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Abstract
John Arbuthnot's celebrated but flawed paper in the Philosophical Transactions of 1711–12 is a philosophically and historically plausible target of Hume's probability theory. Arbuthnot argues for providential design rather than chance as a cause of the annual birth ratio, and the paper was championed as a successful extension of the new calculations of the value of wagers in games of chance to wagers about natural and social phenomena. Arbuthnot replaces the earlier anti-Epicurean notion of chance with the equiprobability assumption of Huygens's mathematics of games of chance, and misrepresents the birth ratio data to rule out chance in favour of design. The probability sections of Hume's Treatise taken together correct the equiprobability assumption and its extension to other kinds of phenomena in the estimation of wagers or expectations about particular events. Hume's probability theory demonstrates the flaw in this version of the design argument.
Keywords:
Notes
1The paper was read to the Royal Society on 19 April 1711. See Royal Society of London, ‘The Journal Book of the Royal Society 1702–1714’, cited in Shoesmith, ‘Controversy’, 134. Useful summaries of Arbuthnot's paper in Shoesmith, ‘Controversy’, Hacking, Emergence, 166–75, and in Gower, ‘Hume on Probability’. The relevant sections of the Treatise are Book I, part iii, sections 11–12 of Hume, Treatise of Human Nature (1738–39), eds. Norton and Norton, and of Hume, Treatise of Human Nature, ed. Selby-Bigge/Nidditch (subsequent joint citation as T by book, part, section, and paragraph numbers (Nortons) and pages (Selby-Bigge/Nidditch). For a summary of the critical treatment of these sections, see Millican, Reading Hume, 435–6. For the observation that Hume's probability theory has been generally neglected by scholars, see Gower, ‘Hume on Probability’, 3 and Collier, ‘New Look’, 21, 31.
The title of the essay refers to speculation about the identity of the recipient (if any) of Hume's letter to a physician (1734). See David Hume to unnamed correspondent [Dr. George Cheyne], [March or April 1734], in Grieg, Letters 1, 12–18. For identification of Cheyne as the recipient, see Burton, Life 1, 42–7. For identification of Arbuthnot as the recipient, see Mossner, ‘Hume's Epistle to Dr. Arbuthnot’. For an argument supporting Burton's (tentative) identification of Cheyne, see Wright, ‘Dr. Cheyne’. For the status of Arbuthnot in Hume's world, see generally Guerrini, ‘Tory Newtonians’ and Shuttleton, ‘Modest Examination’. For a suggestive historical discussion of the very public relations in the early eighteenth century of Arbuthnot, Cheyne, Newton, De Moivre, and ‘the Scottish faction within the Royal Society’, see Chapter Four, ‘Scotica Mathematica’, in Bellhouse, Abraham De Moivre, 50–61. See also Wright, ‘George Cheyne’, 140, n. 50, for an additional relation in this context to the Leibniz–Newton dispute about the calculus.
I am grateful to Daniel Campos for reviewing a draft of the essay and for his assistance in understanding and framing Arbuthnot's argument, and to members of the New York Early Modern Philosophy group for comments on an earlier version of the paper.
2Most notably, Derham incorporates the argument into his Boyle lecture of 1711–12 (published 1713); see Derham, Physico-Theology, 175–7, n. 8 and accompanying text; see also text at note 15 below. Nieuwentyt treats the argument extensively in the first volume of his tract against atheism, translated into English by John Chamberlayne in 1718, with an analysis of Arbuthnot's argument attributed to ‘sGravesande; see Nieuwentyt, Religious Philosopher, 1, 315–25. Arbuthnot produced an edition of Huygens's De ratiociniis in 1692, with a substantial preface (hereafter ‘Preface to Huygens’), in which he previews a recognizable version of the birth ratio argument. See Arbuthnot, Laws of Chance, xiv–xvii. This edition of Huygens, with Arbuthnot's preface, is the most influential in English in the first half of the eighteenth century. See Pearson, History of Statistics, 330. In the first edition of his Doctrine of Chances (1718) De Moivre refers to Arbuthnot not by name but as ‘a very ingenious Gentleman’ who is the author of the preface to the English translation of Huygens' De ratiociniis. De Moivre proposes a generalized but entirely recognizable form of Arbuthnot's argument as a test to distinguish events owing to chance from those produced by design, a text unchanged in the second edition of 1738; see De Moivre, Doctrine of Chances (1718), v–vi and De Moivre, Doctrine of Chances (1738), v. The argument appears without mention of Arbuthnot in Part II of Mandeville's Fable (1729); see Mandeville, Fable of the Bees, 2, 256–8. See also Patey, Probability and Literary Form, 70.
On the question of whether Hume was thinking of Arbuthnot's argument, or ever saw it or read about it, I reserve judgement: this is not an argument for direct historical influence, but rather an interpretation assigning a historically significant and philosophically provocative as well as readily available target to Hume's probability theory. There are other aspects of the reception of Arbuthnot's argument in the early eighteenth century and parts of Hume's own corpus that suggest that Hume may have been aware of Arbuthnot's work, and I note them below (see note 28), but they do not themselves make an argument.
3The 1665 edition of Graunt is in the catalogue of the (nephew) Baron Hume library. See Norton and Norton, The David Hume Library, 94.
4On ‘the primacy of expectation’ for seventeenth- and eighteenth-century probability, see Daston, ‘Probability and Evidence’, 1123–4; Daston, Classical Probability, 29–30; Sylla, Art of Conjecturing, 71–2. For discussion of the historical context for the equiprobability assumption, see Campos, ‘Framing’, 405 ff.
5Here, in outline, is the argument of the 1710 paper: Arbuthnot argues that the birth ratio is either the effect of chance or of divine providence. He assumes without argument that if the birth ratio is the effect of chance, the two possible outcomes, male and female, would be equally likely (this is the equiprobability assumption). Arbuthnot then claims that if the outcomes were equally likely, the chances would be very small that in one year an equal number of males and females would be born, that in no one year an extremely unequal number of males and females would be born, and that in each and every year a nearly equal number of males and females in a constant proportion would be born. Arbuthnot then notes that in eighty-two years at London, the number of males and females born was nearly equal and in constant proportion in each and every year, so that outcomes for the birth ratio are not equally likely, and concludes that the birth ratio is not the effect of chance but of divine providence.
6Pearson puts it most succinctly in his summary of Nieuwentyt: ‘the weakness of Arbuthnot's argument lies in the assumption that ½ is chance’. Pearson, History of Statistics, 301–2. See also the final section of the essay, below.
7Daston sums up the motives for the Newtonians: ‘The Royal Society natural philosophers caricatured Epicureanism in order to purge their own brand of atomism from any taint of atheism’ (Daston, Classical Probability, 248).
8For attribution of Whole Duty of Man to Allestree, see Rivers, Reason 1, 18. For the influence on Hume of Whole Duty, see Hume to Francis Hutcheson, September 17, 1739, in Greig, Letters 1, 34, and Boswell, Journals, 256. For Hume's view of Whole Duty, see Hume, Enquiries, 319, n. 1 and Rivers, Reason 2, 250.
9Compare Lucretius, On the Nature of Things 2, 1058: ‘[T]he atoms themselves collided spontaneously and fortuitously, clashing together blindly, unsuccessfully, and ineffectually in a multitude of ways, until at last those atoms coalesced’.
10For the historical context of Newton's relationship to Bentley on the issue of atomism, see Schliesser, ‘On Reading Newton as an Epicurean’, § 2.2.
11For extensive discussion of the improbability argument, see Rivers, ‘“Galen's Muscles”’, 577–97.
12See also Shoesmith (‘Controversy’, 137; emphasis added):
In developing the implications of the atheistic theories about fortuitous creation of the Earth and its inhabitants, in order to demolish those theories, Bentley was apparently thinking in quantitative terms about chance and probability. But he was able only to sketch his models of supposed ‘chance’ coincidence (of organs, particles, or ships), and therefore limited himself to very broad comments on the orders of magnitude of some of the relative probabilities that might be involved.
13On the seventeenth-century emergence of a fundamental probability set of equiprobable outcomes, see generally Campos, Framing.
14See the comprehensive historical discussion in Daston, ‘Probability and Evidence’, 1136 ff.
15First emphasis added. See note 2 above. Derham's language, especially that standing in for the mathematics, echoes Bentley's. Derham cites Graunt's numbers as well as the anti-polygamy purpose of the ratio that Arbuthnot makes the ‘corollary’ to his own argument in the 1710 paper. Derham also explicitly discusses the evils of Epicureanism. Derham, Physico-Theology, 431 ff. See also Daston, ‘Probability and Evidence’, 1132.
16For the importance of Arbuthnot's paper to Royal Society members and sympathetic Christian apologists, see note 2 above. See also Hacking, Emergence, 169. For the design argument in Britain, see generally Stewart, ‘British Debate’.
17For Hume's motives here, see Baier, Progress of Sentiments, 83 ff. Compare Fontenelle on Jakob Bernoulli:
[I]t is not so glorious for the Spirit of Geometry to reign in Natural Philosophy, as it would be in matters of Morality, which are very complicated, casual and changeable, and the more obstinate and stubborn the Matter it has to deal with, is the greater Honour it would acquire in overcoming it.
On November 14, 1705, Bernard le Bovier de Fontenelle read a eulogy of [Jakob] Bernoulli at the Académie Royale de Sciences in Paris’ (Sylla, Art of Conjecturing, 49). See also Huygens's original declaration in De Ratiociniis in Ludo Aleae (1657), as it appears in Part I of the Ars Conjectandi (1713): ‘[T]he more impossible it seems to subjugate to reason those things that are fortuitous and uncertain, the more admirable will be judged the art that achieves it’ (Sylla, Art of Conjecturing, 131).
18In the same passage, Hume observes that these dogmas have subjected modern philosophy to ‘pertinacious bigotry’ and ‘calumny and persecution’.
19In what follows I refer to Arbuthnot's argument in the 1710 paper as the object of criticism in the probability sections of Hume's Treatise. By this I mean the argument articulated first by Arbuthnot and later adopted, by name, by both fans and critics. I would remind the reader of both the prominence of Arbuthnot's argument in early eighteenth-century discussions of probability and the disclaimer about a direct historical influence on Hume's thinking of the 1710 paper above (see note 2 above).
20For a recent discussion and historical overview of the attribution, see, for example, Collier, ‘New Look’, 21–2.
21Gower suggests that a notable example of the authority Hume cites here is Arbuthnot's 1710 paper. Gower, ‘Hume on Probability’, 5. Note that while the circle of people ‘forming calculations concerning chances’ is sizeable, it is tightly connected both in private correspondence and public citation and discussion, in all of which except the work of Jakob Bernoulli Arbuthnot's argument looms large.
22Note that Hume is claiming that the notion of a superior hazard, not its mathematics, is easy.
23Bernouilli's objection is summarized in Shoesmith, ‘Controversy’, 141 and Hacking, Emergence, 168–9.
24For the history of this edition of Montmort, see also Bellhouse, ‘Banishing Fortuna’, 572 ff. Shoesmith observes that Bernoulli's objection amounts to saying that Arbuthnot's is ‘too restrictive an interpretation of “chance”’ (‘Controversy’, 141–2).
25Shoesmith interprets Bernoulli's counterargument this way:
[t]he fair two-sided die could be replaced by a multifaceted die, with rather more faces marked M than F, without taking away the element of ‘chance’. If tossed a large number of times, such a die would, Bernoulli maintained, yield ratios of M's to F's within the limits exhibited by the christenings data for London, with M's consistently predominating over F's. No appeal to divine providence was therefore necessary to account for either the persistent superiority of male over female births, or the narrow variation in the male: female birth ratio. Bernoulli's perception in this respect has been rightly commended by historians of probability [Todhunter 1865, 130–31; Hacking 1965, 77; Hacking Citation1975, 168, 170; Pearson 1978, 161–161].
26‘What differentiated Bernoulli's work on mathematical probability from the work of Huygens and Montmort was his effort … to apply mathematics to the resolution of civil, moral, and economic questions’ (Sylla, Art of Conjecturing, 350). See also 17 and 107 ff. See generally Daston, Classical Probability, 227 ff.
27De Moivre's Doctrine of Chances and the influence of Arbuthnot's own edition of Huygens are the main authorities in the period for Arbuthnot's extension of probability calculations to natural events. De Moivre's influence on mathematics and its applications, especially, was augmented by his close friendship with Newton. See, for example, Pearson, History of Statistics, 143.
28In the Preface to Huygens, Arbuthnot recognizes that calculations about events for which data is available must be framed differently than those for games of chance:
There is likewise a Calculation of the Quantity of Probability founded on Experience, to be made use of in Wagers about any thing; it is odds, if a Woman is with Child, but it shall be a Boy; and if you would know the just odds, you must consider the Proportion in the Bills that the Males bear to the Females: [in] The Yearly Bills of Mortality …
In 1710 Arbuthnot acknowledges the difference but argues that the near-equality of births is just as unlikely as their equality if one assumes that the outcomes are equally likely. The slight preponderance of males, which must be the result of providence, is to supply the deficits due to war and the dangers of hunting for food, in order to keep the balance that ultimately serves to prevent polygamy. (Arbuthnot, ‘Argument’, 187–9).
Hume makes an observation about such events in general terms in an undated discussion of the possibility of determining the proportion of good to evil:
Did a Controversy arise whether more Males or Females are born; cou'd this Question ever be decided merely by our running over all the Families of our Acquaintance; without the Assistance of any Bills of Mortality, which bring the Matter to a Certainty? (…) Pains & Pleasures seem to be scatter'd indifferently thro Life, as Heat & Cold, Moist & Dry are disperst thro the Universe; & if the one prevails a little above the other, this is what will naturally happen in any Mixture of Principles, where an exact Equality is not expressly intended. On every Occasion, Nature seems to employ either.
Stewart discusses dating the fragment and notes its ‘strong links’ with Treatise probability sections (Stewart, ‘Early Fragment’, 161), and observes that Hume's reference to ‘the Bills of Mortality’ points to Edinburgh's record-keeping practices as regards christenings (Stewart, ‘Early Fragment’, 168, n. 8). Elsewhere Stewart notes that the passage also refers to the subject of Graunt's work on the Bills of Mortality. See Stewart, ‘Hume's Intellectual Development’, 39.
I think it is possible that Hume may also be echoing Arbuthnot's language in the Preface to Huygens.