Abstract
The objectivist approach of probability cannot be applied to all feelings of uncertainty, but only to events liable to occur in identical conditions, during repeated trials. Different types of axioms were proposed during the beginning of the twentieth century and led to the general acceptation of Kolmogorov’s ones. However, as the probability of a hypothesis is a meaningless notion, we can only test with this approach the probability of obtaining the observed sample if the hypothesis is true. We then give examples of application of this approach to some social sciences (political arithmetic, epidemiology and sociology). Finally we try to identify in more details different problems raised by this analysis and show that, in fact, they are often interlinked.
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Notes
- 1.
Tam possum proiicere unum tria quinque, quam duo quatuor sex. Iuxta ergo hanc aeqalitatem pacta constant, si alea sit iusta; & tanto plus, aut minus, quanto a vera aequilitate longius distiterit.
- 2.
Inquirendum nimirum restat, an aucto sic observationum numero ita continuό augeatur probabilitas assequendae genuinæ rationis inter numeros casuum, quibus eventus aliquis contingere & quibus non contingere potest, ut probabilitas hæc tandem datum quemvis certitudinis gradum superet.
- 3.
While Fisher may be viewed as a frequentist in most of his writings, he gave a different definition of probability at the end of his life, noting that ‘no sub-set may be recognizable having a fraction possessing the characteristic differing from the fraction P of the whole’ (Fisher 1960). This concept of probability is generally regarded as unclear and has been little used since (Savage 1976).
- 4.
Ita ex. gr. noti sunt numeri casuum in tesseris; in singulis enim tot manifestè sunt quot hedrae, iique omnes æquè proclives; cùm propter similitudinem hedrarum & conforme tesseræ pondus nulla sit ratio, cur una hedrarum pronior esset ad cadendum quàm altera, quemadmodum fieret, si hedræ dissimilis forent figurae, aut tessera una in parte ex ponderosiore material constaret quàm in altera.
- 5.
eæque faciles.
- 6.
quod sane in usu vitæ ciilis, ubi moraliter certum pro absoulte certo habetur.
- 7.
Indeed, Fréchet (1951) remarks that this principle, while attributed to Cournot, ‘seems to have been already stated more or less clearly by D’Alembert’. But we should note that D’Alembert, while a great mathematician, made a number of errors in his reasoning on probability (Bertrand 1889; Delannoy 1895; Maupin 1895). However, as we shall see later, D’Alembert’s occasionally subjective stance in his reasonings gave rise to some of the criticisms directed against objectivist probabilists.
- 8.
The acronym stands for the Zermelo-Fraenkel theory, formulated with the axiom of Choice.
- 9.
This consistency is essential for the constructivist mathematical school, of which Gauss, Borel and Lebesgue are the best-known representatives: for them, mathematical objects exist only if there is a precise method that tells us how to construct them. By contrast, for the formalist school, of which Moritz Pasch and David Hilbert were the most famous representatives, the lack of contradiction in a system of axioms is a sufficient precondition for accepting that system. Despite siding with the constructivists, von Mises was not unduly troubled by these criticisms. He actually claimed that ‘collectives are in a sense ‘the rule,’ whereas lawfully ordered sequences are ‘the exception’.’
- 10.
For consistency with the notations already used in this chapter, we have modified those used by Kolmogorov.
- 11.
- 12.
Ever since the Observations appeared, it has been claimed that their true author was William Petty. Petty himself claimed authorship when applying for a political office in Ireland. Some observers, such as Le Bras (2000), use this argument to prove that demography never was and never will be a science—contrary to the thesis advocated by Graunt’s supporters. Rather, because of the possibility that Petty might have founded it, demography should be viewed as a political instrument in the hands of political authorities. Le Bras’s contentions—particularly on the determination of the number of deaths—and his attacks on some Graunt specialists hardly allow us to take his demonstration seriously (for more details, see Reungoat 2004).
- 13.
Hacking offers the following argument: ‘Graunt assumes a uniform death rate, that is, that there is a constant chance p of dying in a given year. If the chance of living 10 years is 0.5, consider a population of size N. The number who survive the first year is N(1−p). The number who survive the second is [N(1−p)−pN(1−p)] or N(1−p)2. The number who survive 10 years is N(1−p)10 = 0.5 N. Now let q be the chance that at least one man in a group of ten dies in a given year; then 1−q is the chance that no one dies. This is just (1−p)10, which, solving the above equation is 0.5. So, as Graunt says, q is also 0.5’.
- 14.
We use the standard demographic notation for a probability (quotient), q, which Hacking writes p.
- 15.
Leibniz (1675) gave the correct solution to this problem.
- 16.
‘quick Conceptions’: ‘live births’ in modern English.
- 17.
Let us again note his misuse of his own table, when he confuses the number of deceased persons and living persons. He gives the percentage of individuals aged between 16 and 56 as 34%, which is in fact the percentage of deaths. According to his life table, the percentage of living persons is in fact 41%.
- 18.
Halley’s exact words are as follows: ‘But the Deduction from those Bills of Mortality seemed even to their Authors to be defective: First, In that the Number of the People was wanting. Secondly, That the Ages of the People Dying was not to be had. And Lastly, That both London and Dublin by reason of the great and casual accession of Strangers who die therein, (as appeared in both, by the great Excess of the Funerals above the Births) rendered them incapable of being Standards for this purpose; [.]’
- 19.
Ignoring Neumann’s table, Jaynes (2003), who describes Halley’s work in detail, regrets that Halley did not supply his data in more detailed form. However, despite Neumann’s detailed table, it is clearly impossible to reconstruct Halley’s tables without additional hypotheses.
- 20.
Interestingly, Neumann distinguished between stillbirths and deaths occurring before the age of 1 year. This would have made it possible to determine separate probabilities of stillbirth and infant mortality.
- 21.
Arbuthnot wrote: ‘in the vast Number of Mortals there would be but a small part of all the possible Chances, for its happening at any assignable time, that an equal Number of Males and Females should be born’.
- 22.
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Courgeau, D. (2012). The Objectivist Approach. In: Probability and Social Science. Methodos Series, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2879-0_1
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