Remarks on the idealist and empiricist interpretation of frequentism: Robert Leslie Ellis versus John Venn

Abstract

The goal of this paper is to correct a widespread misconception about the work of Robert Leslie Ellis and John Venn, namely that it can be considered as the ‘British empiricist’ reaction against the traditional theory of probability. It is argued, instead, that there was no unified ‘British school’ of frequentism during the nineteenth century. Where Ellis arrived at frequentism from a metaphysical idealist transformation of probability theory’s mathematical calculations, Venn did so on the basis of an empiricist critique of its ‘inverse application’.

Notes

¹ See, for instance, Gillies (1973, 1), Gillies (2000, 88), Hald (2007, 77), Galavotti (2008, 419), Galavotti (2011, 154), Lang (1964, 298), Schafer (1993), Wall (2006a, 596), Zabell (2005, 186).

² For more or less detailed accounts of Venn’s views on probability see Keynes (1921, chapter 3), Kilinc (1999), Verburgt (submitted for review a), Verburgt (submitted for review b), Wall (2005), and Wall (2006b). The only comprehensive analyses of Ellis available are Kilinc (2000) and Verburgt (2013).

³ This essay was read to the Cambridge Philosophical Society on 14 February 1842 and published in its Transactions in 1844.

⁴ For instance Porter, in his The rise of statistical thinking, 1820–1900, mistakenly remarks that Ellis actually entertained a ‘sensational philosophy’. This may have to do with the fact that Ellis’s writings are, at some points, somewhat opaque—consider, in this context, his statement that ‘I [Ellis] shall be satisfied if the present essay does no more than call attention to the inconsistency of the theory of probabilities with any other than a sensational philosophy’ (Ellis 1844, 2).

⁵ I will return to Ellis’s views on ‘Bernoulli’s theorem’ in subsection Venn’s ‘anti-Bernoullian’ frequentism.

⁶ In Verburgt (2013) this characterization of Ellis’s reinterpretation of probability theory is qualified in light of his renovation of Bacon’s empiricist theory of induction—as put forward under the direct influence of William Whewell. In brief, it is claimed here that both Ellis and Whewell used ‘idealist’ views to supplement or perfect, rather than overcome or dismiss, the Baconian method.

⁷ For this point see, especially Daston (1994), and Verburgt (2013).

⁸ In the preface, Venn makes a similar remark when he criticizes ‘the belief that Probability is a branch of mathematics trying to intrude itself on to ground which does not belong to it’ (Venn 1866, vii).

⁹ Venn writes that ‘[t]he difference […] between them would not appear in the initial stage, for in that stage the distinctive characteristics of the series of Probability are not apparent; nor would it appear in the subsequent stage, for the real variability of the uniformity has not for some time scope to make itself perceived. It would only be in what we have called the ultimate stage, when we suppose the series to extend for a very long time, that the difference would begin to make itself felt’ (Venn 1866, 18).

¹⁰ It is all the more remarkable that, after having introduced the notion of ‘substitution’ as a necessary precondition for reasoning about series of natural events, Venn discusses an example where the substitution is ‘accidentally’ unnecessary—namely a game of chance! ‘In most cases a good deal of alteration is necessary to bring the series into shape, but in some—I refer of course to games of chance—we find the alterations […] needless’ (Venn 1866, 58). Here, Venn thus treats ‘artificial’ series as less artificial than natural series.

¹¹ For the meaning of Part IV of Jakob Bernoulli’s Ars Conjectandi for the philosophy of probability see, for instance, Hacking (1971), Schafer (1996) and Hald (2007, chapter 2).

¹² See, for instance, Hald (2007, 77).

¹³ For an account of the intricate connection between Ellis’s ‘idealism’ and ‘realism’ see Kilinc (2000), and Verburgt (2013).

¹⁴ This was, of course, exactly Bernoulli’s move when he, in a letter to G W Leibniz, argued that—put in modern terminology—there is not only a ‘fundamental probability set’ for games of chance but also, for instance, for diseases. Bernoulli thus wrote: ‘If now in place of the urn you substitute the human body, young or old, which contains the tinder of diseases like an urn contains stones, you can in the same way determine how much nearer the one [the old man] is to death than the [young man]’ (Bernoulli quoted in Hacking 1971, 220).

¹⁵ Recall Venn’s introduction of ‘ideal’ or ‘substituted’ natural series.

¹⁶ For this point, see Verburgt (forthcoming a).

¹⁷ My gratitude goes to Gerard de Vries, who first suggested pursuing the research on the topic of this paper. I would also like to thank Mees van Hulzen and Maite Karssenberg for their friendship and the editors for their help in preparing the manuscript.