Abstract
For aesthetic, strategic and pragmatic reasons, Jaynes (Probability: The Logic of Science, Cambridge University Press, Cambridge, 2003, Appendix A) objects to Bruno de Finetti’s founding of probability theory on the basis of the notion of coherence. In this paper an attempt is made to diffuse this critique, as well as to point out, briefly, that these, and the remarks on a variety of foundational issues in mathematics and metamathematics (op.cit, Appendix B) are misguided.
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Notes
Keynes’ ‘penciled marks or comments’ (Keynes 1973) at the end of his personal copy of Hayek’s review of the Treatise on Money, which appeared in the August, 1931 issue of the Economica.
The remarks on generalized functions, Dirac’s introduction of the delta function, the role of Laurent Schwartz, George Temple and James Lighthill [whose book, (Lighthill 1958), by the way, is dedicated to Dirac, Schwartz and Temple], seem to be symptomatic of Jaynes’ deficient or, at least, severely partial interpretations (cf, Jaynes 2003, Appendix A & B).
Negative aspects have a curious way of having positive effects on clarifying the issues in discussion. This is amply demonstrated by the development of algorithmic probability as a result of unsympathetic mathematical attitudes towards von Mises’ noble attempts to found the frequency theory of probability on a rigorous definition of place selection functions, taken by ‘orthodox’ probability theorists like Fréchet (see van Lambalgan 1987, especially §2.6). A concluding theme in this essay is that there is, after all, a strong affinity between the mathematics of de Finetti’s subjective theory of probability and the algorithmically founded frequency theory of probability of von Mises.
In this effort I am also greatly indebted to the fine ‘de Finetti scholarship’ exhibited in von Plato (1974, chapter 8).
‘There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy’, Hamlet, Act. 1, Sc. 5. Hamlet, it seems, is pointing out to Horatio that what can be explained is limited.
Dante’s advice, in Inferno (III, 51) comes to mind: Non ragioniam di lor, ma guarda e passa!
Among whom I am one, albeit a very minor member of a noble set! However, as a Foreign Member, of the Moral Sciences Division of the Juridical, Political and Economic Sciences, Istituto Lombardo, it gives me unreserved pleasure to note that many of de Finetti’s early and fundamental contributions (for. eg.,de Finetti 1928, 1930a, 1930b, 1930c) were published in the Rendiconti del Reale Istituto Lombardo di Scienze e Lettere.
I do not refer to measurable in its formal, mathematical, senses—but in its normal, engineering, geometric and craftsman’s ways of usage. de Finetti, like most Italian mathematicians of the time, was a maestro of geometry (cf. Guerraggio and Nastasi 2000).
I should like to add that the parallel dichotomy between complete and incomplete, though, for example, referred to in the von Neumann and Morgenstern classic (1953, see, in particular, p. 30—these notions are, however, referred to in the context of information), seems not as well understood even in serious theoretical contributions to economic theory and probability theory, outside the particular field of algorithmic probability theory, or one of its related variants, as algorithmic information theory or Kolmogorov complexity (cf., Li and Vitanyi 1997).
This is made explicit, in particular, in Ramsey (op.cit), when he departed from Keynes’ reliance on partial ordering of choice over alternatives, to his own advocacy of total ordering.
I can do no better than refer a Jaynesian, who is tempted to indulge in ethical or aesthetic strictures against the normal, ‘age-old established way of measuring a person’s belief’ by proposing a bet, and see what are the lowest odds which he will accept’, to Bernard Shaw’s classic study of the The Vice of Gambling and the Virtue of Insurance (Shaw 1960).
The notion of an algorithm is taken as a primitive in most forms of constructive mathematics; it is underpinned by some notion akin to a Church-Turing Thesis in all the known—at least to me—variants of computability theory.
This should be read as ‘computability’, in the various contexts in which ‘calculation’ and ‘consistency’ are explicated in Jaynes (op.cit).
Remarkably, the genesis of Dirac’s innovative idea of the delta-function in his early engineering education, where Oliver Heaviside’s operational calculus was crucial, is not mentioned (cf. Farmelo 2009 and Kragh 1990, pp. 206–207). In my own engineering education I was introduced, first, to the operational calculus of Heaviside and only much later, in fourth year undergraduate classes on quantum mechanics, to the Dirac δ-function.
Strangely, the spelling of this great German analyst is consistently (sic!) given as Weierstrasz (Jaynes, ibid, p. 665, ff)!
To the best of my knowledge this should be 1958.
But this is not the sense in which Jaynes dismisses de Finetti’s Dutch Book Method as ‘vulgar’.
It should be: \( f\left( x \right) = \mathop \sum \nolimits_{n = 0}^{\infty } a^{n } \cos \left( {m^{n} \pi x} \right). \)
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I have been deeply influenced by the comprehensive ‘de Finetti scholarship’ of Professor Eugenio Regazzini (just three of his relevant contributions are: Berti et al. 2007; Regazzini and Bassetti 2008; Regazzini 2012) of the University of Pavia and the interest in de Finetti’s many technical and expository writings shown by my friend, Professor Maria Carla Galavotti of the University of Bologna. Moreover, the important way in which Professor Daniele Mundici has harnessed de Finetti’s notion of coherence, especially to elucidate the variety of metamathematical ways of explicating the richness of the Dutch Book Method, has been a rich source of inspiration for me (cf. especially, Mundici 2008). None of them except—I mean, ‘not even’ (pace Robertson 1936, p. 168)—my friend and colleague Stefano Zambelli are responsible for the deficiencies in this essay.
In the title of this paper, I use the term ‘Jaynesian’ to refer to E. T. Jaynes and his critique of de Finetti. Being an economist, with a Cambridge background, I am not uninfluenced by being a sort of Keynesian, to coin the word ‘Jaynesian’!
The final version of this article has benefitted greatly from the positive and generous criticisms of two anonymous referees of the earlier version of the paper. I have endeavoured to revise the paper to meet their most pertinent suggestions. An important suggestion by one of the referees, on deepening the basis of my vision of de Finetti’s analytical foundations of subjective probability in terms of computability theory, would have required, if it is to be dealt with in any kind of rigorous adequacy, considerable expansion of the present version of this paper. I have, therefore, referred to my companion piece, to this paper, where that theme is more fully incorporated within the same framework as this one.
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Velupillai, K.V. de Finetti’s theory of probability and its Jaynesian critique. Econ Polit 32, 85–95 (2015). https://doi.org/10.1007/s40888-015-0005-z
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DOI: https://doi.org/10.1007/s40888-015-0005-z