umstände deutlich auseinander zu setzen...”

  66. 66.

    Altmann, Mendelssohns Frühschriften, 247–50.

  67. 67.

    Altmann, Mendelssohns Frühschriften, 248.

  68. 68.

    Altmann, Mendelssohns Frühschriften, 250. This is based on a January 6, 1758 letter of Mendelssohn to Baumgarten: “Aus dem Schreiben an Baumgarten geht weiter hervor, dass Aepinus, um seine Sache nicht gänzlich aufzugeben, sich dazu hatte verleiten lassen, die von Bernoulli und anderen längst ausser Zweifel gesetzte Wahrscheinlichkeitsberechung anzufechten, was Mendelssohn, wie er sagt, mit Leichtigkeit zurückweisen Konnte.” The letter is in JubA 11:172–74. Because I did not have this volume at hand, I checked the text of the letter in Moses Mendelssohn’s gesammelte Schriften, Fünfter Band, Leipzig, 1844, 414–18. Hans Lausch spoke on this topic in 1993: “The controversy on probability between Aepinus and Mendelssohn of 1756–1757 – a ‘Bayesian’ issue?” – a paper delivered at the Third Australian Conference on the History of Mathematics, which I have not seen. Hans Lausch, “Moses Mendelssohn und die zeitgenössiche Mathematik,” in Moses Mendelssohn im Spannungsfeld der Aufklärung (eds. Michael Albrecht and Eva J. Engel, Stuttgart/Bad Cannstatt: Frommann-Holzboog, 2000), 119–35, does not provide any more information about what Aepinus’s arguments may have been.

  69. 69.

    Mendelssohn, Philosophical Writings, 302.

  70. 70.

    Mendelssohn, Philosophical Writings, 291–92.

  71. 71.

    Mendelssohn, Philosophical Writings, 304–6.

  72. 72.

    Mendelssohn to Alexander Gottlieb Baumgarten, January 6, 1758, Moses Mendelssohn’s gesammelte Schriften, 5:414: “Ergreife ich die Gelegenheit, Ew. Hoched. zuvörderst für die Gütigkeit zu danken, die Sie gehabt, mir einige meiner Zweifel aufzuklären, mich auch insbesondere vor einer allzu grossen Neigung zu Subtilitäten zu warnen, weil diese zu weit führen können und uns öfters dem Hohngelächter der Welt aussetzen.”

  73. 73.

    Bernoulli, Art of Conjecturing, 320–21.

  74. 74.

    Mendelssohn, Philosophical Writings, 233.

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  1. Department of History, North Carolina State University, Raleigh, NC, USA

    Edith Dudley Sylla

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    Reinier Munk

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Sylla, E.D. (2011). Mendelssohn, Wolff, and Bernoulli on Probability. In: Munk, R. (eds) Moses Mendelssohn's Metaphysics and Aesthetics. Studies in German Idealism, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2451-8_3

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