Elements of Probability Calculus

Abstract

Editors’ Foreword: In 1914, one year after graduating in Mathematics, Diogo Pacheco d’ Amorim submitted his doctoral thesis Elements of Probability Calculus to the University of Coimbra. In the author’s Preface it is said that “the title—An Essay Towards Rationalizing Probability Calculus—would perhaps be more appropriate”. In fact, Pacheco d’ Amorim’s endeavour seems to be an attempt to build axiomatic Probability Theory, as requested by Hilbert in his seminal address to the Paris 1900 International Congress of Mathematicians, avoiding embarrassing paradoxes, as exposed in Bertrand’s Calcul des Probabilit´es. To fulfill this task, he first defines the standard model: someone performs the selection of an element from a sample space which is qualitatively and quantitatively known (for instance, extraction of one ball from one urn where n1 balls of colour C1 ,. . . , nr balls of colour Cr have been thoroughly mixed). This individual knows whether his extraction has been random or not (i.e., in the above example, whether any ball in the urn has or hasn’t the same chance of being selected). Thus, in this standard model, contrarily to Poincar´e’s opinion, the concept of randomly selecting an element from the sample space has a clear meaning for the one performing the extraction, which can be used as a “primitive” concept in the construction of probability. This leads to the concept of equipossibility. The consideration of chains of hierarchically dependent extractions is then used to build up a wise and elegant solution to the main problem of constructing stochastic models in which elementary events are no longer equiprobable. For Pacheco d’Amorim probability is always conditional probability, and in some aspects is construction anticipates R´enyi’s work on the foundations of Probability. In 1909, Borel had published a remarkable paper on continuous probability, that surely influenced Pacheco d’Amorim’s construction of “randomly throwing geometric objects” in continuous sample spaces. In chapter III of his thesis, he gives a solution to one of the celebrated Bertrand’s paradoxes (a solution that in our view has a serious flaw, cf. the editorial note (13)), and in chapter IV the discussion of “image points” — an ingenious construction of the probability measures of functions of random variables, lacking the concepts of random variables and of distribution functions — effectively solves another class of Bertrand’s paradoxes, namely questions arising from using equiprobability models both for choosing a number in [0, 100] and in [0 2 , 100 2]. Pacheco d’Amorim’s believed that he had solved Bertrand’s paradoxes in the standard model, in which the subject performing the extraction knows whether this was or wasn’t done at random. His next step is an anticipation of pseudo-randomness: he deduces Bernoulli’s and de Moivre’s limit results, and from them he judges whether or not a (long enough) sequence of trials performed by someone else, or even by a mechanical device, imitates closely randomness. In the wealth of ideas discussed in the closing chapter, the main ideas of significance and hypothesis testing are clearly shaped. Pacheco d’Amorim thesis is not a mature work, and there are some blunders in the text, that we discuss or at least unveil at the appropriate places. The long and cumbersome discussion of “random figures” is the weak point of this thesis, and we have been unable to understand clearly what the author meant in the last section of Chapter III (if you think that our translation is difficult to understand, you are right: we couldn’t agree on the original’s meaning). But, on the other hand, it has many strong points, it anticipates some influential ideas in Probability and Statistics, and surely deserves a fair opportunity to have international recognition. In this translation, we corrected obvious typos (and we hope we didn’t introduce other typos); figures have been redrawn, and we adopted symbols that, in our view, improve the readability of the text. We are thankful to Prof. José Pacheco d’Amorim, who authorized this edited translation of his father’s thesis.

Sandra Mendonça, Dinis Pestana, Rui Santos Lisbon, 2007 August 08