Are Statistics Only Made of Data?: Know-how and Presupposition from the 17th and 19th Centuries (Methodos Series, 20) [2024 ed.] 3031512537, 9783031512537

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Methodos Series  20

Éric Brian

Are Statistics Only Made of Data? Know-how and Presupposition from the 17th and 19th Centuries

Methodos Series Methodological Prospects in the Social Sciences Volume 20

Series Editors Daniel Courgeau, (INED), Institut National d’Etudes Démographique, Paris CX 20, France Robert Franck, Inst Supêrieur de Philosophie SSH/FIAL, Université catholique de Louvain, Louvain-la-Neuve, Belgium Editorial Board Members Peter Abell, Department of Management, London School of Economics & Political Science, London, UK Patrick Doreian, Department of Sociology, University of Pittsburgh, Pittsburgh, USA Sander Greenland, Dept of Epidemiology BOX 951772, UCLA School of Public Health, Los Angeles, CA, USA Ray Pawson, School of Sociology & Social Policy, Leeds University, Leeds, UK Cees Van De Eijk, University of Amsterdam, Nottingham, UK Bernard Walliser, Ecole Nationale des Ponts et Chaussées, Paris, France Björn Wittrock, Uppsala University, Uppsala, Uppsala Län, Sweden Guillaume Wunsch, Departement de Demographie, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium

This Book Series is devoted to examining and solving the major methodological problems social sciences are facing. Take for example the gap between empirical and theoretical research, the explanatory power of models, the relevance of multilevel analysis, the weakness of cumulative knowledge, the role of ordinary knowledge in the research process, or the place which should be reserved to “time, change and history” when explaining social facts. These problems are well known and yet they are seldom treated in depth in scientific literature because of their general nature. So that these problems may be examined and solutions found, the series prompts and fosters the setting-up of international multidisciplinary research teams, and it is work by these teams that appears in the Book Series. The series can also host books produced by a single author which follow the same objectives. Proposals for manuscripts and plans for collective books will be carefully examined. The epistemological scope of these methodological problems is obvious and resorting to Philosophy of Science becomes a necessity. The main objective of the Series remains however the methodological solutions that can be applied to the problems in hand. Therefore the books of the Series are closely connected to the research practices.

Éric Brian

Are Statistics Only Made of Data? Know-how and Presupposition from the 17th and 19th Centuries

Éric Brian Centre de recherches historiques École des hautes études en sciences sociales Paris, France

ISSN 1572-7750 ISSN 2542-9892 (electronic) Methodos Series ISBN 978-3-031-51253-7 ISBN 978-3-031-51254-4 (eBook) https://doi.org/10.1007/978-3-031-51254-4 The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Foreword1

The Peculiar Meanings of Data Here is a case that I used to introduce a book on scientific projects of demographic collections and their relationship with the beginnings of the analytical theory of probabilities, notably associated with the name of the mathematician Pierre-Simon Laplace (1749–1827). Its value as an example is equally valid for the present work. This conjunction took place shortly before the French Revolution.2 The volume of the history of the royal academy of sciences for the year 1784,3 i.e., the annual summary of the works of this learned company, offers, on page 592, a “Table of births, marriages and deaths [...] in the kingdom including the Island of Corsica from 1770 to 1783”.4 This table summarized some 20 years of work carried out in the royal administration, in Paris and in the provinces. It was published as an appendix to an article signed by three members of the Academy: Condorcet 1

This book was mainly written during the academic year 2022–2023. Fruitful presentations and discussions took place in the more than 40 years old “History of the Calculus of Probability and Statistics” seminar at the École des Hautes Études en Sciences Sociales, this past year or a longer time ago that helped to clarify some of the points argued here. The author would like to thank colleagues and students for their fruitful and dynamic contributions, in particular this year: Prof. Thierry Martin, Prof. Laurent Mazliak. In addition, the book would not have been possible without the trusting support of Series Editors Daniel Courgeau and Robert Franck, of Evelien Bakker, Editor of Population Studies, and the kind and attentive assistance of Bernadette Deelen-Mans, Senior Editor in charge of the Series at Springer Nature. The original manuscript written in French, including the quotations, have been especially translated for this edition and checked by the author. 2 Brian, La Mesure de l’État. Administrateurs et géomètres. Paris, Albin Michel,1994 et id., Staatsvermessungen. Condorcet, Laplace, Turgot und das Denken der Verwaltung. Wien. Springer, 2001. 3 Histoire de l’Académie royale des sciences pour l’année 1784, avec les mémoires de mathématiques et de physique, pour la même année, tirés des registres de cette Académie. Paris, 1787. 4 The royal administration did not govern Corsica until 1769. v

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Foreword

(1743–1794), Laplace himself, and Dionis du Séjour (1734–1794). According to the rules of the learned society, the quality of the authors as academicians was an indispensable condition for publication in the annual collections of memoirs presented to the Academy by its members. However, Jean-Baptiste-François de La Michodière (1720–1797), the real organizer of the recapitulation of which this table and a few others gave an account, was not an academician. The fact that the mathematician Condorcet was one of the nominees, and that he was also the Academy’s Permanent Secretary, i.e., the one responsible for its publications, undoubtedly contributed to this breach of the rules in other circumstances. A handwritten version of this table had circulated in the administrative sphere and among the academicians: it can be found today in the archives of the Parisian learned society. On this preliminary manuscript, as on the printed version of 1786, we read the number of births in 1783 as 947,941. Half a century later, the retrospective statistical series of the new official French office was inaugurated, which would be continued until the 20th century.5 And more than two centuries after its first publication, this figure, which has all the appearances of a number that may be calculated, was compared with the estimates of historical demographers at the end of a major survey to reconstitute the population indicators of the Ancien Régime, a survey based on methods proper to the middle of the 20th century and on the ignorance of the know-how used two centuries earlier, this time in exhaustive penand-ink surveys of nearly 33,000 towns, villages, and parishes of any sizes. Considering from the outset that the compilations of the 1780s needed to be revised, the demographic historians of the 1960s and 1970s applied corrections to the 1786 figures, and then compared their extrapolations with the results they themselves had obtained by recent procedures until they arrived at what they considered to be a generally satisfactory agreement between the old sources and their calculations. The tables published in 1975 to support this finding are hardly convincing.6 To recapitulate, in manuscript form and under cover of the secrecy of the finances, a former intendant, keen on population calculations, had the numbers of baptisms, marriages, and burials compiled from the parish registers, collected by the intendants in the provinces and then sent to Paris, and he ended up with a manuscript intended, according to the expression of the time, to be placed before the eyes of the king. This manuscript and the figure of 947,941 that it gave for the total number of births in the kingdom in 1783 were in no way official, public, or scientific. However, it was inserted, at the cost of a breach of the rules, in a volume officially published by the Royal Academy of Sciences and introduced by means of an impressive series of mathematical memoirs. 947,941 thus became, in 1787, not only an official figure, but also the result of particularly advanced works in the calculation of probabilities. In so doing, the handwritten figure, which the very composition of the table made it

5

Ministère des travaux publics, de l’Agriculture et du Commerce, 1837, Statistique de la France. Paris, imprimerie royale, table 65 p. 286. 6 Blayo (Yves), 1975, « Le mouvement naturel de la population française de 1740 à 1829 », Population, 30e année, n°1, Table III, p. 31.

Foreword

vii

possible to relate to the administrative know-how in force at the end of the Ancien Régime, was abstracted from it. When, 50 years later, this figure reappeared in the ministerial compendium, it gave the public, in the most official way, the prototype of new counts made possible by the establishment of post-revolutionary civil status and consecrated, with layout effects, a break with the skills of the previous century: but the old figure, thus transported, was henceforth read as if it had come from, let us say, modern administrations. In the golden age of historical demography, towards the end of the 20th century, this same figure was considered so abstractly that it was subjected to all sorts of calculations quite alien to its original meaning. Thus, the question of the historical, or even epistemological, approach to data can be summed up as the need to wage a permanent struggle against the abstract reading of figures with a singular meaning, which are imprudently considered as numbers that are themselves supposed to be suitable for abstract calculations and therefore considered as data, in other words against the abuse of abstraction. We will discuss the historical transformation of this term of data later. There have been data for a very long time, and the abstraction that we can grant them leads us to read them in an anachronistic way. In the same way, these data which today circulate intensively by electronic means, if we reduce them to their singularity, will appear for what they are, less fluid and less powerful. The aim of this book is twofold. The first is to bring to a readership of all horizons, unfamiliar with the developments in the history of abstraction over the last decades, what has been achieved by some of this work. The second is didactic: it is a question of explaining by which methodological ways one can, without mobilizing an exceptional erudition, fight the abuses of the readings carrying uncontrolled abstractions and the anachronisms that these abuses involve.7 In the first chapter, the notion of data is taken up to extract the most common preconceptions. Then, offering a more general approach than the discussion of the number of births in 1783 that has just been mentioned, we attempt to synthesize the work on the history of abstraction and to draw from this synthesis a methodology. The two central chapters deal with skills in use on the one hand in the 17th and 18th centuries and on the other in the 19th century. The sharpness of the critical vocabulary of the philosopher Francis Bacon (1561–1626) makes it possible to specify the objective of these chapters. The aim is to track down two “idols”, as he put it in his Aphorisms concerning the interpretation of nature and the kingdom of man in his Novum Organum. “The Idols and false notions which are now in possession of the human understanding, and have taken deep root therein, not only so beset men’s minds that truth can hardly find entrance, but even after entrance obtained, they will again in the very instauration of the sciences meet and trouble us.8 There are four classes of Idols which beset men’s minds. To

7

Same intention in Ehrhardt (et. al.), Le sens des nombres. Paris, Vuibert, 2010. Bacon (Francis), 1620, Novum Organum scientiarum. [Leyde], Wijngaerde, 1645 (1st ed. 1620), Aphorism XXXVIII.

8

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Foreword these for distinction’s sake I have assigned names – calling the first class, Idols of the Tribe; the second, Idols of the Cave; the Third, Idols of the Marketplace; the fourth, Idols of the Theatre.9 The formation of ideas and axioms by true induction is no doubt the proper remedy to be applied for the keeping off and clearing away of idols. To point them out, however, is of great use”.10

For Bacon, the Idol of the Tribe is peculiar to human nature as the need to believe in a supreme intelligence; the Idol of the Cave is peculiar to an individual who closes his search on himself; the Idol of the Marketplace comes from exchanges and trade between men; finally, the Idol of the Theatre is in his eyes derived from various dogmas or philosophies and false demonstrations. He calls them theatrical (we would say today that they are postures) because they come from previously acquired intellectual systems that are played in retrospect in the manner of theatrical roles.11 Thus, Chap. 2 targets an idol of the demographic theatre: the ratio, whose technical competences will be necessary to examine and on what scholarly and philosophical foundations it was formed in the 16th century. This will be an opportunity, by way of example and in depth, to put an end to an Idol of the Theatre that is particularly tenacious in the historiography of the calculus of probabilities: the fable according to which the rise of this calculus was due to the importance of maritime trade in the sixteenth and 18th centuries. Chapter 3 focuses on another idol of the Theatre, this time a statistical one, formed in the 19th century: the average, which will call for similar treatments by applying them not to the scientific literature of the Renaissance and the early modern period, but to that of its own time, which may falsely appear to us to be less foreign, even though the difficulty lies in the international scope of the literature concerned. In both cases, as today, but on other media, data were everywhere. In addition to this omnipresence of data, it will be observed that these ancient know-how and presuppositions persist, in a world where digital material, their supports, calculation capacities, and their extrapolations coexist. Thus, the investigation carried out throughout the book will make it possible not to confuse these various aspects and consequently to ward off the modern superstitions that are attached to data and to see something that has been difficult to assume: we live among earlier spectres of abstractions and we must avoid confusing them. Centre de recherches historiques École des hautes études en sciences sociales Paris, France September-November 2023

9

Id. ibid. Aphorism XXXIX. Id. ibid. Aphorism XL. 11 Id. ibid. Aphorism XLI–XLVI. 10

Éric Brian

Contents

1

Considering Data: Critique and Method . . . . . . . . . . . . . . . . . . . . . 1.1 Statistics and Data: Two Avatars of Numerical Abstraction . . . . . 1.2 Data from Euclid of Alexandria to Norbert Wiener . . . . . . . . . . . 1.3 From Circulation to Know-How and Presupposition . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and 18th Centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Working Out Data in the 17th Century . . . . . . . . . . . . . . . . . . . . . 2.2 A Modern Starting Point: The Liber Abbaci of Leonardo of Pisa . . . 2.3 An Obscurity in Euclid’s Elements and Its Consequence . . . . . . . . 2.4 Approximations and Mechanical Reasoning . . . . . . . . . . . . . . . . . 2.5 The Legend of Maritime insurance at the Beginnings of the Classical Calculation of Probabilities . . . . . . . . . . . . . . . . . 2.6 The Legendary Meeting of Political Arithmetic and Probabilities . . . 2.7 Conclusion. The Historical Corroboration of Ratios . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

. . . . .

Analytical Probability, Averages and Data Distributions in the 19th Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Working Out Data in the 19th Century . . . . . . . . . . . . . . . . . . . . . 3.2 A Legacy of the 18th Century . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Average: From Science to Mysticism . . . . . . . . . . . . . . . . . . 3.4 Averaging the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Know-How and Presuppositions from the Quetelesian World . . . . 3.6 Cracks on the Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusion: The Historical Corroboration of Averages . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 13 15 19 19 24 29 37 44 55 63 64 71 71 75 80 112 119 126 137 138

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Contents

Idols, Paradigms and Specters in Data Sciences . . . . . . . . . . . . . . . 4.1 The Inebriation of Abstraction and Its Misdeeds . . . . . . . . . . . . . 4.2 From Know How to Presuppositions and Idols . . . . . . . . . . . . . . 4.3 Living in a Scientific World Cleared from Idols . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

145 145 147 153 161

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

List of Figures

Illustration 1.1

Illustration 1.2

Illustration 1.3 Illustration 1.4

Illustration 2.1

Illustration 2.2

Illustration 2.3

Illustration 3.1 Illustration 3.2 Illustration 3.3

Ludwig Wittgenstein’s rabbit-duck (1892). (Source: anonymous engraving, Zeitschrift Fliegende Blätter, n°2465, Vol. 97, p. 147) .. . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . .. . . . Detail of a punched card (1936). (Source: Carmille, 1936, plate XIV, fig. 15: example for human resources; author’s copy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Euclidian data: AD and BD (1632). (Source: Frontispice of Henrion; same date) . . . . . . . . . . . . . . . . A view of the Old-Market at Louvain (1665). (Source: Edward van Even, Louvain dans le passé et dans le présent. Louvain, Fonteyn, 1895, p. 227) . .. . .. . . .. . . .. . .. . . .. . . .. . .. . Extract from an accounting learning notebook from the end of the 18th century. The crossed out figures suggest a galley method division. (Source: Anonymous, ca 1790, © Archives of the Paris Academy of Sciences) . . . . . . . . . . . . Gear ratios for transmission mechanisms by Leonardo da Vinci (1478), Sketch-book, Biblioteca Ambrosiana di Milano. (Source: Database by Wolfgang Lefèvre and Marcus Popplow, MPIWG, Berlin. http://dmd.mpiwgberlin.mpg.de/; reference LdVCA022) . . . . . . . . . . . . . . . . . . . . . . Measurement of the elevation of the polar star with the proportion compass. (Source: Apian, 1548, (unpaged) ca p. 40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of Laplace’s second law (Elementary mathematical construction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency histogram published by Quetelet (1846, p. 103) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A box from the Edinburg Medical Journal of Edinburgh (1817) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

11 11

12

22

39

40 81 97 99

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Illustration 3.4 Illustration 3.5 Illustration 3.6

Illustration 3.7 Illustration 3.8

Illustration 3.9

Illustration 3.10

Illustration 3.11

Illustration 3.12

List of Figures

The inclination to crime according to Quetelet (1848a, p. 14) (Letters a, o, b, i added for readability) . . . . . . . . . . . . . . The Law of accidental causes according to Quetelet (1854, p. 56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the frequency curves for the measurements of Scottish conscripts according to the figures published in the Medical Journal of Edinburgh (1817) and according to Quetelet from 1846 Horizontally: scale of chest sizes measured in inches Vertically: frequency for a given size, in ten-thousandths Grey histograms: distribution for the figures published in Edinburgh (1817) Black histograms: distribution for those published by Quetelet (from 1846) . .. . .. . .. . . .. . .. . .. . .. . .. . . .. . .. . .. . .. . . .. Chest size of Scottish soldiers representing column [5] of Table 3.1, according to Quetelet (1869a) . . . . . . . . . . . . . . . . . . . Curve related to the COVID pandemic reproduced in various media. (Source: Centres for Disease Control and Prevention) . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . Size of conscripts from Doubs according to L.×Ad. Bertillon Horizontally: sizes of conscripts in feet and inches, grouped by one-inch intervals Vertically: frequencies of recorded conscripts Continuous curve with one bump: frequencies [inappropriately called here in Quetelet’s vein: “probabilité”] for each interval of sizes for France as a whole Dot curve with two bumps, same frequencies for the district of Doubs) Source Bertillon (1876a, b) . . .. .. . .. .. . .. . .. .. . .. .. . .. . .. .. . .. “Curve that seems to have gained the sympathy of the creator” (A. de Foville) Horizontally: Axis of sizes Vertically: number of men for each size. “M” indicates the mean Source: Foville (1907) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual abilities and social utilities (accordin to A. de Foville) Captions from top to bottom: Talent, Mediocrity, Weakness, Limit of social uses [sic]. (Source: Foville (1907) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthropometric card of Mr. Alphonse Bertillon Source: La Vie illustrée (November 7, 1902), n°212, p. 70 Source: https://criminocorpus.org/fr/ref/114/79368/ . . . . . . .

102 103

108 110

120

122

123

124

131

List of Table

Table 3.1

Comparing Quetelet’s source (1817) and his report (1846) . .. . .. . 107

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Chapter 1

Considering Data: Critique and Method

“The history of abstraction brings up in each occurrence differentiated types of social and cultural practices. The approach is therefore entirely upstream of society [...]: it is constitutive of social history itself; it does not receive its meaning from a preconstructed picture of society. It organizes its own sociohistorical space, its own circulation of ideas”. Jean-Claude Perrot, 2021.1

1.1

Statistics and Data: Two Avatars of Numerical Abstraction

Today, it is a good collection of what can be considered as common knowledge on a global scale. This is the online collective encyclopedia Wikipedia™: its articles come from the most diverse authors and they are checked, and even corrected, by this community of internet users. However, variations sometimes appear on the same subject from one language to another. Thus, the French version of the article “Data” specifies “Data is a term used in French as a synonym for the word “données”, particularly in the computer field”.2 This acceptance and the precision regarding this field call for comments that will appear later. The English version says more: “In common usage and statistics, data [...] is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data”.3

1

Perrot, 2021, p. 498. https://fr.wikipedia.org/wiki/Data consulted on July 11, 2023 at 1:54 pm. 3 https://en.wikipedia.org/wiki/Data consulted on July 11, 2023 at 2:02 pm. 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 É. Brian, Are Statistics Only Made of Data?, Methodos Series 20, https://doi.org/10.1007/978-3-031-51254-4_1

1

2

1

Considering Data: Critique and Method

These two definitions start from the state of the art in contemporary technologies for which the last decades of the 20th century marked a kind of apotheosis with the generalization of digitizations and their large-scale circulation. In doing so, the exploration and management of this vast corpus created new needs: it was then a matter of associating the digitized objects - let’s say, these data - with descriptors or indexes: these were the metadata, according to the ancient Greek prefix (μετά) which designates the accompaniment of the prefixed thing. Thus, to take an example, a scientific article is no longer simply disseminated as in the past in a printed journal, but in the form of an electronic document where, on the one hand, each word can be indexed singularly and which, on the other hand, a series of often coded descriptors aim to facilitate its identification. So much so that above the ocean of data float clouds of metadata. These are not recent innovations: metadata are as old as library catalogs. It just happens today that they are recorded in as electronic a manner as the objects they describe. A word has even been found in science fiction novels to designate the virtual mode where this ocean and these clouds meet: the “metaverse”.4 The universe in question is fortunately within a click’s reach on any computer connected to the Internet. Thus, the identification of specialized publications is greatly facilitated and the systematic nature of the digitization campaigns of printed works preserved since the 16th century means that rarities once inaccessible can be downloaded by all and form an unparalleled personal electronic library compared to physical libraries. What’s more, tools have been designed that allow historians to highlight the frequency of the use of words or expressions over several centuries.5 Should it be made known to anyone who would engage in historical research today that the author of this book began his work, a student in the early 1980s, by manually going through the paper catalogs of the French National Library for six months only to discover later that the preservation of collections on the shelves of the Butler Library at Columbia University then offered a much more convenient thematic access to the books themselves? What’s more, manual compilations of handwritten or printed metadata led to sharply decreasing returns? Most of the references annotated in this book had been consulted in libraries since the 1980s, sometimes at the cost of expensive international travels. However, everything discussed in this work written during the year 2023 was obtained through electronic means and worked on under the most favorable conditions of tranquility. Therefore, it would be futile, from the point of view of historical research and its teaching, to renounce digital resources, metadata, and the compilations they promote. Moreover, in the times of historical documentation only in manuscript or printed form, our research would not have been possible. If previous works have often been confined to routine issues, it is undoubtedly because the old resources

4 5

Péquignot and Roussel, 2015. Mainly: Google Books Ngram Viewer, https://books.google.com/ngrams/used here in July 2023.

1.2

Data from Euclid of Alexandria to Norbert Wiener

3

induced, to break away from them, insurmountable costs in consultation and processing time.6 In particular, one can easily query Google™ Books Ngram Viewer on the relative presence of the word “data” (a fortunately general spelling) in English and French among the digitized documents published since the beginning of the 19th century and accessible on Google™ Books. The results are clear and everyone can verify them on their own: it is during the last three decades of the 20th century that the relative frequency of this word has jumped in both languages. Therefore, the term is typical of this brief period and contemporary electronic technologies. Thus, the two definitions offered by Wikipedia have taken for granted this historical condition of possibility, this particular “sociohistorical space” to use the terminology of JeanClaude Perrot (1928–2021) recalled in the epigraph. In doing so, these definitions have amalgamated previous states of what is now considered data. From such an abstract point of view as a superficial approach undoubtedly allows, these definitions confuse previous states that we need to dissociate and consider as old avatars of data. It remains to distinguish the components of this amalgam around the word data in order to better understand what it is about and, for example, that it is not relevant to confuse data and statistics.

1.2

Data from Euclid of Alexandria to Norbert Wiener

The word “data” is in Latin the plural of “datum”, which is the past participle of the verb “do” which mainly covered the meanings of the actions of giving, assigning or affecting. The Latin term was retained in English while in French, the past participle “given” was substantivized in the singular as “donné” or in the feminine plural as “données”. This genealogy of words has been of great importance since antiquity in the learning of mathematics in the wake of Euclid of Alexandria who lived around 300 BC and was himself a compiler of previous mathematical works. Everyone knows his Elements. They were, for more than two millennia, the basis of the teaching of arithmetic and geometry all around the Mediterranean. They were accompanied by a complementary work intended for the learning of demonstrative reasoning: the “Δεδoμε
να” [Dedomena], or in ancient Greek: “The data”. The scholar, engineer and mathematics professor Didier Henrion (who died in 1632) was one of the early French translators of Euclid’s Elements . He accompanied them with the “Book of the data [donnez] by the same Euclid ” (1632). This covers pages 623 to 689 of the whole, or a tenth of its volume. At the beginning of the 17th century, French spelling, even in the north of the kingdom, was not yet sealed. When it was, notably in the 18th century, these “données” became the “Data”. From the first page of the supplement thus titled, the translator has taken up the definition of the Alexandrian where the past participle had mattered:

6

Brian, 2011.

4

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Considering Data: Critique and Method

“Definitions. 1. The plans or spaces, the lines, and the angles, to which we can find [others] equal will be said to be given by magnitude. 2. A reason [ i. e . the ratio of two magnitudes] is said to be given, when we can find a same, or [another that would be] equal”.7

The objects of arithmetic or geometric reasoning were thus likely to be considered given on the sole condition that they are known by means of a comparison with another analogue, comparison which founded the magnitude of the object. Reading the complementary treatise to the Elements, allows us to clarify this definition. Thus we find, on page 627: “Proposition I. Of magnitudes being given [ etc .]”; page 637: “XXI. If there are two given magnitudes [ etc .]”; or again, for example on page 644, this statement in view of the demonstration of the theorem of Thales of Miletus (6th–5th centuries BC): “XXXIV? If from a given point we draw [towards] straight lines given by position, a straight line; it will be cut in a given ratio”.8 Thus, the substantiated and generic title “the data”, covered everything that the student and the teacher had to take into consideration in view of a demonstration to be established. Therefore, in the Euclidean corpus, the term “given” presupposed on the one hand the comparison of what is considered with an analogue and on the other hand its installation upstream of the demonstration to be undertaken. These presuppositions are nowadays bypassed by authors who comment on their contemporary avatars as they are blinded by the current state of technologies. The important point here is that the Euclidean data is prepared to be adequate for its demonstrative treatment whether it is arithmetic or geometric. In the 18th century, mathematicians remembered this Euclidean conception repeated for more than two millennia in the learning of the basics of mathematics. The mathematician and encyclopedist Jean Le Rond D’Alembert (1717–1783) was thus quite Euclidean when he published in 1755 the article “Donné” in the Encyclopedia of which he was one of the animators with Denis Diderot (1713–1784): “Donné, [adjective], term often used by Mathematicians, to mark what is supposed to be known. So when a magnitude is known, or when we can assign another one that is equal to it, we say that it is given in magnitude. [. . .]. (O)”.9

His disciple, the geometer and himself an encyclopedist, Condorcet (1743–1794), inclined towards what after Immanuel Kant (1724–1804) we will call a theory of knowledge10 has devoted about 20 years to the writing of a Historical Picture of the Human Mind of which unfortunately the posthumous success of its Sketch,

7

Henrion translating Euclid, 1632, p. 623. (We stressed the two words). (Id.)That is to say in a modernized expression: “Let there be a triangle ABC on a plane, and two points D and E, D on the line]AB[and E on the line]AC[, such that the line]DE[is parallel to the line] BC[, the following segments are proportional: [AD]/[AB] = [AE]/[AC] = [DE]/[BC]”. 9 D’Alembert, 1755, where Cyclopedia of Ephraim Chambers (1680–1740) is mentioned here as a source: it indeed served as a model for many articles in the initial French edition animated by Diderot and D’Alembert. “(O)” was D’Alembert’s abbreviated signature at the bottom of his articles, here for id. 1755, in Diderot et al. (1751–1772), vol. 5, p. 51. 10 Crampe-Casnabet, 1988. 8

1.2

Data from Euclid of Alexandria to Norbert Wiener

5

published in 1795, has long overshadowed the scope and motifs.11 However, the manuscript of the Tableau contained Fragments designed to be developed. The fourth of these is an attempt to form a “universal language” whose signs represented the operations of the mind in various sciences. While he considered the universal language of algebra, Condorcet devoted a few manuscript pages to what he designated as its “general operations”: “We necessarily start from a given proposition, that is to say, whose proof is previously acquired and whose truth we assume in order to draw consequences from it [...]. Therefore, a sign is first needed to indicate that a proposition is given, for example this one □”.

For the geometer, a text written in this universal language would have to be understood immediately in each vernacular language. In French. He proposed to read the sign □ .as: “Let ...”, in the manner of the starting points of Euclidean exercises. This sign, in the fragment on the universal language has known variations: □ – for “result” or –□, for “show that...”, being understood that for the author the signs had a hieroglyphic character. He thus coded all the demonstrative operations, direct or intermediate. The three variations just mentioned responded to the needs of the deployment of demonstrative presentations. But other signs were used to distinguish data according to their nature. On the initial rectangle, the addition of a central vertical line marked the “given facts”, as in “I assume such quantity”. Still on the same rectangle, a double vertical line in its center, marked this time an “operation data”, as in: “I assume that: z = 2cosx”,12 not forgetting the initial simple sign that marked the antecedents of the demonstrations. The systematic attention that Condorcet paid to the development of different signs in his attempt at a universal language and the fact that this language aimed to represent by a particular sign each of the operations of the mind that he intended to distinguish shows that this geometer clearly conceived the difference between the antecedent of a reasoning, on the one hand; an element taken for granted, on the other hand and, finally, an arbitrary operation adopted during a demonstration. In any case, since the Euclidean data, we are a thousand miles away from recent definitions of data and especially engaged in distinct registers of operations. To stick to definitions that are part of a common state of knowledge at the beginning of the 21st century would therefore consist in letting oneself be carried away by the abstraction inscribed today in electronic records while skirting what precisely the historian must study: “different types of social and cultural practices”. This would be to surrender both to anachronism and to amalgamation, two unforgivable mistakes for historians, or even for an epistemologist, as the term “data” has covered uses and developments that it is important to distinguish from a critical point of view. As for the less scrupulous commentators, compared to the demands that thus impose themselves on historians as well as epistemologists, they are henceforth seen as charlatans.

11 12

Condorcet, 2004a, b. Condorcet, 2004b.

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Considering Data: Critique and Method

As much as the preceding data in the sense of Condorcet – “□” – and what he held as operational data refer us today to either formal or logical operations, his factual data announced what in the English definition of “data” on Wikipedia is referred to as “statistics”. However, until recent decades, the formal or logical operations in question were undertaken by hand, even though our idea of statistics comes from the 19th century and its technical developments from mechanography to computer science of the 1970s–1980s and ours. During these recent decades, a field of mathematical statistics has taken off: “data analysis”.13 By scrutinizing the pages that Condorcet devoted to “technical methods” in the manuscripts of his Tableau historique14 we find the description of the construction of numerical tables and an invitation to their descriptive analysis. These pages have foreshadowed, in a scholarly world armed with paper and pens, some of the treatments developed only one hundred and eighty years later but with more powerful tools15 But, this historical trajectory has offered and offers specialists anything but a continuous evolution of their scientific horizon. Over the past three centuries, indeed, numerical compilations have known very different approaches such as probability or matrix calculation, two mathematical approaches that are not easy to reconcile. This structure of the relations between the processing of data thus understood and the state of the technical methods provided by mathematics has been changing and is changing today. The mathematician and historian of science Glenn Shafer recently echoed recent expressions of questions raised by these mutations and recalled comparable doubts expressed in the 19th century,16 at the time of what the philosopher and historian of science Ian Hacking (1936–2023) boldly qualified as an “avalanche of printed numbers”.17 From a comparative point of view, if the metaphor is relevant, the avalanche no longer carries printed collections today but flows of electrons. Let’s dare the comparison once again, but going back in time. Between Euclid and Condorcet who in the register of mathematical operations remains in the universe of the Alexandrian, there was of course neither question of electrons nor of prints. Here are three epochs of what is covered today by the ambiguous term data, appeared according to the principle proposed by the material historian of abstraction Jean-Claude Perrot, the one recalled in the epigraph of this chapter: they orient the historian towards as many types of differentiated social and cultural practices. This historical epistemology critique being posed. It is important to recall what was, in the times of the Thirty Glorious, the critique spontaneous data. If a protagonist in a meeting used the word “data”, an antagonist immediately jumped in: “data is never given” (“do not take data for granted”). These were the mantras of those

Lebart et al. (1984). It was under the direction of the first of them that the author of this book conducted his doctorate in mathematics, Brian (1986). 14 Condorcet, 2004a, b. 15 Condorcet thus indicates how to construct what was later called a “complete disjunctive table”. 16 Shafer, 2022. 17 Hacking, 1990, p. 33. 13

1.2

Data from Euclid of Alexandria to Norbert Wiener

7

years that targeted the positivist enthusiasm of calculators. It is true that with the rise of electronic means, they hardly questioned the conditions of data production they seized. The injunction was nourished by a mixture of implicit or explicit references to criticism stemming from the Kantian theory of knowledge (1724–1804),18 to the critique of political economy by Karl Marx (1818–1883)19 and to the works of sociology of sciences governed by the motif of “social construction”.20 Due to a lack of method and in-depth investigations, the slogan remained as vain as others launched during the 1970s–1980s in the field of history and sociology of sciences.21 Therefore, in concluding this chapter, it is necessary to indicate how the “material history of abstraction” could be implemented in order to make more tangible the differences between conceptions of data too often confused. But before doing so, it is important to characterize the conception of data around which the previous ones are now aggregating, in other words, the current avatar so often and conveniently confused with its predecessors. The author who, without a doubt, forged this contemporary sense of the word data was the mathematician born and active in the United States: Norbert Wiener (1894–1964) whose main works were published in the mid-20th century.22 Wiener considered having developed a revolutionary theory and he avoided taking sole credit for it. The introductions to his works published in 1948 and 1950 thus painted a picture of a network of physicists, mathematicians, doctors, computer scientists, and engineers who have developed together and in stages this new discipline that the facilitator of this fruitful network called “Cybernetics” thinking about the etymology of the word as it referred to the idea of command on a maritime rudder whose principle has been since the earliest times to take into account the conditions of navigation and to correct this command once the effects of the initial action were observed. The use of the new word, if we stick to the testimony of the founder himself, is no older than the summer of 1947.23 The historical fact that this network of researchers welcomed computer engineers and that their perspectives were broadened by the rapid development of electronic computing after the Second World War has often confused Cybernetics and Computer Science (without even thinking about the explorations of science fiction on this subject). But, following the retrospective narratives of Norbert Wiener himself, it is easy to see that Computer Science was not first in this adventure. The starting point was indeed Mathematical Physics and particularly the work of Josiah Willard Gibbs

18

Kant, 1781. Marx, 1857. 20 Berger and Luckmann, 1967; Knorr-Cetina, 1981. 21 An examination of these failures would divert us from our purpose, it is undertaken in Brian, 2023. 22 Wiener, 1948; 1950, quickly made known to the French readership by Couffignal and Schützenberger, 1957 and later Couffignal, 1963 which takes up the main lines of the previous one. On the reception in France of Wiener’s work, see Le Roux, 2018. 23 Id., 1948, p. 19. 19

8

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Considering Data: Critique and Method

(1839–1903), a mathematician and thermodynamicist from Yale. Wiener admired him, judging him more revolutionary than Albert Einstein, and especially attributing to his work a singular metaphysical interpretation which he indicated was the starting point of his own work.24 Now, these physicists, mathematicians, computer scientists, and engineers shared the same scientific and technical culture and the idea that data came upstream from such operations and transformations, postulating that they were therefore naturally suitable for calculations. But, could one calculate at will? The Cyberneticians thought so. However, even as they wove the Bayeux tapestry of the conquests of new science, others, contemporaries, have woven a complex debate, that of the philosophical questioning of calculability.25 The presupposition of cybernetics was explicitly indicated by its founder: “A complex action is one in which the data introduced, which we call the input, to obtain an effect on the outer world, which we call the output, may involve a large number of combinations”.26 The characteristic cybernetic cycle has therefore been: the assimilation of inputs; then the calculation of the initial action (or command) based on them; to result in the assimilation of outputs qualified as feedback; and the calculation of a corrected command, all steps foreshadowed by maritime piloting since antiquity. As for the use of electronic calculation, it was not the starting point but a necessity imposed by the volume of data considered.27 When these works met those carried out in physics and information electronics, the word data took on a new dimension, as it covered all possible information records in the sense of the theory of information developed by the engineer and mathematician Claude Shannon (1916–2001). Wiener himself generalized it by writing “communication”. Thus, step by step, an enterprising scientist surrounded by dynamic skills driven by the technical renewals of their time and by the support of funding from Defense sectors as well as private industry, for example in the sector of telephony, sealed the universe of data from the 1950s–1980s and cybernetics, which today the mere names are enough to frighten those who do not know their workings. The extension of this universe to the global sphere of the Internet has only amplified the impact of this vocabulary and the misunderstandings attached to it. The works of Norbert Wiener were permeated with a prophetic enthusiasm that attracted many specialists familiar with the skills gathered around him. His vision, the metaphysics he drew from Gibbs’ work, the announcement of a coordinated and informed tableau, all this was enough to seduce them. Does this mean that mathematicians, physicists, or engineers followed him as if he had been Pied Piper? It is true that the founder had given himself the means to extrapolate wildly on all subjects. He indeed expressed the creed of the scientific fertility of “no man’s

24

Id ., 1950, his introduction; notably, 19542, p. 12. This extensive corpus is provided by Mosconi and Bourdeau, 2022. 26 Wiener, 1950; 19542, p. 23–24. Italics at least in this second edition. 27 Id., 1948, p. 15–16. 25

1.2

Data from Euclid of Alexandria to Norbert Wiener

9

land” freed from established competence controls.28 This was then the best way to equip oneself with a wide intellectual latitude, but it was of course also the opportunity to develop, strictly speaking, senseless considerations in the fields of metaphysics, psychology, sociology or anthropology, if not medicine, all sectors where this jack-of-all-trades intervened. Seventy-five years after the publication of the works that made its author famous, it is undoubtedly high time to submit its findings to rigorous historical epistemology and rational examinations, now that the initial charm has been broken. Isn’t this, after all, the fate of any work of science properly speaking? In the balance of this examination of the links between the data according to cybernetics and their predecessors, and particularly with the concept stemming from the Euclidean tradition, a parallel appears between the oldest avatar and the most recent one. This is not to say that we should confuse the data of the Alexandrian and that of the Harvardian. Indeed, their natures differ radically, in the first case it was the elements of the reasoning to be undertaken and in the second the coded material for subsequent calculations. But, these two data are structurally homologous: Euclid’s data is prepared for... and lends itself to... arithmetic or geometric demonstrative treatment just as Wiener’s data is coded for... and lends itself to... computer, mathematical or physical calculation, such is their only common point - this structure - besides the denomination. The primary question remains: how to study each avatar of the data and what it entails? A comparison with a philosophical object of reflection that provides helps to clarify the problem to be addressed for our investigation. A Bavarian satirical newspaper, Fliegende Blätter [Flying Leaves] published in 1892 an anonymous drawing that could evoke either the head of a rabbit or that of a duck (see Illustration 1.1).29 This drawing questioned psychologists of the late 19th century and particularly Joseph Jastrow (1863–1944) who commented on it in 1900 in a notable article Illustration 1.1 Ludwig Wittgenstein’s rabbit-duck (1892). (Source: anonymous engraving, Zeitschrift Fliegende Blätter, n°2465, Vol. 97, p. 147)

28 29

Id., 1948, p. 8. Fliegende Blätter, n°2465, 1892, Vol. 97, p. 147.

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Considering Data: Critique and Method

since then.30 Later the philosopher Ludwig Wittgenstein took it up and commented on it in chap. XI of the second part of his Philosophical Investigations.31 The illustrator gave an animal to see, this point was widely admitted, but the drawing was ambiguous and its identification unstable. The philosopher concluded that without prior experience, without conceptually presupposing one or the other of the two animals, one could not see either a duck or a rabbit. He called these preliminary conditions for the readability of the image Voraussetzungen, presuppositions.32 “Data”, today, is like “animal” in the case of the duck-rabbit of illustration no. 1.1, something ambiguous whose interpretation is unstable. Two pitfalls that alone would be enough to ruin any investigation to be conducted rationally, so we will have to reconstruct the presuppositions (Voraussetzungen) that accompanied them or that accompany them, as in the most recent case, the concept of input. Failing to undertake such an investigation, one would comment on the data in disregard of these possibly changing presuppositions and therefore in disregard of the actual conditions of the knowledge they entailed. We would therefore renounce, may be for convenience, taking seriously the historicity of the data and that of the abstraction they entail, thus dodging the work to be undertaken which must fall under the history of science or historical epistemology. Illustrations 1.2, 1.3 and 1.4, below, will provide flesh to this theoretical argument. In no. 1.2, here is an example of data input perforated on a card intended for a Hollerith brand machine around 1936 (the model of IBM cards used in subsequent decades is recognizable). The reproduction comes from the work which, in France, in 1936 and then in 1942, promoted this technology in the early days of Wiener’s cybernetics on the other side of the Atlantic.33 In no. 1.3, here are two segments marked on an engraving placed at the frontispiece of a translation of Euclid’s Elements in 1632: AD and BD because it was a matter of representing the measure of the angle ADB: the segments AD and AB were in no way cybernetic inputs but two things considered known before proceeding to the geometric construction. Finally, in illustration no. 1.4, this view of Louvain shows disparate house heights which we will see in Chap. 3 that the astronomer and statistician Adolphe Quetelet (1796–1874) wondered whether they lent themselves to calculations of averages, relevant or not. Adopting a point of view that would consist in pretending that Henrion’s segments, the heights of houses considered by Quetelet and Wiener’s inputs would be

Joseph Jastrow, “The mind’s eye”, Fact and Fable in Psychology. Boston, Houghton, Mifflin and Co, 1900 (fig. 19, p. 292). 31 Wittgenstein, 1953 and 2004. 32 The German term is used by Wittgenstein himself, like the English term it appears in the 1953 edition-translation: the reference French translation retains “presuppositions”, id ., ibid ., 2004, especially p. 274–299. Glock, 2003, comments on the Austrian philosopher in the article “Perception of aspect”, p. 427–434. 33 Carmille, 1936. 30

1.2

Data from Euclid of Alexandria to Norbert Wiener

11

Illustration 1.2 Detail of a punched card (1936). (Source: Carmille, 1936, plate XIV, fig. 15: example for human resources; author’s copy) Illustration 1.3 Two Euclidian data: AD and BD (1632). (Source: Frontispice of Henrion; same date)

the same thing, that is data, would be like pretending that the duck-rabbit would be one single animal and being content with it. However, this is certainly not the case, except to want to confine oneself to the ambiguity of the satirical drawing.

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Considering Data: Critique and Method

Illustration 1.4 A view of the Old-Market at Louvain (1665). (Source: Edward van Even, Louvain dans le passé et dans le présent. Louvain, Fonteyn, 1895, p. 227)

It is therefore important not to be trapped by the ambiguities of the most recent notion of data, the one elicited by the abstraction of the technical notion of input. As for the Voraussetzungen – the presuppositions – at work in cybernetics, we have seen above that they came from the technical conceptions of computer engineers in terms of inputs. What was it like in the 17th century when scholarly or commercial calculators were trained on Euclid’s books (strictly speaking on them as copies of translations printed in the 17th century were covered with handwritten calculation exercises)? That time, indeed, was that of pen calculations and compilations of the same ink. While in the 19th century, as Ian Hacking noted, numbers were massively printed. Chapters 2 and 3 will present the results of an investigation that focused on each of these two centuries. How to find lost Voraussetzungen? These losses are as much due to the renewal of know-how as to the fact that from era to era the holders of new skills have annexed the old ones by concealing the peculiarities of the previous ones that they have considered obsolete. Thus, even today, data science asserts itself as a new world to the detriment of the old statistics, as statisticians and demographers for a century had relegated commercial or political arithmetics to the prehistory of their discipline.34

34

Dupâquier & Dupâquier, 1985.

1.3

1.3

From Circulation to Know-How and Presupposition

13

From Circulation to Know-How and Presupposition

It is against this spirit of relegation that has long prevented understanding what political arithmetic was in the 17th and 18th centuries and which obscured the difficulties actually experienced by mathematicians at the end of the second of these centuries facing compilations subsequently familiar to us that we had to reconstruct the know-how of these scientists in a previous work, that is to say, these geometers versed in mathematical analysis in the second half of the 18th century. To acquire this competence, it was therefore necessary to return to the textbooks in use at that time. In this way, a gesture appeared characteristic of this mathematical competence: the decomposition of formulas for a demonstrative purpose. The model of Erwin Panofsky’s work gave us the key to the result of such an investigation: at stake was the habitus of these mathematician geometers. Faced with numerical compilations, there were well-trained scientists at the time who saw nothing that seemed to lend itself to science: such was the case of the mathematician Jean Le Rond D’Alembert, who was resistant in principle to the use of calculations that would consider human beings as comparable, in his eyes always characterized by particular circumstances; or that of the chemist Antoine-Laurent Lavoisier who, faced with tables of political arithmetic, saw only “added sums”.35 The problem that Panofsky pursued throughout his work was that of the intelligibility of works in their time. Consequently, for the art historian, it was each time the deployment of intense erudition at the end of which he reconstructed the presuppositions once elaborately developed and then shared. They had once given meaning to the works and their reconstruction ensures their readability today. Thus, at the end of his investigation conducted in the times of the initial formation of the Gothic style, the principle of the composition of architectural elements appears as a homologue of that used in the composition of scholastic theological sums. Indeed, the scholarly thought of this time for purely theoretical and theological reasons was characterized by an imperative of clear and luminous expression.36 Panofsky had read the same kind of psychological studies as Wittgenstein and much more of the scholastics which allowed him to specify the object of his exploration. All wanted to grasp what the Viennese philosopher finally called the perception of aspect. As for the art historian, he questioned it by trying to grasp the “mental habits at work” in the production of this perception: the “habit-forming forces” that had produced it.37 For an 18th century mathematician, the art of solving a problem involved such a habitus specific to mathematical analysis as it was conceived.38 Moreover, for the art historian, each product of a habitus, such as a piece of

35

Brian, 1994, p. 49–111 and 258. Panofsky, 1951. 37 What his French translator, Pierre Bourdieu, has sealed in the concept of habitus, Panofsky, 1951 (1967), p. 142. 38 Brian, 1994. 36

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Considering Data: Critique and Method

architecture for example, is characteristic of a specific historical time and a particular geographical space, inseparable from each other. The examples provided by the works of Erwin Panofsky on Gothic architecture (1951), Lucien Febvre on the religion of Rabelais (1942) and Jean-Claude Perrot on the political economy of the 17th and 18th centuries (1992) show that such investigations require a significant amount of erudition.39 Without a doubt, this austere perspective is enough to discourage the hasty thinker who would be content with an overview of the remnants of old works or the conformist student who would be satisfied with a collection of fragments carefully recorded by his predecessors in a history of specialized thought, nourished by selected pieces and without any properly historical attachment, in a word, a mythology of specialized thought. It is no doubt at this stage of the presentation that it is appropriate to work in the wake of these three authors. The great historian Lucien Febvre has outlined the program of the investigation to be undertaken: “The task of tomorrow: first to inventory in detail, then to reconstitute for the period studied, the mental material available to the men of that time; by a powerful effort of scholarship, but also of imagination, to reconstitute the universe, the whole physical, intellectual, moral universe, in the midst of which each of the generations that preceded it moved; to get a clear and assured feeling of what, on the one hand, the insufficiency of factual notions on this or that point, on the other hand, the nature of the technical material in use at a certain date in the society under study, necessarily engendered gaps and distortions in the representations that [such a historical community] was forging of the world, of life, of religion”.40

Furthermore, in order not to lose oneself in scholarship, Jean-Claude Perrot has more recently added fruitful criteria. It is then indeed a matter of highlighting the diversity of even contemporary abstract expressions. Here is an example: after a few paragraphs on ancient or exotic authors, a breviary of the history of economic thought will generally offer for the 17th and 18th centuries a few keywords: the “precursors”; the “mercantilists”; the “physiocrats”; the “classics”, or about 15 names of authors, whereas for the French language alone, from the beginning of printing and until 1789, nearly 4800 titles have been identified and consulted.41 The history of economic thought therefore rests on a documentary pinhead (a metaphor that the author of The Wealth of Nations would undoubtedly not have disowned). JeanClaude Perrot has set himself the task of highlighting and exploring this diversity. This was of course to do justice to a legion of authors. But, it was also for him, at the time of his doctoral research, the possibility of reconstructing what the protagonists of the urban history of the city of Caen, which he was then studying, had seen and perceived and the objects of their reflections. This approach, by radically distinguishing itself from what would have been provided by the use of a uniform reading grid resulting from the simplisms of the history of thought, offered the

39

On its validity, see Febvre, 2023. Febvre, 2023, p. 199 where this passage from one of his articles from 1938 is quoted; it is understood here that this is the program that he followed in Febvre, 1942. 41 Hecht and Lévy, 1956. 40

References

15

historian a range of questions that served to guide urban history research. The concrete history of abstraction promoted by Perrot against the history of thought has enriched historiography in two parallel and interdependent registers, that of the history of ancient knowledge of an economic and social order and that of the history of phenomena contemporary to them.42 When Perrot completed his doctoral thesis, another factor favored the amalgamation of old economic knowledge. It was not only about the harmful effects of the formative habitus of the history of specialized thought but, during the Cold War, the empire of a simplistic Marxist-inspired theory of knowledge that homogenized its expressions according to socio-economic qualities of the authors. That’s why Perrot proposed not to start from these qualities considered homogeneous and determining but to start from the attested abstract expressions to go towards “differentiated social and cultural practices”, and consequently to analyze the diversity of these abstract expressions and their circulation to thereby reveal what their history was.43 As for what concerns us: in the 21st century, data are inputs recorded electronically and therefore carried by the global circulation of this matter. On the other hand, in the 19th century, the statistics were massively printed, the collections and specialized journals exchanged between the libraries of the Central Statistical Offices or national learned societies. Finally, in the 17th century, handwritten calculations hardly went beyond the cabinet of a political arithmetician or the back shop of a merchant, and if they left it was by moving on a narrow network of trust. Each of these three centuries is therefore characterized by particular conditions. It is also characterized by specific skills: today, it is those of data engineering; around 1860, it was those of administrative statisticians; and two centuries even earlier, that of account keepers. The next two chapters will scrutinize the cases of the 17th and 19th centuries. Each of these two periods had its own habitus in the matter, so many presuppositions some of which, we will see, have persisted until now.

References Berger, P. L., & Luckmann, T. (1967). The social construction of reality. A treatise in the sociology of knowledge. Doubleday. (trad. fr.: La construction sociale de la réalité, trad. by Pierre Taminiaux, Paris, Klincksieck, 1986). Brian, É. (1986). Techniques d’estimations et méthodes factorielles. Exposé formel et application aux traitements de données lexicométriques. Orsay, Université Paris-Sud. Brian, É. (1994). La Mesure de l’État. Administrateurs et Géomètres au XVIIIe siècle. Paris, Albin Michel. Brian, É., 2011, L’horizon nouveau de l’historiographie expérimentale, Revue d’histoire moderne et contemporaine, vol. 58-4, n°5, p. 41–56.

42 43

Brian, 2021. Perrot, 1992 and id., 2021, p. 498.

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Considering Data: Critique and Method

Brian, É., 2021, Comment l’histoire intellectuelle peut-elle servir une réforme de l’histoire économique et sociale?, Revue de Synthèse, vol. 142, n° 3–4, p. 290. Brian, É. (2023). Un problème que le XXe siècle n’a pas résolu, dans Febvre, 2023, pp. 179–229. Carmille, R. (1936). La Mécanographie dans les administrations. Paris (19422, ibid.). Condorcet, M.-J.-A.-N. de C. de. (2004a), Tableau historique des progrès de l’esprit humain. Esquisse, fragments et notes. INED, (éd. J.-P. Schandeler et P. Crépel, dir.). Condorcet, M.-J.-A.-N. de C. de. (2004b). Fragment 4, Essai d’une langue universelle, dans id., ibid. (pp. 947–1029) (fragment établi et commenté par É. Brian). Crampe-Casnabet, M. (1988). Condorcet : Une théorie de la connaissance. Revue de Synthèse, 109, 5–12. Couffignal, L. (1963). La Cybernétique. Paris, Presses universitaires de France (a development of this author’s contribution to id. et Schützenberger, 1957). Couffignal, L., & Schützenberger, M.-P. (1957)., La Cybernétique in Encyclopédie française. Paris, Société nouvelle de l’Encyclopédie française (update of volume 1, 21 p., dated on the 2nd of February 1957, p. 9). D’Alembert, J. Le Rond (1755). Donné, In Diderot et al., vol. V p. 51. Diderot, D., et al. (1751–1772). Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers (28 Vols). Paris, Briasson etc.. Dupâquier, J., & Dupâquier, M. (1985). Histoire de la démographie. La statistique de la population des origines à 1914. Perrin. Febvre, L. (1942). Le problème de l’incroyance au XVIe siècle. La Religion de Rabelais. Albin Michel. Febvre, L. (2023). Histoire et Sciences. éd. EHESS. Glock, H.-J. (2003). Dictionnaire Wittgenstein. Gallimard. Hacking, I. (1990). The taming of chance. Cambridge University Press. Hecht, J., & Lévy, C. (1956). Économie et population. Les doctrines françaises avant 1800. II. Bibliographie générale commentée. Presses universitaires de France (preface by J. Cain, presented by A. Sauvy). Henrion, D. (1632). Les quinze livres des éléments d’Euclide, [. . .] plus Le livre des donnez du même Euclide [etc.]. Vve Henrion (puis : 1676. Rouen, Jean Lucas). Kant, I. (1781). Kritik der reinen Vernunft (transl. Critique of Pure Reason, ed. and trans. by P. Guyer and A. W. Wood). Cambridge University Press, 1998. Knorr-Cetina, K. (1981). The manufacture of knowledge. An Essay on the Constructivist and Contextual Nature of Science. Pergamon Press. Lebart, L., Morineau, A., & Warwick, K. M. (1984). Multivariate descriptive statistical Analysis. John Wiley & Sons. Le Roux, R. (2018). Une histoire de la cybernétique en France : 1948–1975. Garnier. Marx, K. (1857). Grundrisse der Kritik der politischen Ökonomie. Dietz, (édition récente : 1974). Mosconi, J., & Bourdeau, M. (Eds.). (2022). Anthologie de la calculabilité : naissance et développements de la théorie de la calculabilité des années 1920 à 1970. Cassini. Panofsky, E. (1951). Gothic architecture and scholasticism. Latrobe, The Archabbey Press (French tr., id., 1967, Architecture gothique et Pensée scolastique. , Minuit. Péquignot, Julien and Roussel, François-Gabriel (ed.), 2015, Les métavers. Dispositifs, usages et représentations. . Perrot, J.-C. (1992). Une histoire intellectuelle de l’économie politique XVIIe-XVIIIe siècles. éd. EHESS. Perrot, J.-C., 2021, Histoire des sciences, histoire concrète de l’abstraction (1998), Revue de Synthèse, vol. 142, n°3–4, p. 492. Shafer, G. (2022). So much data. Who needs probability? Have we been here before? International Journal of Approximate Reasoning, 141, 183–189.

References

17

Wiener, N. (1948). Cybernetics or control and communication in the animal and the machine. John Wiley & Sons (éd. fr. Ronan Le Roux, 2014, La Cybernétique. Information et régulation dans le vivant et la machine. Paris Seuil. Wiener, N. (1950). The human use of human beings. Cybernetics and society. Houghton Mifflin. (19542. Boston, the University Press Cambridge). Wittgenstein, L. (2004). Recherches philosophiques. Gallimard (tr. of Philosophische Untersuchungen et Philosophical investigations. Oxford, Blackwell, 1953).

Chapter 2

Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and 18th Centuries

“The sciences found in the capital of Egypt a refuge that the despots who governed it might perhaps have refused to philosophy. Princes who owed a large part of their wealth and power to the combined commerce of the Mediterranean and the Asian Ocean, had to encourage sciences useful to navigation and commerce”. Condorcet, Esquisse.1

2.1

Working Out Data in the 17th Century

Let’s consider the figures and abstract numbers “as “things” after Jean-Claude Perrot’s Durkheimian-inspired invitation.2 The question to ask is therefore: who worked this material in the 17th century? Immediately, what could have appeared to us as a homogeneous and immediately readable documentary mass decomposes into corpora tied by different intertextualities, most often foreign to each other, even if their intersections are not negligible. Moreover, three historiographical registers intersect: on the one hand the history of mathematics and particularly that of the mathematical theory of numbers,3 then the history of algebra which actually meets that of logic and law,4 and finally the history of merchant or financial calculations.5 It is in this third area that we find number handlers. It is not unfounded to compare them to the data compilers of the early 21st century. In each of these three registers two things attract attention: it is first of all the incredible skill shown by calculators as soon as they juggle with these numbers. If

1

Condorcet, 1795, p. 103, about Alexandria in Roman and Byzantine times. Perrot, 2021. 3 Goldstein, 1989; 2012. 4 Cifoletti and Coquard, 2022. 5 Benoît, 1989; Perrot, 1992; Brian, 1994; Hoock et al., 2001. 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 É. Brian, Are Statistics Only Made of Data?, Methodos Series 20, https://doi.org/10.1007/978-3-031-51254-4_2

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Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and. . .

this virtuosity is different here or there, it is always dazzling, especially since familiarity with these ancient calculation methods has been lost. A second peculiarity concerns the documents, manuscripts or printed, prior to the 19th century, the use of fractional notation is the rule for numbers that were not simple additions of units, those that were therefore not integers, either for numbers then called “broken”. Thus 25/3 was not noted 8.333333 but 13 8 that is to say: “one third and eight units” (this order of writing, it will be discussed later). The adoption of decimal notation, developed on the European side in the 16th century and then taught from the turn of the 18th to the 19th century made lose the immediate readability of this fractional notation. In France, after the Revolution, decimal notation was imposed in schools.6 Thus a very beautiful acquisition of the Archives of the Academy of Sciences of Paris, during the years 1990 – a manuscript notebook of a student accountant – bears this title “Way of calculating before the establishment of the metric system”.7 Neither the ancient explorers of numerical methods whether they were passionate about epistolary challenges or attached to establishing demonstrations in the manner of a Pierre de Fermat (1601–1665),8 nor the jurists and logicians the first contributors to the conception of symbolic algebra,9 had spent months to learning the hard way to calligraph numbers and arrange them on a page, no more than they have dealt with as many numbers as the accountants and charter merchants of the ships that have crisscrossed the Mediterranean since Antiquity and then the Oceans from the Renaissance. One of these merchants based in Reims, Jean Maillefer (1611–1684), left a reason book published two centuries after his death. He staged under the heading “Of commerce and merchandise” a dialogue with his own son, François Maillefer (1650–1692) about the learning of merchant know-how.10 “The father. – There are also many disadvantages and a lot of risk in doing big business, now, because of the infidelity and impotence that is encountered among merchants; and it is quite difficult to give good advice and to take it, we are not here in a city of great traffic or profits like the seaports11 or the big city.12 To be a merchant, you have to buy and sell and keep good books by double entries. The son. – I would like to know how to keep [accounts in] double entries. The father. – It’s a very beautiful science and very necessary for a merchant [. . .] You clearly see in a few hours your receipts and your expenses, your purchases and sales. It requires daily practice which teaches more than the masters can show; you will know better from the books I use and from an inventory that I will add to this chapter, and believe that you need hardly less application, study and attention, than if you had to learn all parts of philosophy.

6

See for example: Chenu, 1798 and Reynaud, 1804. Anonymous, ca 1790. 8 Goldstein, 1989. 9 Cifoletti, 1995. 10 Jadart, 1890, p. 95 and following, commented by Lemarchand, 2001. 11 Reims is not a port. 12 Probably about Lyon. 7

2.1

Working Out Data in the 17th Century

21

The son. – I am already eager to [. . .] know this science. It seems to me necessary to have a lot of books to put it into practice. The father. – That is true; the will you show, it’s already half the journey done. This science is like all others, it is learned, and practicing it every day [it] becomes very easy. I have known cashiers in Lyon who have good wages as clerks in finance”.

This last remark is of great historical interest, it testifies that, under absolutism, a specialist in merchant calculation was aware that he could assert his skills with the royal administration. It confirms an established observation: the main authors of political arithmetic works of 18th century France, high magistrates, were assisted by secretaries from the merchant world (and sometimes their works were published under the name of the clerk because of royal censorship). This was the tandem formed by Auget de Monthyon and Jean-Baptiste Moheau,13 or the one composed of La Michodière and Louis Messance.14 Father Taillefer continues the dialogue: “The son. – Since I am not yet capable of learning, at least as soon as I would like this beautiful science, in the meantime tell me the books I need to have. The father. – I want to and it will give you a start, a disposition and an entry to instruct you. Try to form your letters in beautiful characters, this is called good painting”.

Indeed, the manuscript preserved in the archives of the Academy of Sciences illustrates this “good painting” through a calligraphy and alignments of writings that could not be more meticulous. Its pinnacle is this operational table whose meticulousness is decorated with the image of one of these ships, the glory of the French navy at the end of the 18th century: the apprentice calculator dreamed of great voyages as much as of great profits15 (see Illustration no. 2.1 below). After advising his son to take care of his writing, the father indicates a list of books that a merchant must keep. Thus, regarding purchase and sale accounts, he mentions the example of his own accounts for a period from December 1, 1664 to March 22, 1666. The sum of the two columns, purchases on the one hand and sales on the other, was then the same 269,529 pounds (livres tournois). The father comments: “This is a rather rare piece and one of the most beautiful I have made (beautiful from a commercial point of view). It is God who guided my brush.”16 The manuals of learning accounting and merchant know-how form, from the Mediterranean to the Baltic a documentary continent.17

13

Moheau, 1994. Brian & Théré, 1998. It should be noted here that La Michodière was the intendant of the province of Lyon. 15 Anonymous, ca 1790, cited above, the examples to which the calculations were subjected were located around Agen. This master of writing seems to have practiced between Bordeaux and Toulouse. 16 One thinks here of the analysis of Jean-Claude Perrot: “The invisible hand and the hidden god [of the Jansenists]”, Perrot, 1992, p. 333. 17 Hoock et al., 2001. 14

22

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Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and. . .

Illustration 2.1 Extract from an accounting learning notebook from the end of the 18th century. The crossed out figures suggest a galley method division. (Source: Anonymous, ca 1790, © Archives of the Paris Academy of Sciences)

At the end of the 17th century, in France, “the method for properly setting up all kinds of double-entry accounts” by Claude Irson18 was the reference title which deliberately inscribed itself in the economic policy of Minister Colbert. From the preface of his first chapter, the author argued the merits of a treaty whose aim was to promote a keeping of accounting books capable of providing accurate calculations and convenient and indisputable legal evidence19: “One may wonder, perhaps, why I deal with the method of drawing up accounts according to the maxims of jurisprudence and the authority of doctors, since it seems [that this method] is arbitrary, and that it has no fixed principles. But, if one considers the trouble it gives when it is involved in lawsuits; how much the Judges, even the most enlightened, find it obscure: and how much the authors, who have written on the subject, have found it cumbersome, one will agree that one cannot take too much care to make it certain and intelligible. As for considering that in this matter common sense alone should suffice, one can certainly say that this common sense is more certain when it conforms to that of those who have preceded us. It is for this reason that architects, painters and sculptors would often be considered reckless to prefer their particular sense to the good maxims of the Ancients”.

18 19

Irson, 1678, whose initial pages are not numbered. Coquery, 2006, follows this dual logic even in the shops of the 18th century.

2.1

Working Out Data in the 17th Century

23

The master accountant claims to have conducted a complete examination of the different ways of keeping accounts and he gives a very complete inventory of them. Three methods stand out: the first is said by “receipt, expense and recovery” and would be used in finance and in justice i.e. in the legal sphere; the second is said by “debits and credits” and would be in use among merchants who would attribute it to their Ancients; the third, relatively modern according to Irson would be the one in “double entries”. He specifies that it would be particularly used by traders and the most useful of the from a merchant and procedural point of view. That’s why he intends to promote it. But he immediately adds: “As all tastes are different, I will give rules for each of these ways and will show that, in essence, they all come back to the same point.” The bulk of the preliminaries in the work focuses on the concrete keeping of books and it is considered as much for the use of a Francophone trade as for that of Italian merchant cities. However, the work contains nearly two hundred and eighty pages of duly indexed examples. Thus, the master shared the opinion of his contemporary merchant when he advised his son. Let’s summarize them: “see these examples, practice, keep your books accurate and readable; learn by practice, put and put the work back on the loom a hundred times”. The example of the Lyon commercial crossroads shows that the production and uses of merchant learning manuals were diverse during the modern period and that they recorded both the effective conditions of commercial activities and multiple European currents of circulation of models and their uses.20 The accounting science of the early modern era thus appears in its historical complexity, a fact which the historian must acknowledge to depart as much from the anachronism of an illusion of familiarity with digital writings as from the feeling of exoticism that the same writings can still arouse in him. With this observation, here is the point where it is appropriate to relearn to count and familiarize ourselves with the old forms of mastery of handling numbers.21 These were: the rules of three and the proportions once constantly invoked to solve problems of exchange, conversion between units of measure, calculation of tariffs or even sharing of profits in a commercial society.22 Two authors have provided the foundation of the corpus from which schools and masters have fed apprentice 20

Bottin, 2021. I encountered the same necessity while working on the geometers of the Academies of Sciences of the 18th century (Brian, 1994). Trained in differential and integral calculus two centuries later, I perceived what it was about, but I did not find my way around. So I went to the analysis courses of the years 1750–1770 in order to be able to learn like them and understand how they understood their calculations. 22 Of course, there were less skilled merchants in calculations. The work of François Barrême, 1673, a handy in-12° that fit in the pocket and opens for use in the palm of a hand, while the other hand flips it to the page titled for example “At 7 pounds the thing.” where a multiplication table by seven appears for quantities ranging from 1 to 10,000. The copy that I was able to acquire thirty years ago from a bookseller still bears today the mark of the hand’s grip. One would thus search in the “Barrême”, and all that remained was to add up. In French language, a double metonymy first made the author’s name that of the book-instrument, then generalized this to all tables that provide “calculations made” or calculation tables: that is to say in French “barèmes”. 21

24

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Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and. . .

calculators: the geometer of Alexandria Euclid (around 300 BC) for the fifth book of his Elements devoted to Arithmetic,23 and Leonardo of Pisa (around 1200), known as Fibonacci, for his abacus book.

2.2

A Modern Starting Point: The Liber Abbaci of Leonardo of Pisa

We now have not only a very careful Latin edition of the Liber Abbaci based on the twenty copies identified in Europe, but also a translation into the English language designed to be rigorous both philologically and mathematically. Thus the major work of Leonardo of Pisa – known as “Fibonacci”, according to the incipit of the manuscripts that designate him as the author of this work dated 1202: “Leonardo son of Bonacci of Pisa” (in abbreviated Latin: Fi. Bonacci)24 – is now accessible and we can find entire sections of the manuals of practical or commercial arithmetic taught for six centuries until the end of the 18th century, or even, in revised forms for didactic reasons instead of the establishment of decimal notation, until today for what concerns for example the calculation of proportions and operations on fractions. There will undoubtedly be readers who will remember the arithmetic Stan Laurel and Oliver Hardy of our childhoods; the least common multiple (LCM) and the greatest common divisor (GCD). We know today that the Pisan was familiar with Indian, Arab merchants and those who operated in the Mediterranean basin during the time of Frederick II Hohenstaufen, this German emperor based in Sicily who was passionate about sciences and encouraged him. So much so that historians cannot consider the calculator as a mere observer of the know-how of his time but as a protagonist in the expansion of arithmetic culture, and particularly of merchant arithmetic culture.25 The prologue given by Leonardo to his treatise shows that he was fully aware of performing several actions. He first brought the writing of numbers which he called “Indian”- what would later be called “Arabic” and today “Arabs” – to the calculators of the European continent where Roman numerals remained in use.26 On the other hand, he offered a corpus designed to promote the formation of a hybrid competence that would mix Arithmetic and Geometry. Finally, he delivered a didactic if not new – indeed, we recognize precedents similar to traces left by the schools of scribes of high Antiquity – at least illustrated by a considerable number of examples and

23 The long journey of these texts has been variously examined notably by Archibald, 1950; De Roover, 1956; Lacoarret, 1957; Bompaire, 2006; Benoît, 1992; Bartolozzi & Franci, 1990; Rommevaux, 2003. 24 Fibonacci, 2002, p. 3. 25 Høyrup, 2014 who joins Chemla et al., 1992; or also Allard, 1992. 26 Portet, 2006, analyzes the diffusion of these processes and Perrot, 2021, reports on them in the 17th and 18th centuries. Bompaire, 2006, studies the fate of the chapter that dealt with metal alloys.

2.2

A Modern Starting Point: The Liber Abbaci of Leonardo of Pisa

25

exercises to which apprentices had to respond primarily from after mental calculus and otherwise to conceive the sequence of calculations to undertake quickly and without error. Published in 2002, the English translation by Sigler provides valuable testimony of this awareness: “I [Fibonacci] presented a full instruction on numbers close to the method of the [Indian figures27] whose outstanding method I chose for this science. And because arithmetic science and geometric science are connected, and support one another, the full knowledge of numbers cannot be presented without encountering some geometry, or without seeing that operating in this way on numbers is close to geometry; the method is full of many proofs and demonstrations which are made with geometric figures [according to Euclide]. [. . .] Whoever would wish to know well the practice of this science ought eagerly to busy himself with continuous use and enduring exercise in practice, for science by practice turns into habit; memory and even perception correlate with the hands and figures, which as an impulse and breath in one and the same instant, almost the same, go naturally together for all; and thus will be made a student of habit; following by degrees he will be able easily to attain this to perfection”.28

As early as 1202, the primacy of practice is expressed here, in the specific case of learning the handling of numbers, as probably at this time for any other artisanal skill to be acquired from specialized masters. As we have seen, this idea has persisted until the 17th and 18th centuries and probably as long as calculations were made in the head or by hand. The rise of mechanical then electromechanical and electronic calculation in the 19th and 20th centuries have made these skills obsolete. Let’s return to the notation of fractional numbers. Leonardo gives the principle and comments on it in the manner of a schoolmaster, so that this passage is precious to us. He takes for example: 21 65 107 ; he describes and then indicates how this notation should be understood: “Under [the] fraction line are 2, 6, and 10; and over the 2 is 1, and over the 6 is 5, and over the 10 is 7, as is here displayed, the seven that is over the 10 at the head of the fraction line [i.e. its right side] represents seven tenths, and the 5 that is over the 6 denotes five sixths of one tenth, and the 1 which is over the 2 denotes one half of one sixth of one tenth”.29

It is actually seven twenty-fourths noted fractionally, from right (« the head ») to left and according to the principle of the Egyptian fraction.30 The Liber thus federates various mathematical cultures today recognized in the highest Antiquity and on a particularly wide geographical spectrum that goes from the Indian world to the Mediterranean.31

27

Sic, that is to say, the Arabic numerals. English: Fibonacci, 2002, p. 15, §1 and Latin, 2020. p 3 (§2-§5). 29 English: id., 2002, p. 50 and Latin, id., 2020. p. 44 (§9). 30 Caveing, 1992; in accordance with Rey, 1942, p. 222–291. 31 Goonatilake, 1998, p. 126. 28

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Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and. . .

However, the hero of the Liber is the Alexandrian geometer Euclid (around 300 BC) whose Elements, themselves already synthetic of several authors and ancient mathematical cultures, are regularly called upon and whose author’s name attributed by tradition is covered with praise as soon as one of his results is invoked, always appropriately. The first five chapters of the Pisan’s work deal with the arithmetic of whole numbers and the next two with fractional numbers. Then come five chapters that apply the previous calculations to a host of concrete problems and described as much for realism as for providing exercises to acquire good habits: the evaluation of goods, barter, the alloy of metals, the setups of companies (that is, societies formed by several investors whose profits were to be shared taking into account both the proportion of investments and sometimes even their duration), the divisions among heirs. Many examples also seem to be qualified as tavern games through which traveling merchants and travelers could measure their calculation skills. This was for example the case of problems exposed through the description of a transaction process and punctuated by the question “how much money did such a protagonist have at the start?”, the so-called divination problems whose resolution did not fail to produce a double effect of admiration: for the maestria in terms of calculation on the one hand, and for this mysterious power of divination that appeared to provide mastery of mental calculation. The last chapters return to questions of learned calculations: square roots and cubic, various geometric problems and the so-called methods of false position: explicitly inspired by an Arab algebra, they allowed to solve problems that we would classify among first degree equations if we could establish that they were then thought at the degree of abstraction that presupposes such a conception, given that Leonardo clearly affirms the general scope of the method but not in these terms. For everything related to the Arithmetic of whole numbers or fractional numbers, the Pisan is in line with the fifth book of Euclid’s Elements. It is difficult to specify how, he himself would have been inspired as the work was known in the 12th century. The fact is that Leonardo’s references to the Euclidean corpus are relevant – a fact verified on the occasion of the recent translation (2020). He certainly had a copy. The hypothesis is plausible for a familiar of the scholarly Palermitan circle of Emperor Frederick II. Moreover, his expression, throughout the Liber gives us something of the Mediterranean mathematical culture beyond such an elite: the work was designed to be widely heard, and the multiplication of its copies in Europe proves that this was the case. What were the gestures of the practice in question? They were attached to two tools: the sheet and the pen. The latter had to be perfectly mastered so that it produced a consistent and readable writing, even at the cost of a certain slowness, especially in the early stages of learning. The sheet, on the other hand, was an organized space and the relative positions of the numerical signs focused the attention of the apprentice. They had to write by attaching the greatest

2.2

A Modern Starting Point: The Liber Abbaci of Leonardo of Pisa

27

importance to the position of the signs and they were asked to read their writings.32 As for calculations on whole numbers or fractions, the examples and their treatments recounted by Leonardo are based on rules learned to be implemented in a procedural manner. These ancient data therefore called for algorithms, in the ancient sense of the word, that is, a series of formal operations that since antiquity specialists in all fields had to learn as part of their profession. An excerpt from Chapter VIII of the Liber provides direct access to his style if only translated, the Latin prosody escapes us. This carries something hypnotic that the volutes of the masters’ commentary undoubtedly maintained and perhaps came from the Vedic prosody where the art of permutations and combinations mattered.33 Moreover, the same passage also shows the importance of Euclidean arithmetic in the baggage of the calculators to whom the Pisan addressed. We must also pay attention to the way in which whole numbers – understood as additions already made of units – and their fixed ratios are inserted into sentences. In Latin, or as below in the translation into modern English which accentuates it by adding in the body of this commentary the article “the” as for example in “the 17″ or “the 36″, the text stages fixed entities, the numbers on the one hand and the ratios on the other. They are sometimes endowed with properties: divisors or a place in a relationship. These numbers thus designated are like the pieces that would be combined during a game of chess: “the 17″ to be multiplied by “the 36″ and the ratio 17 29 kept in reserve. Thus, each number and each ratio is taken as a sign itself considered in block until Euclidean arithmetic offers a rule to transform it within a proportion. “You have multiplied the 17 by the 36, then divided by the 29 then you have taken the thirty34 29 36 sixths of 17 29 . Indeed 29 is equal to 36 . [. . .] When IIII numbers are proportional, the product of the second and the third is equal to that of the first by the fourth as demonstrated by Euclid”.35

32 By analyzing the copies of compilations resulting from a survey from the 1780s on baptisms, marriages, and burials of French parishes, we found that during this large-scale copying operation, the clerks who drafted the tables proceeded as follows: one had an original in hand and dictated the numbers to the one who held the pen to take note on a table drawn up in advance. If, for example, the dictated line indicated 98 baptisms, 12 marriages and 86 burials, the reader dictated: 9–8 (pause) 1–2 (pause) 8–6. During our recent study, the intermediate totals at the bottom of the pages allowed us to uncover some misplacements of numbers. This method of dictating and then collating was effective: for the entire survey in question: nearly 99,000 numbers of baptisms, marriages or burials were thus reproduced and the error rate was one erroneous number for ten thousand transcribed, see Brian, 2001. This “dictate collate” method was still in force in the notarial world of the mid-19th century. It was staged in Melville, 1853. 33 Goonatilake, 1998, cited above. 34 Here The late Roman notation IIII appears in place of “four” because the Arabic notation 4 could have interfered with the calculations commented on. The status of the four terms of the proportion is solemnly distinguished. On this effect of solemnity while the two writings coexisted, see Perrot, 2021, p. 497. 35 English: Fibonacci, 2002, p. 125, §2 and Latin, id., 2020. p. 137 (§172).

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Data Arithmetic, Ratios and Mechanical Reasoning in the 17th and. . .

Besides the sheet and the pen, calculations were conceived mentally. The considerable part of examples involving proportions or the rule of three, reveals a vast field of exercises which today would seem pointless to students equipped with calculators. A crowd of merchants and traders bustle between the lines of the Liber often in a stylized manner. The learning aimed to give a mental mastery of figures and numbers sometimes composed in a proportion. It is not a single isolated calculation that the writer asks his reader, it is most often a series of them of the same kind that follow one after the other. It was then necessary to show good gymnastics in mental calculation endowed with concentration and a good memory. Even in the 18th century, a mathematician such Condorcet designates this quality by the expression “strength of head”.36 Thus, until the 17th century at least, commercial arithmetic carried a vast posterity of Euclidean arithmetic not just in formulas but also in mental habits. In mathematics, as in any other field where the humanism of the Renaissance was deployed, scholars wanted to draw directly from the works of the Ancients.37 That’s why, in terms of learning arithmetic, we find on the one hand books intended for the initiation of merchants, if not for the training of their children and their clerks which abound in examples while delivering calculation rules and on the other hand treaties which present themselves as contributions to the European movement of translations of Euclid‘s Elements into vernacular languages. Neither the merchants, nor the rulers were indifferent to this blossoming of manuals obviously favored by the extension of trade and by the success of printing. The addresses of the works are quite often “at the author’s”. With the frequency of reissues they show that a good calculation teacher had every interest in publishing a manual. Several dedications to the main ministers of the time remind that then the freedom of the press was not in force and that it was necessary to have a protector and a privilege to print, that is to say have his work examined by a commission appointed for this purpose. In this intense and heavy climate, we saw that the merchant from Reims Taillefer was wary of established teachings and preferred to entrust his son the example of his own accounts so that he could practice. To understand the know-how involved in handling figures in the 17th century, we must therefore follow these two corpora: the models printed by masters in account books on the one hand and the translations of Euclid on the other hand. On the first track, the manual of Irson (1768) like, a century later that of Caignat de l’Aulnais (1771), have instructed to keep a journal of transactions and books summaries intended to provide evidence in disputes or trials.38 The writings were there of the order of legal acts and not a matter of arithmetic or Euclidean geometry. But to excel in the calculations they involved, such as for obtaining the conversion of sums acquired in a foreign currency, or for fixing the volumes of goods acquired or sold

36

Condorcet, 1795, for example: p. 351, 354 or 383. Febvre, 1935, reprinted in Febvre, 2023. 38 This is a singular work, due to its subject, the accounting of a typical slave expedition. It is commented on in Lemarchand & McWatters (2006). 37

2.3

An Obscurity in Euclid’s Elements and Its Consequence

29

taking into account the variety of local measures, or even for distributing down payments or debts, one had to be a skilled calculator. It is at this point that the two corpora intersected, that of accounting models and that of translations of Euclid. These, however, had remained in the long duration of the world of scholars and erudites even for those who sought to reconstitute the corpus of the Alexandrian and to transmit it in the languages being formed at the Renaissance and suitable for didactic uses. Among these erudite scholars, the rigor of calculations, the quality of the language and the care given to the art of thinking went hand in hand.39 It is therefore necessary to scrutinize in this movement of translations, the producers of reference works in Arithmetic in the 17th century where the translations of the Euclidean corpus would have stumbled and incorporated presuppositions that could characterize the treatment of numbers at that time.

2.3

An Obscurity in Euclid’s Elements and Its Consequence

For the English and French languages, the history of translations of Euclid’s Elements has been known for quite than half a century.40 On the other hand, we also have a recent translation in French whose author has scrutinized his predecessors.41 However, Book V, which deals precisely with Arithmetic, contains in the eyes of this recent translator an embarrassing obscurity: “Definition 3 in Book V is very general and, from a modern point of view, more metamathematical than operational. [. . .] A ratio [has thus been considered as] a relation between two terms and proportion as a ‘unification’ of two or more of these ratios. A symbolic representation of these notions can be the following: a ratio between two terms (A, B) is represented by R (A, B): the ratio that exists between A and B is not limited to what we designate by A: B”.42

The translator has extended this observation: “Euclid’s Elements does not develop a theory of fractions; [. . .] Given a ratio of two numbers, Euclid establishes certain results concerning the [let’s say irreducible] expression of this ratio. However, it has long been noted that he does not explicitly state a definition of the ratio between two numbers. In any case, what is important here is that these ratios are considered as relations rather than as objects, and consequently Euclid does not operate on these ratios as he does with integers, or a fortiori as we operate with our rational numbers. What interests him is [on the one hand] the conservation of ratios: [and therefore] finding the manipulations that preserve a certain ratio, in other words, which establish a proportion; and [on the other hand] the production of new ratios by composition, what the terminology

39

Cifoletti and Coquard, 2022, cited above. Archibald, 1950; Lacoarret, 1957. 41 Vitrac, 1990: he presented his conclusions in id.: 1989, 1993. 42 Id., 1990, p. 58. The notation R (A, B) reflects a current way of thinking about this relationship. This is where the difficulty lies, we can speculate here that this may be a reminder of the Egyptian concept of fractions, see Rey, 1942, p. 236. and 243–247. 40

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designates additively and which, in no case, can be confused with our multiplication of fractions”.43

During the defense in which Bernard Vitrac presented his work, this translation and its expectations: “the way of considering The Elements themselves and their meticulous examination leading to a rehabilitation of their coherence” was praised.44 It is true that the design and uses of proportionality revealed by Vitrac met the perplexities of his evaluator regarding the imprudences of commentators on proportionality in Aristotle’s doctrine of nature.45 The obscurity of the third definition of Book V has therefore darkened entire sections of the history of philosophy as, we will see, it has confined the arithmetics of the 17th century to presuppositions specific to that time. A 19th century French translator delivers the Greek text of this third definition in a restitution held since as a reference. His translation will be discussed later. γ´. Λóγoς ἐστὶ δύo μεγεθῶν ὁμoγενῶν ἡ ϰατὰ. πηλιϰóτητα πρὸς ἄλληλα πoιὰ σκε σις.46

Since the 16th century, this Λóγoς has been translated as “reason” or “ratio” and its definition has included the obscurity identified by Vitrac. Indeed, for its first translator into French, Pierre Forcadel (1500–1572), reader of the King at the University of Paris, it was necessary to understand: “Reason is a mutual habit of two magnitudes of the same genre according to quantity.” [that is to say homogeneous in modern terms]”. [Forcadel immediately adds, inserting his name to clearly mark that he was then speaking:] “This regard that is between two magnitudes of the same genre is called reason, like when we consider the equality or inequality of equal or unequal magnitudes and when they are unequal how much the greater contains the lesser [. . .]. Proportion is the similarity of reasons. Like when there are several habits or comparisons [offering the same ratio] all this is called proportion”.47

The 21st century reader would be inclined to think of such a proportion by thinking of an algebraic notation unthinkable among our arithmeticians and absent as such from the considered corpus: a c = b d However, we can use this notation to reconstruct Forcadel‘s mental process. Indeed, the numbers represented by the letters here were written in this order: first a, then b, next c and finally d. They said “the first, the second, the third, and the fourth” in the

43

Id., 1992, p. 150–151. Caveing, 1994b. 45 Id., 1994a. 46 Peyrard, 1814–1818, vol 1, p. 235. 47 Forcadel, 1564, p. 123–124, our emphasis. 44

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31

order of writing. Thus, these arithmeticians wrote, just like Leonardo of Pisa, not as we were taught in our childhood “the product of the extremes (a×d) is equal to the product of the one of the means (b×c)” but: according to Euclid, “the first by the fourth is equal to the second by the third” Once it was established that the quantities were pairwise homogeneous,48 the focus of the calculators was not so much the values of the numbers themselves, the magnitudes, as their respective positions noted below in parentheses: it was with these positions that they juggled. Thus, definition 4 on Book V was understood in this way: ðð12ÞÞ = ðð34ÞÞ . By using this notation in parentheses, we want to emphasize that, in the reasoning, the position of the numbers was paramount. In doing so, we are following the example of Simon Stevin who designated the position of numbers by circled order numbers: ① ② ③ ④.49 Armed with this ad hoc transposition and returning to the passage from Forcadel that has just been quoted, we can understand that if he considered a ratio to be a “mutual habit of two magnitudes”, he did not intend to read the reason as a quantity,50 a number sometimes whole, sometimes fractional or “broken”, two concepts that he mastered perfectly otherwise. Thus, for example, ① ② was thought of as a block, the so-called “mutual habit”, a dependency between what intervened in the first and second position. For example, if the numbers occupying these two positions were respectively 297 and 11, he would not say “their ratio is 27”, but: 297 351 11 and 13 have the same ratio and are proportional. The ratio, as this word was coined by translations, thus establishes a relationship considered sealed between the two terms. This conception, free of conceptual devices that will be found in the physics of the following centuries, did not leave the Ancient or Renaissance scholars blind to dependencies that will be analyzed more finely only after the Galilean revolution. This is evidenced by the translation of Archimedes given by Forcadel a year later where several results established arithmetically express interdependencies between physical quantities.51 The same Forcadel also published a Complete and Abridged Arithmetic.52 It opens with a definition of this field followed by two definitions in the manner of Euclid: the first holds a number for an aggregate of units, and the second holds it for a magnitude whose elementary measure is the unit (in modern terms: “the unit is a divisor of every integer”): “Definition of Arithmetic. Arithmetic is the science of numbers.

As above, English: Fibonacci, 2002, p. 125, §2 and Latin:, id., 2020. p 137 (§172) or again 2002, p. 31; and id., 2020, p. 23 (§57) 49 Stevin, 1585, p. 9 or id., 1625, p. 13 where the order of appearance of numbers is symbolized by such circled numbers. On this mathematician, see Coquard, 2022. 50 Here, Forcadel appears in line with the interpretation given by Maurice Caveing of proportionality in Aristotle’s Physics, which concluded: “it follows that modern interpretations are at least inaccurate and imprudent”, Caveing, 1994a. 51 Forcadel, 1565b. 52 Id., 1565a. 48

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Definition of number. Number is the multitude composed of several units. Or, number is a multitude compared to the unit [. . .]. To number is to name and put the number in figures.53 It is necessary to know well recognize two things to subject the numeration [. . .]: the first is to recognize the figures [the digits] in which the numbers are put, their values being alone. And the other is the places of figures and the value or their values according to the places”.54

The work then deviates from the Euclidean text to move on to an extensive exposition of calculation procedures that are also found in his own translation of the first six Books of Euclid published a year earlier as well as in the Liber of Leonardo of Pisa. This Arithmetic, as its title indicates, responded to the dual demand of the time of apprentice calculators, oriented on the one hand towards commerce and on the other hand towards the calculation of scholars, but not towards Euclidean scholarship to which a previous work had contributed. The first decades of the 17th century then saw the flourishing of several translations or adaptations of the Euclidean Books and therefore of the first lines of the fifth among them. Here first are the six first books translated by Jacques Peletier (1517–1583), published in 1611.55 The third definition of Book V then became “The reason, or ratio of two magnitudes of the same kind, is a certain habit between them.” This new adaptation from a mathematician particularly attentive to language shows that the qualification of the habit in question was delicate. As rigorous as Peletier was, he immediately writes “reason or proportion“and reserves the word “proportionality” to designate the equality of reasons (or therefore of proportions).56 In this way two things were saved, on the one hand the singularity of each reason (or proportion in the sense of Peletier), and on the other hand the latitude to proceed with Euclidean calculations in case of proportionality between two reasons. Like his predecessor Forcadel, the mathematics professor Didier Henrion (born in the 16th century – died in 1632) has produced a work intended for practical teaching and a translation of the Alexandrian. His Practical Geometry is rich in Euclidean calculations, as the proportions intervene almost everywhere such as in the proofs related to the theorem of Thales or because of this definition: “Geometry is the art and science of measuring well [. . . either] of considering the nature or quantity of one or more measurable things, and by comparing them with each other [to] recognize what proportion they have with each other, and their difference”.57 Further on, the same followed, paragraph by paragraph, the first definitions of Book V.

53

This old sense of the French word, identical to the English word, is that of the ten numerical signs. Forcadel, 1565a, p. 5–6; the translator of Euclid and Archimedes uses the vocabulary of the Neoplatonist Plotinus, see Amado, 1953. As for the question of “places”, it is that of the location of the digits of tens, hundreds, etc. in the notation in what we call base 10. 55 Peletier, 1611; on this author and his scholarly horizon, see Cifoletti, 1995. 56 Peletier, ibid.; p. 269–270, our emphasis. 57 Henrion, 1620, p. 4. 54

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“Definitions: [1.] A part is a smaller magnitude drawn from a larger one, when the smaller one measures the larger one. [2.] A multiple is a larger magnitude than a smaller one when the larger one is measured by the smaller one. [3.] Reason is a habit of two magnitudes of the same kind, compared to each other according to quantity”.58

The end of the third definition stipulates that the two magnitudes had to be related to the same measure. Moreover, the notion of magnitude, as in the Greek text, is very general here: it is valid both for numbers, for “lines” (that is, segments), surfaces or volumes for example. In the case of surfaces, the example of similar triangles in the Euclidean geometric sense relates to such comparisons. If, in his Vulgar Arithmetic (1621), published a year later, the first definitions of Book V do not appear, the vocabulary of the third among them is used in one of the many exercises discussed: “If 15 give 12, how much will 5 give R59 4”. [Then comes the moment to verify the validity of the calculation:] “I conclude that the operation was well done, that is to say that there is the same ratio and habit of 4 to 5 as of 15 to 12 [sic]”.60

From the same author, we also have a translation of the fifteen books of Euclid’s Elements. It is therefore possible to verify his interpretation of the first definitions of the fifth among them. In this translation, published several times even after his disappearance, Henrion translated the third definition of the fifth Book in these terms: “ratio is a habit of two magnitudes of the same kind, compared to each other according to quantity.”61 The qualification of this “habit” is then glossed over, and this supports his understanding as a block that should only lend itself to calculation after having been put in proportional equivalence with another equal one. The Dutch financier, engineer and mathematician Simon Stevin (1548–1620), in 1585, also proposed to untangle the links between “ratio” and “proportions”: “Proportion, to speak a little generally [. . .] is the similarity of two equal ratios. Ratio is comparison of the two terms of the same kind of quantity.”62 His Arithmetic, published posthumously forty years later, is very explicit: “Arithmetic ratio is the mutual habit according to quantity between two or more terms.”63 [And further on:] “being [. . .] notorious that ratio is not a number, but the mutual habit of numbers; it follows that the ratio cannot be multiplied by ratio, but well by number”.64

58

Id., ibid., p. 25, our emphasis. For “Answer to verify”. 60 Henrion, 1621, p. 62. As we have seen, since Euclid, in the reading of proportions, it is the order of the terms that mattered. It seems therefore that the editor or more likely the typographer (put to a tough test) made a mistake: it would have been more relevant to write: “there is the same ratio from 4 to 5 as from 12 to 15”. 61 Id., 1632 (second ed., 1676), vol. 1, p. 249; “same kind”, for example: two numbers, two segments two triangles, our emphasis. 62 Stevin, 1585, p 56. 63 Stevin, 1625, Definition LIX, p. 56. 64 Id., ibid., p. 713, §1, our emphasis. 59

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Regardless of the later developments of the ancient corpus, this reluctance to extrapolate ratios numerically has remained in popular French language. An old unit of measure for liquid volumes was the setier, whose definition depended on its use. For transport or transactions between merchants, a setier was nearly three hundred of our liters from one place to another, but at the tavern it was about half a liter.65 Still at the tavern, the custom was to order “a half-setier” by calling out to the owner or the waiter: “un demi!”, which is about a current British pint. However, the expression “un demi” remains in common use long after the establishment of the units of the revolutionary metric system: it then designates a mug or a glass of 25 centiliters: it is the usual order in bars today. Assuming that four people are sitting around a table and they all want to order “un demi”, they would not ask for a liter (four times a quarter of a liter), but if for only one consumer, for a one liter “formidable”; or for the four rogues, for “quatre demis!” neither multiplied nor shared. The singularity of the Basque-born mathematics professor Pierre Hérigone (1580?-1643?) offers us a useful sieve to restore the way ratios were understood in the 17th century.66 Cardinal and Minister Richelieu had called him to Paris and we have from him a mathematical course (or Cursus mathematicus67) and a translation of the first six books of Euclid‘s Elements.68 Undoubtedly because he was multilingual (Basque, Spanish, Bearnese, French), Hérigone conceived that Euclid’s Books, then known to him in Greek, Latin and French, could be elaborated in an abstract language written with partly new signs. His Cursus is thus published in three languages, Latin, French, and the new formal language of his invention, such is the reason for its title: “Mathematical course demonstrated by a new, brief and clear method, by real and universal notations that can be easily understood without the use of any language”. This universal language was meant to speak directly to the mind. That a mathematician aims for an art of reasoning whose improvement would bear as much on formal procedures as on their literal or oral expression is not exceptional among his contemporaries.69 The same spirit of seeking a universal language will be found later, for example, in Leibniz and Condorcet. The first volume of the Cursus, after the dedication, an address to the reader, and some prolegomena offers an explanatory table of notations (called “notes” in the work). This is the case for ratio: “raŏ, rao, ratio, reason”. The course later resumes the succession of the fifteen books of Euclid, and the third definition of the fifth among them appears at the bottom of page 185: given in Latin “Ratio est duarum magnitudinum eiusdem generis mutua quaedam, secundum quantitatem habitudo.” and in French (here translated in English) “Reason is a habit of two magnitudes of

65

See the French National Centre for Textual and Lexical Resources: https://cnrtl.fr/definition/ setier. 66 For this exceptional mathematician, one can consult Chapter IV in Cifoletti, 1990. 67 Hérigone, 1634–1642. 68 Id., 1639. 69 Cifoletti, 1995.

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35

the same kind, compared to each other according to quantity.” Two pages later, the notation is used in an example: “hyp. | raŏ. aπb 2|2 raŏ.cπd” [which should be understood as:] By hypothesis: the ratio of a to b is equal to the ratio of c to d.70

Then comes this clarification on vocabulary “In every ratio the quantity that refers to another is said to be the antecedent of the ratio (this is our numerator written above the fraction line: the first to be written, which explains this old terminology). But that to which another refers is said to be the consequent of the ratio (this is our denominator, then written second), as in the ratio of 6 to 4, the antecedent is 6 and the consequent 4. [. . .] The quantity of the ratio of 12 to 4 is 3, because this number shows how many times the antecedent 12 contains its consequent 4.” The use of two words confirms that in a ratio the position of the two terms takes precedence over the value of the numbers. Moreover, the words “antecedents” and “consequents” express here no logical or causal order, but only a scriptural order.71 The following pages bring other details. Thus, in the first paragraph on page 188: “Reason is the habit of two magnitudes” and on page 190: “The habit can be “greater“ or “smaller” depending on whether the ratio is larger or smaller.” The translation of the fifth Book of Euclid published a few years later is consistent with the content of the Cursus. Thus the formulation in the manner of the 20th century proposed by Vitrac meets the so abstract translation of Hérigone conceived three and a half centuries earlier, even though it remained buried in the mathematical vestiges of the 17th century. Our analysis of how ratios were then thought is thus strengthened: a ratio was then the sealed relationship of two homogeneous magnitudes, the first written being compared to the second, and it will lend itself to calculations only on condition of being inserted into a proportional equality. The arithmetic know-how of the 17th century therefore had a tool suitable for highlighting comparisons. However, its use was then much more rigid than we might believe today while we have more flexible calculation methods. However, we will see, this ancient know-how has left its mark on the interpretations given today of ratios even when they are used in a scientific context. Similarly, the lack of current familiarity with this conception specific to the early modern era has induced historiographical errors, marked by anachronisms. The analysis of two of them will conclude this chapter. Before that, let’s see what happened to the translations of Euclid’s Elements in English. The scholar Isaac Barrow (1630–1677) clarifies the meaning of ratio in the direction just indicated, but he does not escape the terminology of the “habit”:

In Hérigone’s language, 2|2 symbolizes equality and 3|2 indicates that the first term would be larger; and raŏ. Aπb is phonetically read as “ratio a pi b” (reason of a then b, “pi” was – and is – the popular phonetic abbreviation for “et puis”: “then”; as in the phonetic of “épi”). 71 The reader can practice writing a fraction: he will start by writing the numerator, then underline it, and finally write the denominator below. The ancient and adequate gesture persists, as subtle as the algebraic structures involved in a division and the calculation methods are today. 70

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“Definition III. Ratio (or rate) is the mutual habitude or respect of two magnitudes of the same kind each to the other, according to quantity”.72

During the second half of the 18th century the French vocabulary was sealed in the columns of The Encyclopedia by Diderot and D’Alembert: “Antecedent of a ratio, is the first of the two terms that make up this ratio. Thus in the ratio of 4 to 3, the first term 4 is the antecedent”.73 “Consequent, this is what we call in Arithmetic the last of the two terms of a ratio, or the one to which the antecedent is compared”.74 “Ratio, in Geometry and Arithmetic, it is the result of the comparison of two quantities with each other, relative to their size. We also use the word reason and even more commonly especially when this word is joined to an adjective, like direct reason, inverse reason”.75

In the 19th century, in both languages reference translations are published that are no clearer than previous attempts. Thus the scholar François Peyrard retains the manner of Peletier: “a reason, is a certain way of being of two homogeneous magnitudes between them, according to the quantity.76 » In 1845, in England, Robert Potts published a collective work conducted at the University of Cambridge which includes this translation: “Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.”77 The numerical know-how, whose main instrument was the ratio of two quantities in the 17th century, thus consolidated over twenty centuries of learning practical arithmetic skills, synthesizing uses and habits of mental or written numerical manipulations formed and maintained in a vast geographical area from India to the Mediterranean Basin. And this consolidation stumbled upon a particularly obscure definition in Euclide’s fifth book that scholars have questioned from time to time during this long period. This means that in the abstraction of these calculations, tied to this obscurity, are crystallized a considerable extent and duration. Thus, retrospectively, these calculation skills seem to us universal and timeless. That’s why we are spontaneously inclined to read them in an anachronistic and ethnocentric way, inserting in there the words numbers, data and ratio, that is to say current and inappropriate numerical skills. But that’s not all, with the uses came presuppositions related to the ratios of the numbers thus manipulated and the preservation of old uses like their confusion with their contemporary counterparts have conveyed to us these presuppositions to such an extent that the accustomed, today, do not take notice. This is for example the case of ratios: let’s remember from our journey at the beginning of the modern era that a ratio between two homogeneous numbers should not be understood, under the pen of

72

Barrow, 1660, p. 91, our emphasis. D’Alembert, 1751. 74 Id, 1754. 75 La Chapelle, 1765. 76 Peyrard, 1814–1818, vol. 1, p. 235. 77 Potts and others., 1845, p. 126. 73

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an author of the 16th and 17th centuries, as a rational number in the current mathematical sense but as a relation between these two numbers, a relation taken as a whole that can be subject to comparisons from ratio to ratio or even calculations of proportions. The presupposition here is based on a possible hypostasis of the ratio: the block formed by the ratio of the two numbers would then be considered as a fact, even though it is only a process. A very current example will be given in the last chapter. It will come from the contemporary demography where ratios are widely commented and sometimes abusively treated in the manner of the 17th century. The simple repetition of old calculations is not only at issue here. Indeed at the end of the 19th century and during the first half of the 20th century, demography was annexed by strong ideological tensions marked by eugenics or natalism or even by various totalitarian inspirations, After the Second World War the discipline opposed calculation techniques to the dangers of such tensions.78 The old calculation technique and its hypostasis were then sealed: the establishment of ratios, was consolidated in an anachronistic way into a process considered generally accepted in the discipline. Such was the outline of the formation of what Francis Bacon called an Idol of the theater,79 in this case of the demographic theater: resulting from various dogmas or false demonstrations, to be then adopted by specialists in the manner of postures.80 However, according to Bacon, “To point them out [. . .] is of great use.81 This is what we have tried to do in this chapter about the formation of the concept of ratios in the 17th century. This theater scene will be discussed in the last chapter.

2.4

Approximations and Mechanical Reasoning

The historian Lucien Febvre (1878–1956) praised the synthesis given by the philosopher and historian of sciences Alexandre Koyré (1892–1964) on reasoning based on approximations during the Renaissance and their importance in the techniques of that time. Indeed, the philosopher himself had relied on a historian’s investigation: “[Koyré] notes that the distinctive feature of the first machines is that they are never ‘calculated’. Executed by judgment, they belong to the world of approximation. And that’s why they are only entrusted with the coarsest operations of the industry. The others, more subtle, are only performed by human hand. [. . .] And “[Koyré writes (p. 813)], I believe it is not even enough to say with Mr. Lucien Febvre that, to do this (meaning to

78

Cahen, 2022, part 4. Bacon, 1645, Aphorism XXXIX. 80 Id., ibid. Aphorism XLI-XLVI. See the foreword in the present volume. 81 Id., ibid. Aphorism XL. 79

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count, weigh, measure), the man of the Middle Ages and the Renaissance lacked material and mental tools”.82

In fact, Alexandre Koyré compared with what he knew of the Greeks. “Never did Greek thought want to admit that accuracy could be of this world that the matter of this world, of our world, of the sublunary world, could embody mathematical beings (unless forced by art.83)”; “When studying the machine books of the 16th and 17th centuries [. . .] one is struck by the approximate character of their structure, their operation, their design. All of them belong to the world of approximation.”84 “The Bible had taught that God had founded the world on ‘number, weight, measure’ [Wisdom, 11: 20]. Everyone repeated it – but no one believed it. At least, no one until Galileo took it seriously. No one [in the 16th and 17th centuries] ever tried to determine these numbers, these weights, these measures. No one thought to count, to weigh, to measure. Or, more precisely, no one ever sought to go beyond the practical use of number, weight, measure in the imprecision of everyday life [. . .] to make it an element of precise knowledge. [. . .] It is undoubtedly true, and of capital importance, that the use of the most common instruments, the most familiar to all and moreover the simplest, always remained unknown to them”.85

In his Rabelais (1942), in book II, chapter III, section 4, Febvre had given the example of a public clock where a regular but approximate construction was seconded by an assistant who rang the main time marks. There is a class of machines where the mechanical process exactly coincided with the arithmetic of the ratio, it is that of gears where the numbers of teeth of each wheel are the two terms of the ratio. This is then the constant characteristic of the multiplication or demultiplication provided by the system of the two toothed wheels. Several reasons have made that the investigations on the mechanistic approach of physical phenomena have not brought to light this affinity between strictly Euclidean arithmetic and the technology of these most often coarse gears formed of wooden wheels. Firstly, our analysis of the design of ratios as a relationship tied between two quantities has not been adopted by the authors who have studied the history of mechanicism according to the terminology of Giorgio Israel (1945–2015).86 Secondly, this history was written under the shadow of the philosophical and scholarly work of René Descartes (1596–1650) and it is based on investigations whose keys are the doctrines and the resources they provided in the sciences, explorations which, certainly, require a lot of erudition. The line we are drawing (on the contrary to the philosophical approach) requires considering the gestures of the arithmetic know-how of the Renaissance and the early modern period and particularly rudimentary technical objects.87 We know a very different field where we find the same protagonists studying the arithmetic

82 Febvre, 1950, p. 26, commenting on Koyré, 1948 (i.e. 1971, p. 341–362) who had referred to Febvre, 1942! 83 Koyré, 1971, p. 343. 84 Id., ibid., p. 347. 85 Id., ibid., p. 349. 86 Dijksterhuis, 1986 Roux & Garber, 2013; Israel, 2015. 87 This is the program followed in Gauvin, 2008.

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Illustration 2.2 Gear ratios for transmission mechanisms by Leonardo da Vinci (1478), Sketchbook, Biblioteca Ambrosiana di Milano. (Source: Database by Wolfgang Lefèvre and Marcus Popplow, MPIWG, Berlin. http://dmd.mpiwg-berlin.mpg.de/; reference LdVCA022)

derived from Euclid and Fibonacci and comparable calculation rigidities, it is that of the history of notations and musical machines.88 Such is the key to building the machines of this era where we recognize those of Leonardo da Vinci (1453–1519), their barns were chosen to multiply movements or forces according to the ratios of the number of wooden teeth that the rudimentary wheels carried (see Illustration no 2.2). After Koyré, the mathematician Georges Guilbaud (1912–2008) defended in Paris the proper and conceptual rigor of reasoning based on approximation.89 The cover of the collection published in his honor offers the reproduction of a particularly suggestive 16th-century engraving about the measures in whole numbers and numbers of teeth of a compass wheel designed for this purpose (see Illustration no 2.3).90

88

For the notations: Busse Berger, 1993, p. 198–210; for the machines: Gauvin, 2008, again. Guilbaud, 1985 and 1988. 90 This engraving, the measure of the Polar star with a compass of proportion, illustrates the cover of the tribute collection to Guilbaud: Guilbaud, 1988. 89

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Illustration 2.3 Measurement of the elevation of the polar star with the proportion compass. (Source: Apian, 1548, (unpaged) ca p. 40)

At this point, the arithmetic know-how of the 17th century in terms of ratios and the presupposition it entailed due to the singular history of the circulation of the Euclidean corpus up to this time, that is to say the conviction that the ratio of the antecedent (first written above the fraction line) to the consequent (then written below) should be considered as sealed forming a rigid comparison which certainly could be seen to have the same magnitude or not as another ratio. This comparison between approximate ratios opened since Euclid the possibility of additional calculations involving not two but three or four of the terms of the proportion: rules of three or proportions. The dated know-how being circumscribed at this point, it will be possible to consider the history of probability calculation by distinguishing three periods. First, the one where this conception of the ratio did not intervene. Then, the one where it prevailed and then those where it will be surpassed. During the first period and especially during the Renaissance, the probable was believed to be Aristotle‘s ἔνδoξoς: it was therefore an attribute of opinion as opposed to what would have been knowledge to be established by way of demonstration (this will be the case of probability as it will be understood during the following periods). Indeed, after the Stagirite, the acceptance of probable was marked by a false sense indicated by the reference translator in French Jacques Brunschwig in a “clarification”. He emphasized that “the term ἔνδoξoς in relation to an opinion or an idea was not, in principle, a property that belongs to it by right by virtue of its intrinsic content, which should prohibit translations by probable, likely, plausible [etc.], but a property that belongs to it in fact.”91 However, the medieval Aristotelian tradition has retained that the probable was what appeared so to all men or to their majority or

91

Brunschwig in Aristotle, 1967, n. 3, p. 113.

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to the wise, to all or most, to the most illustrious or to the most credible. In the first case, it was the verdict of the doxa, in the second that of an authority and in both always a matter of opinion.92 Historians have much debated what I indicate here as a second period that would cover the 17th and 18th centuries. It is indeed characterized by the novelty of resorting to calculations. Until then, many were the scholarly or non-scholarly texts, and in various fields of expertise, where the word “probable” appeared, to designate a criterion of reflection and action of their protagonists.93 This probable was part of the register of opinion in the Aristotelian sense and as it imposed itself on the reasonable and fair man, it nourished a science of conjectures but not their geometry or arithmetic.94 The radically new came from a few authors among whom the writings circulated: Blaise Pascal (1623–1662), Pierre de Fermat (ca 1605–1665), Christian Huygens (1629–1695) and Jacques Bernoulli (1654–1705), who all, of course, worked in the scholarly and general climate analyzed by Coumet. This state of affairs forbids the historian from attributing to such an accidental encounter this moment of innovation. With this new probable, it will no longer be a question of opinion.95 The new calculation, in the manner of these four scholars, was based on a principle: probability was defined as the ratio of the number of favorable cases to the number of possible cases, often a combinatorial analysis made known by inventory both of the two numbers. This characteristic of the calculation is well known, but the analysis that we propose of the conception of ratios in the 17th century gives it a particular flavor. According to us, the probability thus calculated was not conceived as fluctuating, but as a ratio proper to that of which the probability was considered. Thus, for example, in the throw of a six-sided dice the probability 1/6 of obtaining any of the six faces was held for the property of the game and of the as long as it was not biased: this characteristic had no reason to change. Thus the so-called “classical” calculation of probabilities was an extension of reasoning according to the Euclidean ratios of mechanical reasoning that characterized the scholarly thought of the 17th century, an extension prepared by the intellectual climate analyzed by Coumet (1970) and the conceptual elaborations distinguished by Hacking (1975). In other words, an ambiguous legacy of Euclid’s

92 Hacking 1975, p. 17–30 notes the long influence of the Aristotelian definition of what was considered probable according to the first paragraphs of Aristotle’s Topics I, 1, 1967, p. 1–2. 93 Such is Olivi, 2012, Treaty of contracts of the 13th century where the word “probable” appears in Latin to qualify reasonably shared opinions but not calculations based on previously numbered cases. Coumet, 1970, has extended the spectrum of ancient specialties where the question of probabilities arose until preparing the conceptual emergences analyzed in Hacking, 1975, and the mathematical emergence scrutinized in Meusnier, 1996. 94 On this period prior to mathematization Coumet, 1970, and Franklin, 2001 agree. 95 Condorcet (1743–1794) will return to questions of opinions and testimonies in the mid-1780s but by other paths than those extrapolated from Aristotle: not to restore this ancient conception but to put it to the test of new developments of integral calculus.

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work has driven from the realm of the probable an equally ambiguous legacy of that of Aristotle. In the 18th century, the development of differential and integral calculus among mathematicians allowed them to break free from strictly post-Euclidean arithmetic and to approach classical probability calculation by importing more flexible processes, even as calculation presented them with embarrassing paradoxes. Thus, one of these thought experiments that characterized the maturity period of classical probability calculation according to the favorable/possible ratio – the St. Petersburg paradox, named after the place of its first publication – came to disrupt the certainties acquired so far among mathematicians.96 It was a hypothetical game of heads or tails whose rules were such that the expectation of gains or losses, depending on the point of view adopted, would tend towards infinity. This is not the place here to account for the perplexity of scholars in the mid-18th century on this subject.97 In Paris, the geometer and philosopher D’Alembert (1717–1783) concluded that the rules of classical probability calculation were unfounded. This radical position earned him the contempt of European scholars, a contempt that the encyclopedist’s quarrel with Daniel Bernoulli (1700–1782) only fueled. Given his prominent place at the Parisian Royal Academy of Sciences, this crisis in the scholarly world placed this Company and the French conception of mathematical analysis in a critical situation. For entirely different reasons, these crises specific to the scientific world of the mid-Enlightenment century encountered another: the Company initially conceived, under Louis XIV, to examine and mobilize knowledge useful to the monarchy and reshaped in this spirit in 1699 had become obsolete during the last decades of the 18th century, both in terms of mathematical sciences for the reasons just mentioned and physical sciences, that is, natural sciences in the terms of the time. Two young academicians close to the reformist minister Turgot (1727–1781) – the first in chronological order in the reign of Louis XVI – the chemist Lavoisier (1743–1794) and the mathematician Condorcet (1743–1794), the latter acting also as perpetual Secretary of the Academy, the former as director of the same learned society, undertook to update its usefulness, leading to a new reshaping in 1785. If, on the side of Lavoisier, it was a mobilization of his followers and the new chemistry within the academic enclosure, on the side of Condorcet, it was a strategy of publishing new works of differential and integral calculus, the usefulness of which the perpetual Secretary advocated in the columns of the annual collections of the learned company. Two mathematicians, trained by D’Alembert, then published a dazzling corpus of memoirs on probability calculation, this time newly founded, no longer on the rule of the ratio of possible cases to favorable cases, but on integral calculus, Condorcet himself and Laplace (1749–1827), it being understood that other European mathematicians also practiced at this time in the same register. The ratios henceforth retained to define probabilities were no longer only integers but most

96 97

See Jorland, 1987 and Brian, 1994, part. I, chap. 4. Brian, ibid.

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43

often integrals, so that the fixity of the ratio was abandoned. In this renewal of probability calculation, of certain aspects of integral calculus and of the academic institution, the conjunction of crises just mentioned was overcome.98 The corpus of these memoirs prepared before the French Revolution was the foundation of the teachings delivered in the schools it founded and that of the major works of Laplace published in the 1810s.99 So much so that to the two previous eras of probability calculation, that of conjectural probabilities and that of ratio calculation, a third has been added here: that of analytical calculation based not on sealed ratios but on the mathematical analysis of dispersion functions which allowed for less rigid ratios. This will be discussed in the next chapter where we will see that this new era itself involved skills and presuppositions that are unique to it. Anyone who would consider the calculation of probabilities from these three eras to today could not only stop there. Indeed, the 20th century marks, for its part, a fourth era, that of axiomatic calculation of probabilities for which the works of Paul Lévy (1776–1971) and Andreï Kolmogorov (1925–1971) marked a turning point. Its study does not fall within the scope of this book.100 As we leave the world of ratios in probabilities, it is important to correct two historiographical legends, one about the beginnings of classical probability calculation, the other about a then very active field: that of political arithmetic. But, before proceeding to these two critical examinations, here is a fact that is not without irony, the main founder of the analytical theory of probability calculation, even in its initial gestures, remained a prisoner of the traditional conception of ratios. Indeed, in the corpus of Laplace‘s memoirs published by Condorcet at the Academy of Sciences in Paris during the 1780s, there is one that aimed at comparing the ratios of the number of boys’ births to the number of girls’ births in London and Paris. Laplace wondered if the calculation of probabilities would allow him to decide on the relevance of a difference between the two capitals in this regard. He explained the principle of his calculation: “I give the solution to some interesting problems in the Natural History of Humankind such as that of the greater or less ease of births of boys relative to those of girls in different climates [. . .]. This phenomenon [births of boys not in a greater number than births of girls] is much less probable in London than in Paris, which comes of the fact that in [Paris], the ratio of births of boys to births of girls is more considerable. The odds [are] more than four hundred thousand to one that births of boys take place with greater ease in London than in Paris; thus it can be regarded as very probable that there exists in the first of these two cities a cause, more than in the second, that facilitates births of boys there and which depends either on the climate or on the nourishment and the customs”.101

98

Id, part. III; and Oki, 2011 These are his works that will become foundational: Laplace, 1812; 1814 and the teachings at the École normale de l’An III, Laplace, 1992. 100 This is the subject of chapter V of Maistrov, 1974, an author particularly well informed on the Russian domain. 101 Extract from Laplace, 1781, read at the Academy in 1778; translation in Brian and Jaisson, 2007, p. 14. 99

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These greater or lesser facilities of the birth of one or the other of the two sexes considered by Laplace are like the characteristics of a coin that would be thrown to play heads or tails. He posed the question in these terms: should we hypothesize a difference between these “facilities” specific to each of the two cities? These ratios of male births reported to female ones, held as constants characteristic of each city and governed by their climates and respective lifestyles, is it more or less probable to distinguish them or to consider one higher than the other. As we can see, Laplace held these ratios as sealed ratios characteristic of each capital. The very one who will build a calculation of probabilities based on the division of one integral by another, therefore on a radically new process compared with that of the ratios of possible cases to favorable cases, as soon as he considers the natural history phenomenon in question assigns it characteristics conforming to post-Euclidean ratios. That the same author holds the two modes of reasoning in the same memoir, comes from the fact that on one side he considers characteristics of the studied phenomenon and on the other hand, it constructs the probability of an observation regarding them. The old ratio characterizes the objects and the new calculation is of an epistemic order. In the next chapter, we will see the know-how inaugurated by Laplace leading to a second hypostasis: the astronomer Adolphe Quetelet (1796–1874) will intend to establish this kind of new calculations in a phenomenon he will consider universal. Before moving on to the 19th century, let’s try to finish with the two legends that we have announced.

2.5

The Legend of Maritime insurance at the Beginnings of the Classical Calculation of Probabilities

That the beginnings of modern probability calculation came from the needs of maritime insurance, here is indeed a semi-learned legend. We find its trace in this collective collection of acquired knowledge, relevant or not, that is the online encyclopedia Wikipedia. It is then amusing to note that the English article says nothing of the sort while its French counterpart goes so far as to specify102: “During the Middle Ages, in the 13th and 14th centuries; exchanges were created to insure maritime transports depending on weather fluctuations; in the 16th century [it is] the generalization of maritime insurance contracts”.

Two articles published in a specialized journal are then summoned,103 but that of Sylvain Piron only deals with doctrinal questions which, certainly, Coumet and Hacking have shown the importance in the formation of probabilistic conceptual devices. Giovanni Ceccarelli, for his part, has examined a vast corpus of contracts and noted the variability of premiums according to the type of the ship, the nature of the cargo, the envisaged course and contexts marked or not by wars or piracy. Having found neither loss register nor explicit calculations, he comes to this conclusion, it was about: “practical and experience-based response to risk”. None of this has foreshadowed a probability calculation despite the author’s expectations expressed in the introduction.

102 Wikipedia, 2023b, “History of probabilities” and Wikipedia, 2023a, “History of probability”, at the date of February 15th, 2023, 17: 00; the second referring to Rüdiger (Campe), 2012, p. 278–288. 103 Piron, 2007 and Ceccarelli, 2007.

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The legend persists because it responds to an ideological prejudice of Marxist inspiration. The historian of probability calculation Leonid E. Maistrov (1920–1982), often quoted, expressed it in 1974 without detour in a paragraph that we must comment on: “Statistics was one of the basic stimuli in the initial development of probability theory [a]. An increase in capitalistic relations constantly posed new problems in statistics [b]. As Marx points out: ‘Although we come across the first beginnings of capitalist production as early as the fourteenth or fifteenth century, sporadically, in certain towns of the Mediterranean, the capitalistic era dates from the sixteenth century.104’[c] In the fourteenth century, the first marine insurance companies were established in Italy and in Holland [d]. Insurance of goods shipped by sea was followed by insurance of freight across continents, lakes and rivers [e]. These companies carried out calculations of chances since larger risks made for larger insurance premiums [f]. For shipping by sea, the premiums amounted to about 12–15% of the cost of the goods, while for intracontinental deliveries the rates were about 6–8% [g]. Beginning with the sixteenth century, marine insurance was introduced in many other countries [h]. Other forms of insurance originated in the seventeenth century [i]. The data collected by insurance companies also served as source material utilized in the development of probability theory”.105 [a] As long as scholars had compilations to which to attribute the qualification of statistics. However, the calculation of combinations has focused the attention of mathematicians before the era of statistical compilations assigned by historians to the 19th century to the second half of the 20th century. [b] Here lies the unproven ideological assertion. [c] It was customary in the U.S.S.R. to place one’s article or work under the aegis of Marx or Lenin, the first of them fulfills this protective role here even though his proposal is only vaguely connected with the rest by its periodization. [d] There were certainly insurance offices where contracts were concluded, and probably kept in the manner of notary offices, but nothing suggests that compilations were made and elements of probability calculation were implemented. [e] The methods then used were indeed suitable for these risky transports. The statement suggests an increase in the use of these insurances while the same non-probabilistic methods remained in force from the Romans to the 18th century. [f] To date, no one has shown that insurance premiums then called for a recourse to probability calculation when they came from rules of jurisprudence of ancient origin which will be discussed later. [g] We will see that these figures are contradicted by sources from the time. [h] The fact is much older: it comes from the Latin foundation of jurisprudence in most of the countries concerned. [i] This will only be the case from the 19th century onwards.

If the first part of Maistrov‘s work is thus marked by ideology and the second is rather anecdotal, his last three parts still deserve attention from anyone who would want to delve into the history of the mathematical developments of probability calculation in the 19th and 20th centuries – a rather arduous field. On the principle of this kind of initial approach, the great historian Lucien Febvre expressed a definitive judgment in 1941 to the new students of the École normale

Here, in brackets, Maistrov quotes: Капитал, [1867], Moscow, 1949, [book I, sect. VIII, chap. 23], p. 715, i.e. Marx, 1976, book 1, p. 680. 105 Maistrov, 1974, p. 5. 104

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supérieure. His lecture was published in Les Annales in 1943 and often reprinted since.106 He then said, about his illustrious predecessor Jules Michelet (1798–1874): “I note in passing, [Michelet] did not establish a hierarchy, a hierarchical ranking among the various activities of man: he did not carry in his mind the simplistic metaphysics of the mason: first layer, second layer, third layer – or first floor, second, third. He also did not establish a genealogy: this derives from that, this engenders that”.

It was a critique of the Marxist-inspired vulgate for which intellectual superstructures would be the reflection of underlying economic infrastructures. Comparing the periodization of the formation of probability calculation based on the ratio of the number of favorable cases to the number of possible cases and that of the historical growth of maritime commerce, here is something that lends itself to the simplicity of the “mason’s metaphysics” and it is by the virtue (or more rigorously the vice) of this simplicity that the legend is maintained. As for the beginnings of probability calculation, Maistrov wrote it, and the ideological environment in which he worked was probably the cause. But others have imprudently repeated it, like Anders Hald: “The industrial revolution, the increasing overseas trade, the growth of the British Empire, the accumulation of capital for investment and speculation [. . .], in short, the increasing capitalist structure of the British economy also led to the foundation of private insurance companies at first for marine and fire insurance [. . .]”.107

There is undoubtedly something pathetic about seeing this legend still circulated fifteen years after the fall of the Berlin Wall,108 that is to say, long after the dissipation of the self-censorship of authors who were under Marxist skies and especially two centuries after the historian of mathematics, Montucla, had given the key to the enigma. Indeed, it was not private capitalism but state finances as we will see later.109 At the end of the 18th century, scholars returned to the concerns of their very distant predecessors who, of course, could not have had access to the processes inaugurated after Pascal and Huyghens. This renewed scholarly interest in the theory of maritime insurance was marked at the Royal Academy of Sciences in Paris, during the decade 1780, punctuated as it was, in 1781, 1783, 1785 by a prize on this theme announced in the press and the first two editions of which were unsuccessful. As perpetual secretary of the Academy, and in this capacity organizer of its prizes, Condorcet approached some insurance chambers of major ports, notably Nantes and Bordeaux, to obtain from them lists of premiums and accidents (these were notarial type instances where contracts between borrowers and lenders were signed and sometimes kept).110 Therefore, during these years, it was as a particularly 106

Febvre, 1943, 1952, 2023. Hald, 1990, p. 508. 108 For example in Pradier, 2006. 109 We indicated the Montucla’s conclusions in Brian, 1994, p. 287. For his part, Stephen Stigler, 1986, p. 62, was cautious in not specifying the objects of insurance calculations around 1700. 110 On the editions of this prize, see Condorcet, 1994, p. 466, and on the survey by correspondence, p. 476. 107

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well-informed author that Condorcet wrote the article “insurance (maritime)” published in the Encyclopédie Méthodique Mathématique (1984). In doing so, he addressed a wide readership. He did not fail to allude to the Aristotelian tradition of the meaning of the word probable while providing the principles of new combinatorial calculations. But, he especially lamented the lack of information collections. The extraordinary thing about this article is its foundation on three epochs of probability calculation, the conjectural, the combinatorial and that of differential calculation on compilations, a process that would then be anachronistic to qualify as “statistical.”111 If Condorcet, without a doubt, mastered these references, or even these allusions to memoirs recently published by the Academy, it is difficult to conceive that his readers clearly perceived them. Thus, the historiographical scheme of Maistrov collapses not only because of the weakness of its ideological engine, but also because of its lack of “statistical fuel”. Condorcet in 1784: “It can also be very useful to know how, in practice, men who are considered wise and whose projects have succeeded have solved the [. . .] problem [. . .]. The solution to this question can be [notably] envisaged [by resorting to] tables for different insurance rates that would contain the number of insured vessels, the number of vessels that have perished, that of vessels that have not experienced accidents [. . .]. We will not linger any longer on this subject. It is enough for us to have exposed the general principles on which the calculation must be based. The application to practice would require too extensive research, and it might even be quite difficult to obtain the necessary data to make this application accurate enough to be useful”.112

That’s not all. Frederick Martin, the first historian of the Lloyd’s – the famous British insurance company – wrote nearly a century after the Parisian mathematician: “What is undeniable is that marine insurance, though in existence centuries before life insurance was even thought of, is still behind the latter, in not resting, as this does, on the firm ground of mathematical calculations, drawing laws of probabilities from the results of experience. At this moment [1876], the business of marine insurance is subject entirely to the exercise of personal and individual experience, fallible in its very nature, even when brought to the utmost possible perfection; while life insurance, on the other hand, stands on the solid foundations of “mortality tables,” and of actuarial computations derived from a vast amount of widely gathered statistics. To construct “mortality tables” for ships, the same as for human beings, is no doubt a matter of greater complication; still there appears no absolute impossibility for accomplishment of the task”.113

Thus, we must refer the promoters of the historiographic legend discussed in this section to the historian of mathematics Jean-Étienne Montucla (1725–1799), particularly well informed on the matter:

The use of the word “statistics“in the sense of accumulations of numbers only became widespread in the 19th century. 112 Condorcet, 1784b, and in id., 1994, p. 485. 113 Martin, 1876, p. vi. In France, such statistical collections will not be published until the end of the century, Turquan (1890). 111

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“There is little theory in mathematics where the resources of the analytical mind shine more than in that of probability [he wrote in opening his article on the history of probability calculations]. Indeed, if there was any subject that seemed to escape mathematical considerations, it is undoubtedly chance. But what can’t the human mind do, aided by geometric spirit and the art of analysis. This kind of Proteus, so difficult to fix, the mathematician has somehow managed to chain it and subject it to his calculations114 [. . .]”. And further on, referring to this incipit: “We said at the beginning of [this] article that the theory of probability is not only one of the most curious, but also the most useful [. . .] This utility appears especially in the application of this theory to a large number of political or economic problems and civil contracts. All the states of Europe have been forced in recent times, by their needs or their political follies, to borrow, both in perpetual annuities and in life annuities”.115

This is the crux of our matter. If the circulation of arithmetic from the works of Euclid and Fibonacci was indeed induced, as we have seen, by the utility of such calculations in the course of commercial transactions on a private contract scale or in merchant accounts, or even during legal disputes that these contracts have no doubt caused since antiquity and up to modern times, these merchant investments – to use the synthetic term employed by Anders Hald, capitalism, at an early or much more mature stage did not produce compilations that would have been indispensable for the development of actuarial calculations necessary for setting premiums for marine insurance, compilations comparable to what were the birth and death registers gathered in the offices managing state annuities, sometimes worked on by great figures of the monarchical administrations versed in this matter or suitably assisted, or even by scholars who responded to the demand of the time.116 It is therefore not to capitalism that we must assign the beginnings of modern probability calculation but to the critical finances of the absolutist European states of the modern era, a diagnosis that the reforming minister Turgot (1727–1781) and the great magistrates of his generation shared, this same Turgot who was the spur of the mathematician Condorcet in the exploration of these questions.117 In his narrative of the progress of probability calculation since the 17th century, in volume III of his History of Mathematics, Montucla mentioned without further comment the unsuccessful prize of the Paris Academy of Sciences on marine insurance.118 However, eager to highlight the usefulness of probability calculation, he also summoned two of its founders Jacques Bernoulli (1654–1705) and his nephew Nicolas (1687–1759) who did not fail to announce the benefits to be expected from the family mathematical heritage in political, legal and for example nautical fields. Long before a 19th century ideology came to seize this selfpromotion, the words too often repeated since were thus employed. What did the Bernoullis actually write on this subject? Historian Norbert Meusnier undertook to 114

Montucla, 1802, vol. 3, p. 380–381. Id ., ibid., p. 406. 116 Id ., ibid., p. 406.Such were Antoine Deparcieux (1703–1768) and his 1746 work, republished in 2003; and Emmanuel-Étienne Duvillard de Durand (1755–1832), 1813, republished in 2010. 117 Brian, 1994. 118 Montucla, 1802, III, p. 423–424. 115

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translate into French the Ars conjectandi of the uncle (1713) and the nephew’s thesis, De usu artis conjectandi in jure (1709).119 In Jacques, the announcement will have touched on the admission of powerlessness. In a letter to Leibniz dated October 3 1703, the patriarch had written “I have already finished most of my book, but I still lack the essential part in which I show how the foundations of the art of conjecturing can be applied to civil, moral and political life.”120 He died in August 1705, and his nephew, Nicolas, initially initiated by his uncle, will publish the work in question posthumously in 1713. The title of its fourth part resumes the program announced to Leibniz ten years earlier: “Fourth part of the art of conjecturing. The use and application of the previous doctrine to civil, moral and economic affairs.”121 This part is presented in the manner of a course where the definitions and principles are first set. At the beginning of Chapter II, the founder specifies the meaning of the title he gives to his work: “To conjecture something is to measure its probability: thus the Art of conjecturing or the Stochastic is defined for us as the art of measuring as accurately as possible the probabilities of things. The goal is that in our judgments and in our actions we can always choose or follow the party that we will have discovered as better, preferable, safer or better thought out. This is where all the wisdom of the Philosopher and all the sagacity of the Politician lies”.122

The fourth part, if it does not answer (besides its title) to our question is of great importance for the history of probability calculation: indeed its fifth chapter offers the demonstration of the theorem associated with the name of the author of the Ars, later reformulated by Laplace and by several mathematicians of the 19th and 20th centuries.123 For the Basel native, it was through combinatorial methods that the convergence of the accumulation of observations of a binomial towards the probability that characterizes it was justified. This result falls into the class of “central-limit” theorems of which several variants are known today. The text of this fourth part is anything but concrete and it is deliberately that its author takes the most strictly mathematical point of view in order to obtain a proof of principle designed to serve the program announced in the title of his fourth part. He indeed observes that in usual combinatorial exercises we start from a probability that he considers known a priori (it would be for example ½, set by principle for a coin toss) whereas for the phenomena announced in the title of this fourth part such a priori probabilities are unknown. However, we sometimes have what we call observed frequencies, something that Jacques Bernoulli qualifies as a posteriori probability. He then intends to demonstrate the usefulness of stochastic by establishing how the

119

J. Bernoulli, 1713; N. Bernoulli, 1709. Cited by Meusnier in J. Bernoulli, ed. 1987, p. 3, who cites the Gerhardt edition of Leibniz’s mathematical writings, vol. III, p. 77. 121 In the terms of Meusnier‘s translation ibid. p. 14. 122 Id. ibid. p. 20, “Stochastic – specifies in a note the translator after his master Guilbaud, duly cited, designates the art of aligning aim and goal, as can be practiced by the javelin thrower” (ibid., p. 76). 123 Shafer, 1996. 120

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observed frequencies would come to the rescue of the ignorance of probabilities a priori. To do this, he focuses on the case of a basic binomial law. This fourth part, despite its enthusiastic announcement, thus did not deal with marine insurance. What’s more, they were all the less within the reach of calculations, even exposed according to such principles, as there were no records of frequencies of fortunes at sea. Moreover, the possible urgency there would have been to approach this field of concrete interests was not mentioned either. On the contrary, the thesis of Nicolas, the nephew, explicitly displayed this application in the title of a sixth chapter on insurance and nautical usury: “De Assecurationibus et Fœnore nautico” (p. 44–47).124 It is announced on the thesis title page as “mathematico-juridical” and the double aegis deserves attention. We indeed remember two historiographical elements: in the 17th century, it is on the one hand the legal background of the reflections held on probabilities towards which Ernest Coumet called attention in 1970 and on the other hand, in the less specialized field of accounting writings, this time, the fact that their arithmetic, as we have seen above, was taken under the double legitimacy of the two expertises: legal and mathematical. Nicolas’ thesis was therefore a gesture of promotion of the family scientific heritage.125 He aimed at a universe where he was not yet admitted. “One can estimate [. . .] the danger that the ship that has not yet left the port must run if one observes on many ships that have made the same journey the number of those who arrived unharmed at their destination. For example, if it has often been observed that out of a hundred there are ninety who arrived unharmed, the danger will be estimated at one tenth of the price that the ship is worth with the goods it contains. From there, we also see how the interest rate in nautical usury should be determined. Because, as Grotius very well observes, for what is composed of a loan contract and an adverse danger, one must estimate the danger that the creditor of the sum of money engaged bears and the nautical interest must be higher than the ordinary interest”.126

Nicolas refers precisely to chapter XII, in book II of the De Jure Belli ac Pacis. (1625) by Hugo Grotius (1583–1645) by succinctly adding, paragraph 5. This chapter indeed deals with contracts and its fifth paragraph defines those which, when composed, cover two distinct objects at once. Further on – in paragraphs 22, 23 and 25 which Nicolas does not indicate – the issue of marine insurance was however explicitly addressed.127 To consider a loan contract for an expedition subject to the fortunes of the seas, Grotius observed that the rates of such loans were higher than those practiced in the absence of perils and he distinguished two operations which were composed in the same contract, on one hand a loan of money (or of particular means) granted at a rate comparable to that of a risk-free loan (let’s

124 Bernoulli, 1709, p. 44–47, or the odd pages between 94–101 of Meusnier‘s translation where the even pages carry the French text. 125 So much so that the master mathematician under whose protection the thesis was presented was Jean Bernoulli (1667–1748), undoubtedly a renowned mathematician, younger brother of Jacques and second uncle of the petitioner. 126 Bernoulli 1709, p. 45, here after the translation by Meusnier. 127 Here we follow Grotius in the old French translation by Jean Barbeyrac, 1724.

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say terrestrial) and on the other hand the counterpart of the risks incurred which explained the practice of rates higher than in simple contracts without however entering into the register of usury. The surplus is then justified by a secondary transaction which consisted of assuming the risk for the partner. It was not therefore a question of usury (the real concern of the jurist) but of a combination in the same act of two transactions which covered two distinct objects: on one side the loan of money and on the other the counterpart of the assumed risk. In paragraph 22, Grotius specified that in Holland mixed contracts of this kind covered perilous loans at the level of eight to twelve percent per year and he added “if this interest does not exceed the compensation for what one loses or one can lose by lending, there is nothing contrary to either Natural Law, nor Divine Law”. He then concluded paragraph 23 in these terms: “The estimation of this danger must be regulated by the common estimation”. Thus Grotius finally mobilized the Aristotelian conception of the probable. In 1650, a work by Johannes Loccenius (1598–1677) was in all respects like an intermediary between Grotius’ collection (1625) and Nicolas Bernoulli’s thesis (1709). He explicitly dealt with marine insurance “with nautical usury, it is a mixed contract in the manner indicated by Grotius, book II, chapter XII, paragraph 5”.128 The mathematician has without doubt found in this compilation something to extend that of their illustrious predecessor. The argument of the jurists served him to decompose the problem of marine insurance into two transactions: on one hand a usual loan contract, and on the other the sale of the potential peril according to the estimation process of which he intended to indicate the principle. Therefore, contrary to what they may have suggested and what has been repeated after them, neither Jacques, nor Nicolas Bernoulli contributed to the calculation of probability applied to marine insurance except at the stage of principle considerations for this simple reason, always the same, that the records of successful or unsuccessful voyages were not then available, a deficiency that Condorcet could deplore almost a century after the initial announcements and that Frederick Martin will deplore even later. The historians of the probabilistic conceptualization and its culture in the Enlightenment century, respectively, Ian Hacking and Lorraine Daston, have been more cautious than Leonid Maistrov and Anders Hald. Questioning the beginnings of probabilistic reasoning, Hacking mentioned “the economic theory of the Russian”,129 then he called for an “undogmatic version of this doctrine” to quickly come to his analysis whose object was the conceptual devices, and not the rise of capitalism. In her famous book, Lorraine Daston mentions marine insurance about ten times but she was both victim of the announcements of Nicolas Bernoulli and the method she has chosen: that of a so-called cultural history that constructs large lists of similar things, seen from afar but never analyzed in their diversity such as, in this case, commercial practices which she obviously cannot say what their mathematical

128 129

Loccenius, 1650, Book II, Chapter V, Paragraph 12, p. 185 of vol. 3. Our translation. Hacking, 1975, p. 4.

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elaboration would have been.130 In an article published a year earlier, she reported on a more in-depth investigation into the case of marine insurance, she specifies that the great French jurist of the 18th century Robert-Joseph Pothier (1699–1772) “characteristically declined to go into details about fixing premiums”.131 The historian further published her conclusions on this issue in 1989 in Les Annales, she was unequivocal: “Marine insurance spread rapidly in Italy and the Netherlands in the 15th and 16th centuries but insurers did not gather any statistics on shipwrecks and they certainly did not create a mathematical method for estimating premiums, it was rather the mathematicians who later, much later, influenced the insurers”.132

The cause, however one takes it, is thus understood. To definitively extinguish the fires of the historiographic legend that attributes to the needs of marine insurance the beginnings of probability calculation at the beginning of the modern era, we must return to the formula of Grotius, a synthesis of centuries of experience: “the estimation of this danger must be regulated by the common estimation”.133: the common estimation was regulated by customs and by the jurisprudence that effectively responded to them. The geographical position of the Eastern Roman Empire – in other words of the Byzantine Empire – made it a crossroads of land and sea routes that saw the bulk of maritime trade before the opening of ocean routes to the East and West Indies during the Renaissance. Commercial maritime law was therefore first shaped by Mediterranean traffic to be then extended to oceanic traffic. As we know, the Eastern Roman Emperor Justinian I (482–565) had compilations of Roman jurisprudence established. They form The Digest.134 Title II of his book XXII is “De nautico fenore” (on nautical usury): “One cannot [in such a case considered] demand stronger interests than ordinary interests [legitima usura]. [The lender] cannot retain neither the pledges nor the mortgages to get paid an interest stronger than the ordinary interests [id.]. “[He will stay] within the bounds of legitimate interests [id.] which are twelve percent, and without this sum being able to surpass the capital, that is to say without being able to demand, including interest, more than double the capital “non ultra duplum debetur”.135

Pothier, the 18th-century jurist, could not have been clearer about the principles: “The insurance contract, which is the most commonly used, is that of maritime insurance; a contract by which one of the contracting parties takes on the risks and fortunes of the sea that a ship or the goods that are, or are to be, loaded on it must run; and promises to indemnify the

130

Daston, 1988, p. 50, 55, 116–120, 136–137, 165, 168 and 174. Daston, 1987, on Pothier, 1775 (posthumous edition). 132 Daston, 1989, p. 715. 133 See above. 134 Justinian I, 533 (Fr. ed. 2017). 135 The Digest here refers to the Responses of Papinian, which take us back nearly three centuries further. 131

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other contracting party, for a certain sum that this one gives him, or undertakes to give him for the price of the risk with which he charges him”.136

The terminology of the time was “contrat à la grosse aventure”. The epic of the image still has its effect today, alas, around a table of historians unfamiliar with this subject (who hasten to add: you see, it is at the origin of the calculation of probabilities!). These contracts are analyzed by Pothier over about sixty pages.137 “These contracts of loan à la grosse aventure, are also called [. . .] contracts à la grosse [or] contracts on return of voyage; because usually the lender runs the risks until the return of the ship and only has the [restitution] of the loaned sum, in the case and at the time of the successful return of the ship, although sometimes the contracts are made for the outward journey only, and not for the return. This contract was in use among the Romans; it is the one known under the names of nauticum foenus, or contractus trajectitiae pecuniae. This contract is allowed [. . .] and it is not usurious; for the usury which is forbidden by civil and ecclesiastical laws, consists in demanding something beyond the loaned sum, for the reward of the loan. The loaned sum, is not the reward of the loan, but the price of the risks which the lender has taken on to the discharge of the borrower”.138

He had previously set out a definition and said what its use was: “[An element] of the essence of the insurance contract [is that] the insured gives or undertakes to give to the insurer for the price of the risks he takes on. This is what is called in the maritime insurance contract, the insurance premium . It is called premium because it was paid primo and before everything: even before the departure of the ship had started the risks. [. . .] The practice has prevailed of no longer paying it in cash; it is usually passed in a premium note, payable at a certain maturity. It is customary for this premium to consist of a sum of money, which the parties agree between them at a rate of so much per cent of the insured sum. [. . .] The premium to be fair, must be the just price of the risks which the insurer takes on by the contract: but as it is not easy to determine what this just price is, one must give this just price a very wide range, and consider as just price that which the parties have agreed between them, without one of the parties being able to be heard by magistrates alleging in this respect [having been wronged]. “Insurers run much more risks in times of war than in times of peace, the premium agreed upon in times of war, is much more considerable than that agreed upon in times of peace”.139

As observed earlier in this chapter regarding the concomitance between arithmetic and legal logics in trade that was not subject to the fortunes of the sea, those who were dependent on it did not escape a similar dual grip and probabilistic calculation could not be deployed due to a lack of centralization (even local in commercial metropolises) of observations of experience, custom and law alone came to regulate these contracts. Pothier clearly indicated the logic of such contracts: “There can be no loan contract at gross adventure, if there is no [226] maritime profit stipulated by the contract, that is to say, a certain sum of money, or some other thing that the borrower undertakes to pay to the lender, in addition to the sum loaned, for the price of the risks which [the lender] has taken on. The maritime profit of this loan contract at gross,

136

Pothier, 1775, p 6. Id., ibid., p 205–264. 138 Id., ibid., p 206. 139 Id., ibid., p 86–88. 137

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consisted among the Romans in a certain interest of the sum loaned, which was called nauticum foenus or usura nautica, [it] ran throughout the time that the risks lasted. Before the constitution of Justinian, the rate was not regulated, and was left to the free disposition of the contracting parties”.140

Ultimately, Pothier in 1775 shares the principles retained by Grotius (1625) and Nicolas Bernoulli (1709): a contract at gross adventure is broken down into two transactions: “This contract being composed of the loan [with mutation of the property that is loaned] and a convention that is added to it, by which the lender takes on maritime risks for a certain price”.141

The Method for properly setting up all kinds of accounts by Claude Irson (1678) which has already been discussed above recalls the same principles. He also provides printed (indexed by means of a very explicit table of contents) that trace some examples of account journals. The premiums indicated on these contract models range from 10 to 40% depending on the voyages undertaken. The most expensive is a departure from Bordeaux for a return to the same port after a stopover in North America. Then come other expeditions in the North Atlantic for whale fishing or cod fishing on the great bank of Newfoundland., the premiums are these times at the level of 25 to 30%. The races on the North Sea, round trips to Amsterdam or to London are then covered by premiums of 10 or 13%. Although the Mediterranean is a perilous sea, everything leads us to believe that the actual experience of oceanic races could not be contained by the Justinian scale. For the rise of oceanic commerce since the Renaissance to have been the engine of the formation of the calculation of probabilities, it would have been necessary for the lenders to have a culture of data or at least registers on a scale other than that of isolated lenders or even insurance offices whose activity was related to the registration of contracts rather than their compilation. The economic interest of the class of investors played no historical role here. Thus, ancient probabilistic reasoning is attested here, well before the 17th century and nautical usury was practiced from Antiquity to the 18th century without these two areas having effectively crossed due to a lack of data on the subject, although this deficiency did not prevent scholars such as Nicolas Bernoulli or Condorcet from announcing (the first as early as 1709 and the second in 1784) such a possibility which they wanted to believe in, driven by the enthusiasm of their science and the dreams of maritime commerce of their times. But, a few centuries later this view of the mind has nourished a simplistic historiography to which it would be high time to renounce in order to better understand the actual conditions in which these calculations and these reasonings were formed, a set of questions for the treatment of which we already have useful explorations. This new historiographical program to which the present book could contribute would reduce the simplistic illusions carried by numerical abstractions and would help to destroy superstitions on the subject.

140 141

Id., ibid., p 225–226. Id., ibid., p 232.

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The Legendary Meeting of Political Arithmetic and Probabilities

Another encounter between numerical compilations and the calculation of probabilities has given rise to a second historiographical legend – numbers make you dream! Admittedly, it is more punctual: it is the argument that it would be to calculate the population, in France at the end of the 18th century, that Laplace would have for the first time considered a survey technique.142 It is remembered that the merchant from Reims, Jean Maillefer, had pointed out to his son, François, that a good bookkeeper could find a decently paid job in the financial administration of the kingdom and that cases are known to historians. Arithmetic competence, in the 17th and 18th centuries, was indeed deployed both in the field of fiscal accounting and in what Laplace still called in 1778 “the natural history of man” and what was called Political Arithmetic after William Petty (1623–1687). The extent of this field is known: it is for the French language alone around four thousand and height hundred identified works143 of which unfortunately the routine tradition of the history of economic ideas stubbornly refuses to consider only around fifteen of them.144 This corpus has fallen into a triple historiographical wasteland. It is firstly the little case that the history of economic thought gives it, denounced by Jean-Claude Perrot: a pride in wanting to dialogue only with a handful of chosen authors. It was secondly, the contempt of historical demographers who did not find before 1800 the basis of the demography of the following century: the institutions of civil and statistical status. No doubt disappointed to have to face sources, uses and methods of other times, they have placed the works of scholars prior to this turning point in the rank of stammerings of later competences.145 During the years 1970–1980 this anachronistic conception could be qualified as positivist but the philosophy of science was not the strength of historical demographers of those years. Fabrice Cahen was right to recently emphasize that it was a naturalistic approach to population phenomena among these demographers in the sense that they would consider them in the manner of a natural science.146 A third bias in the historical study of political arithmetic came from the success of Michel Foucault’s Archaeology of Human Sciences: of Kantian inspiration, this approach can be qualified as critical.147 In this analytical framework, the human sciences of the 19th century were struck with the anathema of “positivism” which characterized the system of relations then maintained between these sciences among

142

Bru, 1988. Hecht & Lévy, 1956. 144 Perrot has lamented it several times, 1992; 2021. 145 Dupâquier & Dupâquier, 1985. 146 Cahen, 2022, p. 15–25. 147 Foucault, 1966. 143

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themselves and with a particular conception of language: their characteristic episteme. From then on, the system of knowledge of the 17th and 18th centuries was considered radically foreign to those of the following centuries as the structure of homologous relations was of a different order and appeared in the eyes of Foucault from another episteme. If Foucault’s propositions have opened the eyes of historians to new questions, they have made political arithmetic incomprehensible to them as if it came from another world. Thus, neither the history of ideas, nor naturalist historiography, nor critical historiography have been able to provide the means to study in all rigor and attention the vast corpus of political arithmetic. Yet, as mentioned above, this corpus is on the order of several thousand titles in French. Some of them highlight particularly elaborate scholarly skills as long as the knowhow involved in these works are taken seriously.148 The article “Political Arithmetic “from the Encyclopédie Méthodique. Mathématique.149 is precious for the historian of sciences. The object is specified as in the initial edition of the Reasoned Dictionary. But a supplement by Condorcet deserves attention for its methodological indications: they tell to what extent the field could actually interest mathematicians who were then developing Analysis and its applications (not bookkeepers nor political economists): “Political Arithmetic: it is that whose operations aim at research useful to the art of governing peoples, such as those of the number of men who inhabit a country; the quantity of food they must consume; the work they can do; the time they have to live; the fertility of the lands; the frequency of shipwrecks, etc. It is easy to see that these discoveries and many others of the same nature, being acquired by calculations based on some well-established experiences, a skilled minister would draw a multitude of consequences for the perfection of agriculture, for both internal and external trade, for the colonies, for the course and use of money”.150

Follows a discussion of authors known since the 17th century, then this methodological addition: “Political Arithmetic in a broader sense, is the application of calculation to political sciences. This branch of Mathematics has three main objects, like all those whose aim is the application of calculation to the knowledge of nature: thus, it can be divided into three parts; the first is the art of obtaining precise facts to which calculation can be applied, and of reducing the particular facts that have been observed to more or less general results; the second aims to draw from these facts the consequences to which they lead; the third finally should teach to determine the probability of these facts and these consequences [. . .]. These researches can only be regarded as a very small part of one of the most extensive sciences and the more useful. In general, geometers have been more concerned with calculation methods than with the examination of the principles according to which each question

148

Some of these works have been published in the collection of Classiques de l’Économie et de la Population, by INED since the mid-1990s, initially at the initiative of Jean-Claude Perrot. 149 The Encyclopedia of Diderot and D’Alembert, or Reasoned Dictionary of Sciences, Arts and Trades, (1751–1772) knew volumes of Supplements. The whole was updated and published until the beginning of the 19th century in a new formula segmented by specialties. There was therefore an Methodical Encyclopedia (Mathematics) to which Condorcet contributed. 150 Condorcet, 1784a, p. 132.

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should be resolved; they have almost only dealt with those for which the necessity and the possibility of applying calculations were felt at first glance, and they have rarely sought to submit to it objects that seemed to refuse it; finally, they did not extend the principles and calculation methods they used to the different questions to which these principles and calculations can be applied: their main goal was the progress of mathematical analysis rather than that of political sciences”.151

How was the question of population calculation posed during the second half of the 18th century? There was no census nor register so the number of inhabitants for a given place or for the whole kingdom – let’s say “P” for “Population” – was unknown. The specialists in political arithmetic resorted to the registers of religious communities (generally, in France, to the registers of Catholicity) to establish numbers of baptisms (B), of marriages (M) and of burials (“S” for “Sépultures”) at various scales: parishes, villages, cities, provinces, entire kingdom. In 1772, the Ministry of Finances (Le Contrôle général des Finances) ordered the intendants, the provincial representatives of royal power in each province, to establish from the copies of the parish registers the numbers of these three acts for each city, town, village an parish of the kingdom. Each year, these compiled copies in a single volume for each province were to be presented to the king in mid-June. The survey, despite ups and downs, was held continuously from 1772 to 1792 and was quite costly in time for the local clerks of the monarchic administration.152 About ten years after the launch, the former intendant Jean-Baptiste François de La Michodière (1720–1797) was in charge of collecting the figures that appeared in the volumes sent from the provincial capitals. Meanwhile, the astronomers of the Academy of Sciences in Paris, notably the dynasty of Cassini, had established throughout the 18th century a map of the kingdom based on geodetic surveys taken on the positions of the high points and most often the bell towers of the churches. It was almost complete at the beginning of the 1780s. La Michodière therefore had on one hand a list of cities, towns, villages and parishes from the administrative origin compilations in the intendances, and on the other hand the names of the bell towers marked on the sheets of the Cassini map (their principle of assembly is the same as that of later topographic maps). La Michodière undertook with a team of assistants to cross these two compilations. He thus obtained, for the first time in France, a relevant and realistic list of local communities, conforming both to the legal states of places and to the physical observations of geodetic measurers in the service of Cassini. The notebooks resulting from this crossing show that sometimes parishes identified by the administrative route had to be distinguished on the territory, or that bell towers sometimes spotted by the geodesists were not the place of an active parish. The innovation fit perfectly into the tax reform policy wanted by Minister Turgot and his successors (this reform aimed at establishing a hierarchy of administrative competences for a stratified collection of taxes and their management at the same levels of scale). The work of La Michodière’s office and the Cassini maps were so

151 152

Condorcet, 1784a, p. 135–136; reprised in id., 1994, p. 483–484. On this survey see Brian, 1994, 2001; Bru, 1988.

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fundamental that they were used in 1789, at the beginning of the French Revolution, to establish the new communes and the new departments. Their framework still exists today as it exists in the hierarchical system of elections to the Senate of the Republic and in the definition of local taxes. The list covered nearly thirty-three thousand “cities, towns and villages”. For each of the nearly one hundred and eighty sheets of the Cassini map, La Michodière had prepared a cartouche where the number of births (more specifically baptisms) in the cities on the one hand and in the countryside on the other hand appeared. At the cost of a transgression to the rules of the Academy of Sciences, a memoir in several parts appeared in the columns of its annual volumes which delivered these cartouches in clusters during the 1780s. The work in question had been the one of the former intendant and his assistants. The Academy’s rules stipulated that only its members were allowed to see their work appear under their name in the volumes Histoire et Mémoires de l’Académie royale des sciences. The Perpetual Secretary, Condorcet, responsible for publications, published the pieces given by La Michodière under his own name and that of two other academicians, notably that of Laplace. A series of his mathematical memoirs and their clever presentation in the History section of the same volumes made the numerical cartouches of the former intendant appear as extensions of the mathematicians’ works.153 The commentators on the writings of Condorcet and Laplace in those years were victims of this sleight of hand.154 Yet the methodological paragraph in Condorcet’s “political arithmetic“could have warned them as it emphasizes the difference in priorities between analytical research and political arithmetic research. The cartouches (one per sheet of the Cassini map) operate in this series of memoirs drawn from the compilations organized by La Michodière as the elements of a table of astronomical observations. They respond to the “first part” of the agenda set by Condorcet in the Méthodique: “to obtain precise facts such that calculation can be applied to them”. The foundation thus laid could lend itself to all sorts of later calculations: the whole therefore offers a kind of printed calculating machine. To understand the mechanism, it is appropriate to consider how at that time one hoped to extrapolate the unknown population (P) starting from indicators of what we now call in nowadays words the movement of the population: the numbers of baptisms (B), marriages (M) and burials (S). In the absence of satisfactory population counting, one sought to estimate the stock (P) by the flows (B, M, S). In this matter as in any other similar one, deducing stocks from flows is no small matter. The authors of the 17th and 18th centuries, driven by the arithmetic know-how that we have analyzed in this chapter, tried to establish all possible relationships between these different quantities while waiting for the constancy of these relationships for a given place to help them in their extrapolations and often hoping that the

On this editorial operation see Brian, 1994, Part III; it is also analyzed in the first part of the same author’s thesis, which bears the same title, defended at EHESS in 1990. 154 See Condorcet, 1994, ed. by Crépel & Bru, and Bru, 1988. 153

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variability of these relationships would be negligible for example at the scale of the kingdom. The most systematic work published in France on this subject is due to JeanBaptiste Moheau: in his book, Recherches et considérations sur la population de la France (1778)155 like contemporary authors, he focused his attention on the population‘s relationship to births (to baptisms, therefore). This ratio was then called “the birth multiplier”, let’s call it “m”, and it was hoped to calculate P by means of the product m.B. One does not understand anything about this methodological choice if one ignores that a ratio was then held to be constant. Hence a new question becomes: to what extent, or more precisely over what range could this multiplier be considered sufficiently constant and suitable for local living conditions. Moheau, like others, examined the question and like some of them, he observed that the proportion of singles in the population concerned had an impact on the multiplier: there are then fewer births. Now, where were there a large number of singles at this time: in cities where religious communities resided and where there was significant domesticity. Among the calculators, it is thus considered necessary to adopt larger multipliers for cities than for the countryside. In particular, in the introductory fragment to the tables of La Michodière, the multiplier 33 is retained for Paris and Versailles and 26, for the rest of the kingdom. The cartouches published in the successive deliveries of this memoir offer the possibility of distinguishing the numbers of baptisms in cities and those of the countryside. The calculating machine begins to operate. Before reading some passages from Condorcet‘s commentary on Moheau’s Researches. It is important from the point of view of historical epistemology to specify that the expectations of political arithmeticians were incomprehensible in the 19th century and subsequently, for two reasons: on the one hand arithmetic calculation had freed itself from the straitjacket posed by the rudimentary diffusion of the arithmetic of Elements of Euclid, and on the other hand, the use of civil status and new demographic calculations led to considering as known by modern censuses the population numbers P and to scrutinize for example the crude rate of natality that is B/P (expressed in 0/00), that is the inverse of the multiplier: P/B. Thus, since the 19th century an object of predilection of demography, the variability of the crude birth rate is the exact inverse (to the factor thousand close) of the birth multiplier that previous calculators intended to consider as much as possible as constant. This is, in the case of social sciences, a perfect example of what the epistemologist Gaston Bachelard (1884–1962) called an epistemological obstacle156: the conception of the ratio, forged on the third definition of the fifth book of Elements of Euclid was an obstacle to the understanding of the variability of birth rates among political arithmeticians. Condorcet reviewed Moheau’s book in the Parisian press. The divergence of their approaches – that of a geometer for the first, that of the foreman of laborious 155 156

Moheau, 1994. Bachelard, 1938.

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calculations for the second – was expressed in these columns.157 The first task that the encyclopedist designated in his article “political arithmetic” was not fulfilled (although it will be five years later at the end of the work directed by La Michodière). Nevertheless, Moheau’s work drew conclusions from the figures he put before his readers. This could, at a stretch, serve as the second task. As for the third “the probability of these facts and consequences”, the intendant had not been sensitive to it. In his review, the mathematician proposed a method that is similar to what today would be called stratification by types of agglomeration: “So if I want, based on observations made on a certain number of men, to determine precisely what should take place in a large country, I will have to choose for the object of my experiment men taken from the different climates of this country, in the different types of air, in the different states, in the different ways of life; and it will also be necessary that all classes of men have among them in my observations approximately the same relationship as they have in the great state to which I propose to apply the rules deduced from these observations. Unfortunately, Mr. Moheau was not able to choose his observations”.158

In the memoir published at the Royal Academy of Sciences in 1783, as a preliminary to the first delivery of La Michodière’s compilations, Laplace wanted to apply the processes of the analytical calculation of probabilities that he had already published in 1781 about the proportions of newborn boys and girls in London and Paris, an object for which, let us here use the terms of Condorcet in his article “Political Arithmetic” of the Méthodique: “the necessity and possibility of applying the calculations was [no doubt] felt at first glance”. In the case of the proportions of the two sexes at birth, the reworking of Jacques Bernoulli‘s theorem by Laplace himself opened the way for this application. And in the second memoir, that of 1783 on the multiplier of births, he wanted to transpose his new calculations at the price of a thought experiment that radically departs from the path designated by Condorcet in his critique of Moheau: “It is easy to apply these results [from the 1781 memoir] to the theory of population deduced from births, for we can consider each annual birth as being represented by a black ball [drawn from an urn] and each existing individual as being represented by a white ball; the first draw will be the count in which it was observed that out of q births the number of inhabitants is p, the second draw will be the population of the whole of France [. . .]”.159

Far from the reality of population calculation operations, Laplace here assumes that all the individuals who would compose it would be like balls to draw from an urn and that two random draws would then be made, one covering all existing individuals, the other births alone. No one will dispute the fact that Laplace had a particularly abstract mind, but in this case to presuppose that a limited number of existing individuals to be considered as been drawn randomly from the total population was to assume that this total population was like a vast set of balls in the urn where the draw would be made, and that obtaining a limited number among them would

157

Condorcet, 1994, p. 130–141. Id., Mercure de France, July 5 1778, id., 1994, p. 132. 159 Laplace, 1783, p. 700, i.e. Complete Works, vol. XI, p. 38. 158

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have resulted from a random draw. However, no homogeneous chance delimited the districts on which calculations were then made. The same is true for what Laplace referred to as “the second draw”. This memoir by Laplace is therefore a pure flight of the mind – certainly, issued by a powerful mind. The author wanted to derive from it a method for calculating the number p in such a way that the estimation of the multiplier was known with an acceptable probability. It is therefore at a particularly abstract level that Laplace would have tried to fulfill the third task indicated by Condorcet in the Méthodique., if such had been his intention. And this memoir was perfectly inadequate for the calculations in political arithmetic of the time. Finally, if one considers the data set that the mathematician had at his disposal, even with the work conducted by La Michodière, it is necessary we can note, to paraphrase Condorcet again, that “Unfortunately, Mr. Laplace was not in a position to choose his observations. [even at random]” Thus, it is only by anachronism that one has believed to see in this of 1783 a prototype of a random survey. A definitive proof of this anachronism would be given if one were to highlight the work of a scholar who would have followed the program of stratification by type of agglomeration outlined by Condorcet in his critique of Moheau’s work. In their bibliography of works on population and political economy published in French from the beginnings of printing press until the Revolution, this directory of quite five thousand titles published in 1956, Jacqueline Hecht and Claude Lévy mentioned (under the item number 202, according to a collection of rarities for the use of bibliophiles) a Political Arithmetic by a certain Baras, published in Paris at the very beginning of the Revolution. This reference remained a mystery for half a century. Various catalogs have identified a Marie Marc-Antoine Baras, born in Toulouse in 1763, in the South of France, and guillotined in the same city on April 13, 1794 during the Terror. This author left, in 1793 (when the Convention was examining proposals on the organization of public education while the country had rid itself of the religious congregations that had hitherto monopolized teaching) a double volume entitled « On Public Education in Free France160 ». In note 1, page 97 of the first volume of the book printed in Toulouse and sold in Paris according to the title page, the author feels the need to specify: “The assessment I have made of the population of cities is not arbitrary at all, and the data I used are certain: one can consult the Theory of Political Arithmetic that I published in Paris, in 1789 which was printed by Didot, the younger. We are getting closer to the goal. Why was the calculation of the population of cities important in this study on public education? Baras argued that educational establishments should be located and sized in the country according to the volume of the population of cities. This principle is still in force today, in anachronistic terms: local demographics govern the supply of education. The 1793 work therefore proposed at the national level a stratification of agglomerations (or to use the terms of the time of “cities, towns and villages” or new communes). Certainly, we must find this theory of political arithmetic. Another reason pushes us there. Baras left a brief autobiographical

160

Baras, 1793.

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work in the manner of justification while he was implicated in the political turmoil of the Terror.161 Born and baptized in Toulouse, he spent his youth in the Dalbade district near the protagonists of the tragic Calas affair in which the father of this family, a Protestant, was condemned and executed for the murder of his son suspected of wanting to convert to Catholicism. The trial of Calas senior was marked by religious intolerance. Voltaire denounced it vehemently in his Treatise on Tolerance (1763). As for Baras senior, he was a well-off lawyer without being rich and the Marie Marc Antoine’s maternal grandfather, one of the magistrates elected in turn to administer the city: a Capitoul. In 1785, the young man went to Paris to study law and jurisprudence. Was he attracted by the young generation of the Enlightenment or by the author of the works that ruined the principle of arguments based on quarters or eighths of evidence summoned at the time of the Calas trial? He writes in his fragment of justification that he was close to the academicians Condorcet, Lalande, Bailly and Lavoisier and shared their political ideas. Back in Toulouse, he was administrator of the District of Toulouse, which means that he held a position comparable to the one of his maternal grandfather, but in the new political era. Accused of federalism, he was caught in the storm of the Terror and executed in the spring of 1794. The title page of his work on the organization of education specifies that he was a member of the Academy of Sciences of Orleans where Lavoisier officiated and of the Free Society of Emulation of Bourg[-en-Bresse] where Lalande probably introduced him. With Lalande, Lavoisier and Condorcet, the young Baras had, at the end of the 1780s, good masters in political arithmetic and his network of scholarly affinities also suggests that he would have been familiar with those among them who frequented Masonic lodges. The final word of the investigation was provided by Giorgio Israel during the publication of his recent edition of the Principles and formulas of probability calculation by Duvillard de Durand (1813).162 It turned out that an Italian library had taken care to catalog not only the works it held but also the different memoirs they could be composed of. By the virtue of the centralized catalog of Italian libraries (ICCU) it was then possible to reach this notice cited by Giorgio Israel: Baras (Marie-Marc-Antoine), 1790, “Theory of political arithmetic, or essay on the means of evaluating the population”, Tribute of the National Society of the Nine Sisters, or Collection of Memoirs on the Sciences, Fine Letters and Arts, and other pieces read in the sessions of this society, vol. 1, p. 17. The Masonic lodge of the Nine Sisters was, during the second half of the 18th century, the one where many scientists and philosophers met. Everything fits together: Baras, intellectually and politically close to academicians about twenty years older than him, was admitted by them to the lodges they frequented and the memoir in question was presented at a meeting of the Nine Sisters perhaps to

161 162

Baras, 1794. Duvillard de Durand, 2010.

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63

foreshadow some beginning of a journey to the Academy. At this point, it is high time to read the memoir in question.163 Baras shows an excellent knowledge of the history of political arithmetic which he traces back in an academic gesture to Antiquity and scrutinizes up to the most recent publications. He finely criticizes the methods to conclude in favor of the validity of the method of the birth multiplier. He acknowledges the variability of the multiplier. He explicitly mentions the work carried out by La Michodière and specifies that it is then necessary to determine several multipliers. He says he was able to have the results produced around the Intendant, and just as the latter distinguished Paris and Versailles on the one hand by assigning them 33 as a multiplier and the rest of the kingdom. Baras proposes to classify the cities according to their size and various characteristic criteria. The memoir ends with a double table where the cities are sorted in different orders, once according to their size, another according to the homogeneity that the author believes to find in the birth multiplier to be applied, these are in total forty-two cities that are divided into six classes understanding that for each class it would be necessary to apply a particular multiplier that goes from 33 to 27, while 26 is reserved for the rest of the country, the country side. Thus, a young scholar, who is also close to the academicians themselves authors of contributions to political arithmetic at the end of the 18th century, had at his disposal the compilations established by the office of La Michodière and proposed a classification of cities that responded to the method outlined in the article of the Mercure de France where Condorcet criticized Moheau. Baras offered additional calculation keys to the cartridges published in the memoirs of the Royal Academy of Sciences. Once the two components are assembled, the population calculating machine is complete. It is clear that the analytical calculation of probabilities and Laplace‘s 1783 memoir served no technical purpose here other than to reinforce the value of the analysis that Condorcet distilled volume after volume in the publications of the Royal Academy of Sciences of which he was then in charge.164

2.7

Conclusion. The Historical Corroboration of Ratios

Since antiquity, merchants have dealt with numbers, their recording and control have been of primary importance for these transactions. That’s why for eleven centuries, from Justinian I (6th), to Louis XIV (17th), through Frederick II Hohenstaufen (12th×13th) sovereigns keen to encourage them have established reference legal collections or encouraged the dissemination of manuals designed for the training of calculators. From a long-term perspective, as we have seen, these encouragements through the medium of know-how, have interconnected a vast world that has

163

Equipped with the recent Italian reference, it can be easily found at the National Library of France. 164 On the line of conduct of the perpetual Secretary of the Academy, see Brian, 1994, Part III.

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covered the Alexandrian space from India to the Mediterranean, the commercial circuits of this sea to the Baltic and then, in modern times, exchanges between Europe and what was called the East and West Indies. Eleven centuries of know-how and learning and a almost planetary extent have thus forged a calculation tool – the ratio – and its counterpart, the presupposition that this ratio was sealed, both forged in the vicissitudes of the transmission of knowledge on this scale and over this extent. Indeed, such transmission is not transparent and it sometimes has roughness that neither time nor space, nor the intensity of exchanges have managed to smooth. The calculator as the layman, today, faced with numbers, overwhelmed by the weight of previous uses will immediately approach the numbers they seem to carry without concern or even awareness of this long duration and planetary extent. What? So many centuries, almost the whole planet and so many transactions would corroborate the abstraction of numbers and the power of the process! Here is, indeed an idea of an epistemologist historian: to take abstraction at the foot not of the letter but of the figure is still simpler. However, as Lucien Febvre wrote in the preface to the seventh volume of the Encyclopédie française: “the enemy is simplism”. In terms of data, it is the same and superstitious extrapolations like technophile dreams believe they can stick to the numbers as they are sealed in the registers, while in each of them resides a fragment of this long global history and that the apparent realism of such extrapolations depends on it. Among today’s calculators, there are particular ones, those whose specialized expertise would be ruined without the belief in the constancy of ratios. The last chapter will come back to this. This belief is corroborated by the same duration and the same extent of collective experiences. This means that it is robust and that it presents itself in the daily life of these specialists as an evidence which, if it was not shared would only call for contempt, or in the terms of Francis Bacon an idol of this theater of competence165: in this chapter we have tried to indicate its principle.166 In terms of theater idol, the 19th century was not left out. This is the subject of the next chapter.

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165 166

Bacon, 1645, Aphorism XXXIX. Bacon, 1645, Aphorism XL.

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Peletier, J. (1611). Les Six premiers livres des Éléments géométriques d’Euclide etc. Genève, Jean de Tournes. Perrot, J.-C. (1992). Une Histoire intellectuelle de l’économie politique XVIIe-XVIIIe siècles. EHESS. Perrot, J.-C. (2021). Histoire des sciences, histoire concrète de l’abstraction (1998). Revue de Synthèse, 142, n°3–4, p. 492. Person de Teyssèdre, A. (1824). Notions élémentaires d’arithmétique, de géométrie, de mécanique, de physique, de dessin linéaire, perspective et architecture (p. 1824). Tourneux. Peyrard, F. (1814–1818). Les Œuvres d’Euclide en grec, en latin et en français, d’après un manuscrit très ancien qui est resté inconnu jusqu’à nos jours. Patris. Piron, S. (2007). Le traitement de l’incertitude commerciale dans la scolastique médiévale. Journal électronique d’histoire des probabilités et de la statistique, 3, n°1. Portet, P. (2006). Les techniques du calcul élémentaire dans l’Occident médiéval: un choix de lectures. In Coquery et al., p. 51. Pothier, R.-J. (1775). Traités des contrats aléatoires, selon les règles tant du for de la conscience, que du for extérieur. Debure. Potts, R., et al. (1845). Euclid’s Elements of Geometry etc. Cambridge, University Press. J. W. Parker. Pradier, P.-C. (2006). II. Le risque probabilisé, dans id. La notion de risque en économie (p. 16). La Découverte. Rey, A. (1942). La science orientale avant les grecs. Albin Michel (2nd ed.). Rommevaux, S. (2003). La réception des Éléments d’Euclide au Moyen Âge et à la Renaissance. Revue d’histoire des sciences, 56, n°2, p. 267. Roux, S., & Garber, D. (eds.), (2013). The Mechanization of natural philosophy. Springer. Rüdiger, C. (2012). The Game of Probability. Literature and Calculation from Pascal and Kleist. Stanford University Press. Serres, M. (1989). Éléments d’Histoire des Sciences. Bordas. Shafer, G. (1996). The significance of Jacob Bernoulli’s Ars Conjectandi for the philosophy of probability today. In Journal of Econometrics, 75, n°1, p. 15. Stevin, S. (1585). L’Arithmétique [. . .] l’Algèbre. Ensemble les quatre premiers livres d’algèbre de Diophante d’Alexandrie, maintenant premièrement traduits en français. Encore un livre particulier de la pratique arithmétique, contenant entre autres, les tables d’intérêt, la dîme et un traité des incommensurables grandeurs; avec l’explication du dixième livre d’Euclide. Christophe Plantin. Stevin, S. (1625). L’Arithmétique [suivie de La pratique d’Arithmétique] revue et corrigée [etc.] par Albert Girard. Leyde, Elzeviers. Stigler, S. M. (1986). The History of Statistics. The Measurement of Uncertainty before 1900. The Belknap press of Harvard University Press. Turquan, V. (1890). Statistique générale des naufrages. Journal de la Société statistique de Paris, 31, 214. Vitrac, B. (1990). Euclide, Les éléments. Vol. II: Livres V à IX. Presses universitaires de France. Vitrac, B. (1992). Logistique et fraction dans le monde hellénistique. In Benoît et al., p. 149. Vitrac, B. (1993). De quelques questions touchant au traitement de la proportionnalité dans les Éléments d’Euclide. thèse de l’EHESS. Wikipédia. (2023a). History of probability, Wikipedia [EN], on 17th February 2023, 09: 50. Wikipédia. (2023b). Histoire des probabilités, Wikipédia [FR], le 17th February 2023, 9: 5.

Chapter 3

Analytical Probability, Averages and Data Distributions in the 19th Century

“It is [. . .] a very convenient prejudice for ignorance and laziness to regard as demonstrated everything that is written in numbers.” Condorcet, 1778. (CONDORCET, Le Mercure de France, dated November 5, 1778, quoted in id. 1994, p. 139, on the occasion of his critique of Moheau, 1994 [1778])

3.1

Working Out Data in the 19th Century

As early as 1784, as we saw in the previous chapter, the geometer and philosopher Condorcet had assigned to political arithmetic a primary task: “the art of obtaining precise facts such that calculation can be applied to them and of reducing the particular facts that have been observed to more or less general results”.1 Before the French Revolution, calculators could only rely on parish registers kept for religious reasons and not standardized according to administrative or legal criteria, let alone scientific ones of course. Thus, when registering marriages or deaths, ages were often indicated from family memory and approximately, and ministers of religion could only know the regulars of their services. The migrant population, often coming from a few tens of miles around, was poorly known to them, and for the sedentary, they relied on the imperfections of local memory. The facts thus collected were therefore anything but certain or precise, and their compilations reflected this. Sweden was an exception in this regard in modern times, as the ecclesiastical apparatus had been integrated into the state as early as the 16th century. In France, high-ranking officials in favor of its reform and often political arithmeticians themselves have long pondered its administrative refoundation on a civil status, it finally saw the light of day with the Revolution. The administration was then newly organized and the Ministry of the Interior attempted to develop through various means a general perspective that could provide Paris on the one hand with a numerical synthesis of civil status records and on the other hand with a collection of departmental collections that would have reported on 1

Condorcet, 1784, p. 135–136, quoted in the previous chapter; quoted in id., 1994, p. 483–484.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 É. Brian, Are Statistics Only Made of Data?, Methodos Series 20, https://doi.org/10.1007/978-3-031-51254-4_3

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local peculiarities and resources. This is what historian Jean-Claude Perrot described during the Napoleonic era as “the golden age of French regional statistics.2 After the collapse of the Empire, the administrative base resulting from the Revolution endured as a modern achievement (municipalities, departments, civil status records). However, in the prefectures, the departmental capitals, the editors of these departmental statistics collections no longer had an administrative hierarchy with which they could have asserted skills once appreciated by the government and they sometimes found themselves under the influence of other models, such as those of the works of local scholarly societies and historians. A calculating intendant from the end of the Old Regime, Jean-Baptiste Auget de Monthyon (1733–1820), the patron of Moheau’s very complete research discussed by Condorcet in Le Mercure de France in 1778,3 emigrated during the Revolution. At the head of a considerable fortune, once the monarchy was restored, he returned to France and devoted part of his wealth to philanthropic works. With the complicity of the mathematician Laplace (1749–1827), he thus created at the Academy of Sciences a so-called “statistics” prize in order to channel the publications resulting from the skills formed under the Empire and to fix the model by means of a widely disseminated program. The principle of the prize was to encourage the collection of reliable observations that could be subsequently reused by scientists. His capital was placed “on the State”, as in the previous century it would have been placed on “the king’s head”. Historical experience proves that, today, despite ups and downs, the State in France has not been ruined or dissolved (while of course the king was mortal) and therefore the prize has endured until today. During the first decades of the 19th century, the members of the commissions responsible for examining the works and possibly awarding this prize were protagonists of all aspects of the observation and calculation skills that had been implemented and consolidated since the Revolution. As for the mathematicians, two names came back among them year after year: those of Laplace himself and Joseph Fourier. (1768–1830). The analysis of the composition of the committees shows that they were carried by a homogeneous generation of academicians, the one that essentially ensured the transition between the end of the Old Regime and the post-Napoleonic period and which, consequently, began to fade from the 1830s. The inventory of awarded works, the rejected ones, and the compilation of reports published each year on the occasion of the award ceremony show that the effect of the Monthyon foundation was the establishment of a scholarly genre: “statistics”, understood as the collection of reliable observations that could then lend themselves to calculation (in other words, the first task that Condorcet had called for in 1784). A new sense of the word “statistics” was thus forged during the first decades of the 19th century, anchored on the French peculiarity of the history of institutions and numerical sciences of that time.4

2

Perrot, 1977. Moheau, 1778 [1994], see the previous chapter. 4 Brian, 1991b and 1994, part. IV, chap. 2. 3

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A young astronomer from Brussels will learn this model from Joseph Fourier and Pierre-Simon Laplace: Adolphe Quetelet (1796–1874), a prodigy from the flat country sent to Paris in 1823 by his government to train with Parisian scholars while there was talk of establishing an observatory in the city where he was already teaching. It turns out that the philosopher Victor Cousin (1792–1867) succeeded Joseph Fourier on November 18, 1830, in chair number 5 of the Académie française. The philosopher was then covered with duties and honors after the revolution of 1830, which led to the fall of the last king of France and Navarre, Charles X (1757–1836), the last grandson of Louis XV (1710–1774), and to the advent of Louis-Philippe (1773–1850), king of the French, or the “July Monarchy” (1830–1848). The customs of this company required that the successor read a eulogy of his predecessor. We therefore have an academic eulogy of Fourier by Cousin, and it is accompanied by additional notes published separately. However, in the 1820s, the philosopher and the mathematician had maintained a friendly relationship during which the former confided to the latter valuable indications about his opinion regarding his illustrious elder: “[Fourier] did not like Laplace much [. . .]. He told me several times what others have also repeated to me, that Laplace had certainly done a lot, but he wanted to have done everything or inspired everything. [. . .] There are no greater barbarians, he often told me, than certain mathematicians”.5

Despite the diversity of moods among the mathematicians he met in Paris, Quetelet will federate the corpus accumulated by them between 1780 and 1820 to implement it in his country, seizing all the opportunities offered by a country reorganized after the fall of the Napoleonic Empire (1815), and especially the creation of the independent kingdom of Belgium (1831) where the administrations of the brand new country were carried by a specific “Gerschenkron effect”: their establishment in a virgin country could benefit from the achievements of large and old countries where the innovations of the turn of the century had taken so long to take shape.6 Beyond Belgium, Europe itself will offer a favorable climate for the work promoted by the young Brussels astronomer: the second half of the 19th century welcomed frequent universal exhibitions where the economic, political and scholarly elites of the continent crossed paths. Quetelet then had the idea of organizing international congresses on their occasion that would bring together the heads of statistical offices of the main European countries once mandated by their respective governments, so that they can pool their skills and jointly set resolutions regarding the organization of administrative statistics. These bureau chiefs, upon returning to their countries, were thus able to take advantage of the resolutions adopted during the congress sessions held from 1853 to 1878 to reform the organization of their administration, and their initial mandate served as prior approval. Over this quarter-century process, a solid European network of administrative 5

Cousin, 1831 p. 39; I owe to Laurent Mazliak for drawing my attention to this important document. Gerschenkron, 1962 who formed the principle of such analysis regarding the development of the Russian economy in the 19th century.

6

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statistics bureau chiefs and specialized academics consolidated, which even the extinction of the congress did not erase: it is indeed found in 1886 gathered on the occasion of the foundation of the International Statistical Institute still established in The Hague today.7 This foundation is the best proof that this network and the expertise it initially carried devoted to administrative statistics have acquired in about thirty years a great level of autonomy relative to the requirements of the various governments for which, strictly speaking, these bureau chiefs worked.8 The nine congress sessions gathered nearly four thousand five hundred congressmen. Three hundred and ninety of them took part in two of its sessions; and about thirty did it five times. They are found in the local organizing committees and during the sessions at the presidency of the meetings. This means that a core of specialists has thus consolidated. They have seen nearly three thousand nine hundred works presented from session to session and listed as such. Nearly a thousand of these titles had been written in the language of the host country of the session for which they were cataloged, nearly three thousand had been in another language. These regular meetings were therefore the occasion of a vast international mixing of people and works: about a quarter of these titles had been published in German, a fifth in French and an eighth in English. This vast international movement was accompanied by the creation, in imitation of what had been established in Belgium, of central (or national) statistical commissions or scholarly associations and specialized scientific journals that closely followed the implementation of the congress resolutions and the circulation of publications from the offices. Never in history have a few dozen specialists produced or mixed so many figures and the collections that carried them.9 Thus, it is a homogeneous expertise that the congress promoted throughout the European continent, in the colonies of its countries and during the second half of the century in other more distant countries such as the United States or Japan. The initial statistical internationalism lasted nearly twenty-five years: from 1853 to 1878.10 In the end, the model of the statistician was no longer the “good observer” promoted by the Monthyon prize, but the official of the “statistics bureau” – clerk, calculator or chief. The highest officials regularly participated in the Congress sessions and collected its voluminous minutes, treasures of expertise, experience reports, recipes and recommendations.11 This is easily seen in specialized journals and in the general press, as also attested by the imposing lists of works signaled to the congressmen on the occasion of their meetings and in their reports.12

7

Heuschling, 1882a, b; Neumann-Spallart, 1886; Nixon, 1960; Horvath, 1972. In other words, administrative dependence did not mean the absence of scientific autonomy. 9 Brian, 2002. 10 Brian, 1989a, b, 1991a, 1992. 11 Brian, 1999, 2002. 12 Brian, 2002. 8

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A Legacy of the 18th Century

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The historian of scientific concepts Ian Hacking (1936–2023) has referred to as an “avalanche of printed numbers,” the unprecedented scale of number production in the 19th century. He himself questioned the hyperbolic nature of this phrase.13 In any case, it has been taken up by many authors after him who have been content with it without delving into the dense conceptual analyses of its author or into the formerly productive competencies of these numbers, or even into the conditions under which they had been consolidated. For our part, in reading it, we navigate between authors and competencies among which it is not possible for us to glean so easily: the concrete history of abstraction requires us to reinsert each statement into the intellectual, technical and conceptual climate in which it was formulated. We concede without difficulty that our method would not have allowed us to address the question addressed by Hacking in his 1990 book in 250 pages. Proceeding in a less airy manner, we give ourselves the means to account for the consistency acquired by the skills that we have chosen to study. Nothing prevents, in a second step, from placing elements highlighted by Hacking in the landscapes of which we will have traced the mountains, the cliffs and the plains, those that have governed the historical conditions of the circulation of scientific concepts. At the beginning of the previous chapter, the question was: who were the data brewers in the 17th century and the answer: the account keepers. Here, it is analogous: where were the data brewers two centuries later? Its answer is provided by the analysis of the activities of the statistics congress and specialized offices: they were the administrative office statisticians. Thus, the scientific, political and administrative changes at the turn of the 18th to the 19th centuries appear as the historical break between two eras during which specialized skills had acquired an autonomy vis-à-vis the objects of the enumerations and the conditions of exercise of these competencies, that of data brewing in the manner of the 17th century merchants and that of analogous operations in the administrative statistics offices of the late 19th century – analogous but quite different in their organization, in their gestures and in their skills, and especially as to the supports almost exclusive of these old data in each case: manuscripts in the 17th century and printed materials two centuries later. We now need to grasp the key to these 19th century data and uncover the presuppositions it has carried.

3.2

A Legacy of the 18th Century

Adolphe Quetelet was its designer and promoter. Following him in his scholarly journey and in his international dissemination effort will allow us to identify the characteristic skill of his time. Analyzing the comments will also reveal the presuppositions that remain to be highlighted. Throughout his scientific career and his international activism, Adolphe Quetelet presented himself as the promoter of the

13

Hacking, 1990, p. 33.

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average, understood as the method specific to statistics which he wanted to promote the development. In political arithmetic, in the previous century, this calculation method had been used based on birth and death registers to establish the average lifespan of cohorts born the same year. This retrospective calculation result (indeed, the cohort must be extinct to undertake it) and the prospective notion of “life expectancy” distinguished by demographers from the mid-19th century were and are often confused. Thus the Annuaire du Bureau des longitudes proposed a “table of the average duration of life at each age, or of the expectation of living in average cities14”. The point to consider here is that the notion of average, even before Quetelet, was not foreign to the natural history of man. It is through another path, that of celestial mechanics, that it has imposed on the mind of the young Belgian scholar, founder in his early days of the Brussels observatory, a city whose altitude – thirteen meters above sea level – is not exceptional. It is true that two centuries earlier, an observatory had been founded in Paris – whose altitude is barely triple – less for collecting observations inaccessible elsewhere than to establish the prestige of the Sun King. Thus, it was the brand new kingdom of Belgium that asserted itself in this foundation. In the first paragraphs of a then noticed paper published at the Royal Academy of Sciences in Paris in 1776, Laplace had expressed his admiration for the English designer of the law of gravitation, a sentiment shared by his contemporaries. “There is no truth in Physics more indisputable, and better demonstrated by the agreement of observation and calculation than this one: all celestial bodies gravitate towards each other. Newton, author of this most important discovery ever made in natural philosophy, found that the observed movements of the planets cannot exist without a tendency towards the Sun, proportional to their mass and reciprocal to the square of their distance to this star. The movements of the satellites gave him the same result in relation to their main planet. He did not hesitate from then on to generalize this idea”.15

By an analytical calculation that started from the law of universal gravitation, the Parisian geometer established one of the main results that posterity recognizes him for in celestial mechanics: the average distance between a satellite and its attractor is constant and the observable duration of the movement of this satellite could be related to this average conveniently: the law was universal and it lent itself conveniently to an empirical test which consisted of measuring from a point located on Earth – and why not Brussels? – rotation angles that would present a beautiful regularity. Thus, at the beginning of the 19th century, in Europe, the astronomers’ agenda was set to the time of theoretical and empirical relations revealed by the Laplacian celestial mechanics where the average distance between a planet and the star around which it orbits or even that between a satellite and its planet had become measurable and calculable.16

14

Duvillard de Durand, 1802, a table of the same kind appears again the following year. Laplace, 1776; or Id., 1891, p. 212. 16 Id., ibid. 15

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At the cost of a tight analysis of the entire early work of the young Belgian scholar, the philosopher Joseph Lottin has highlighted that this conception of Laplacian celestial mechanics had been the model of the idea of a general scientific law in Quetelet,17 and just as in the Parisian geometer the properties of the average distance of the satellite to its attractor have been the guarantee of the universality of the law of universal attraction,18 in his Brussels disciple, the average of an identified magnitude and its possible stability will in his eyes be the proofs of the manifestation of a universal law. However, the same Lottin has highlighted a significant difference between the two scholars. For the elder, indeed, the process of the average provided, through an analytical calculation, the morphology of a phenomenon whose cause was the universal law of Newtonian gravitation. But if the younger has relentlessly exhibited comparable morphologies, he has established no formal demonstration that would have based these regularities on causes posited by principle in law: “Quetelet calls law the simple statistical regularity as it conforms to the formula given by the calculation. Quetelet suffered, in [this] terminology, the influence of mathematicians Laplace and Fourier [. . .]. One can call a simple regularity of fact a law, in accordance with the theory of probabilities; this is a question of terminology on which one should not insist; but one cannot claim to have discovered a law, until one has determined the causes and their mode of efficiency”.19

In 1793, Condorcet had argued for the general utility of social mathematics in terms that could only touch young man who had gone through historical trials, for example those, revolutionary and military, of the years 1790–1810: “When a revolution ends, this method [social mathematics] of treating political sciences acquires [. . .] a new degree of utility. Indeed, to quickly repair the disorders inseparable from any great movement, to restore public prosperity, whose return can only consolidate an order of things against which so many interests and various prejudices rise, stronger combinations are needed, means calculated with more precision, and they can only be adopted on proofs which, like the results of calculations, impose silence on bad faith, as well as on prejudices. Then, it becomes necessary to destroy this empire usurped by speech over reasoning, by passions over truth, by active ignorance over enlightenment. Then, as all the principles of public economy have been shaken, as all the truths, recognized by enlightened men, have been confused in the mass of uncertain and changing opinions, it is necessary to chain men to reason by the precision of ideas, by the rigor of proofs, to put the truths that are presented to them beyond the reach of the eloquence of words or the sophisms of interest; it is necessary to accustom minds to the slow and peaceful march of discussion, to preserve them from this perfidious art by which one seizes their passions to lead them into error and crime; of this art which, in times of storm, acquires such a fatal perfection. Now, how much this rigor, this precision, which accompanies all the operations to which calculation applies, would add strength to that of reason! How much would it contribute to ensure its progress on this ground covered with debris, and which, long shaken by deep shocks, still experiences internal agitations”20!

17

Lottin, 1912, p. 378–385. For example: Laplace, 1846, XV, p. 354. 19 Lottin, 1912, p. 111. 20 Condorcet, 1793, p 108–110 or id., Oeuvres, 1847, 1, p. 542–543. 18

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Further on in this same program article, the geometer found something of the methodological program he had exposed fifteen years earlier in Le Mercure de France while commenting on Moheau’s political arithmetic: “Whatever the object that this science considers, it contains three main parts: the determination of facts, their evaluation, which includes the theory of average values and the results of facts. But, in each of these parts, after having considered the facts, the average values or the results, it remains to determine their probability. Thus, the general theory of probability is at once a portion of the science we are talking about, and one of the bases of all the others.21 [. . .] The same individual fact, if observed several times, can also present with differences which are an error of observations; it is therefore necessary to seek, according to these same observations, what is believed to be most suitable to represent the real fact, since, most often, there are no reasons to prefer exclusively one of these observations to all the others. Finally, if we consider a large number of facts of the same nature from which different effects arise [. . .,] it results [. . .] a common value of these effects [. . .]. The determination of this unique fact, which represents a large number, that can be substituted for these facts in reasoning or in calculations is a kind of assessment of observed facts or considered as equally possible; and this is what is called an average value. The theory of average values should be considered as a preliminary to social mathematics; [but] it is not limited to this particular application of calculation. In all physical-mathematical sciences, it is also useful to have average values of observations or of the result of experiments. [. . .] The science we are dealing with here must naturally be preceded by [a theory] of average values”.22

Such was expressed, even before the end of the French Revolution, the agenda of a new method and Quetelet explored it. This program has synthesized the research conducted by Condorcet for fifteen years. Laplace had not ventured deeply into moral and political sciences. This was the domain of Condorcet, and among the works he left at his death, were on the one hand the program of this social mathematics and on the other hand the Fragment on Atlantis or combined efforts of the human species for the progress of sciences, designed to be inserted in the Xth epoch of the Historical Picture of the Progress of the human mind,23 a fragment where was traced the project of a society of scholars organized to observe facts, record these observations and study them: all this was published and known after the French Revolution and governed the work of his successors, such as Fourier, the commissioners of the Monthyon prize and Quetelet himself. Condorcet thus launched the word and the methodological concern that characterized the mixing of data in the 19th century: “the average”: this concept and this process were to become what were the “ratios” in the 17th century, the know-how specific to a competence situated in its time. As in the previous chapter it is important to show how this know-how was formed and how it has acquired an almost global impact.

21

Id., 1793, p. 112 or, id., ibid., 1847, 1, p. 545. Id., 1793, p 115–117; or 1847, p. 548–549 (the underlines are ours). 23 Id., 2004, p. 871–919; it was first published in the editions of the Esquisse from 1804, 1822, 1823, 1829. 22

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Then, it will be necessary to show what presuppositions it carries, their validity and their limits. Several authors have wanted to sketch the formation of the conception of the average promoted by the Brussels astronomer.24 The mark of Condorcet’s recommendations on his work is indisputable: it is found both in his predilection for a method based on the calculation process of the average, and in his international activism oriented towards the coordination of the work of astronomers, meteorologists and administrative statisticians. He believed that science was meant to scrutinize the World and the Universe, to collect the facts that would result from it and to preserve the calculations thus made possible.25 Thus he wrote about the coordination of maritime observations: “[It] has proved that we can [. . .] arrive at the most vast system of observations that the human mind has ever conceived: that of covering the entire globe, in all its accessible parts, with a vast network of observers, spaced so that no natural phenomenon of any importance can manifest itself without having been seen and observed with care, without having the means to follow it and study it in its progress so that the eye of science remains so to speak incessantly open on everything that happens on the surface of our planet”.26

Much has been glossed over about statistical panopticism, denouncing implicit or explicit attacks on human dignity. However, here the object of the astronomer’s somewhat delirious omnipotence is not only made up of human targets, or even the entire species: but it is about the largest and most uncontrollable universe since maritime phenomena where relatively rare observers would draw the relevance of their panorama from a common coordination to the skies scrutinized by mutually informed astronomers since the earliest times, and finally, the populations established on Earth, implicitly assigned, directly or not, to states where newly coordinated administrative statistics would perform the same functions. Thus, the critique of panopticism proceeds from two errors: on the one hand, ignorance of its history and on the other hand, anthropocentrism, if not philosophical egocentrism: the panopticism from which statistical know-how has proceeded has not primarily targeted humans but first events considered inaccessible, maritime or celestial. Condorcet had not gone so far, but his Brussels disciple, carried by the success of his Belgian and international enterprises, let himself be carried away by a pathetic enthusiasm that is not uncommon to see shared by the promoters of supposedly new global approaches in social and historical sciences: the hybris of the point of view from Sirius in short, encouraged by satellite images.

24

Notably Lottin, 1912; Halbwachs, 1912; and recently Armatte, 1991. Brian, 1992. 26 Quetelet, 1867, p. 22–23. 25

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The Average: From Science to Mysticism27

From the end of the 18th century, it was the distribution of observations that became important among astronomers, this was already apparent in the passage extracted from Condorcet’s program. For a long time, it was known that a single observation could not be considered definitive, and attempts were made to combine several to establish a value that would be considered better than those obtained separately.28 Therefore, a method of calculation was sought, exploring for example the relevance of arithmetic averages of measured values, or sometimes even those of extreme values resulting from observations. In two metonymic steps, this method became the “average”: first to designate the arithmetic process and then its result. To judge the quality of such a calculation, it was important to represent the dispersion of the density of the measures, that is to say, to reason in a graphic way, the curve that would be drawn using two axes, the horizontal to place the measured values, the vertical to mark the quantity or frequency of observations made of each of these values. It was quickly noticed that in terms of measurement and observation, the dispersion curve showed a maximum around which the other observed values, lower or higher, were distributed. Several figures were proposed: triangles, semicircles or compositions of such elements. As soon as the study of these dispersion schemes was thought of as that of curves whose equation would have to be established, it entered the field of geometers such as Laplace, experts in differential and integral calculus. The average value appearing more frequently than those at its deviation, less frequent, the use prevailed to qualify these graphs as “probability curves” or “probability law”, frequent expressions under the pen of Quetelet as noted by Lottin (1912). In a memoir from the early 1780s where he had questioned the application of the analytical calculation of probabilities to the “approximations of formulas that are functions of very large numbers”, Laplace, had seized this question. To tell the truth, the integral calculation equations to which he had arrived were quite inextricable, except to deploy great skill to simplify their reduction and this was indeed the case. This had already preoccupied him and he arrived at a new formula in the memoir published in 1786.29 This was his second attempt in this area, which is why this result is sometimes called “Laplace’s second law”. The mathematician’s tortuous path involved justifiable mathematical approximations and simplifications, the consequences of which were not measured until more than a century later. Still on the same graph, if x, the horizontal coordinate, is the measured value and f(x), the vertical coordinate, is the frequency of this observation (then thought of as its probability), this second law has the analytical form and the distribution curve that appear in Illustration no. 3.1:30 27

Some elements of this section were presented in Brian, 2020. Armatte, 1991. 29 Laplace, 1786, p. 434–448. 30 Kahane, 2009, it being understood that these two elements, equation and figure are common knowledge. 28

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Illustration 3.1 Diagram of Laplace’s second law (Elementary mathematical construction)

Equation of Laplace’s second law

1 - x2 f ðxÞ = p e 2 2π The curve accounted for the fact that the observed values most often distribute symmetrically on either side of the central value (in the equation and illustration above, this center is located at x = 0): the average value. The frequency of measures x slightly different from this central value then appeared less and less probable, this regardless of the sign (because, positive or negative, it is squared: x2), which ensures the symmetry of the curve. If this deviation x moved very far from the center, its probability would tend towards zero (a characteristic of this exponential) and the ends, conceivable as they would be from a mathematical point of view, would fall outside the graphical scheme. The formula first proposed by Laplace was later confirmed by the German mathematician Carl Friedrich Gauss (1777–1855),31 posterity has attached their two names “Laplace-Gauss law” and several other denominations: mainly the “bell curve”, if not for reasons specific to 19th century French clothing the “policeman’s hat” and, once its reception was accomplished, the “normal law”. The scientific and more general contexts of these researches are known and have been discussed earlier.32 The problem was the dispersion of observation errors in astronomy. Laplace considered that this form was adequate because the area under the curve between two values taken on the horizontal, that is to say the probability that the considered measurement would be between these two values, is quite easy to calculate numerically, the exponential function lends itself indeed to numerical developments which, tedious as they were (they were of course done without machine by hand) did not scare a good calculators of that time. Thus, Laplace suggested that numerical tables be established. It is the relative simplicity and 31 32

Gauss, 1809. Brian, 1994.

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convenience of the calculations that led Laplace to his second law. These tables the statisticians active at the end of the 20th century remember them so much then the electronic calculation did not replace their consultation. The Alsacian astronomer Christian Kramp (1760–1826) published them in 179933 as did an actuary close to Condorcet and later active in the Parisian office of Longitudes: Emmanuel-Étienne Duvillard de Durand (1756–1832) in 1813.34 Laplace assumed that the attention for this distribution curve of the variability of observations had proceeded from a predilection for two types of simplifications: firstly the hypotheses called by the integral calculus to arrive without trouble at a simple formula and the fact that this integral, at the cost of the known development of the exponential, lends itself to the establishment of tables. The formula has since been dedicated to a resounding success. The young and ambitious Brussels astronomer was its craftsman. During his training stay in Paris, in 1823, he frequented the circle of Parisian geometers of the Academy of Sciences: their elder, of course, Laplace and the next generation: Sylvestre-François Lacroix (1765–1843), disciple of Condorcet; Joseph Fourier or Siméon-Denis Poisson (1781–1840), he then remained in epistolary contact with them and other Parisian scientists.35 Laplace had proposed his “second law” from a strictly mathematical point of view: he had aimed at the approximation of formulas functions of very large numbers and not proposed a theory of the average whose puzzle however included among its pieces this memoir while other mathematicians will accumulate others until the 1830s. From the point of view of intellectual history or the history of science, it is the variety of these proposals and the climate in which they circulated that matter.36 As for the climate, a very old scheme made one imagine the compensation of highs and lows. As Jean-Claude Perrot pointed out, the theme of the balance of charges is frequently found in political economy works of the 17th and 18th centuries. A recent thesis in the history of tax law has shown its origin and importance in specialized French jurisprudence where the adage “the strong carrying the weak” served as a principle of equalization.37 Returning to the mathematicians of the turn of the 18th to the 19th centuries, Condorcet did not mention, in the 1790s, the law derived by Laplace when he undertook to give the public this theory of the average that he had called for in 1793. His writing is inserted in his Elements of the calculation of probabilities published posthumously in 1805: “The average value [. . .] exclusively possesses [these properties]: 1°. That the sum of the positive and negative differences between it and the values given by observation, or the sum of the differences between this value and the equally possible true values, is equal to zero [. . .].

33

Kramp, 1799. Duvillard de Durand, 1813. 35 Droesbeke, 1991. 36 See on this question of method: Perrot, 2021 and Febvre, 2023. 37 Perrot, 1992; Le Gonidec, 2022. 34

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3°. That by taking, under similar circumstances, this average value for a true value, the most probable event will be the one where the differences more or less between reality and the hypothesis, will compensate [. . .] We can therefore see that, in all cases where we can substitute an average value for the true value, the one given by the ordinary method, is the only one we should choose. Otherwise [. . .] we would not choose the value for which it is most probable that the errors will compensate”.38

Lacroix’s treaty, Lottin notes its subsequent importance for Quetelet, offered in 1816 a synthesis of the work in this register accumulated over nearly thirty years. This student of Condorcet, who became a professor at the École Polytechnique, immediately placed himself in the wake of his master.39 He reviewed the protagonists of the European analytical scene and when he approached Laplace’s work, he presented one of its starting points as a culmination: a collective memory of mathematicians rearranged in the 19th century as a statue of a Commander began to emerge. “To finish going through the main applications of probability calculation, I would still have to talk about how to take the middle between several results or observations, taking into account the various probabilities of errors, or to determine the most advantageous corrections that very close values should undergo to best satisfy a large number of observations. This research started by Lagrange (1736–1813), clarified by Euler (1707–1783) was pushed very far by Mr. Laplace; but it mainly relates to Astronomy, it goes beyond the limits that I had to set for myself. I will only say that the best rule, that of making [minimal] the sum of the squares of the errors [. . .] was established by Mr. Legendre (1752–1833), in a very simple way; as for the evaluation of the probability of the result to which it leads, it can be found in [Laplace’s] Analytical Theory of Probabilities”.40

With Fourier, the apprentice astronomer from Brussels who came to train in Paris made his honey in another way of synthesis. Indeed, in 1823, the one whose scientific posterity has mainly retained a dazzling analytical theory of heat (1822) and a way of decomposing periodic functions of primary importance for the sciences and techniques of subsequent centuries (that is, the series that bear his name) had experienced all the variants of the statistical skills of the early 19th century. He received the teachings of Laplace at the École normale de l’An III.41 Having then become a teacher at this École Polytechnique, he had been designated in 1798 to take part in Bonaparte’s Egyptian Expedition42 and he had ensured its scientific, geographical and cultural coordination. Back in France, Napoleon appointed him prefect of the Isère department,43 a position that Louis XVIII had preserved for him in 1814 at the Restoration of the Monarchy. Upon his temporary return from exile on the Island of Elba, the emperor made Fourier his prefect of the Rhône department. In

38

Condorcet, 1805, p. 109–111. It is to be noted that this principle of substitution with an average value is the one in force today in the processing of big data in the event of missing information. 39 He mentions from the first pages the programmatic writings of 1793 and 1805: Lacroix, 1816, p. IV. 40 Id., ibid., p. 255–256, this refers to Laplace, 1812, Chapter IV, p. 304–348. 41 Laplace et al., 1992. 42 Fourier, 1809: Bret, 1999. 43 The prefect, even today, represents the government in his department.

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his prefectoral duties, he had to respond to the demands of the ministers of the Interior, and notably, in 1803, to those of Chaptal who had ordered on the one hand local surveys on the relationship between the numbers of births and the population (modeled on the latest work published on the subject by the Academy of Sciences under the Old Regime but this time based on civil status and the post-revolutionary administrative foundation),44 and on the other hand systematic inventories of each of the departments.45 After the ultimate collapse of the Empire, without resources, he was called in 1817 by one of his former students at Polytechnique who had himself become prefect of the Seine department to direct its statistics office,46 it being understood that this department was then much larger than it has been since its reform in the mid1960s.47 On the scientific level, the mathematician Jean-Pierre Kahane (1926–2017) was right to highlight three passages from the Analytical Theory of Heat (1822) whose title alone shows how much its author had hoped to don the clothes of his 2 1 elder probabilist. Thus, the ninth chapter presents the formula x e - t dt in such a way that it could not have escaped an astronomer.48 Moreover, it was used to establish “average temperatures” important for various aspects of heat diffusion. But, Kahane, he points out, two excerpts from the preliminary discourse of the theory offer, like its counterpart in Laplace’s 1812 work, a theory of science and the relationship between mathematics and physics: “The primary causes are not known to us, but they are subject to simple and constant laws that can be discovered by observation, and the study of which is the object of natural Philosophy. Heat penetrates, like gravity, all substances of the universe; its rays occupy all parts of space. The purpose of our work is to expose the mathematical laws that this element follows”.49 “[. . .] We will observe, in various places around the globe, the temperatures of the ground at various depths, the intensity of solar heat and its effects, whether constant or variable, in the atmosphere, in the Ocean and lakes; and we will know this constant temperature of the Sky, which is unique to planetary regions. The theory itself will guide all these measurements and will assign their precision. It can no longer make any significant progress that is not based on these experiments; for mathematical Analysis can deduce, from general and simple phenomena, the expression of the laws of Nature; but the specific application of these laws to very complex effects requires a long series of accurate observations”.50

44

Bru, 1988. Perrot, 1977; Bourguet, 1989. 46 Fourier, 1826 and 1829. 47 This refers to the area covered today by the departments of Seine (75), Hauts-de-Seine (92), Seine-Saint-Denis (93) and Val-de-Marne (94). 48 Fourier, 1822, in articles n°365 and n°366 where the editor of Works of Fourier, in 1888, p. 418, argued for the simplicity of the calculation by referring to Kramp, 1799 “this work contains a table 2 1 of values of the integral x e - t dt “, as Poisson will do, see below. 45

49 50

Fourier, 1822, incipit of the “preliminary discourse”, p. XV of the 1888 edition. Fourier, 1822, clothing of the “preliminary discourse”, p. XXVIII of the 1888 edition.

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Here we easily recognize the program that Quetelet will assign thirty years later to international meteorology congresses or his line of conduct regarding the maintenance of relationships between analytical developments and the accumulation of empirical observations in astronomy and administrative statistics. Thus, the young man was able to return to Brussels, equipped with a mathematical apparatus and an empirical line of conduct of planetary ambition. Before following his work step by step, it is useful to consult Fourier’s memoirs published in introduction to the statistical collections of the department of the Seine and a major memoir published at the Academy of Sciences in Paris, in 1830 by another mathematician of the same generation: Siméon-Denis Poisson. A few years after his Analytical Theory of Heat, Fourier gave a series of methodological articles as an introduction to the Statistical Research on the city of Paris and the department of the Seine. In 1826, he reported the success of average calculations and proposed a technique for their testing of disconcerting simplicity, as long as we admit the homogeneity of the cases considered and the theorems of Jacques Bernoulli and Laplace (we would call them today “central-limit”): “The study of the properties of the climate, that of the population and of the agricultural or commercial wealth, most often require that we determine the average numerical value of a certain quantity. We observe, or we extract from public registers, a large number of different values of this quantity; we add all the numbers that express it, and we divide the sum by the number of values that have been measured [. . .] There is no one who does not know this simple process by which we determine an average number [. . .] That’s why it is very useful to examine carefully the consequences it provides, and the degree of approximation to which we arrive [. . .]. Before resolving this question, we will point out that we can acquire a fairly accurate knowledge of the precision of the result without resorting to mathematical theories. It is enough, for example, to divide into two parts the set of observed values, whose number is supposed very large, and to take for each of these parts the value of the average result; for, if these two values differ extremely little from each other, we are justified in regarding each of them as very precise. Nothing is more suitable than this kind of tests to highlight the accuracy of statistical results, and it is almost unnecessary to present to the reader consequences that are not verified by these comparisons of average values”.51

One can catch the petition of principle in which the great scientist got caught by imagining that each of the two parts held separately for the need of our argument for homogeneity but diverse, have been characterized by clearly central magnitudes distinct from each other. Even precise, each calculation, after Bernoulli’s theorem, would lead to different arithmetic average. The simple test proposed by the heat theorist to the reader – undoubtedly for didactic reasons – focused both on the process and on the homogeneity of the set that was submitted to him. From then on the test became ambiguous. However, on homogeneous populations it is not devoid of quality, even though the mathematician did not justify it with some analytical calculation in Laplace’s vein for instance.

51

Fourier, 1826, p. 525–528 in the 1890 ed.

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“To apply this first remark fruitfully, we must go back to the principle from which it is deduced, and clearly conceive that the repetition and variety of observations are sufficient to discover the constant relationships of effects whose cause is unknown. This conclusion [. . .] applies to the most diverse objects; and there is no more general and more important notion in the subject we are dealing with”.52

Thus, we owe to Joseph Fourier the early expression of what I would like to call “pan-normality”: the repetition of observations would be enough to reveal the constancy of cause-effect relationships in any domain. This idea is not Jakob Bernoulli’s one: indeed the early probabilist considered a known cause, for instance the number of face on a regular dice, and its many throws concluding on the frequencies, from a known cause to the unknown effects: this is the direct problem. But there is – and clearly was – an inverse or converse problem, from the regularity of effects how to conclude on the causes? This was the object in a well-known posthumous memoir (1764) of Thomas Bayes’ (1702–1761), a paper at the origin of the first works by Condorcet and Laplace in probability. In Fourier above, the direct and the convert problems are confused: the average is then established as a metaphysical principle. His disciple, Quetelet, will take the lesson and he will adopt it in the most general way possible, without any other attestation, as we will see. The Brussels astronomer will present it as a revelation. The mixing of numbers produced in an avalanche has thus met its mystique. Fourier himself was ultimately more cautious: the rest of his 1826 memoir consists in proposing, after Legendre, to measure the errors and deviations from the average by the sum of the squares of these errors or deviations.53 The same Fourier gave a second memoir “on average results and measurement errors”.54 It is a very systematic presentation that unfortunately did not lead to a convenient calculation rule to implement. Nevertheless, it remained famous because its author used as an example a problem he had thought of during the Egyptian Expedition: determining the height of the Great Pyramid of Giza.55 However, as early as 1821, Fourier had written about calculations on population: “We first recognize that [the population] is subject to accidental and fortuitous variations that compensate each other over several years. [. . . The] indefinite repetition of events that are considered fortuitous eliminates all their variability; in the series of an immense number of facts, there remains only constant and necessary relationships, determined by the nature of things. [. . .] But what is the exact measure or the value of precision, and how does this quantity depend on the number of observations?”.56

To tell the truth, these methodological introductions to the statistical collections of the Seine department57 were so many mathematical blows struck at the statue of the

52

Id., ibid., p. 528. Id., ibid., p. 537 and following; the same process was proposed by Gauss. 54 Id., 1829. 55 Id., ibid., p. 569. 56 Fourier, 1821, p. XXXVI–XLI. 57 Chabrol de Volvic, 1821, 1823, 1826 and 1829. 53

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Commander of a seventy-seven-year-old Laplace: indeed, Fourier then opened a breach in the Laplacian monopoly of the transmission of analytical probability calculation formed at the end of the 18th century in reactivating the empiricism of his predecessor, Condorcet, who died prematurely during the Revolution. In these 1820s, between the statistical office of the Seine department led by Fourier (called by Chabrol de Volvic, one of his former students at the École Polytechnique and his companions from the Egyptian expedition) and the office of longitudes where Poisson worked, there was a kind of competition on both empirical aspects and the analytical construction of calculation rules, and this parallelism even went as far as the publication by Poisson of a response to Fourier’s question just mentioned and a mathematical theory of heat where the exponential of -x2 intervenes.58 Fourier published in 1824 an article in the columns of a periodical closely followed by the young generation of scientists trained in the schools resulting from the French Revolution: this piece synthesizes his introductions to the statistics of the Seine, he openly delivers the principle of calculating the probability that an unknown value is greater than a given value g (that is to say between g and 1) by means of the 2 1 integral g e - t dt.59 This provides the area under the bell curve from the abscissa g as long as we admit that this curve restores the distribution of probabilities from this reference point; the integral then offers the cumulative probability of all possible cases. During these years, while he was publishing his statistical methodology memoirs in various forms, the heat theorist was elected, precisely on November 18, 1822, to the perpetual Secretariat of the Academy of Sciences for Mathematics. He was then able to exercise control, at least in principle, over the volumes published in the name of the learned society, like the one exercised by Condorcet in his time and which Laplace had suffered from in his early days.60 His successive memoirs sparked a flurry of articles written by Poisson and his close associates in recognized publications but located out of reach of the new perpetual Secretary.61 In February 1829, that is to say nearly two years after Laplace’s death (March 5, 1827), Poisson submitted to the Academy of Sciences a memoir in which he reformulated the analytical calculations of Laplace, Legendre and Fourier on the dispersion of errors. By its style partly already announced by those of Fourier as these two authors have distinguished themselves from the laborious demonstrations of the late 18th century, it is held by historians of probability calculation as the starting point of the contemporary mathematical statistics.62

58

Poisson, 1835b; the scientific situation is analyzed in Bachelard 1928. Fourier, 1824, p. 88. 60 See on this subject: Brian, 1994, part I, chapter 5. 61 Poisson, 1824; Poisson & Nicollet, 1824 and the following years. 62 Poisson, 1830, extended by id., 1835a; as for historians, see Stigler, 1986; and 1999 and Hald, 1990 and 1998. 59

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The turning point that the publication of Poisson’s research on probability calculation represents is marked by the presentation to the Academy of Sciences in Paris of two decisive papers. The first, published in 1830, was based on the statistics of births of boys and girls that the mathematician was dealing with at the Bureau of Longitudes. Poisson reasoned about a binomial where the boy vs girl alternative lends itself to modeling according to a random draw of the type of heads or tails. In the mathematical scheme of the game, as in the approach to births at that time, one of the possible cases would have a probability p (measured by the frequency of male births over all births) and the other q, it being understood that each recorded birth is certain, regardless of the sex of the newborn, that is to say that p + q equals 1. The mathematician then deployed an integral calculation which he simplifies in the manner of Laplace in 1786 duly paying him tribute. He arrived, to evaluate the probability that the proportion of one of the two sexes exceeds a threshold arbitrarily 2 1 set, u, at the integral: u e - t dt. He immediately mentioned the table to be found in the work of Christian Kramp (1799).63 A few pages later,64 he offered a numerical application resulting from the compilations of the Bureau of Longitudes. The total number of births recorded for the years 1817 to 1826 inclusively was then 9,656,135 and that of boys: 4,981,566, this leads him to propose for the probability of a male birth 0.5159 ± 0.0007, or what we would write between 51.52% and 51.66%. In 2024, this reminds us of a confidence interval around a central value. And without using this vocabulary, Poisson asserted that 0.999978 was the probability that the probability of a male birth is between 0.5152 and 0.5166, the certainty to which the calculation led him did not concern the probability of the birth of boys itself, but the probability that this probability is strictly circumscribed. From a conceptual point of view, this second-order probability has served as a model until today, even if it is often established by other means. But the same Siméon-Denis Poisson brought another thing promised to great success to the calculation of probabilities: the designation of this reasoning scheme. This was the case in his “Research on the probability of judgments mainly in criminal matters”, presented to the Academy of Sciences and published by it in the first volume of his new Weekly Reports which replaced, starting from 1835,65 the previous annual volumes, too slow in their distribution. The table of contents of their thick collection for the year 1835 indicates, p. 591: “Numbers (large), what can be called the law of large numbers (p. 478)” and “This law, according to Mr. Poisson, is the basis of all applications of probability calculation (p. 481)”. Regarding statistics of court decisions, Poisson presented himself as a continuator of probabilistic research on judgments inaugurated fifty years earlier by Condorcet,66 then in the meantime approached by Laplace in his Analytical Theory.67 He thus stifled any 63

Poisson, 1830, p. 262. Id., ibid, p. 283. 65 Id., 1835a, b. 66 Condorcet, 1785. 67 Laplace, 1812. 64

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tension between the supporters of one or the other of the old generation. As for the law whose name he then forged, Poisson took care to show, and it was relevant, that it extended both the theorem of Jacques Bernoulli and its reformulation by Laplace himself. However, his argument is more of an assertion than a demonstration: “Things of all kinds are subject to a universal law that we can call the law of large numbers. It consists in the fact that, if we observe very large numbers of events of the same nature, dependent on causes that vary irregularly, sometimes in one direction, sometimes in the other, that is to say without their variation being progressive in any determined direction, we will find, among these numbers, ratios that are almost constant. [. . .]. This law of large numbers is observed in the events that we attribute to blind chance, for lack of knowing their causes, or because they are too complicated [. . .]. Now, this universal law of large numbers, of which we have just given examples of all kinds,68 which we could have, if necessary, multiplied and varied even more; this law, we say, is the basis of all applications of probability calculation. Now, it is obvious that it also applies to moral things that depend on man’s will, his enlightenment and his passions; for it is not a question here of the nature of the causes, but indeed of the variation of their isolated effects and the number of cases necessary for these irregularities to balance out in the average results”.69

As for moral phenomena, Poisson, then empirically and precociously observed regularities that would feed the German Moralstatistik of the second half of the 19th century one, a field that will mobilize administrative and descriptive statistics rather than probability calculation,70 and of course French Durkheimian sociology which will reconnect with probability calculation only with the work of Maurice Halbwachs (1877–1945) in the early 20th century.71 In other words, Poisson was grasping consistent empirical phenomena. But his argument oscillates between elements of mathematical demonstration and fragments of generalized empirical observations. This confusion of mathematical and empirical arguments will be the matrix of the success of the expression “law of large numbers” for two centuries, and that of a mythology that historians unfortunately amplified by straddling it without critical reflection.72 However, as early as 1912, the mathematician, physicist, philosopher and professor Henri Poincaré (1854–1912) had not hidden his reservations: in his course on probability calculation he had posed what could have been an analytical formulation of a law of probability of errors and immediately observed the ambiguity of the expression forged by his predecessor Poisson: “This would not teach us much if we had no data on ϕ and ψ [two functions that intervened in this analytical expression]. So a hypothesis was made about ϕ, and it was called law of errors. It is not obtained by rigorous deductions; more than one demonstration that has been wanted to give it is crude, among others the one that relies on the assertion that the probability of deviations is proportional to the deviations. Everyone believes in it however,

68 Poisson, had commented earlier on the regularity of meteorological observations (such as tides), astronomical and judicial (such as the regularity of conviction rates). 69 Poisson, 1835a, b, pp. 478–481. 70 Such will be the case of Georg von Mayr (1841–1925). 71 This time it’s the work of Émile Durkheim (1858–1917). 72 Such were Desrosières, 1993 and Cole, 2000.

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Mr. Lippmann told me one day, because experimenters imagine that it is a theorem of mathematics, and mathematicians that it is an experimental fact”.73

We will now follow the fate of this so-called law in the double sense of the mathematical formula and its synthetic expression. Quetelet, admirer of Parisian mathematicians, learned from them, Lacroix, Fourier or Poisson, what could be the calculation of probabilities and what should be its power in its relations to empiricism. In this respect, Fourier and Poisson, a few years apart, expressed the same conviction about the universality of the law of errors for the former and the law of large numbers for the latter. It was indeed a conviction as it was not based on any rigorous and complete mathematical demonstration as Poincaré observed a century after them. We have previously referred to it as “pan-normality”. It was forged in Paris during the half-century that bridges the 18th and 19th centuries. It will now operate among specialists as a prism through which they will see their methodological as well as empirical programs. As such, it is promised to a global success during the second half of the 19th century, its first conception by Adolphe Quetelet has been tracked by various commentators. It came from his first strictly analytical works. His beginnings at the Royal Academy of Sciences in Brussels were marked by mathematical analysis: as soon as his thesis was defended, he drew from it memoirs that he submitted to this learned Company where they were read and commented on by his masters. For what interests us here, the most important is the one on conic sections.74 Quetelet was then following in the footsteps of Newton and, as we know, ellipses and parabolas, as many conic sections, are the curves of the trajectories of celestial objects. In particular, as mentioned above, the observation from Earth of chosen parameters for planets or their satellites allowed to calculate their mass or their distance to the Sun or to their focal planet. This is where the analytical calculation of the near-constancy of angular measurements and average distances to the focus of the considered conic could meet the test of experience and observation. In the wake of this publication and another one published in the same volume, the young man was elected member of this Academy at the beginning of the year 1820.75 Jean-Guillaume Garnier (1766–1840) – who would be, with his junior by thirty years, the co-editor of the Mathematical and Physical Correspondence published in Ghent, a periodical of information and scientific emulation – was within the Company the referee of this memoir on conic sections. He was favorable but he asked for corrections before any publication in the volumes of the Academy, so much so that the text finally published in 1822 is not, strictly speaking, the initial document submitted by Quetelet. We only know it indirectly. Its author, indeed, has given in 1825 a brief text entitled “Statistics” in the periodical he animated with his

73

Poincaré, 1912, p. 170–171. Gabriel Lippmann (1845–1921), Franco-Luxembourgish physicist was awarded the Nobel Prize in Physics in 1908. 74 Quetelet, 1822. 75 New Memoirs of the Royal Academy of Sciences and Fine Arts of Brussels. Brussels, Mat, vol. II, p. XLIJ, February 1st, 1820.

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elder.76 It opens with the announcement of a scientific ambition worthy of that expressed three years earlier by Fourier in his Theory of Heat (1822). “The observer, by closely following the course of nature and by subjecting to rigorous calculation even its smallest effects, has managed to unravel some general laws and to grasp distant relationships, which must make him desire to extend further the circle of his knowledge”.77

An annotation signed by Garnier punctuated this short article which led to the formulas proposed by Johann Heinrich Lambert (1728–1777) for the mortality curve.78 It specifies: “The two terms [to the right of the formula, each in e-x2] have a lot to do with the expression of the cooling of bodies. In a memoir entitled: “New theory of conic sections”,79 Mr. Quetelet has shown that if one cuts obliquely a right cylinder, so as to produce an ellipse, the surface of this cylinder developed, gives a sinusoid or a curve which is similar to it; now, if one considers, at the same time, the laws of growth in populations, one can represent the development of generations, like the development of a roll of cylindrical paper, which has for bases on one side a circle, and on the other an ellipse, each turn representing the revolution of a year”.80

This is the only trace we have left of this initial memoir by Quetelet. Compared to the version published in 1822 at the Academy, it is clear that the purely geometric view that Garnier attributes to his junior is governed by the agenda and Joseph Fourier’ formal theory, an author perfectly known to the two Belgian scientists and especially to the elder of them: he had assisted him in his teachings at the École polytechnique. Fifty years later Late in life, Quetelet, then seventy-seven years old – and having been struck eighteen years earlier (in 1855) by a stroke that according to the description of one of his collaborators reported by Lottin seems to have been a temporary ischemic accident from which he recovered quickly – gave a series of testamentary writings in the form of scientific or autobiographical assessments.81 One of them is very revealing as it gives the short text of 1825 commented by Garnier as the key to his work: “Still young, and in the midst of the narrow sphere where I lived, I already held some of the threads that were to guide me in my journey. [note:] When, in 1825, I began, with Professor Garnier, the publication of the Mathematical and Physical Correspondence, from page 16 of the first volume of this journal, an article on statistics appeared, which carried in note the following remarks from my collaborator [i.e. Garnier]: ; [follows the passage quoted above from ‘a memoir entitled’ to ‘the revolution of a year’, and Quetelet to add in 1873:] one can notice, today, that this theorem serves, in a way, as the basis for the theory that I have given since, on the development of man and living beings, which are part of the three kingdoms of

76

Quetelet, 1825. Id., ibid., p. 16. 78 On Lambert’s calculations, see Barbut, 2006. 79 See: Quetelet, 1822, the one examined by Garnier for the Belgian Academy. 80 Garnier, 1825, p. 18. 81 Quetelet, 1873a, b, c. 77

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nature. This is, I believe, one of the most fruitful propositions contained in the physical and mathematical sciences”.82

It is remarkable that the Belgian scholar, in his later years, did not feel the need to formally and explicitly cite either Fourier or Poisson. At this point in his career, he was probably sufficient to himself, but the key is here: by the play of a particularly abstract geometric vision the same mathematical laws applied to the three kingdoms of nature, their common characteristic was the law in e-x2, that of the diffusion of heat established analytically by Fourier, that which Poisson will propose to call “law of large numbers”. In his enthusiasm, Quetelet confused the calculation process and the morphology of phenomena, prepared for this confusion by what we have recognized as an affirmation of “pan-normality” under the pens of Fourier and Poisson, on which Poincaré will be ironic. In the mid-1820s, his research, initially governed by the model of his Parisian elders, met the interest his British counterpart and almost contemporary, John Frederick William Herschel (1792–1871). He indeed testifies to this in the autobiographical note he presented to the Royal Academy of Belgium in 1873: “Sir John Fr. William Herschel, who was always for me a sure friend and an enlightened judge, strongly encouraged me in my work, during his scientific stay at the Cape of Good Hope [ca 1825]. He asked me, rightly, the formula expressing the conditions of growth for the sizes of man and the law of his proportions at different ages. He had truly perceived the delicate point of the theory that occupied me and he asked for the solution. I replied that in the absence of necessary documents on man, I found myself in the impossibility of determining this formula earlier that he wished to know: but I was fortunate enough to be able to give it soon after. It can be found in the second edition of the Social Physics.83 [In note:] it is remarkable that this formula is precisely that of the famous curve which was already of such keen interest due to the work done, for over a century, by the school of Leibnitz and Newton.84 This same formula also lends itself, for all phenomena on size, weight, strength and in general, on all living beings, whether humans, animals or plants, to provide the most difficult values to calculate”.85

In an article published in Edinburgh in 1850 where he commented on a publication by the Belgian mathematician, dated 1846, the same Herschel offered a dazzling overview of the mathematical journey that went from the Parisian founder to the Brussels astronomer: “Laplace gave a rigorous demonstration, based on the comparison of equipossible combinations, in infinite number. But his analysis is excessively complicated: and although presented in a more elegant manner by Poisson; and stripped of all superfluous difficulty by Mr. Quetelet in the work we are dealing with,86 reduced moreover to the most elementary and simplest form we have ever seen, it must nevertheless, of necessity, remain incomprehensible to all those who a knowledge of higher analysis has not perfectly familiarized with

Quetelet, 1873a, p. 199, the word “statistique“is in italics in the original. Id., 1869a, b. 84 Quetelet here evokes with confusion the theorem of Jacques Bernoulli. 85 Quetelet, 1873a, p. 200. 86 Herschel was indeed commenting on Quetelet, 1846. 82 83

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the delicate considerations by which we pass from finite differences to ordinary differentials”.87

This particularly elegant synthesis is followed, a few pages later, by a very simple intuitive justification for the appearance of -x2 in the equation of the curve of the distribution of probability of variations of observation or of “the curve of probability” to use the terminology of the early 19th century. With the same finesse, Herschel justifies that this equation is that of an exponential and, in passing, he refers to the tables of Kramp (1799).88 Undoubtedly the quality of this exposition led Quetelet himself to place at the head of the second edition of his great book (1869a) a translation of this enlightening commentary published nineteen years earlier. A note at the beginning of this introduction suggests that the Belgian astronomer did not initially identify his laudatory commentator. But, in any case, he had his probabilistic apparatus before 1850 as we will see, a baggage made of calculations, the epistemological ambitions of his predecessors may be refined in his discussions with his British counterpart. For twenty years, from the mid-1820s, Quetelet will mobilize all the sources he can reach to strengthen his initial conviction, however, derived from a purely geometric view. This will also be a period of publication of didactic and methodological works. So much so that, in 1850, in the eyes of his friend from across the Channel: “[Mr. Quetelet] is a master well worth listening to: he can claim attention for the excellent reason that he himself has studied his subject in a practical way, having undergone a long and rigorous apprenticeship, formed a collection of documents on very varied branches, and concluded from these data precise results of undeniable value and importance by means of the rules and principles he teaches”.89

Thus armed with his doctrine and carried by such a reputation, the Belgian scholar, during the second half of the 19th century, will demonstrate a formidable spirit of international organization: he will extend to the whole world accessible to him the results of his Belgian experimentation. Before following the average in the assault on the world, let’s see how the mathematician has consolidated his statistical arsenal, the keystone of which has been the process of the average – a particular know-how, therefore – which has gone hand in hand with a series of presuppositions about the distribution of observations around it. During these years and until the middle of the century, he published numerous articles in the Mathematical and Physical Correspondence, then the Bulletins of the Royal Academy of Belgium or the Bulletin of the Central Statistical Commission of the Kingdom of Belgium, all periodicals that were at his disposal, as he directed the first, was the perpetual Secretary of the Belgian Academies and the President of the Central Statistical Commission.90 His empirical works were compilations of these Herschel, “Introduction”, in Quetelet, 1869a, vol. 1, p. 30. Id., ibid., p. 31–33. 89 Id., ibid., p. 21. 90 Quetelet, 1826, 1827, 1847, 1848a. 87 88

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studies hitherto delivered separately.91 Between these publications appeared books or articles with a methodological vocation.92 In 1828, Adolphe Quetelet published in Brussels a small work, his Popular Instructions on the Calculation of Probabilities: nineteen lessons in all that start from the elementary notions of probability calculation, such as those posed in the first chapters of the homologous works of Condorcet93 (1805) or Lacroix (1816). Laplace is cited with respect and the Brussels professor acknowledges for the 12th and 13th lessons his debt to Fourier’s introductory memoirs to the volumes of statistics of the Seine department. Thus, the way of taking the average results is the subject of the 13th lesson and, of course, a few years before Poisson coined the expression, there is no explicit question of the law of large numbers. From the first pages the author affirms the omnipotence of a wellregulated probability calculation: “This small work, which I present to the public, is the summary of the lessons I have been giving for several years at the Brussels Museum, to serve as an introduction to my physics and astronomy courses. It seemed to me that the calculation of probabilities, unfortunately too neglected, should, according to the current state of knowledge, serve as the basis for the study of all sciences and particularly observational sciences. Most of our knowledge indeed rests only on more or less strong probabilities, which the common man appreciates vaguely and as if by instinct, but which the philosopher or at least the man who aspires to deserve this title, must know how to appreciate according to sure rules”.94

The virtues of large numbers are presented as self-evident: “It is obvious that the average value is known with more precision, the more observations are used in this research” because: “accidental variations compensate each other and thus form an average and general result.” In these pages, chance, variability, and errors are confused to be reduced “in a large number of experiments, [where] the multiplicity of chances makes what is accidental and fortuitous disappear, and only the certain effect of constant causes remains; so there is no randomness [hazard] among natural facts considered in large numbers.” (p. 143–144). Reading these pages, where the author repeats without formal demonstration his conviction has something hypnotic, at least one guess that during the courses the professor must have been particularly persuasive. He has, however, taken care to indicate the method of calculating averages on a partition of the total number of observations in the manner of Fourier (p. 144) and, a little further on. he also proposes to measure the degree of approximation of an average result according to the principle of least squares of deviations from the average – a process that elsewhere in the equations reveals the x2 of the exponential of the Laplace-Gauss, Fourier or Poisson formula (p. 155). Then the author states this principle: “this precision (calculated according to the square of the deviations) increases as the square of the number of observations [sic].” (p. 161). 91

Quetelet, 1835, 1848b, 1869a, 1870. Quetelet, 1828, 1845, 1846, 1848a, 1854. 93 If the Bernoulli theorem is indicated in a somewhat confused way, there is an indisputable trace of the reading of Condorcet: the use of the expression “motives to believe”, QUETELET, 1828, seventh lesson, p. 71–72. 94 Id., ibid., p. i-ii. 92

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The expression is clumsy, because it is in fact “like the square root of this number”; an example given a few lines later corrects the clumsiness. The last chapters provide several applications of probability calculations: to annuities, mortality tables and insurance; the probability of testimonies, that of decisions rendered in court and that of elections. No graph illustrates the lessons. The Instructions recapitulated the Parisian probabilistic learning of the Brussels astronomer. The foundations of his method, the conviction of its omnipotence, already expressed by his predecessors and the author’s persuasive force are already present in 1828. The young professor, crowned with his local reputation and some benevolence from abroad, had the honor of being the tutor of Ernest and Albert of Saxe-CoburgGotha, the sons of the first Duke of Saxe-Coburg and Gotha and nephews of the King of the Belgians Leopold I er (1790–1865). The younger of them, Albert (1819–1861) was destined to marry Queen Victoria of Great Britain and Ireland (1819–1901) and he will be therefore Prince Consort from 1840. As he had done for his courses at the Brussels Museum eighteen years earlier, Quetelet published this teaching in 1846 but this time giving his work a genre and a title that placed it under the aegis of the young prince: Letters to His Highness the reigning Duke of SaxeCoburg and Gotha, on the theory of probabilities, applied to moral and political sciences.95 By doing so, the Belgian scientist also placed himself in an aristocratic lineage, but scientific this time. Four years earlier in Paris, the philosopher Augustin Cournot (1801–1877) had given a new edition of the Letters of Leonhard Euler “to a princess of Germany” (1775)96 many times published again along the XIX century, those that Condorcet had himself already published in Paris in 1787–1789 and which, in the eyes of this mathematician philosopher, were to be extended by an introductory work on probability calculation, a project that took the form of his Elements (1805). Regarding the genre of Letters, the British astronomer Herschel made a criticism, it would obscure the demonstrations by the interposition of agreed dialogue elements.97 In 1850, it is clear that Quetelet put all the resources he had into his promotional endeavor. What was his goal? Unlike the elementary instructions of 1728, didactic, the Letters promoted a scholarly agenda: the application of probability theory to moral and political sciences. Stripped of its courtly style woven throughout the work both to please the prince and to satisfy less aristocratic readers, it contains the basics of calculation set out in the Popular Instructions and developments aimed primarily at providing a theory of the average and then highlighting a range of examples drawn mainly from previous publications in the volumes of the Academy or the Central Statistical Commission. The tutor explains in the preface:

95

Quetelet, 1846. Euler, 1842. 97 Herschel, 1869, article that reuses a publication from 1850. 96

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“Particular circumstances [. . .] put me almost ten years ago,98 in the necessity to focus all my attention on the application of probability theory to the study of moral and political sciences. I then felt how desirable it was to make this theory more elementary, and to bring it down from the high regions of analysis, to make it accessible to those who are most often obliged to use it. [. . .] This work [was] started in 1837”.99

There are forty-five letters in total: a first part deals with the basics exposed to the popular public eighteen years earlier but this time given in a more elegant style. The second part deals exclusively with averages and dispersion around them, the third focuses on the study of causes and the fourth on statistics of all kinds. The whole is accompanied by about twenty additional notes assigned to some of these letters. The work is relevant for appreciating the methodological doctrine of the Belgian scholar in the mid-1840s, and it was read as such by his successors, his critics, and his commentators. The body of the doctrine, the formation of which we have been able to observe since the Parisian years, was exposed a year earlier in a methodological article published in the Bulletin of the Central Statistical Commission100 that the philosopher Lottin was right to take into consideration, even if, probably due to a lack of a mathematician’s perspective, he did not perceive that the binomial law occupied a notable place in it,101 admittedly in the somewhat heavy form of combination number tables. On averages, Quetelet first proclaims the creed of “pan-normality” that he conceived during his stay in Paris, but he adds this time a crucial point: the extent of dispersion around the central value deserves as much attention as this one. The story of the bell curve has thus taken shape until it found its place in that of the progress of the human spirit: “The theory of averages serves as the basis for all observational sciences; it is so simple and so natural, that we may not appreciate enough the immense step it has made for the human mind. We do not know to whom it is due; this is how all the great discoveries have been established, without knowing their inventors. [. . .] Let us note from now on that by focusing on the idea of the average for quantities that can vary, we may have overlooked the limits within which variations occur. Wherever we can say more or less, we necessarily have three things to consider, an average state and two limits. Without resorting to science, habit gives us a vague appreciation of the average and the limits of variations that belong to each variable element that nature or the social state presents to us; it is, according to this appreciation, that we are guided in our reasoning. But it is suitable for the progress of enlightenment to substitute precise ideas for vague notions”.102

Then comes, on pages 65–66, a crucial distinction in the eyes of the tutor: “When taking an average, one can have in mind two very different things: one can seek to determine a number that truly exists; or to calculate a number that gives the closest possible idea of several different quantities, expressing homogeneous things, but

98 Around 1835 because this introduction, p. i–iv, is signed “Brussels, December 18, 1845”; ten years earlier the first edition appeared in Paris that Quetelet, 1835. 99 Around the time when Quetelet was the tutor of the young princes of Saxe-Coburg. 100 Quetelet, 1845. 101 Lottin, 1912. 102 Quetelet, 1846, p. 60–61.

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variable in magnitude.” He thus distinguishes the average of several measurements of the height of the same building and the average of the heights of several different buildings aligned on the same street: “There is a very notable difference between these two examples that one may not have grasped at first glance, but which is nonetheless of great importance. In the first, the average represented something that really exists; in the second, it gave, in the form of an abstract number, a general idea of several things essentially different, although homogeneous [. . .]. This distinction, [. . .] I will even use different words to better establish it: I will reserve the name of average for the first case, and I will adopt that of arithmetic mean for the second, in order to make it clear that this is a simple calculation operation between quantities that do not have essential relations”.103

Then come examples of meteorological and astronomical observations likely to reveal frequency curves similar to those resulting from random binomial draws whose good old paradigm is the drawing of black or white balls among a thousand of them distributed equally in an urn: the physical phenomena considered provide graphs thus commented; “With a few exceptions, we see that this line rises [to express that it is increasing] in a fairly symmetrical way to reach a maximum value and then to lower itself”, (p. 103) and he immediately adds “I have chosen not to represent [the extremities] in the figure, for even a magnifying glass would not have been enough to appreciate them” (Illustration no 3.2). These pages on the distinction of averages vis-à-vis arithmetic means and on the resemblance of observed frequency curves with a theoretical curve constructed using a binomial or analytically aim to prepare the putative recipient of the letters – taken to witness from time to time at the cost of rhetorical formulas and some chatter – to the heart of the Quetelet doctrine: the apologue of the Gladiator. This name refers to a famous Hellenistic marble from the Borghese collection, once found in the excavations of Nero’s villa at Antium and now preserved. at the Louvre Museum:104 Illustration 3.2 Frequency histogram published by Quetelet (1846, p. 103)

103 104

Quetelet, 1846, p. 66–67. The apologue is discussed on p. 133–137 of Quetelet, 1846.

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“The Gladiator is undoubtedly one of the most beautiful works of ancient sculpture. It is with good reason that artists have studied its noble and unencumbered forms, and they have often measured the main dimensions of the head and body, to grasp their relationships and harmony. Measuring a statue is not as easy an operation as one might think at first glance, especially if one wants to obtain it with great precision. By measuring the chest circumference ten times in a row, one is not sure to find two identical results. It almost always happens that the values obtained are more or less distant from the one we are looking for; and I even suppose the most favorable circumstances, those where one would have no tendency to take measurements too large or too small. If one had the courage to start over a thousand times, one would end up with a series of numbers that would differ from each other according to the degree of precision that one had put into collecting them. The average of all these numbers would certainly deviate very little from the true value. Moreover, by classifying all the measurements in order of magnitude, one would not be moderately surprised to see the groups succeed each other with the greatest regularity. The measurements that deviate the least from the average general would make up the most considerable group; and the other groups would be all the smaller, as they would contain measurements more in disagreement with this same average. [. . .] If one had to measure the chest of a living person instead of that of a statue, the chances of error would be much more numerous [. . .]. Despite this disadvantage, the thousand measurements grouped by ranks of magnitude would still proceed in a very regular manner. The line that would represent them, would always be the curve of possibility, but dilated in the horizontal direction, proportionally to the probable error. Let’s modify our hypothesis again, and suppose that a thousand sculptors were used to copy the Gladiator with all the imaginable care. [. . .] The thousand copies that will have been made [will not] each reproduce exactly the model, and [. . .] by measuring them successively, the thousand measurements that I would obtain [would not] be as concordant as if I had taken them all on the statue of the Gladiator himself. To the first chances of error would be added the inaccuracies of the copyists; so that the probable error would perhaps be very large. [However,] the experiment is all done. Yes indeed, more than a thousand copies of a statue that I will not assure to be that of the Gladiator, but which, in any case, deviates little from it: these copies were even alive, so that the measurements were taken with all the chances of possible error: I will add, moreover, that the copies could deform due to a host of accidental causes. One should therefore expect, here, to find a very sensitive probable error. I come to the fact. One finds, in the 13 e volume of the Edinburgh Medical Journal, the results of 5,738 measurements taken on the chests of soldiers from different Scottish regiments. These measurements are expressed in English inches and grouped by order of magnitude”.

Before going any further, a historiographical comment deserves to be formulated. The apologue was so convincing and the figure of 5738 so notable that all historians of statistics have copied them, victims of this hypnosis of printed figures designated by Condorcet as early as 1778 in the passage placed as an epigraph to this chapter. No doubt the Scottish source was not easy to find.105 It was a simple news article in a medical current affairs journal.106 It was published in 1817107 and summarized two anthropometric measurements taken on nearly six thousand soldiers from eleven local regiments of the Scottish militia (Annan, Argyll regiments, Lanark, Eskdale, Fifeshire, Edinburgh, Peebles-shire, Kinross-shire, Stirling-shire, Kirkcudbright,

105

Fortunately, we can now access the digitized collection of this Journal through the page: https:// www.ncbi.nlm.nih.gov/pmc/journals/2495/#edinbmedsurgj 106 On this periodical, see Coyer, 2015. 107 And not in 1835 as indicated by Lottin, 1912 (p. 176), who must confuse with Marshall, 1835.

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Illustration 3.3 A box from the Edinburg Medical Journal of Edinburgh (1817)

Lanark-shire,): their height and the circumference of their chest. For each of the counties, a cartouche presented the cross-table of the number of soldiers according to their height and chest circumference, both measurements being given in inches. Each time a table of five lines by four, and one and sixteen columns delivered the distribution of the local headcount according to sizes (four columns), the marginal totals on the fifth and the chest circumferences (on the remaining sixteen) (see an example in Illustration n° 3.3). It is clear that this source contained errors: thus in the first line of the cartouche relating to the regiment of the Kirkcudbright militia (the one at the bottom of page 262, and reproduced in Illustration n° 3.3), the total number of soldiers whose chest measurement is distributed between 35 and 44 inches is 35 and not 34. Moreover, the table itself is inconsistent because the sum of its fifth column is equal to 595 while the number of soldiers indicated in the title is 596, indeed the total of all the staff from the sixth to the twenty-first column. It is true that on the first line, the total of conscripts measured is 34 in the fifth column, while the sum of the staff distributed on the scale of chest circumferences, always along this first line is 1 + 1 + 1 + 10 + 7 + 7 + 6 + 1 + 1 = 35. Even before the publication of the Letters (1846), the Belgian scholar had relied on this Scottish article in a methodological article published in the Bulletin of the Central Commission for Statistics in 1845.108 He had given there what Lottin (1912) rightly considered the first expression of the theory of the average man. However, it is amusing to note that the tutor announced each time he mentioned this source a total of 5738 measured militiamen whereas the figures published in the original, once compiled, amount to 5734. Since these cartouches distribute, in each county, the soldiers according to the measurement of their chest circumference, it is not forbidden – using the convenience of a current spreadsheet – to calculate the actual distribution of the whole of these 5734 soldiers on the sixteen measurement marks from thirty-three to forty-eight inches. In principle, to establish the table he published in 1845 and 1846, Quetelet should have compiled the content of the cartouches himself if he did not have it done by a close associate. The frequencies 108 The source is Ed. Med. Journ., 1817, p. 260–264 (now digitized). The previous mention is in Quetelet, 1845, p. 258.

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displayed by Quetelet unfold according to the shape of the binomial curve or those of a distribution according to the Laplace and Gauss formula. But on closer inspection, we discover that the relative differences between the frequencies published by the astronomer-statistician and those we can reconstruct vary greatly, sometimes reaching nearly ±30%. The promoter of the average man has therefore somewhat forced the figures to make them look good. Another difference between the initial publication and the use made of it by the Belgian scholar deserves attention: the latter presupposed that the Scottish conscripts would form a homogeneous population whereas the last paragraph of the report on anthropometric measurements specifies that their author (“an army contractor, a gentleman of great observation and singular accuracy”) hoped to deepen his research to highlight the local variability of the measured sizes according to the characteristics of each county.109 In this affair, as in another that we have already commented on elsewhere, Quetelet therefore proves to be quite a fabulist.110 But in his defense, even a superficial glance at the cartouche reproduced above suggests to anyone who would insist on distributions centered on a maximum frequency, first increasing then decreasing, that the distribution of the size of the conscripts (the fifth column) does indeed have such a look, and that for a given size, the distribution of chest sizes also does (it can be read on each line). As a year earlier, he presented in the Letters the same table extrapolated from the Scottish source.111 The tutor then drew this conclusion: “The example I have just cited [. . .] shows that things happen absolutely as if the chests that have been measured had been modeled on the same type, on the same individual, ideal if you will, but whose proportions we can grasp through a sufficiently prolonged experience. If such was not the law of nature, the measures would not group together, despite their defects, with the astonishing symmetry that the law of possibility assigns them”.112

Here again we must stop because, like a skilled magician, the tutor has played a sleight of hand. He considered a proposition, let’s call it A (the hypothesis of the average man), and the shape of the distribution of the measures considered (confusing error and variability along the way), it is here a shape, let’s call it S. He first posited that property A, and resulted in shape S. From a logical point of view, this would mean that “A implies S”, or “A => S”. In such a case, the contrapositive is logically true: without S, no average man A “~S => ~A” (~ here to mean: non. . .). But, he wrote erroneously to appeal to the common sense of his student and his reader: without an average man A, no pleasing curve S: “~A => ~S”, most probably thinking the contrapositive of this assertion “S = > A”. Thus for Quetelet A and S were equivalents. Quetelet’s common sense argument expresses nothing more than

109

Ed. Med. Journ., 1817, p. 264. See Brian and Jaisson, 2007, Chap. 3, p. 66–69: it is about the thesis of the influence of the parents’ age difference on the probability of the birth of a child of one sex or the other that Quetelet promoted under the name of the “Hofacker-Sadler” law, a speculation completely unfounded neither in the facts nor in a relationship between the two cited authors. 111 Quetelet, 1845, p. 259 and Quetelet, 1846, p. 400. 112 Quetelet, 1846, p. 137. He writes elsewhere “law of probability“. 110

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his faith in the average man and in the average: confusing A ) S and S ) A (i.e. necessary condition and sufficient condition). The reasoning failed and the mystique of the average is here caught in a trap that the tutor set to himself. The doctrine was sealed in the mid-1840s and the statistician had another opportunity to express his convictions the following decade before his academic colleagues, then in a popularization book that renewed the Instructions of 1728. The communication to the Belgian Academy published in 1848 responded to a wind of criticism issued by defenders of free will who targeted the determinism attributed to the statistician.113 The question of free will has indeed been vigorously debated after the publication of the astronomer’s theses on the application of calculation to moral sciences: it involves, from a theological point of view and after Augustine of Hippo (354–430), the individual responsibility of each one. From the first pages, the scholar has designated his intention and mobilized an analogy that is unlikely to have convinced his opponents: “Individual observation is baseless because of free will. But this free will, which makes individuals, taken separately in their sphere of action, escape all our conjectures, does it extend its action far enough to also make impossible the predictions that would concern a more or less large number of men? This is what needs to be examined. Only experience can enlighten us. on this delicate point. [And a few lines later:] From the various researches I have undertaken, I believe I can deduce, as a fundamental principle, that man’s free will fades and remains without noticeable effect when observations extend over a large number of individuals. It would therefore result that the effects of all individual wills neutralize or destroy each other, absolutely as the effects that would be produced by purely accidental causes. [. . .] It is appropriate to proceed here as the physicist who, for electrical phenomena, can also only give relative values, and is reduced to judging causes by effects. We do not perceive any more what gives birth to the moral phenomenon than what produces the electrical phenomenon. We only see the effect itself, and it is this effect that we seek to appreciate “[. . .] I will admit this fundamental principle of all observational sciences that effects are proportional to causes”.114

As for calculation methods, he referred to the Letters to arrive at this pictorial explanation of his analysis of the inclination to crime (Illustration n° 3.4): “In summary, it is conceivable that there exists in all men a certain possibility of being in hostility with the laws and of committing some reprehensible act. This possibility, however minimal it may be, admits of lower degrees to the point of becoming absolutely null, just as it can grow, in some, to become equal to certainty. Thus, on the one hand, some men will certainly not oppose the laws, while on the other hand, this opposition will manifest without any doubt. The other men, in greater number, will approach more or less the first ones. The following figure may make this distribution visible to the eyes”.115

113 On the Belgian scientist’s side, it is about Quetelet, 1848a, which we comment on here; the memoir is contemporary with the philosophical critique of Gouraud, 1848. The Benedictine Joseph Lottin will address the question in Lottin, 1908; many authors have revisited this theme; a state of the art has recently been traced in Seneta, 2003. 114 Quetelet, 1848a, p. 4–7; the italics are in the original. 115 Id., ibid., p. 13–14.

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Illustration 3.4 The inclination to crime according to Quetelet (1848a, p. 14) (Letters a, o, b, i added for readability)

a

o

b

i

“At point o, the probability of crime is absolutely null. The probability increases as one moves away from point o, to move to the right, and it converts into certainty at point i. The curved line oai, by the magnitude of its deviations from the straight line oi, indicates the number of people corresponding to each probability. Thus, the maximum number, represented by the ordinate ab, has for it the probability ob of committing a crime. The line of the inclination to crime here takes the form of the possibility curve; I must warn, however, that I present it as such only by induction. The identity, moreover, matters little for the object that concerns us. It must be understood only that the curve has been given here for a certain age of man, and that, while remaining a possibility curve for other ages, its limits and consequently its form can change”.116

As before, the ends that lead to infinity are not represented, an absence that the physical limits of the page may explain but who has accustomed readers not to take into consideration extreme events on the horizontal axis, whose characteristic magnitude would be very far from the center of the curve. The latest popular science book published by the Belgian scientist in 1854 is part of a scientific information encyclopedia. From the first paragraphs of the foreword, the author places this new opus in the flow of his other publications: “This small work is written on the same plan as the Instructions [1828] that have been translated into different languages. The first edition has been out of print for over twenty years; the new edition contains the results of various writings that I have inserted in the Memoirs of the Royal Academy of Belgium and in the Bulletin of the central statistical commission of the kingdom. However, I could not develop as much as I would have liked the part that concerns moral and political sciences; those who would like more information on this branch so interesting and at the same time so new to science, can find them in my Letters [1846]. Those who would like more mathematical developments can consult the French treatises of Laplace, Poisson, Lacroix, Cournot; the German and English works of Gauss, Hagen, Fischer, Galloway, de Morgan, etc.117” [Then comes the creed]: “The theory of probabilities should serve as the basis for the study of all observational sciences; I would consider myself fortunate if this booklet could contribute to propagating the taste for it and to appreciating its importance”.118

116

Id., ibid., p. 14. Quetelet, 1854, p. 5–6. 118 Id., ibid., p. 6. 117

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103 0⬘

c 32

28

24

20

16

12

b

a

b⬘

a⬘

8

4

0

4

8

d 12

16

20

24

28

32

Illustration 3.5 The Law of accidental causes according to Quetelet (1854, p. 56)

If the whole work indeed follows the plan of the Instructions, the reasoning according to the average and the symmetrical shape of the distribution of errors and variations is the subject of a clear exposition accompanied by a graphic illustration offered without further ado (Illustration no 3.5): “[. . .] if it is true that the errors of the measures have occurred under the influence of purely accidental causes, and without there being a tendency to take measures either too large or too small: 1° The groups of positive errors and those of negative errors will be equally numerous [sic, understand in equal numbers] and essentially the same on both sides of the average; 2° The numerical value [understand the effect] of each group decreases as one moves away from the average; 3° These numerical values are calculable in priori and depend on a formula that expresses the law of possibility or of accidental causes. (note italics in the original) [. . .]. The law of accidental causes can be made more apparent by the following figure [. . .].” “The numbers placed on the horizontal line cd, indicate, by their distance from the [central] point o, the respective magnitude of positive and negative errors; and the perpendiculars that correspond to them represent, by their length up to the curve, the number of errors. Thus, the perpendicular aa’ indicates, by its length, how many positive errors of 6 millimeters there are comparatively; and the entire surface aa’ oo’, between this perpendicular and the average oo’, represents the sum [sic, for the number] of errors from 0 to 6 millimeters. The perpendiculars bb’ and aa’ serve as limits to the probable error, and the portion bb’a’a of the surface that they include between them, is equivalent to those aa’d [p. 57] and bb’c placed externally and representing the [number] of positive and negative errors greater than 6 millimeters. This curve assumes that the accidental causes of error, more or less, are perfectly equal and independent. It sometimes happens that the accidental causes that tend to exaggerate the quantity sought do not have the same probability as those that tend to diminish it, and for example that the errors in addition have a tendency to be larger than the errors in subtraction. The curve then loses its symmetry”.119

In the following pages, Quetelet intended to introduce the way of reasoning derived from the calculation of probabilities subsequent to Laplace: integral calculus led to posing the problem of the analysis of the distribution of observations in a richer way

119

Id., ibid., p. 56.

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than the old calculation of the ratios of the numbers of favorable cases to the numbers of possible cases. The curve represented on the schematic graph restored the integral 2 of a function of the kind of e - x : that identified by Laplace, Fourier and Poisson. The graphical representation made visible the surface under the curve, if necessary before or after the verticals aa’ and bb’, that is in terms of integral calculus, the integral of this function from c to b’, then between b’ and a’, finally from a’ to d. As in other illustrations of this kind the ends are indicated elliptically, here by the points c and d. But we can also note that Quetelet arbitrarily proposed a magnitude for the probable error interval (what we could understand today as some confidence interval): beyond this interval he expects that the errors will be equally probable on both sides; and on each side, the probability of more and that of less should be equal to that admitted for the probable central error. That is to say that under the curves from c to b’, then from b’ to a’ and finally from a’ to d there would be three equal surfaces, thus three equal probabilities which would cover each of these three palettes of values. Their total being equal to one, each portion of the graph would therefore correspond to a probability of one third. However, Quetelet knew perfectly well how to calculate these surfaces using Kramp’s table (1799) but he said nothing about it. It was therefore for aesthetic balance reasons that he schematized an equality between the probability of the known central value with a margin of errors, and those of excess errors and default errors. Nothing, neither in the analytical theory of probability calculation nor in the practice of observations justified such a balance between these three sectors. The didactic concern therefore prevailed over mathematical rigor. Translated into an anachronic confidence interval of a Laplace-Gauss distribution, such a probable error interval, if it covered a third of the area under the bell curve would have an amplitude of approximately 0.45 standard deviation. This means that, without resorting to tables or an analytical calculation, Quetelet envisaged a balance between the numbers of observations close to the mean and those of their counterparts on either side by presupposing a strong concentration of observations around the center. By comparison, in the mathematical statistics of the 20th century, the practice is to take into consideration confidence intervals not at 33% but at 95% whose deviation is approximately equal to 2 standard deviations and no longer at 0.45 standard deviation. This observation obtained by a calculation based on the example published in 1854, p. 56–57, clearly shows to what extent the statistician wanted to see observations fit into the mold of the process he was promoting. It must be acknowledged that in order to make himself accessible to the reader and avoid having to define an integral, Quetelet used a confusing vocabulary: numbers, frequencies, and probabilities are here summoned indiscriminately and these blunders remind us of the confusion of Quetelet’s passages on the theorem of Jakob Bernoulli. It is as if reading his predecessors had sealed his approach to what he long called the “law of possibility” and now the “law of errors”, and to direct and converse problems. In any case, after the Letters of 1846 and the manual of 1854, Quetelet’s conception of the law that will be called normal has reached its maturity as the new terminology reveals.

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The large book published in 1835 and its second edition, given in 1869 are each formed of two volumes organized in the manner of the same continuous treatise. In addition to additional results, the last edition during the author’s lifetime includes as a preface the laudatory article by the astronomer John Frederick William Herschell which had been a commentary on the Letters (1846) – first published anonymously in 1850 and which has been discussed above.120 The first pages written by Quetelet clearly state his objectives and his ambitions: “The work that I present to the public is in a way the summary of all my previous work on statistics. It consists of two distinct parts; the first three books contain only facts; the fourth contains my ideas on the theory of the average man and on the organization of the social system”.121

Quetelet wanted to make an impression with a formula that aimed to forcefully designate the numerical regularity of statistics from the French Ministry of Justice. The formula prepared, a few pages later, a methodological profession of faith of which the astronomer was not unaware of the boldness price: There is a budget that is paid with a frightening regularity, it is that of prisons, penal colonies and scaffolds.122 [. . .] Moral phenomena, when we observe the masses, would in a way fall into the order of physical phenomena; and we would be led to admit as a fundamental principle in research of this nature that the larger the number of individuals observed, the more individual peculiarities, whether physical or moral, fade and allow the predominance of the series of general facts by virtue of which society exists and is preserved.123 [. . .] In what pertains to his physical qualities, but even in what relates to his actions, man is under the influence of causes, most of which are regular and periodic; and have equally regular and periodic effects.124 The man I consider here is, in society, the analogue of the center of gravity in [physical sciences]; he is the average around which social elements oscillate: this will be, if you will, a fictitious being for whom all things will happen according to the average results obtained for the society. If one seeks to establish, in a way, the foundations of a social physics, it is him that one must consider, without stopping at particular cases or anomalies.125 [. . .] The nature of the research I am engaged in this Work and the way I view the social system, have something positive that [can] at first frighten some minds: some will see a tendency towards materialism; others, misinterpreting my ideas, will find an outrageous pretension to enlarge the domain of exact sciences and to place the geometer on a ground that is not his own; they will reproach me for engaging him in absurd speculations, by occupying him with things that are not susceptible to being measured.126 [. . .] Certainly, the knowledge of the wonderful laws that govern the system of the world, which we owe to the research of philosophers,127 gives a much greater idea of the

120

Herschel, 1869. Quetelet, 1835, vol. 1, p. i (preface dated April 15, 1835). 122 Id., ibid., p. 9. 123 Id., ibid., p. 12. 124 Id., ibid., p. 13. The expression of this conviction retrospectively suggests how Quetelet could have progressed from his thesis on conics to his beliefs about human statistics: the key here is the hypothetical periodicity of the phenomena considered. 125 Id., ibid., p. 21. 126 Id., ibid., p. 26–27. 127 Here Quetelet refers to the works of Newton and Laplace. 121

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power of the Divinity, than that of this world that a blind superstition wanted to impose on us. If man’s material pride has been frustrated in seeing how small the place he occupies on the grain of dust that he made his universe, how much his intelligence must have rejoiced to have carried its power so far and to have delved so deeply into the secrets of the heavens128! -

On the methodological level, Quetelet repeats from page to page that it is important to gather “exact observations in sufficient number to arrive at results that deserve some confidence.” To achieve this goal he has provided the reader with everything he could find in specialized periodicals from public institutions, independent learned societies or isolated observers (whom he designates in the post-Aristotelian manner by the term “authorities”129), covering a vast European perimeter, and sometimes even sources collected in the colonies or Asia. All figures churned, all these quantitative material, he has properly made them enter into the scheme of what he calls “the law of probability” or “the binomial curve”, a scheme that he synthesizes 2 by repeated graphs of exponential distributions in e - x . The second edition is quite similar except that other material are added.130 Quetelet, who, as we have seen, is a prince of persuasion, proceeded in 1869 in a less direct manner than thirty-four years earlier: he insinuated step by step the distribution model he was promoting, proceeding as before with a petition of principle in which he trapped his reader. “If I were allowed to conjecture with the little information we have, I would be inclined to believe that here [in terms of birth measurements] the same law would be observed that is recognized in all phenomena related to man [followed by the call of this note: I indicate here by a figure the presumed state of science. Experience, I hope, will later tell whether I was wrong or right in my conjectures131!]”

The curve in question, offered in small format as it appears inserted in a note, is that of the distribution of frequencies of the duration of human gestations according to Belgian statistics. It presents a very sharp peak for nine months (!) and minimal deviations around this mode. The reading of this narrow bell curve will have been shaped by common knowledge about pregnancy. But here is that the second volume of the 1869 edition, which had been announced from the first pages of the set as methodological in nature, sees reappear, contrary to the first edition, the chest measurements of the Scottish conscripts already commented on by Quetelet in the Letters.132 The source of the Edinburgh Medical and Surgical Journal is indicated (without the year being specified). The table is offered to the reader as in the Letters with the same differences from the source – one still thinks of what Condorcet wrote about “everything that is written in numbers”. Assuredly, Quetelet read it assiduously.

128

Id., ibid., p. 27–28. Id., ibid., for example, p. 136. 130 Id., ibid., p. 27–28. The recent reissue, 1997, indicates the variations between the two initials. 131 Id., ibid., 1869a, vol. 1, p. 160–161; and 1997, p. 56. 132 Id., ibid., vol. II, p. 56; and 1997, p. 343. 129

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Table 3.1 Comparing Quetelet’s source (1817) and his report (1846)

[0] 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Totals

107

[1]

[2]

[3]

[4]

[5]

[6]

3 19 81 189 409 753 1063 1082 935 646 313 169 51 17 3 1 5734

5 33 141 330 713 1313 1854 1887 1631 1127 546 295 89 30 5 2 10000

3 18 81 185 420 749 1073 1079 934 658 370 92 50 21 4 1 5738

5 31 141 322 732 1305 1870 1880 1628 1147 645 160 87 37 7 2 10000

5 31 141 322 732 1305 1867 1882 1628 1148 645 160 87 38 7 2 10000

0% -5% 0% -2% 3% -1% 1% 0% 0% 2% 18% -46% -2% 24% 33% 0%

Titles of columns of Table 3.1 [0] Chest circumference measurements in English inches [1] Total by size recalculated in 2023 starting from the 1817 source [3] Total by size given by Quetelet in 1846 and later [4] Actual frequencies in ten-thousandths calculated in 2023 [5] Frequency in ten-thousandths given by Quetelet in 1846 and later [6] Relative differences between columns [1] and [3]

As for the total general and the calculation of frequencies for chest measurements far from the central value, the following table (Table 3.1) will allow the reader to have a clear heart. The errors of copies or calculations are manifest as a certain negligence towards the largest busts. Illustration no 3.6 shows the frequencies of conscripts on the scale of chest sizes according to Quetelet’s own compilation and after a new one of the tables published in Edinburgh in 1817 for the preparation of this work. It is true that a variation of a few units on the total would not radically alter the shape of the curve. However, the demonstration of the astronomer, from a numerical point of view, does not come out here strengthened, especially if one pays attention to the significant deviations that appear when returning to the source. It remains nonetheless acceptable from the schematic point of view that Illustration no 3.6 restores. The promoter of the average man was probably no longer concerned with the calculations made from the initial source to which he referred in a vague enough way to discourage critical readers. He reproduced them as he had published them in 1846 – and this until 1870 – to illustrate the principle of his theory. His readers since then sometimes criticized the principles but not the initial empirical basis around

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Illustration 3.6 Comparison of the frequency curves for the measurements of Scottish conscripts according to the figures published in the Medical Journal of Edinburgh (1817) and according to Quetelet from 1846 Horizontally: scale of chest sizes measured in inches Vertically: frequency for a given size, in ten-thousandths Grey histograms: distribution for the figures published in Edinburgh (1817) Black histograms: distribution for those published by Quetelet (from 1846)

which Quetelet had aggregated all other phenomena according to the profession of faith he affirmed in the first pages of the initial edition of the Social Physics (1835). He never stopped asserting that the profile of the distribution was general. and to introduce the figures that his statistical activism put within his reach. This was the case, for example, with the measurement of 1516 chest circumferences of Potomac conscripts mobilized to serve during the Civil War, the compilation of which had circulated during the fifth session of the International Congress of Statistics held in Berlin in 1863. In doing so, he betrayed the hybris that drove him: “we could not resist [. . .] the desire to give as an example, the table of the chest circumference of the soldiers of the Potomac army, which can be compared with that concerning the chest of the soldiers of the Scottish army”.133 The distribution of their frequencies, recalculated by Quetelet, indeed resembled that which he had constructed on the Scottish conscripts. In Quetelet, the conviction that phenomena presented themselves according to the scheme of a Laplace-Gauss distribution governed the way he looked at compilations of figures, his determination to fit any source into this scheme, and the intensity of a persuasion more assertive than demonstrative. In the 1980s, I had the opportunity to work with the quantitative historian Charles Tilly (1929–2008) on a project conducted at the Maison des Sciences de l’Homme in

133

Id., ibid., vol. II, p. 59–60.

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Paris, during which we exchanged databases resulting from the computer input of publications from the statistical offices of the early 20th century. On this occasion, he entrusted me with a magnetic tape (the medium used at that time) produced by his team of students at the University of Ann Arbor in Michigan where he taught. When I asked him about the checks carried out after the input of sources, he replied: “Well, you know, we sample the input and if we find less than 5% errors, we consider that everything is fine.” This 5% criterion came from a rough application of the LaplaceGauss law to the students’ input errors. Thus, the scheme that Quetelet forged has even entered the finest minds until the end of the 20th century.134 Let’s stop on pages 54 to 61 of volume II: Quetelet finally asserts (p. 61): “after the complete development of individuals of both sexes, the weights are approximately like the squares of the sizes”. This is the birthplace of the “Body mass index” (BMI) known as “Quetelet’s”, probably today his most widely known result as the World Health Organization recognizes the BMI, P/T2, as a relevant criterion revealing individual under or overweight. However, it is true that the preference for this index is also criticized by some specialists. Further on, the statistician comes to moral phenomena. He first indicates the methodological hypothesis he adopts, then he specifies his definition of the propensity to crime.

134 The tolerance for data entry errors in the manner of Charles Tilly was undoubtedly characteristic of the division of labor in data processing at a time when operations were performed on large computers, such as the IBM™ 370 (from the 1970s): historians, statisticians, programmers and data entry personnels then took turns around the same documentary source. I was fortunate to benefit from two of the first IBM-PC™ microcomputers given in this case to the Maison des Sciences de l’Homme (Paris) and Columbia University (New York). So much so that I resolved to systematically take advantage of the flexibility of programming on the spreadsheets available from these 1980s on these microcomputers (Lotus™ and Excel™) with which a single user could combine the range of skills previously divided. In such a case the internal consistency of the sources to be entered (they almost always present marginal totals) can indeed be verified in real time, data entry error notifications and calls for corrections issued even as the manuscript or printed source remains within reach. This is an effective technique for detecting old errors underlying the original transcribed electronically (copies, typographies or calculations dating from the time of the sources). On these empirical issues in contemporary economic history, see Brian, 2001 and 2011. Here, the use of a recent version of the last mentioned spreadsheet allowed me to piece together the tables published in 1817 being certain to conform to the figures then collected. We are so accustomed to the potential of such use barely more than forty years old that few readers were probably shocked to read that I had corrected the copy of the original given by Quetelet, an unthinkable gesture before 1960. Everything suggests that the variations from the source observed in the versions published by Quetelet from 1846 come from the necessary recourse, in his time, to uncertain copies and less rigorous calculators than contemporary machines. Yet, Quetelet was an astronomer and in this field, calculations were not taken lightly. In the past, they were tedious and the statistician probably delegated them contenting himself with the sketch of a result that he considered general and convincing.

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Illustration 3.7 Chest size of Scottish soldiers representing column [5] of Table 3.1, according to Quetelet (1869a)

“Certain moral qualities are roughly in the same case as physical qualities and can be appreciated by admitting that they are proportional to the effects they produce [p. 98]. Assuming men placed in similar circumstances, I call propensity to crime , the more or less great probability of committing a crime [p. 160]”.

The general summary given at the very end of the second volume is a recapitulation of the theory of the average man and his method: the examination of the distribution of frequencies of the measures taken into consideration. It is illustrated by the curve reproduced in Illustration no 3.7, but modified: the axis of measures (AC) is no longer horizontal but vertical (p. 441), the scheme having pivoted a quarter of a circle in the anti-clockwise direction. It has thus become freed from its initial empirical motivation, as if cut off from the measures it synthesized, becoming the mark of a principle of reading the data in the digital sense and the statistical knowhow of the mid-19th century. The book Anthropometry was dedicated to Herschel and published in 1870. It is made up of the array of results published the previous year and a state of the reflections maintained by the astronomer since his youth about the proportions of the human body in the arts.135 As we can see, the groundwork had been prepared for him long ago for the formulation of the apologue of the statue of the Gladiator (1846). As for the defense and illustration of the doctrine, one more year has not changed anything. Some new sentences, sometimes, are enlightening: they reveal the endpoint of the Queteletian doctrine. Thus for example:136

135 Thus the table of frequencies of chest measurements in Scotland and in the Potomac appears on p. 289 in Quetelet, 1870. 136 Quetelet, 1870, p. 415, the italics are in the original.

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“Each binomial line has three points that must be considered separately: these are the two extreme points through which it will touch the axis of the abscissas,137 then the maximum intermediate point, the highest above this same axis.138 This maximum point, in the order of growth, reveals, by its height, the number of men of average height; the two extreme points represent, one the dwarfs , and the other the giants. In the true state of science, it is no longer enough to give the average height of the man of a country, to know what concerns his size: it is necessary to also give the extreme or maximum or minimum size;139 it will also require the surface of the curve and its center of gravity.140 These elements, until recently, have been consistently neglected in research related to anthropometry”.

Along the way, he confided to the reader:141 “What especially deserves our attention, what has been the constant object of our studies [. . .] is that human sizes, while appearing to develop in the most accidental way, are nevertheless subject to the most exact laws; and this property is not particular to size: it is also noticeable [. . .] in everything that relates not only to its physical qualities, but also to its intellectual and moral qualities. For more than forty years I have been striving to highlight it, I have had the satisfaction of seeing it spread successively in different countries, especially in Germany, Italy, England, Scotland, and America”.

Carried by his pride, in this last sentence, the statistician betrayed a strategic vision. Indeed, one of two things: either this regularity was a general property that would have been widely observed around the world, or the average man remained in his eyes a doctrine that he had forged and which now spread across the planet for his satisfaction. Thus, by his own admission, Quetelet forged a mystique and made himself its proselyte. In July 1855, Quetelet collapsed from a temporary stroke.142 He progressively recovered and resumed his Belgian and international activities. As Lottin observed, his publications from then on were quite repetitive. Quetelet died in February 1874, between these two dates, besides his active participation in sessions of the International Congress of Statistics, he endeavored to publish retrospective studies,143 summaries of his own work,144 those of the Congress,145 and updates on methodology.146

137 Certainly not: the trend towards the x-axis is indeed asymptotic and these points, strictly speaking and thus defined, would be at infinity on either side of the center. It is the necessities of the material conditions of the reproduction of graphics that create the illusion that these points would be at a finite distance from the center. 138 That is to say, in more elaborate terms, “the mode” of the distribution. 139 Illustration no 3.4 with which Quetelet had argued for the “inclination to crime” and his commentary already followed this principle. 140 Still on illustration no 3.4, we saw that the areas under the frequency curve, and therefore the integrals of the represented function obeyed a balance on either side of the central value. 141 Id., ibid., p. 257–258, the italics are in the original. 142 Lottin, 1912, p. 84. 143 Quetelet, 1867, 1868. 144 Id., 1869a, 1870, 1872, 1873a. 145 Id., 1873c. 146 Id., 1873b.

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Averaging the World

In 1853, armed with the doctrine set out in 1846 in the Letters, Quetelet formed around him the arena of an international statistical congress and that of its double, the international meteorological congress, while astronomers still maintained mutual exchange relations formed over millennia. Thus, the second half of the 19th century saw the global triumph of the Queteletian conception which henceforth covered the States, the Seas, and the Skies. His doctrine’s imprint can be traced in the minutes of the statistical congress meetings for twenty-five years. From the first session, in 1853 in Brussels, the concern for comparisons of countries through their official statistics was displayed. Agricultural censuses are an example at a time when agriculture remained one of the main indicators of wealth production in a country: “It is necessary that agricultural censuses be carried out, in all countries, in such a way as to establish comparable results, and, therefore, that they relate everywhere, either to the same agricultural year, or to an average year expressing a normal situation. However, while leaving the choice of the time [of the censuses] to the discretion of the various governments [. . .], the congress expressed the wish that preference be given, as far as possible, to the last quarter of the year, considered as the most suitable time.147 [Further on are discussed] the annual averages or those related to numbers of inhabitants”.148

Two years later, at the Paris session, it was not only the combinations offered in results that were sifted through the average, but the initial measures themselves, as in the following excerpt about the statistics of transport infrastructure. On this subject, which was currently undergoing major changes in Europe, the call for physical descriptions even understandable borders on the absurd: “[Regarding the operation of railways according to various categories, each time]: “average length operated in the year,149 and for inland navigation [. . .] record for the “remarkable points of the course [of a] river forming section [. . .] the average width of the bed and the average slope per kilometer”.150

As for the factors that contributed to the success of the congress, one is tempted to paraphrase about Quetelet what the great historian Lucien Febvre (1878–1956) wrote about this almost contemporary of the statistician, the historian Jules Michelet (1798–1874): “An entire century conspired with him151”. From the first decades of the 19th century, indeed, states in Europe, sometimes disrupted by recent wars, sometimes loyal to one of them or competitors, have undertaken to build specialized offices in the regular production of administrative statistics. In the case of continental empires, Austria, Germany, Russia and in the national case of Italy of the Risorgimento, as in the case of the colonial rival expansion of France and Great Britain, the 147

CIS, 1853, p. 149–150. Id., ibid., p. 149–150; 188; 190;245. 149 Id., 1855, p. 54–58. 150 Id., ibid., p. 62, note that the minimum width would have provided a more relevant element: the template boats likely to use the considered navigation route. 151 Febvre, 2021. 148

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formation of such offices went beyond the construction of administrative updating: these were instruments of unification of these empires or these nations.152 Less extended independent countries, such as the Scandinavians or the states fragmented in the Germanic world until the Franco-Prussian war, and nations in gestation such as those that formed within the Austrian empire have formed comparable offices with the dual purpose of marking a national authority and in the latter case of inserting themselves into the Austrian statistical edifice. To these both general and local factors, is added the fact that the congress developed in the years that followed the European revolutions of 1848. So much so that a profile emerges in the core of its most active members: that of statisticians and economists most often liberals, sometimes nationalist activists within the Habsburg empire but volens nolens artisans of the restoration of a so-called neo-absolutist order after the revolutionary year, during the reigns of Franz Joseph I er (1830–1916) in Austria (he was crowned in 1848); Victoria (1819–1901) in the United Kingdom, (crowned in 1838), or Napoleon III (1808–1873) in France (first President of the Republic in 1848, then after a coup d’État emperor in 1852).153 Among the most active members of the congress, there are thus astonishing trajectories such as that of the Hungarian Keleti Károly (1833–1892) – or Charles Keleti in the documents published in French –, head of the central statistical office of Budapest and organizer of the ninth session of the congress in this city. However, Keleti was a Hungarian nationalist hero of the revolution of 1848 and the office he directed was integrated into the administration of the dual monarchy, this statu quo resulting from the revolution.154 Today, the main train station in Budapest is named Keleti, like one of its main arteries of the city, even though the names of the streets around the Central Bureau of Statistics and the station are those of its statistician collaborators. This case directs historical analysis towards other factors that supported the congress: it was the expansion of railway networks across the continent and the equipping of metropolises with mainly hotel infrastructures capable of accommodating the thousands of visitors from frequent universal exhibitions and congresses that crisscrossed the continent, for whose use opportunistic publishers published special guides of the inviting city. It was the time of splendid congress palaces, luxury railway lines, imposing stations, comfortable grand hotels, all things that are easily found today in the topography of European capitals and which were then the places of grand inaugurations and copious banquets which the local press diligently reported. For instance, if at the end of 1853 there were 29,190 km of railways operated in Europe,155 and by the end of 1883 this cumulative length reached 152

There are many regional historical works on this subject. The Age of Nationalism and Reform (1850–1890), subject of Rich, 1971. 154 During the dual monarchy everything that fell under it was qualified as “kaiserlich königlich “[imperial (for Austria) royal (for Hungary)] or everywhere “k. k “, an abbreviation that the Austrian writer Robert Musil (1880–1942) ironically referred to as “Kakania” in Der Mann ohne Eigenschaften. Berlin, Rowohlt, 1930, book 1, chapter 8. 155 French Ministry of Agriculture and Trade and Public Works, Statistical Documents on Railways. Paris, Imperial Printing House, 1856, p. XCIII and 142. 153

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183,186 km,156 an average increase of 6.3% along thirty years. These were external factors, but we must also add internal factors generated by the congress itself. From the first sessions, it was decided that in each country a central statistical commission would be organized, made up of senior administrators and concerned academics on the model of that first established in Belgium. Such a decision could only meet the approval of the government delegates present at the sessions because it extended the action of international meetings. Similarly, the creation of specialized learned societies was encouraged, following the example of the Royal Statistical Society of London already existing and, later, of the Société statistique de Paris. The offices, commissions and learned societies then published journals, bulletins or specialized journals which were, with the minutes of the sessions, collections of methods, debates and results.157 While the French language was predominant, German and English languages were allowed during discussions. The congress also recommended that in each national office and in universities where the subject was taught this extensive literature should be preserved and measures should be taken to reduce postal costs, no doubt high for the thick volumes in question.158 It is true that the post was developing then and that the statisticians made it both an instrument and an object of enumerations. On this occasion, here is the return of the method of the average pushed to the absurd: during the third session held in Vienna, it was indeed discussed and the congressmen agreed to demand everywhere “statements of letters and printed matter sent, the average distance of the journey, the number and total value of orders etc.”. In terms of school statistics, it will be: “the maximum and average distance from the school seat of localities within the school radius.”, and for agriculture: “average harvests and average grain prices, by provinces”. In the summary of the Congress’s work that he has published in 1873, Quetelet discussed a conversation held in Berlin in 1863, the motive of which had 156 International Commission for Railway Statistics, Statistics of the Railways of Europe [. . .] for the year 1883. Vienna, Imperial Royal Printing House, 1885 p. 631. 157 Until the 1990s, access to the reports was particularly difficult in libraries as they were sometimes catalogued under the name of the publication director, that of his affiliated institution or even the name of the congress itself. Only a few deposits concentrated all the volumes, most often due to their place in the statistical world of the 19th century, such as the Swedish, Dutch or French national offices, or university libraries, due to acquisition policies at the beginning of the 20th century, as was the case at Columbia University, in New York; or institutions recipient of donations from certain members, themselves once present at the congress. This was the case in France for the Library of the Institut de France because several members of the Academy of Moral and Political Sciences were from the Quetelian congress. See Brian, 1991a and 2002. Today, the clutter that such a series can represent and the poor quality of its paper have led some of these libraries to include these volumes in digitization campaigns and when this has not been the case, online catalogs facilitate research. So much so that the deepening beyond the articles that I published around 1990 are greatly facilitated. 158 The resolutions and debates from the congress were so numerous that after about a decade the organizers of the new sessions published summary tables of them: CIS, 1863 and id., 1866. The founder himself gave a general overview in Quetelet, 1873c. Summaries, not resolution by resolution, but synthetic ones were delivered by Heuschling, 1882b, Neumann-Spallart, 1885 and Nixon, 1960.

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been the validity and comparability of the numerical productions of different countries: “It would be a huge advantage if everywhere we could have the same measurements: it’s an advantage that we can undoubtedly hope for, but that we should not expect anytime soon. As for us, we are all striving for it. The life of a calculator would be saved by at least three quarters, if, taking statistical works, one could, thanks to the uniformity of measures, compare them immediately [. . .]. We can nonetheless, from now on, examine the value of the documents. Science must try to keep pace with what current society demands. The value of a statistical document depends on many elements, and we must not disdain those that belong to science. There are two in particular that it is important to consider: one, for example, concerns the duration of time over which the observations are made. Everyone will prefer the average of ten years of observations to the average of a single year, just as everyone will prefer, all other things being equal [elsewhere], to consult the average mortality rate of the whole of France to that given by one of its departments; but it is important to calculate the probable values of these results”.159

In 1869, in The Hague, the report of the seventh session shows some indications that the success of the measures adopted during the previous sessions had resulted in such a vast numerical production that new skills and new questions had emerged. Thus Quetelet himself was rebuked by statisticians, heads of offices or university professors who had come to want to distinguish on the one hand the production of figures for administrative needs or their favorite discipline and on the other hand the scientific objectives of the founder. From this date his pleas took a defensive turn. Had the initial scientific project given birth to bureaucrats and new professors who had become intellectually too independent from the science that had inspired them? This is why the founder concluded his review, published in 1873, with a text entitled: “The future of Statistics”. It opened with a self-satisfaction pronounced a year earlier by an authority, then returned to the driving conviction of the work of the Belgian astronomer to finally reaffirm his methodological dogma – the thesis of “pan-normality” – to the point of calling for a new concentration of efforts towards the initial objective: “At this time, all governments have recognized the value of statistics, and no longer shy away from the means to improve its institutions, nor to broaden the sphere of investigations of this science. These are the noble expressions used in the opening session of our last congress, [His Imperial Highness] Grand Duke Constantine [. . .]. Some [of the following observations] have been stated by me, for nearly half a century, without anyone noticing their novelty, and perhaps because of this novelty itself. I like to believe that no one will refuse to take note of them: especially today, if I add that several of my results have been admitted and verified by the most skilled calculators and observers: I will cite, among others, with a feeling of gratitude, the learned Sir John Herschel, whose science still mourns the loss. For a long time I have shown, with the feeling of the deepest conviction, that human sizes, although appearing to develop in the most accidental way, are nevertheless subject to the most exact laws and that this property is not particular to size: it is also noticeable in everything that concerns the weight, strength, speed of man, in everything that relates, not only to his physical qualities, but also to his moral and intellectual qualities. This great principle that governs the human species, and which, while diversifying the effects of its

159

Quetelet, 1873c., p. 53.

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qualities, gives these enough play to show that everything is regulated without the intervention of the will of the man, to us, seems one of the most admirable laws of creation. It is already a great advantage to be able to reduce the universality of methods and studies to a single principle, which forms, so to speak, the key; and to avoid, in this way, this disparity of systems that constantly threatened to divert statistics from its true purpose”.160

The final account, in the founder’s report, is concluded by the publication of a comparative table of male and female mortality rates for eight European countries. Indeed, international statistics had become the new horizon of the congress in 1869, which Quetelet was quick to report to the Brussels Academy. From the outset, his presentation was as much a scholarly self-examination as a personal showcase to his colleagues at the Brussels Academy: “Laplace, Fourier, Poisson, Bienaymé, Lacroix, etc., who had promoted, in France, the calculation of probabilities in the assessment of physical phenomena, vainly sought to maintain the authority of this science for social phenomena. These latter phenomena were observed; but their numerical assessment was entirely neglected: one became a statistician without prior study, and the most serious errors multiplied, without one having, so to speak, the will or the power to combat them”.161

Quetelet then mentions a discussion held during the fourth session organized in London in 1860: “a member [. . .] put forward the idea he had already formulated previously, to create a general statistic for the whole civilized world [sic: for Europe in the title of his communication].” This member was told that he would be happy if he could provide an example. A few pages later, it appears what one might have guessed: this member had been the astronomer himself, later the co-author of a prototypical collection published in 1865.162 The founder therefore practiced the tactic of moving forward. “Around 1870, as we have seen, the Quetelet project experienced a kind of growth crisis: solidly built statistical collections had multiplied and they circulated widely in the countries whose nationals had participated in the congress. The libraries of specialized offices, central statistical commissions, relevant scholarly associations and universities were full of tables or to put it anachronistically were drowned under the avalanche of numbers or the flow of data from this time: the printed figures. But, the science derived from the founder’s lineage did not follow. He himself repeated and fled into international statistics. However, during this time, calculators in France, in England, in Italy and in Germany were building new theories without resorting to the Quetelet dogma, nor placing themselves under his aegis. Thus we can see that at the very moment when the Quetelet doctrine was thus found to be a victim of its own success, the calculators took hold of this numerical treasure to examine it afresh: by renouncing the focus on the average and sometimes 160

Id., 1873c., p. 129–130. The Grand Duke Constantin, brother of the Russian emperor and liberal reformer, had presided over the eighth session of the congress, held in Saint Petersburg. As for Herschell, it is remembered that Quetelet had published his analysis of the Letters from 1846 in the introduction of the second edition of the Social Physics. 161 Id., 1869b. 162 Quetelet and Heuschling, 1865.

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turning their attention to variations. Thus the know-how forged by the astronomer did not dissolve, on the contrary, once the usual average was calculated, for example, concern was given to measuring the fluctuations around them until wanting to establish an average of deviations from the average. We remember the already commented passages where Quetelet discussed the measurement of deviations from the average in the vein of the work of mathematicians Legendre and Gauss on least squares. British mathematical statistics deepened this vein”. However, in 1869, at The Hague, the congressmen were not yet there. For example, Georg von Mayr (1841–1925)163 argued on the morning of September 8 for a resolution thus adopted: “The congress is of the opinion that it is desirable to calculate not only averages, but at the same time numbers of oscillations to make known the average deviation of the numbers of a series from the average of that series itself.164” Now, if, starting from the average, we calculate the average of the raw deviations – positive or negative – from this value, by construction we will obtain a null value, a result that only confirms the observation that the average is at the center of the distribution. But, if we now took the average of the absolute values of the deviations (either by removing their sign) or that of the deviations squared, each term would then be positive and the result would give an indicator of the extent of the surface around this central value. The mean square deviation from the average, is what has later been called the variance. If the average is provided with a given unit. This variance, coming from squares, will be expressed in squares of this unit. By taking the square root of the variance, one will get an homogeneous indicator measure. This is what has been called later on the standard deviation. It appears thus that the German statistician von Mayr initiated the construction of this process. The statisticians of this time, in seeking to break away from the Queteletian dogma of the average, did nothing other than turn this know-how on its favorite results. The object has moved, but the know-how has endured. Although mathematical statistics is generally considered a British current, it must be noted that twenty-five years before the reference article by Karl Pearson (1857–1936) on the standard deviation,165 the way had been opened in continental Europe, without even mentioning here the genealogical elements derived from the works of Legendre and Gauss.166 Moreover, the scheme of an average of deviations from the average remains today locked in the Queteletian conception for which what is submitted to calculation is considered item by item. This is a very arbitrary presupposition: indeed it is a question of finding an index of variability, and to construct it, one fixes a reference 163 “With [Émile] Levasseur, [Jacques] Bertillon, [Anders] Kiaer and others, [Georg] von Mayr belongs to the breed of these statisticians who have decisively influenced the preparation and decisions of all international statistical congresses and meetings”, excerpt from a tribute for his seventieth birthday published in the Journal of the Statistical Society of Paris, Volume 52, 1911, p. 167. 164 Quetelet, 1873c., p. 193. 165 Pearson, 1894. 166 Perhaps in the middle of the 20th century, references to German statisticians, such as Gauss or von Mayr, disturbed British pride.

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point, the initial average, and then considers the distances to this reference. It’s a bit like if to measure the extent of a village we started from the town hall and then measured its bird’s eye distance to each house. Why privilege a starting point? We can feel that there is something arbitrary here. However, to apprehend the same extent another operation would consist in measuring all the bird’s eye distances between the houses without privileging any particular initial place. From a mathematical point of view, it is possible to show that the two processes lead to the same result. And we can even propose a more realistic process by not adopting the point of view of the birds but by considering that some streets are impassable and by excluding from the calculation the distances between two houses in these inaccessible conditions. This last process provides a measure of variability freed from the presuppositions coming with the initial average and can know various applications.167 Still at The Hague and concerned with variability, the congressmen understood that they were prisoners of the constraint of having to offer a simple numerical index. They then sought to free themselves by exploring the possibilities offered by the surface of the leaves and graphic statistics. Obreen, director of the map depot at the Dutch Ministry of the Navy, gave a report at The Hague, a preparatory report on this method. He concluded that “the question [was] not yet sufficiently matured [. . .].” and proposed this resolution: “The congress, considering that the graphical method is very suitable for teaching and popularizing statistical sciences, expresses the wish that the main official statistical documents be accompanied by maps and diagrams.168” It was only in 1876 that the congress examined the contributions of statistics to teaching.169 And from the international meetings that took place during the universal exhibition in Paris in 1878, the network of international statisticians has deliberately engaged in the promotion of graphical statistics. The itinerant meetings of the International Congress of Statistics and the commission that extended it for a few years were prolonged by the activity of the International Statistical Institute, from 1886 in Rome, then in The Hague, where the same active international network of official bureau chiefs and mathematicians is found. In 1874, at the death of Quetelet, this network was solidly established mainly in Europe but also beyond the oceans. The colonial policy of the European powers helping, its works have practically covered the entire planet. At the end of the 19th century, the teachings of statistics, in addition to the contributions of mathematical developments mainly British, consisted of summarizing the corpus accumulated in the libraries of offices and specialized learned societies over nearly half a century of international documentary exchanges.170

167

Brian, 2017 presents the mathematical demonstrations and various examples. Quetelet, 1873c., p. 193. 169 CIS, 1876, p. 281. 170 For example, Bertillon (J.), 1895. 168

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Precisely in 1874, the Congress commission held a session in Stockholm in 1874. All the Swedish press echoed it up to the Söndags-Nisse (the Sunday elf, “Illustrated weekly for Fun, Humour and Satire”) of September 6, 1874 where the following epigram appeared, showing the perplexity of its readership towards the last fires of the Quetelesian enterprise: A Congress of Statistiscs [sic171]. – Say, Daddy, prayed little Beda what is a Congress of Statistiscs? – Mr Daddy, who was anxious to avoid all questions until he had had time to read the editorial in the newspaper’s densely printed columns,172 answered, not exactly for the sake of sniping, but because the coffee was getting cold: – Well, it’s a group of people who find out how things work everywhere. And Beda, a perceptive girl, took the answer at face value and left; and sat next to her neighbour Maricka, as soon as she saw her: – Now I know, you, what do aunt Lina and aunt Theresa and old aunt Ulla Kristina, Mrs Pilqvist and Miss von Hess, now I know what they are. they are a Congress of Statistiscs!173

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Know-How and Presuppositions from the Quetelesian World

Just as the know-how forged on ratios in the 17th century required only a basic calculation process, its counterpart, two centuries later, tied to averages did not require sophisticated arithmetic skills from calculators: it was only a matter of adding a certain number of quantities and dividing their total by this number. The characteristic of the 19th century, resulting from the encounter, at the end of the previous century, of probability calculus with integral calculus, was to associate the value thus obtained with a dispersion scheme around it, this one representing the probability in principle of observing quantities of the same kind around the average or further from it. Quetelet promoted this scheme – later known as the “bell curve”. With the specifics clarified, let’s look at the presuppositions it entailed. The most obvious of these, so often suggested or affirmed by the astronomer and prepared in him by the reading of Fourier and Poisson, is the doctrine of what we have here called “pan-normality”: the measurement of all natural and human phenomena would present to calculators the same appearance, that of a distribution of its values according to a curve centered on their average, first gently ascending then

An evocation of a childish understanding of “statistics”. As opposed to illustrated ones as The Sunday elf. 173 En Statistik kongress. – Säg, Pappa så bad lilla Beda hvad är ren statistik kongress? – Herr pappa, som mån var att Freda sig för alla frågor till dess – att han hunnit ledaren läsa i tidningens tätt tryckta spalt, gaf svar, ej precis för att snäsa, men derför att kaffet blef kallt:– Jo, det är en hop, som ta reda på hur det går till öfverallt. /Och Beda, en uppfattlig flicka, / tog svaret ad notam och gick; /och sad’ se’n till grannens Maricka, så fort henne träffa hon fick:– Nu vet jag, du, hvad faster Lina och moster och mosters Thérèse och gamla tant Ulla Kristina, Fru Pilqvist och fröken von Hess, nu vet jag, hvad de ä’ för ena. de ä’ en statistisk kongress! ». Epigram signed “Paul-Louis” which we have been unable to identify. Copy of the newspaper consulted in the microfilm collection of the Carolina Rediviva Library at the University of Uppsala. 171 172

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Illustration 3.8 Curve related to the COVID pandemic reproduced in various media. (Source: Centres for Disease Control and Prevention)

symmetrically descending until it fades on either side without even being able to represent these extinctions on the graphs, even though, since the 18th century, analysis provided rigorous reasoning up to mathematical infinities. This presupposition about the shape of the curve was underpinned, as we have seen, by the principle that causes would be proportional to effects whose probability was being studied. The book in which this chapter appears was written in 2023 and its project stems from the startling observation that during the Covid pandemic, from 2019 to 2022, commentators, first epidemiologists, then journalists – too happy to resort to graphic illustrations supposed to draw on common knowledge – have continually demonstrated their adherence to the bell curve scheme: the curve had to be lowered. No doubt this objective, backed by a vague statistical culture, seemed convincing enough to mobilize readers and viewers. We have already analyzed these “stories” at greater length elsewhere.174 Their principle was illustrated by the scheme of Illustration no 3.8 disseminated for instance by The New York Times175 and CNN.176 These acts of devotion to the doctrine of the bell curve were nevertheless irrational even if they appealed to memories of mostly practical teachings or routines shaped by known calculation softwares. This irrationality comes from mathematical reasons that will be discussed later and from drifts that could have alerted those who held these views. Indeed, as soon as the peak of the curve in question was passed, commented on during the first months of 2020, another “peak” appeared, and we heard about a “second wave”, etc., up to a fourth wave if not more, as the disease could become endemic and seasonal. But, let’s be rigorous, one of two things: either the intensity indicators would obey a bell curve and in this case, there would be only one wave because beyond the central value of the intensity would reduce almost

174

Brian, 2020, the following paragraphs are based on a point made in this article. Siobhan, 2020. 176 Sepkowitz, 2020. 175

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totally, or there would be several waves and in this case the bell curve could not be suitable: it would be necessary to resort, as for other seasonal phenomena, to oscillators.177 In short, the bell curve offered only the illusion of predictability unless it simply provided its commentators with the relief of warding off the anxiety caused by confronting a hitherto unknown pandemic. Waving the bell curve has therefore not been a rational response but the invocation of a fable to ward off the unknown threat. It is at this point that a mind alienated from the Queteletian doctrine reacts: “Come on, my brave man!” He will be indignant, “it was simply a combination of bell curves”. The argument saves the doctrine rather than the phenomenon: it saves the idol of the theater. The history of statistics has been the occasion of a precedent. At the end of the 19th century, the Queteletian tropism was indeed expressed by Dr. Louis-Adolphe Bertillon (1821–1883), then the head of the Statistical Bureau of the City of Paris, active member of the international congress and promoter of its methods and doctrines. He was also one of the founders of the Anthropological Society of Paris, a young discipline whose part of the calculation techniques came from the works of the Belgian scholar. The sons of Louis-Adolphe Bertillon were first Dr. Jacques Bertillon (1851–1922), later his successor at the Parisian Bureau and one of the promoters of demography, then Alphonse Bertillon (1853–1914), French promoter of anthropometry and dactyloscopy in criminal police. Bertillon senior had his year of glory in 1876 when he published on the same subject two reference articles where statistical methodology was exposed. One was titled “Average” and published in the Dictionnaire encyclopédique des sciences médicales178 addressed to his original corporation; and the other, “The theory of averages in statistics”, in the Journal de la Société de Statistique de Paris.179 Both were illustrated by a table of sizes collected from French conscripts and by a graph announced as built on this table (it is reproduced in Illustration no 3.9, below). A curiosity reported by the author was about the young men measured in the department of Doubs (now located in the region of Burgundy-Franche-Comté in the East of France, near Switzerland and Germany). Unlike the general population of their French counterparts, they presented a curve of the frequencies of their sizes where two bumps appeared. The author immediately recognized two quetelesian distributions where he believed he discerned the types, and therefore two different races supposedly present in the region: Celts and Burgundians. The reading proposed by Louis-Adolphe Bertillon takes its meaning in a then intense controversy that opposed the monogenists to polygenists.180 Intense among naturalists and anthropologists, it was played simultaneously in scientific and 177

At this point I spare the reader the discussion of the delirious outbursts of pathetic commentators, anxious in the face of the rise of the pandemic, or imagining themselves responding to the anxiety of their listeners, who were waving the specter of exponential growth, a pattern that in no way fits with the bell curve, nor with a periodicity. Here again the Condorcetian principle of the authority of the written figure is illustrated. 178 Bertillon (L.-A.), 1876a. 179 Id., 1876b. 180 Blanckaert, 1981.

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Illustration 3.9 Size of conscripts from Doubs according to L.×Ad. Bertillon Horizontally: sizes of conscripts in feet and inches, grouped by one-inch intervals Vertically: frequencies of recorded conscripts Continuous curve with one bump: frequencies [inappropriately called here in Quetelet’s vein: “probabilité”] for each interval of sizes for France as a whole Dot curve with two bumps, same frequencies for the district of Doubs) Source Bertillon (1876a, b)

theological arenas: all humans would have a common origin and should we then consider the unity of the human species in calculations as the monogenists and Quetelet himself had defended; or was the species composed of races that anthropometric criteria could differentiate, this was the position of the polygenists and most often that of the proponents of physical anthropology. For Bertillon senior, therefore, an empirical curve that presented two peaks illustrated two underlying bell curves and therefore two racial types. The historians and archaeologists of this time had prepared this reading by considerations on how the region had been populated: the Celts would have appeared in the region first in the Bronze Age and the Burgundians later in the 5th century. Bertillon senior, riveted to his curve and driven by his interpretation, combined these different findings without taking into account the circulation and mixing of populations since these ancient times. His doublehumped curve is reproduced in Illustration no 3.9 above.

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Illustration 3.10 “Curve that seems to have gained the sympathy of the creator” (A. de Foville) Horizontally: Axis of sizes Vertically: number of men for each size. “M” indicates the mean Source: Foville (1907)

In the tense context of the controversy, this curve and its interpretation did not fail to cause perplexity. Only three years after his father’s death, Jacques Bertillon, his eldest son and successor, published a new article and a revised graph in a later issue of the same reference journal of the Parisian statistical society: “The height of men in France”.181 The raw data were the same, no correction was added and the smoothed curves were abandoned: only raw numbers and broken lines like successive triangles. The two peaks were preserved but not the ideal of two groups of average men in their types: the two races had disappeared. The same Jacques Bertillon published in 1895 a manual of administrative statistics which has long been used for the training of statistician clerks. In this reference volume, he again used the broken line and abandoned his father’s overinterpretation.182 Today, everyone can understand how the obsession with the bell curve led the patriarch on the path of an empirical justification of racial differences: he would have projected onto a curve shaped in an adequate way a doubling of the bell curve scheme. Recognizing on the intensity curves of the recent pandemic a series of waves, is to let operate without epistemological control the same projection of the quetetian scheme.183 Twenty years later, in the same specialized Parisian journal, another manifestation of a stunning adherence to the doctrine of the bell curve is found, almost an act of faith: In an article published in 1907, Alfred de Foville (1842–1913) – former president of the Paris Statistical Society (in 1886) – expressed his deep attachment to the totem of the average man “homo medius”.184 He commented on the Illustration no 3.10, reproduced below, in these terms which express the success of the doctrine of his Belgian predecessor in the language of a recognized specialist from the beginning of the 20th century:

181

Bertillon (J.), 1886. Id., 1895, p. 116. 183 Brian, 2020, where this case has already been exposed was submitted, before its publication to anonymous referees. One of them expressed a feeling of indignation as if a trial for racism had been initiated against the statistical corporation. This reaction calls for two observations. Firstly, the case study is about a predecessor and not the profession; secondly this reaction shows that the very fact of discussing this case from a new angle is appeared shocking. The direction of the journal showed wisdom and did not abound in the sense of a critique governed by emotion more than by reason. 184 Foville, 1907. 182

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Illustration 3.11 Individual abilities and social utilities (accordin to A. de Foville) Captions from top to bottom: Talent, Mediocrity, Weakness, Limit of social uses [sic]. (Source: Foville (1907)

“What was for Quetelet a revelation and what, for his disciples, has become a traditional notion, is the tendency that many variables in man and outside of man, have to group, to cling, so to speak, around a central point that seems to call them to him. [. . . We must salute] this symbolic curve that seems to have gained the sympathy of the creator. Its appearance is less simple than that of the ellipse or the parabola; but the supple eurythmy of its alternating bends shows us how the spirit of authority and the spirit of tolerance can be reconciled here below. Nature does not wish for uniformity [. . .]. A minimum, a maximum and, between the two limits, an average, that’s what constitutes the framework and the guiding axis of many phenomena. The variable could have been allowed to move at will between the two barriers that it is forbidden to cross. But no: it is clearly the median region that attracts it and, when it does not reach the marked place, it must at least be as close as possible. Large deviations are only allowed on an exceptional basis and it is true to say that the exception confirms or presupposes the rule”.185

A few pages later, adding the sketch reproduced on Illustration no 3.11, this former president of the Paris Statistical Society moves on to moral statistics and criminology. He asserts on this occasion: “Should we [. . .] admit that human mentalities, from the most powerful to the most incomplete, should be ranked like the numerical combinations obtained by throwing a handful of dice a thousand or ten thousand times on a table? This is how Francis Galton [1822–1911] or Otto Ammon [1842–1916], reasoned, and, without wanting at this moment to delve into the details of their theses and their hypotheses, I can at least reproduce, as is, the schema that summarizes them.” [followed by a reference to the curve that we reproduce in illustration n°11186]”.

Foville, always so revealing of the mindset among statisticians of his time adds: “Here is a silhouette that one does not forget once one has seen it. It makes one think of a kite, or a spinning top, or – to take up the comparison of Mr. Otto Ammon himself – a tulip bulb. But it is enough to look at this image from the side to find, in duplicate, [the famous

185 186

Id., ibid. p. 323–324. Id., ibid. p. 328.

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bell curve]. In this respect, the lottery of intelligences would not be organized differently than the lottery of weights or sizes. And Quetelet would triumph”.187

The admirer of the Belgian astronomer nonetheless expressed some perplexity regarding the argument of his mentor: “The plea which, in [the book] La Physique sociale, leads to the apotheosis of the Homo medius is quite disconcerting. It rolls on a double equivocation, first of all no longer seeing in the idea of average anything but the idea of balance, and then considering the word balance as synonymous – or almost – with the word perfection. The sophism denounces itself. However, let us acknowledge that Quetelet could invoke in support of his interpretation numerous precedents. Proverbs are for him. The French adage “Excess in everything is a defect” is only a useless truism; but in Latin the established expression ‘in medio stat virtus’ implies, if taken literally, a whole debilitating doctrine. . .”.188

Today, even if the expression of Foville’s imagination seems ridiculous, it found a replica in psychology where the distribution curve of the intelligence quotient (IQ) remains a legitimate reference, even to the point of feeding an extravagant social psychology in a recent work that has sparked debate.189 But beyond this case, the Queteletian presupposition remains solid because the law of Laplace-Gauss is intensively used to model noise – the residue left to uncertainty – in the construction of econometric and epidemiological models in particular. Thus, at the beginning of the 2000s, we asked economists active across a wide spectrum of theoretical orientations to write an article where they had to define the concept of “fundamental value” according to their preferences: the theory of informational efficiency on financial markets, that of rational bubbles, that of noise in stock prices or even that of collective beliefs and mental representations at work on these markets. Spontaneously, all wrote their article in such a way that this fundamental value appeared as a balancing center of values more or less close but not very distant. These authors of all economic obedience, whether it is said orthodox or heterodox, expressed that they had in common a Queteletian representation of the distribution of uncertainty of financial phenomena.190 Where does such unanimity come from in a specialty where relations are so tense? More than one hundred and fifty years after the publication of the main works of the Belgian astronomer, his conceptions of randomness and variability have deeply entered the teaching of the economic discipline, in the processes of econometric calculations and in the software for monitoring and forecasting prices. The economic world of the years 2000 is therefore a vestige of the Queteletian world. We will discuss later the reasons, all mathematical, that have made these models have found their limits on the occasion of the stock market crashes of the end of the 20th century. A revision of the fidelity to the Queteletian doctrine is necessary.191

187

Id., ibid. p. 328–329. Id., ibid. p. 329. 189 Herrnstein & Murray, 1994. 190 Brian and Walter, 2008. 191 Brian, 2009 and 2015; Le Courtois & Walter, 2014. 188

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Cracks on the Bell

If Quetelet could rightly express his self-satisfaction in the assessments he published in 1873, the stumbling block of his doctrine was the determinism attributed to his statistics and the defense of the free will of each one, a theological register since the 4th century after Augustine of Hippo. One of the most determined in this criticism was the philosopher and man of letters Charles Gouraud (born in 1823) who published at the age of twenty-five a history of the calculation of probabilities nourished by previous works in this genre and careful readings of the scientists of the 17th, 18th and 19th centuries. The historical work of about a hundred pages is punctuated by a short thesis of philosophy particularly severe and surprisingly accurate in its conclusions, “on the legitimacy of the principles and applications of this analysis [the calculation of probabilities]”, according to the indication of the title page. The thesis does not claim a method: its author submits in a few lines to the judgment of the Faculty of Letters of Paris which finally approved it on July 12, 1848.192 The object is clearly stated: “it is proposed [. . .] with eyes fixed on its history, to discuss the solidity of the definitions on which [this calculation] is based, of the principles it uses, of the applications where it goes, and of the mathematical theory that he created.” At the end of about ten pages, Gouraud asserts six “general propositions” that he intends to have demonstrated: FIRST PROPOSITION. The philosophical theory conceived by Jacques Bernoulli and on which, since this great geometer, the calculation of probabilities is based, is false.193 SECOND PROPOSITION. The principle invented by Jacques Bernoulli and perfected by de Moivre, which bears in science the name of principle of indefinite multiplication of events: the principle glimpsed by Bayes and analytically demonstrated by Laplace, which consists in concluding the probability of causes and their future action from the simple observation of past events; the principle finally discovered by Mr. Poisson and called by him, law of large numbers: are only useful in a supposed system of things, where: 1° Numeration is possible;194 2° The most absolute fate, and not just a simple observed and probable constancy, chains effects to causes and causes to effects;195

192

Gouraud,1848, p. 149: the thesis itself only covers pages 139–149. It had been briefly discussed in the historical part, in our opinion without sufficient critical distance towards the questionable review given by the Belgian astronomer. 194 Understand here that the cases taken into consideration must be countable, a concept that will only be developed on a mathematical level later and specified in terms of probabilities in Borel (1909), see on this subject Bru and Bru (2018). 195 This time it is the measurement of causes by effects that is targeted. Henri Poincaré will propose to assume the hypothesis of a non-proportionality of causes to effects; this is indeed the principle of complex phenomena in the strictly mathematical sense of the word. He even went so far as to find a definition of chance cases in it, see Poincaré, 1912, p. 4: “A very small cause, which escapes us, determines a considerable effect that we cannot fail to see, and then we say that this effect is due to chance.” This passage is remarkable because it resembles the expression of Laplacian determinism while it is its exact opposite! Gouraud was therefore right to draw attention to the implicit presuppositions that tied causes and effects. 193

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3° where finally, events do not only succeed each other at unequal intervals, according to certain pre-established and observable laws, but where they appear regularly under the confirmed influence of these laws, at the same periods and in the same way.196 THIRD PROPOSITION. In principle, the calculation of probabilities is applicable in the order of physical things; – however, this application ceases to be of any use, when the universal enumeration of physical events, the object of calculation, cannot be carried out either absolutely or approximately; – secondly, there is an exception for medicine or therapeutics, to which the algebra of chances is completely inapplicable.197 FOURTH PROPOSITION. The application of the calculation of probabilities to moral sciences, and notably to historical criticism, to jurisprudence, to legislation, to social economy, to metaphysics, is one of the greatest errors into which the human mind has fallen.198 FIFTH PROPOSITION. The analytical theory of probabilities, considered in itself, and abstracting from the philosophical data it presupposes and the subject on which it operates, is as correct as ordinary algebra, and its invention is one of the greatest efforts of mathematical genius; – but it only leads to useful results in the small number of subjects where the likelihood of human opinions is appreciable [by] calculation.199 SIXTH PROPOSITION. The calculation of probabilities is a science whose definitions are still to be written, principles to be explained, applications to be restricted, and the entire organization to be founded: only the mathematical part deserves to be fully preserved.200

These conclusions, issued by a young philosopher who has not displayed any method, although uneven, are remarkable as they announce the surpassing of Laplacian conceptions by specialists in the mathematical theory of probabilities in the following century. What was, in Gouraud, the driving force of his analysis? An epigraph reveals it: “The geometric science of the Universe differs from the moral science of Man: the latter has other principles more mysterious and more complicated, before which Geometry stops.201” This sentence was expressed, the young philosopher specifies, on the occasion of the reception at the Académie française of Pierre-Paul Royer Collard (1763–1845).202 To grasp the meaning of this epigraph, which is considerably oriented against Condorcet and Laplace, it is important to quickly trace the journey of Royer-Collard, a young lawyer who had enthusiastically attended the events at the beginning of the revolution of 1789. In 1792 he had supported the Girondins and in 1793 fled the Terror in the provinces. In the spring of 1797, under the Directoire, he was elected deputy. But, after Bonaparte’s coup d’État, in September of the same year, he was

196 This third paragraph touches on the way in which the cases considered in the calculation are defined. In a much later vocabulary, it would be said that it touched on the definition of the probability space. 197 The first part of this proposition joins the previous ones and its second part addresses a theme already discussed by D’Alembert (1717–1783), then by Claude Bernard (1813–1878). 198 Launched without any further ado, this fourth proposition is a principle attack against the program displayed by Condorcet in the 1780s and 1790s. 199 Compared to the first propositions, this one disappoints as it is a truism. 200 This time, Gouraud, shows himself to be visionary: this will be the agenda of the mathematicians of the 20th century, as we have begun to see in the cases of Poincaré and Borel. 201 Gouraud, 1848, p. 141. 202 Royer-Collard, 1827, p. 9.

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excluded from the Assembly. He then approached the pretender to the crown, the future Louis XVIII, until 1804. Appointed professor of history and modern philosophy at the Sorbonne in 1810, his teaching marked Victor Cousin, himself undoubtedly the most influential philosopher in 19th century France. After the Restoration, he became the principal speaker among the “doctrinaires”, that is to say, the liberal royalists in favor of a constitutional monarchy who opposed the “ultra”, in favor of the return of the Ancien Régime. After the revolution of 1830, and with the July monarchy, the institutions were granted to his opinions. From the restoration and for the last thirty years of his life, Royer-Collard, a practicing Catholic faithful to the Jansenism of his childhood, became a prominent professor, was the reference in modern philosophy. Gouraud, by placing his work under this epigraph, that is to say in a lineage of a well-known Catholic philosophers, thus opposed to the Queteletian mystique of the bell curve the mysteries of faith. Condorcet, Laplace, Fourier, Poisson and Lacroix disappeared, Quetelet himself undertook to respond on the ground of free will: “In the face of such a set of observations (regularities in marriages, crimes, etc.), should we deny man’s free will? Certainly, I do not believe so. I only conceive that the effect of this free will is confined within very narrow limits and plays, in social phenomena, the role of accidental cause. It then happens, that by abstracting from individuals and only considering things in a general way, the effects of all accidental causes must neutralize and destroy each other, so as to leave only the true causes by virtue of which society exists and is preserved. . . Man’s free will fades and remains without noticeable effects when observations extend over a large number of individuals”.203

The question of free will resurfaced to extricate itself from this sectarian dispute in a series of articles by the German economist Georg Friedrich Knapp (1842–1926),204 whose work had been mentioned as early as 1863 during a discussion held at the fifth session of the international congress in Berlin and who participated in its ninth session, in Budapest, in 1876. In the background of his position, in 1871 in favor of what he held to be a new school of statistics in the German style was, as early as 1867 and in a Lutheran context this time, the work of the German astronomer, mathematician, philosopher and logician Moritz Wilhelm Drobisch (1802–1896)205 who had split the protective mathematical armor of the Belgian astronomer’s doctrine. Breaking with the unifying spirit of the congress and in the aftermath of the Franco-Prussian war, Knapp argued during a solemn university conference the divergence in statistical matters, between what he designated as an old French school and a new German school:206 the French one, in the lineage of Laplace and Quetelet would focus its attention on numerical regularities while the German one, under the name of Moralstatistik (moral statistics) would scrutinize with the same figures not the formal regularities but the “binding external [moral] laws”; because “it cannot be

203

Quetelet, 1848b, p. 69–70. Knapp, 1871a, b, 1872. 205 Drobisch, 1867. 206 Knapp, 1871a. 204

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denied that if such binding external [moral] laws existed, there should be a regular recurrence of crimes, marriages, suicides.207” Surprisingly, Knapp, to distinguish himself from the doctrine of the average man, used the same figure of speech as Quetelet: without the hypothesis (in the Belgian astronomer the average man and in the German economist, the external constraints) no observable regularity (the gesture aims to save the empirical phenomenon but in vain): “It is [. . .] agreed [among statisticians] that the very great regularity of the number of annual actions does not allow belief in free will. We can no longer believe in free will, if by free will we mean a will without motivation. If we consider acts as products of the will that is not hindered by anything, there is no free will, determined will and moving itself, so that actions would be isolated from any causal relationship, then statistics experimentally demonstrate that this conception is false because with [this definition] of the action of the will, the regular repetition of an almost identical number of acts [. . .] is quite incomprehensible. Absolute arbitrariness, free will in the vulgar sense, is therefore rejected by all schools”.208

Knapp did not therefore invoke the mysteries of faith: but he distinguished two rational approaches to statistical regularities: one, technical and mathematical where he wanted to recognize a French characteristic, and the other, conceptualized around the notion of moral constraint that he claimed as properly German. In doing so, he took over the statistical production from specialized offices and therefore from the congress by ordering it around the idea of external constraint. In this way, he outlined the perimeter of a domain actually studied in the German universities of his time: moral statistics. We know that the young French philosopher, Émile Durkheim (1858–1917), was trained during a tour in Germany, of which he gave a report published in a French reference philosophical journal, an article that announced the program of his sociology, understood as the implementation of the positive method in philosophy; a reference to the typically French positivism that was not on the horizon of the Moralstatistiker. In this early document, the future founder took note of the works of German moral statistics and if he did not mention the name of Knapp himself, he discussed the question of “external constraints” in relation to law.209 Durkheim, made the same gesture as Knapp, which consisted of seizing the statistical production of the late 19th century, organizing it around conceptual approaches that the office calculators had not put in. This is not the place here, but it would be possible to show that the founding concepts of Durkheimian sociology allowed him to escape the two sophisms by which Quetelet and then Knapp wanted to save the phenomena.

207

Knapp, ibid., p. 9 of the offprint that we were able to consult and translate here. Knapp, ibid., p. 7–8, (id.). 209 Durkheim, 1887. Knapp, neither 1871a nor 1871b are mentioned by the founder of sociology but his successor Maurice Halbwachs, indicated it in his supplementary thesis on “The theory of the average man”, Halbwachs, 1912, p. 1 even though p. 130–135 he commented more extensively on Drobisch, 1867. 208

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The historian of science and sociologist Libby Schweber has designated this gesture as “Disciplining statistics” by comparing the formation of demography among French authors and that of vital statistics among their British counterparts.210 Thus, at the end of the 19th century, the scientific legacy of Quetelet, if it has not lost anything from the breadth of quantitative production, saw its conceptual substance transmuted into new disciplines: in Germany, it was moral statistics notably promoted by Knapp, undoubtedly the finest among his peers; in France, it was demography which developed its concepts and organized itself through international meetings around Louis-Adolphe and Jacques Bertillon, two regulars of the CIS and chiefs of the Paris City statistical office; in France again, as we have just seen, it will be Durkheimian sociology; in Great Britain, it was on the one hand vital statistics animated by William Farr (1807–1883), one other regular of the congress; and on the other hand the British school of mathematical statistics which focused, always dealing with the same matters, and sometimes adding other sources, on the measurement of variability after Francis Galton (1822–1911). The mention of the names of Bertillon and Galton, allows to indicate three other branches derived from the Quetelet project: anthropometry, eugenics and. . . scientific police. Indeed, the French Police in the mid-19th century encountered the greatest difficulties as soon as it came to identifying repeat offenders for whom, in France, the law provided for increased penalties. It is on a bell curve scheme that Alphonse Bertillon, the youngest of the family, built an identification process: Anthropometric measurements were taken and classified on a scale of five categories: median, moderately smaller than average, moderately larger, strongly smaller and strongly larger. The boundaries of these intervals were established according to the principle of an equal distribution of the theoretical frequencies of these five groups calculated using the Laplace-Gauss law. A square wooden table made up of twenty-five boxes (five by five) had been set up in the basement of the Police Prefecture. As soon as a suspect was arrested, anthropometric measurements were taken, along with front and profile photographs and the recording of his fingerprints. The measurements allowed to locate the box where to place the new card (an example is given with Illustration no 3.12). Other cards were there that corresponded to the same measurement intervals and the photographs were compared. The suspects already arrested were recognized and suspicions of recidivism established if the suspect’s criminal history lent itself to it. Thus, the same corpus of doctrine based on the accumulation of mass observations also resulted in an individual identification technique. In each of these cases, the numerical matter, the data accumulated during what was termed the avalanche of printed numbers, underwent a new conceptualization, a particular disciplinary process, and these numbers, as soon as they were considered from the perspective of one or another of these new disciplines, could not be held as common without making unjustifiable amalgamations: indeed, it was the conceptual constructions specific to each of the disciplines that gave them a particular meaning. Thus, the data mine resulting from the collective activity initially driven by the

210

Schweber, 2006.

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Illustration 3.12 Anthropometric card of Mr. Alphonse Bertillon Source: La Vie illustrée (November 7, 1902), n°212, p. 70 Source: https://criminocorpus.org/fr/ref/114/79368/

Belgian scientist fragmented into as many particular stocks as it would be abusive to consider as a whole. Around 1900, from the Queteletian enterprise, there remained primarily extensive libraries of numerical collections; then several distinct disciplinary legacies whose protagonists wanted to distance themselves from the initial aims of the Belgian astronomer in order to assert the uniqueness of their new field of expertise; and a common knowledge induced by the know-how of the average and its presuppositions, the graphic scheme and interpretation scheme of which are the bell curve. Also remained a series of purely mathematical problems that touched on the foundations of probability calculation and were addressed by specialists in the 20th century in the German-speaking world, Italy, Russia and France,211 in this regard Charles Gouraud was a prophet. 211 Maïstrov, 1974, particularly well informed, devotes to these researches, including his last two chapters, IV and V, p. 188–264.

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Paul Lévy (1886–1971) explained in his mathematical synthesis of 1937 why one could encounter distributions conforming to Quetelet’s expectations, and why there could be some other types.212 The same Paul Lévy comments on the maturation of abstraction illuminates both the rush of the Belgian astronomer and the time it took to untie the Gordian knot he had tightened for decades. Regarding the theory of measure, which Lévy specifies that a letter from Gauss to Laplace indisputably offered a precursor case, he notes that neither of the two correspondents, despite their mathematical acuity, saw the interest there could be in systematically studying the sets of which an example was under their eyes, so that the fertile theory of measure had to wait another century to experience a real development, notably with the work of Émile Borel (1871–1956), Henri Lebesgue (1875–1941) and Maurice Fréchet (1878–1973).213 In other words, this delay of nearly a century was the sanction of the uncontrolled Queteletian extrapolations that had blinded mathematicians for decades to the strictly mathematical interest of foundational developments perceived later. Before coming to the culmination of this particularly enlightening mathematical critique in the work of Paul Lévy, let’s see what happened to the last fires of the defense of free will at the beginning of the 20th century. They were rekindled by the neo-Thomist Benedictine philosopher Dom Odon Lottin, born Joseph Lottin.214 In his summarizing work on Quetelet’s statistics and sociology, he showed that he had made his honey from the readings of Gouraud and especially that of Knapp, of whom he was probably the intermediary to a more French-speaking than German-speaking readership.215 During his preparation, the Benedictine published an article in the Neoscholastic Review which he then communicated to the Paris Statistical Society. One of its most active members at the time, Gaston Cadoux (1857–1930), immediately reported on it, outlining the main points and not hiding this paraphrase from the readers. The Parisian commentator emphasized that in his view, the importance of Lottin’s study laid in its philosophical approach and in its critique of Quetelet’s causality, of which Laplace had been the inspirer: Cadoux sneered in passing: “The notion of Quetelet’s average man has been much criticized, even ridiculed.” (p. 327). Lottin’s neo-Thomism had led him to try to reconcile the science of his time and theology, that is, on the one hand, Augustinian free will and verse 11, 33 of Paul’s Epistle to the Romans “His ways are inscrutable” and on the other hand, the accumulation of numbers as long as one renounced to deduce causal laws from them. The last paragraph of the review repeated word for word that of Lottin’s article.

212

Lévy, 1937. It will be discussed just before the conclusion of this chapter. Lévy, 1966. 214 Van Steenberghen, 1965. 215 Lottin, 1912, where Gouraud is mentioned p. 347 and the analysis of Knapp largely restored p. 107; p. 120; p. 135–137; p. 149–150; p. 190; p. 244–245; p. 249; p. 393; p. 438–430; p. 461–462; p. 478–481; p. 488–502 and p. 553. 213

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Therefore, no proof, by statistics, of a natural link between causes; no proof either of a true causality. What remains? Only, a set of relatively constant influences; resulting from the relative stability of the social environment, to which most individuals who find themselves in the position to perform the acts recorded by moral statistics will normally acquiesce. This formula leaves full and legitimate role to freedom, within the permitted limits, the statement of social laws; it allows the establishment of sociological inductions as far as the results are applied only to the mass, and that, taking into account the mutability of the social environment and the always possible individual influences, one takes care not to want to predict with certainty the future course of events.216

As for statistics and one of its derivatives, demography, the question was closed, their specialists could, from then on, work without qualms. With the rise of empirical sociology in the 20th century, the question of free will reappeared, stirred up against what was again referred to (not without the simplism already indicated by Knapp) as sociological determinism. This controversy no longer disturbs today except minds preoccupied with theological questions. From this point of view, Queteletism, in its original form, is indeed extinct but remain, in various ways, in many authors the attachment to the average and the obsession with distributions that would present a bell curve shape. These followers of the mystique of the bell curve, or of the pannormality affirmed by Fourier, Poisson and then Quetelet, forget that a necessary condition for the induction of causes starting from the recognition of the shape in question was the hypothesis of the proportionality of effects to causes, while Poincaré showed that it did not have to be held as a prerequisite for all science.217 Similarly, these forgetful authors, no doubt carried by an enthusiasm comparable to that of Quetelet in his prime, do not take note of the fact that the question of the universality of the shape of phenomena according to a bell curve, is definitively resolved in the negative in the state of mathematical knowledge. The fact is that in contemporary statistical, econometric and epidemiological models, this curve is used to express the equation of the probability distribution of the deviations between the observations and their modeling. We then speak of a Gaussian noise which, added to the model, would restore the original observations. The success of this process in practice and in scientific teaching comes from the fact that it is based on a minimization of the sum of the squares of the deviations between the observations and the result of the model. We have already seen the principle of this calculation consolidate since the works of Legendre and from Gauss to Quetelet’s proposals. In other words, the principle of most statistical models is to consider observations as random fluctuations around a constructed model, these fluctuations being assumed to be Gaussian. This is the reason for the practical attachment of calculators to Gaussian residuals at the beginning of the 21st century.

216

Lottin, 1908, then Cadoux, 1908. Maurice Halbwachs was the first to observe that the concept of complexity expressed in Poincaré, 1912, could be opposed to Quetelet‘s theory, Halbwachs, 1912, p. 51–55. On the mathematical and physical developments of the concept of complexity in Poincaré, see, Ghys, 2023. 217

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To understand how Paul Lévy provided a definitive mathematical answer to the problem of the validity or not of the bell curve, it is necessary to distinguish between calculators on the one hand and mathematicians on the other hand, although these two skills are often confused due to a lack of knowledge of their characteristics. The former, the calculators, have an already established toolbox, most often computerized, where they find the technique that they consider most convenient and most adequate for the concrete problem posed – among them are the statistical modelers, econometricians, and epidemiologists; their initial question is: “which tool to choose?”. The latter, the mathematicians, implement a completely different approach because they immediately question the properties of what is given to them and ask themselves this question: “given these variable values (in a word these variables), what are their properties and consequently what is it justified to do to them?” Or, to reach out to the calculators, which method to use in a mathematically justifiable way? The former devour the data, the latter wonder under what conditions they could savor them with taste. From a distance, one is tempted to say: “well! all these numbers (all this data) are established values because they are written on tables or recorded in databases and therefore, they do not fluctuate randomly”. But, the specialist in probability calculation thinks differently: for him, each of these values is the occurrence of a variable subject to random fluctuations, which he therefore calls random variables. While it is true that since political arithmetic (if not commercial accounts) we had added or subtracted established numbers,218 the theory developed by Paul Lévy, is not about the accumulation of fixed numerical values but, to use the title of his reference book, it is a theory of the addition of random variables:219 each variable can randomly fluctuate in a way that remains to be qualified. Here is now a serious problem: from a practical point of view, proceeding to sums value by value or doing it with intermediary subtotal, we must obtain the same results. Of course, for determined values, it comes from the basic associativity rule for addition, but random variables are not so docile. It is for example a minimum requirement for statistics established by districts: these districts added, we hope to obtain the same result as in aiming directly at the total general. At this point, we say that we want random variables invariant by addition. Clearly, such a request could not be conceived in the realm of the avalanche of printed statistics. This property characterizes sets of adequate random variables, understanding that the others subjected to calculations would lead to inconsistencies. This is where mathematical competence comes into play and asks: what are the probability laws invariant by addition? And what about the known laws? Let’s first clarify a concept and then go to the conclusions that I will express in the countable case (understanding that it would be possible to generalize them).

218

It was the perplexity expressed in 1772 by Lavoisier upon reading Hume’s Essay on Commerce was therefore justified: “I only see sums added one to another” (on this point by Lavoisier, see Brian, 1994, p. 258). 219 For current mathematical axiomatic construction: Neveu, 1970.

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THE CONCEPT OF MOMENT: taking the average of a random variable X whose N values are X1, X2. . . XN, is to take its first order moment and assuming that each of the N considered values has a probability of N1 , we calculate X =

1 N

N

Xn (where 1

N

Xn is the sum of the successive values Xn, for n ranging from 1 to N); If we 1

now consider the average of the deviations from the mean, it is its second order moment, something like: 1 ðN - 1Þ

N 1

X - XN

2

1 N

N

X - XN

2

and indeed the more adequate:

1

. X presupposes a first summation Σ (it is the first order) and the

average of the squared deviations, a second summation Σ(Σ). The order number is the number of summations to be performed. FIRST CONCLUSION after Lévy:. The Laplace-Gauss laws are invariant by addition and this explains most of the results exhibited by Quetelet although they become immediately tautological. Based on Gaussian distributions, they showed, once added together, bell curves (that is, Gaussians). SECOND CONCLUSION. The condition for non-Gaussian laws, when added, to form quite a Gaussian is that one must be able to calculate their first two moments (in fact their computer calculation based on fixed values will always be practically possible, but a non finite number of records could lead to an unstable result if this calculation were repeated on other occurrences of the same random variable, for instance actualized stock exchange prices). In common terms, they need a stable mean (strictly speaking a mathematical expectation) and a stable variance. In the formalism used today, after carefully constructing the mathematical space that will be probabilized, let’s abbreviate it as Ω, we write X 2 L2(Ω) –X is as an element of L2(Ω) (the exponent 2 evoques here the double sum) it has a second order moment and implicitly a first order one. The set of Gaussians whose equations are in e-x2 is only a subset of L2(Ω). However, the addition of variables belonging to L2(Ω) due to their invariance by this operation reveals Gaussian distributions, which is why this set is referred to as the Gaussian attractor (the image is that repeated addition attracts such variables into the subset of Gaussians).It was Quetelet’s realm. THIRD CONCLUSION. Sticking to L2(Ω) excludes vast sets of random variables. And consequently, the Queteletian scheme is not universal. On this point, Fourier, Poisson and Quetelet gave in to their enthusiasm: the Gaussian attraction domain is not universal and pan-normality is a false principle. This has been demonstrated for nearly a century today. FOURTH CONCLUSION. Outside the Gaussian attraction domain, there are however random variables invariant by addition. They therefore offer properties compatible with statistical calculations even though they are quite exotic considered from the world of the Belgian astronomer. Then, for instance we may speak of Lévy law or “power law” because their equations are not exponentials in the manner of Laplace and Gauss laws but powers of the considered coordinate. Consequently, the probability of values that would be at the ends of the horizontal axis in the usual graphical

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representation, instead of being almost null becomes no longer negligible: this happens precisely at these ends that the graphs of the Belgian astronomer threw a modest veil as it was difficult to represent them if not to think them in his scheme. This is not to say that the calculation of a power law, its empirical equation, is difficult.220 But this is to say that if a phenomenon obeys such a law, we will not be able to calculate the average of deviations from the mean, its variance, and that its second order moment will be inaccessible. It is also called Pareto’s law because it was empirically demonstrated by the economist Vilfredo Pareto (1848–1923). He concluded from his analysis of taxation in various European countries that 20% of the population contributed to 80% of tax revenues, hence it is also referred to as the “80/20 law” or it is about a supposed principle according to which 20% of the causes would produce 80% of the effects, which is far from the underlying presuppositions of the Belgian statistician’s reasoning. The psychologists’ extrapolations on the 80/20 law are hardly worth more than those developed on the bell curve. These are abusive generalizations where the use of the term law and a mathematical-looking expression are epistemological smokescreens. As for equations, it is not difficult to express an “80/20 law” in the form of a power law. The mathematician Georges-Théodule Guilbaud (1912–2008) compiled a list of areas where such laws have been observed: “the distribution of city populations, industrial concentration, various geographical, linguistic statistics, etc., have the same characteristics as the statistics originally compiled by Pareto.221” It can be seen in terms of individual wealth, formation of agglomerations, size of islands or lakes, that is to say both economic phenomena and natural phenomena, power laws seem inherent to accumulations. Another approach has recently considered that power laws were induced by networks where scale logics, constrained or not,222 matter, and these two classes of explanations are not incompatible if we consider that the circuits of accumulations are most often reticular. Furthermore, the field of finance is where the most massive effects of the inadequacy of Gaussian statistical models to concrete phenomena are manifested. It is remembered that Quetelet’s graphs, both due to the material necessity imposed by the printed medium of his works and in the literal sense due to narrowmindedness, did not highlight the ends of the dispersion curves he offered to his readers. However, if a random variable presents non-negligible probabilities at the extremes of the scheme, on both sides, the calculation of its variance will be compromised, it will exit the Gaussian attraction domain and the models built on this hypothesis will become inadequate. This is precisely what happened during the stock market crashes around the year 2000, so much so that neither the traders nor the automata, all programmed according to the principle of Gaussian predictability, could apprehend what was happening. It is not difficult to show that the intensity of

220

Barbut, 1998, who provides elegant methods. Guilbaud, 1996, commenting too the work of George Kingsley Zipf (1902–1950). 222 Clauset et al., 2009 and Broido & Clauset, 2019 providing new insight; and for a general overview: Zajdenweber, 2009. 221

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the rhythms of transactions affects the shape of the distribution of the uncertainty observed on the indices of major stock exchanges.223 And for several years now, specialized researchers have been exploring models based on taking into account extreme risks, in the sense of their position on the already commented graphs.224

3.7

Conclusion: The Historical Corroboration of Averages

Throughout the 19th century, statisticians, administrative or scientists, have churned numbers deliberately accumulated on a quasi-planetary scale, that is, on a new scale in such a short time. From a historical point, this global scale and the coordination of the production and circulation of digital documents draw attention. Under the impetus of the astronomer Adolphe Quetelet and through the action of the international congress of statistics, a know-how has thus been forged and promoted, a calculation tool – the average – and its counterpart, the presuppositions according to which this tool was suitable for measuring the probability of causes supposed proportional to their effects, a dispersion of this probability of effects symmetric around its maximum, the average, first increasing then symmetrically decreasing, deviations from the central value remaining relatively infrequent, and the probability of extreme values negligible. The calculator as the layman, today, facing numbers, overwhelmed by the weight of practices formed in the Golden Age of international statistics, they will immediately approach the figures and abstract numbers they seem to carry, without concern, or even awareness of the particular conditions of the formation of this know-how and the presuppositions induced by the hypostasis of a particular process, as was the case for the ratio in the 17th century, this time, in the doctrine of the founder as in use, the process has become the object. But, such a vast enterprise as that of the Belgian scholar, how to keep in mind the actual conditions! Here is another idea of a historian epistemologist. We must again summon here Lucien Febvre and his slogan of 1936: “the enemy is simplism!”. In terms of figures and data, it is the same and unfounded extrapolations like technophile dreams believe they can stick to the figures as they are sealed in the registers or databases, while in each of them resides a fragment of this long global history and that the apparent realism of such extrapolations is dependent on it. Among today’s calculators, there are particular ones, those whose specialized expertise would be ruined if they had to give up the credit they give to the bell curve. This belief is corroborated by the extent of collective experiences of the 19th century. This is to say that it is robust and that it presents itself in the daily life of these specialists as an obvious fact which, if it were not shared, would only call for contempt, either in the terms of Francis Bacon an idol of this theater of compe-

223 224

Brian, 2009. Le Courtois & Walter, 2014.

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tence:225 in this chapter it was question of indicating the principle, as the English philosopher invited us to do long before this idol was formed.226 There remains an epistemological problem, do we live in the world of ratios restored in the previous chapter or in the world of averages visited here, or even in a third world? In which world do we live in this regard? Answering this question is the object of the next chapter.

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225 226

Bacon, 1645, Aphorism XXXIX. Bacon, 1645, Aphorism XL.

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Chapter 4

Idols, Paradigms and Specters in Data Sciences

“What will [...] the philosophy of science be? It will be a phenomenology of the studious man, of the man strained in his study and not just a vague balance of general ideas and acquired results”. Gaston Bachelard, 1951.1

4.1

The Inebriation of Abstraction and Its Misdeeds

Digital traces lend themselves to abstract calculations and everyone can therefore conceive of extrapolations whether they are prompted by fears or by ignorance; or carried by enthusiasm tested in the face of new possibilities attributed to these computing; or finally even based on reasoned constructions. At this point, the superstitious, charlatans and attentive scholars are touched by a common syndrome: the inebriation of abstraction. As soon as a level of abstraction is reached, they cannot help but want to surpass it in order to envisage an even more abstract level. There were schools of true scholars where such a disposition of mind was held as a heuristic method: this was, for example, the case in France with the mathematical school known as Bourbaki.2 But, for a few hundred high-level mathematicians, how many apprentice scholars or novice philosophers are lost, especially among those passionate about the philosophy of mathematics who, rather than seeking to cautiously take their place, according to the old philosophical adage, on the shoulders of giants, cannot stop themselves from climbing onto those of dwarfs themselves perched on the backs of other dwarfs who ride other commentators of whom we no longer even know if they are themselves dwarfs or giants. After the true scholars

1

Bachelard, 1951, reprinted in id. 1972. The great mathematician Szolem Mandelbrojt outlined the principles in 1946, in a text recently published: Mandelbrojt, 2020.

2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 É. Brian, Are Statistics Only Made of Data?, Methodos Series 20, https://doi.org/10.1007/978-3-031-51254-4_4

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and the half-skilled, here finally are the ignorants who, carried away by the inebriation of abstract conjectures, the speculations of the second or fascinated by technical powers are taken with vertigo and panic. This is where contemporary mistrusts are expressed as they feed on themselves and their repercussions. Hence, the priority is the fight against the inebriation of abstraction. By principle, philosophers and true scholars cannot surrender to the phobias of the moment: the path of superstition is therefore to be discarded. We observe everywhere today those taken by charlatans carried away in their technical enthusiasm: it is the path of geeks and opportunists, promoters of so-called cutting-edge techniques, or sometimes their commentators who under the appearance of a critique actually forge its inevitability, not without perversity. The path of this last hypocrisy is hardly better than that of simple superstition. Remains the rationalist path and the controls and the weapon it provides to guide the sciences, epistemology and the commentary of technical innovations.3 But how to proceed in the face of abstraction carried by ancient documents and that of their contemporary counterparts. The historical history of abstraction promoted by Jean-Claude Perrot (1928–2021) offers a second weapon.4 This consists of focusing attention on the materiality of abstract traces. Hence, a figure taken in the 17th century in the turmoil of merchant accounting as we saw previously in the second chapter, although it lends itself to calculations like another, is not to be considered in the same way as a counterpart printed in the 19th century and once carried away in the avalanche of numbers designated by Ian Hacking (1936–2023),5 the one considered here in the third chapter, no more than to such another digitized whose materiality does not come from manual tracing, nor from printing, but from an electronic recording. These three materialities have gone and go hand in hand with radically different conditions of production, use and circulation. Here are three regimes that the material history of abstraction allows to distinguish and of which it analyzes the effects. It is thus only because of the tropism aroused by the inebriation of abstraction that these figures seem to be able to be chained in the same calculation operations largely subsequent to their production. The numbers they carry because they are abstract, certainly, lend themselves to calculations. But from the point of view of their history and of their own historico-epistemological necessity, to confuse these three numerical states, their materiality and their potentials, would be both an amalgamation and an anachronism. On the contrary: the relationship of figures to numbers appears from then on as an object of historical investigation.6 Thus the three epochs, all carriers of abstract traces distinguish themselves according to characteristic material supports: manuscript registers in the 17th century, printed collections in the 19th century and electronic files today. Their confusion, once sobered from the 3

Bachelard, 1949. Perrot, 1992, 2021. 5 Hacking, 1990, p. 33. 6 This is the starting point of Brian, 1994. 4

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inebriation of abstraction is nothing less than the worst of errors if it is a question of thinking as a historian or an epistemologist.7 It is still acceptable that professional calculators are satisfied with such anachronistic artifacts, the only thing that matters to them is to consume data. In doing so, they despise meticulous historical work but they play with their sources without rhyme or reason. In this respect, such unscrupulous calculators are hardly historians in the proper sense and they thereby step outside the scope of exercise of controlled reasoning, even though the material history of abstraction could open up new horizons for them.8 The historical problem of the relationship of figures to numbers once properly posed, the two periods examined in the previous chapters appeared as characterized by specific skills and presuppositions.

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In the times of the profusion of manuscript accountings, these skills came from the teaching of arithmetics derived from Antiquity and several times reformulated all around the Mediterranean basin. In this particularly explored, translated and extrapolated ancient corpus, the definition attributed to Euclid of the ratio between two quantities of the same kind, their ratio, has had a particular fate due to its obscurity. Here is the very type of an ancient state of knowledge that a contemporary calculator, or even the programmer of the machine he would implement, would not suspect. So much so that the ratio between two magnitudes, once held for a “habit” in them, has moved from concept to substance and has been hypostasized. The know-how, the arithmetic process of the ratio, in the long run, thus forged a presupposition: the idea that well-chosen ratios should be constant. We remember the idols that Sir Francis Bacon (1561–1626) targeted in his time: these false notions that he held to be deeply rooted in human knowledge.9 He divided them into four classes to which he gave evocative names: the idols of the tribe inherent to human nature; the idols of the cave maintained by an individual entrenched in his reasoning; the idols of the market aroused by exchanges between humans; and the idols of the theater10 arising from ancient scholarly or philosophical dogmas, acquired in the manner of systems and played by their promoters in the manner of theater roles.11 Sir Francis intended to rid the sciences of such idols but four centuries after the publication of his Novum Organum scientiarum, thus named to mark his desire to break away from Aristotelianism that had preceded him, it must be acknowledged

7

Lucien Febvre did not have harsh enough words to condemn anachronism in the introduction of his great book on Rabelais: Febvre, 1942. 8 Brian, 2021. 9 Bacon, 1620, Aphorism XXXVIII. 10 Id. ibid., Aphorism XXXIX. 11 Id. ibid., Aphorism XLI–XLVI.

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that it would be appropriate to undertake some new idol hunt. Thus, the idea that a ratio would be constant runs in specialized corpora under the pen of authors who suspect neither its origin nor its history, that is to say the duration and extent of human activities from which it proceeds. Precisely, this extent and this duration, out of reach of the recent promoters of this theater idol, make it so they cannot conceive of a ratio otherwise and they therefore maintain a particular orthodoxy not conducive to peaceful scholarly exchanges. A case in point is offered, in Demography, by the proportion of the two sexes at birth.12 Since the 18th century, calculators have taken pride in the fact that they had predecessors in the golden age of classical science: the 17th century. Among them, John Graunt (1620–1674) has been considered one of the main founders of political arithmetic, then of statistics and demography. With disciplinary piety, excerpts from his works have been repeated and his calculations have been reiterated without paying attention to the fact that, today, this understanding of arithmetic, if it is about doing science, has become completely obsolete after two fundamental subversions: the development of statistical calculations and probabilities based on integral calculus since the work of Laplace at the beginning of the 19th century and their axiomatic basis developed in the 20th century. The master statistician Jacques Bertillon (1851–1922) concluded the second chapter of his Cours élémentaire de statistique administrative (1895) in these terms13: “The motto of statistics is therefore: “Noli me tangere”. Do not touch my definitions, my methods, my frameworks! This motto is wise, no doubt, but it is good not to remain too constantly faithful to it”.14

He meant by this that the permanence of these frameworks was the guarantee of the quality of international and chronological comparisons of statistical work and he left a door open for minor improvements. Thus, the ideal of statistical competence proclaimed by one of the founders of demography at the end of the 19th century and the presupposition carried by a procedure specific to political arithmetic have mutually reinforced each other to make demographers still hold the sex ratio at birth as a constant. This ratio of the two sexes at birth, the first to have studied it in depth and who drew the attention of scholars to his results was the draper John Graunt (1620–1674). A passage from his 1662 work thus inaugurated a type of calculation that is still alive

12

Brian and Jaisson, 2007, analyze the relationships maintained between natural, mathematical and moral sciences for four centuries on this subject. 13 Jacques Bertillon, head of the statistics office of the City of Paris and regular participant in the sessions of the International Statistics Congress, had succeeded in this role his father, LouisAdolphe Bertillon (1821–1883), promoter of demography; as had been his grandfather Achille Guillard (1799–1876). 14 Bertillon, 1895, p. 12: address attributed to the resurrected Jesus in front of Mary Magdalene according to the Gospel of John (20,17) in the Latin tradition stemming from the translation of Jerome of Stridon.

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today: the comparison of the numbers of boys and girls among newborns for a given place: “[I]n [the] Parish [of Romsey] there were born 15 Females for 16 Males, whereas in London there were 13 for 14, which shows, that London is somewhat more apt to produce Males, then the country. And it is possible, that in some other places there are more Females born, then Males, which, upon this variation of proportion, I again recommend to the examination of the curious”.15

As we can see, the draper merchant Graunt, like his contemporaries, reasoned in terms of ratio and considered that the one calculated for such a locality was characteristic. Its century and the next did the same, even simplifying comparisons by using the number of boys per hundred girls as a criterion.16 Even today, specialized studies use the index of the number of boys per hundred girls. And specialists, in a recent reference work, even wrote: “the figure of 105 boys to 100 girls [is] one of the rare demographic parameters that are almost constant.”17 The statement is obviously flattering for the demographic discipline, which is thus implicitly elevated to the rank of sciences where universal constants such as Planck’s constant or the speed of light reign. But it is simply false, for two reasons, one empirical and the other methodological. Indeed, since Maurice Halbwachs’ research on this subject, it has been established that, during the First World War,18 the proportion of male births in France was certainly higher than 105 boys for 100 girls (or 51.22%). Moreover, it is possible to show that this ratio can vary greatly depending on various social and historical conditions.19 As for the methodological argument, it comes from an extension of the paper on the question presented by Siméon-Denis Poisson (1781–1840) to the Academy of Sciences in Paris in 1830.20 This mathematician abandoned the old ratio (the number of boys per hundred girls, formally 100.G/F if G is the number of boys and F is the number of girls). He approached the calculation as a probabilist studying a binomial law, in short, scrutinizing not 105 but 51.22%, i.e., the frequency G/(G + F), it being understood that the latter has the same probabilistic properties as its complement F/(F + G). To test the possible variability of the proportion of the two sexes at birth starting from Poisson’s scheme, we will first consider the central value s of a hypothetical binomial distribution of births of one sex or the other (for example s = 50% if we were to verify the hypothesis of equiprobability, or s = 51.22% if we wanted to test the constancy affirmed by the demographers cited above); then secondly the empirical frequency S = G/G + F); and finally a threshold for which we would consider the gap between s and S to be sufficiently improbable (the practice here is to retain for example a threshold of 95%). As soon as this threshold 15

Graunt, 1662 [1975, p. 71; 1977, p. 109 and 138]. Brian and Jaisson, 2007, chap. 1. 17 Caselli & Vallin, in Caselli, Vallin, & Wunsch, vol. 1, 2001–2006, p. 57. 18 Halbwachs, 1933. 19 Brian and Jaisson, 2007, chap. 6. 20 Poisson, 1830. 16

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is set, an interval between s and S is deduced: that for which we can consider, at the retained threshold, this gap as very unlikely. Thus we can rule on the gap between the hypothesis and the empirical results. It is by this means that we can establish, following Poisson and starting from the most diverse sources, that the proportion of sexes at the birth is not constant. Let’s then denote α as the measure of the magnitude of this interval: indeed, we have here been interested in the probability that |S - s| ≤ α. If we now kept the same hypothesis about the probability of births of both sexes, and if we calculated the ratio 100.G/F. We could also reason according to a principle of confidence interval. However, as in this empirical ratio the random fluctuation affects both the numerator and the denominator and this in a combined way, the calculation is less elementary. It is not very complicated though.21 Its result leads to the conclusion that for the same test threshold the magnitude of the deviation is 4 times α! This means concretely that to decide at the same level of probability it would require 16 times more observations. Therefore, two conclusions can be drawn: from a metrological point of view, the sensitivity to variations of the phenomenon of the instrument that constitutes the old ratio is clearly worse than that of the proportion adopted since Poisson; and from a demographic point of view, this poor quality of the instrument could only reinforce the idea of a quasi constancy of the traditional ratio. Why then, do specialists of all kinds persist in the use of such a bad indicator when since 1830 another, better one, was within their reach? By tradition and routine: any demographer to whom one mentions a ratio of 97, 102 or 110 imagines the phenomenon, however inadequate this index may be.22 It is high time to discard the proportion of boys per hundred girls, this idol of the demographic theatre. Keeping in mind the comparison of the last three odd centuries – the seventeenth, nineteenth and twenty-first – and the fact that each time the figures were elaborated, exchanged and consumed being carried by different material supports, respectively: manuscript documents, printed collections and finally electronic supports, what can be said about the present times? The first observation is that the skills associated with ratios and averages and the presuppositions that accompanied them came from extensive circulations according to a time scale of several millennia in the first case or several centuries in the second and in very vast spaces at least Mediterranean in the first and global in the second. The experience of today’s data does not fit into a long homogeneous duration, at most a few decades. On the contrary, their circulation is undoubtedly global. Do we have works that would synthesize their experiences, comparable to what were in their time the works of Leonard of Pisa or Adolphe Quetelet? Probably not. From a more technical point of view, it is in the analysis of the properties of networks and in the adaptation of methods already known during

21

It is discussed in Brian and Jaisson, 2007, Appendix D, p. 221–229; and in Brian, 2007. The calculation tests that have just been summarized were published in Brian and Jaisson, 2007. The work mentioned in the reference list of Duthé et al., 2012. However, in this article dealing with a relevant question, the authors are content with the traditional index of the number of boys per hundred girls among newborns.

22

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the last decades of the 20th century to treat gigantic collections of data – to use a current term: “big data” – that new skills are being formed without any of them becoming the emblem of the new century or, already, presuppositions spreading among specialists or laymen. In a count of newborns, the traditional instrument of the number of boys reported to a hundred girls has undergone a hypostasis in the demographic field. Similarly, since the 19th century, the concept of average and the process of its arithmetic calculation have together taken on the form of substances in various fields of expertise: this is how corpulence measured by the BMI (Body Mass Index) in the field of nutrition or intelligence measured by the IQ (Intelligence Quotient) among psychologists. Quetelet’s average, however, seriously subjected to scientific criticism between the middle of the 19th century and the First World War,23 has therefore experienced various hypostases to which experts are as much attached as some demographers to the number of boys per hundred girls in a set of newborns. Today, the concept of network in the field of data is an abstract concept that finds its origin in computer technology and in mathematics, likewise that of algorithm is a concept just as derived from the multi-millennial history of mathematics. In two or three decades today, these two concepts, consolidated by the global scale of electronic circulations and by the intensity of global exchanges of skills in their regard have already reached the stage of their particular hypostases: most authors or speakers use the terms network or algorithm considering them as permanent and existing by themselves, that is to say as substances, blind that they are or that they want to remain to the formation and transformation of these know-how and concepts, in the way that, under the influence of statistical conceptions formed in the 19th century, we have long held, in the last century, the enumerations for things given of themselves, or under that of the demographic tradition the number of boys per hundred girls among the newborns of a given place is envisaged as analogous to a physical constant. The three centuries considered had or have therefore each nourished their hypostases of abstraction and it is up to rationally conducted investigations to analyze their formation if only to offer tools to fight against contemporary superstitions apparently so difficult to unmask as they thrive on the ignorance of ancient or recent abstract operations. At this point a question arises: to reach these conclusions it was necessary to break with the effects of the inebriation of abstraction. For this purpose we have distinguished three regimes of its numerical expression: that of handwritten traces, that of imprints left in statistical collections and that converted into electronic codes. Would these be three distinct worlds so to speak incommensurable? And the knowhow that were the property of the two main ones studied here as ideal-types, are they the indices of two scientific paradigms that it would be important to distinguish before cautiously entering a third? The first question, that of incommensurability can be considered in various ways, but if we consider it from a material and technical point of view, it is clear that the

23

Gouraud (1848); Lottin (1912) or Halbwachs (1912).

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compilations of political arithmeticians have been copied and transported up to the times of statisticians, something that can be seen notably in the columns of the first collections of the Statistique générale de la France or in the publications of the Swedish Statistiska centralbyrån. What’s more, the compilation techniques of the 19th century often found their origin in works from previous centuries. That’s why the heads of offices as regular members of the sessions of the International Statistical Congress were keen to inform each other of the national history of their competence. Jacques Bertillon has summarized these narratives by affirming their methodological interest in his Cours élémentaire (1895). Thus, between the seventeenth and the 19th century, it was not only numerical traces that circulated, but elements of specialized know-how that circulated too. Second argument: between the world of state offices statistics on one hand and contemporary computer systems on the other hand, there is indeed a continuity. The engineer Herman Hollerith (1860–1929), the inventor, in the United States, of electromechanical machines with punched cards has put implemented this innovation during the constitutional census of 1890. His company later became the International Business Machine Corporation. Its acronym IBM says enough about what subsequent technologies owe to the shift from paper and printed calculations to electromechanical, then electronic. Thus, as long as we pay attention to the material and technical aspects, the three regimes that we distinguish are not separated by chasms of incommensurability. Does this mean that we should reason in a continuous manner throughout five centuries? Certainly not, because between the political arithmetic of the 17th century, the statistics of the two following centuries and the data science of today, several epistemological breaks have occurred: the one made between Euclidean origin calculations on the one hand and differential and integral calculus on the other hand; similarly, between these two classes of mathematical works now dated and the nonlinear methods or algorithmic processes implemented today, another epistemological break has occurred. These breaks are not a reason to lose sight of the continuity of the material elements just described or the technical lineages attested at this point in the survey.24 The question therefore arises of how we can implement the rational activity of science – to use a Bachelardian tone – once the scholarly horizon is cleared of idols, especially those of theater according to Sir Francis’ principle: “To point [the idols] out [...] is of great use; for the doctrine of Idols is to the Interpretation of Nature what the doctrine of the refutation of Sophisms is to common Logic.”25

24 25

On the concept of technical lineage, see: Simondon, 1958. Bacon, op. cit., Aphorism XL.

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Living in a Scientific World Cleared from Idols

The philosopher Jacques Derrida (1930–2004) invoked the notion of specter to analyze the imprint of the theories of Karl Marx (1818–1883) and Sigmund Freud (1856–1939) on thought in the 20th century. The historian Élisabeth Roudinesco showed how Derrida himself followed the founder of psychoanalysis, in whom this concept had been built on the principle of a repression of the spectral object.26 The philosopher Charles Alunni, a student of Derrida, used it to account for the scope and potential of Gaston Bachelard’s epistemological work despite deliberate or unintentional eclipses.27 As far as we are concerned here, should we write “the specter of ratio” or “the specter of the average”, or even “the specter of networks and algorithms” in the sense elaborated by Derrida and Alunni? Probably not: the hypostases of calculation processes respectively specific to the 17th, 19th and 21st centuries have not experienced repression: they have on the contrary been highlighted for the first two of them and they are widely known for the most recent ones. Here, it is rather the etymological meaning of the word “specter” that suits them: the appearances of ratios in the manner of Graunt and of the average man in that of Quetelet are indeed ghosts, idols, capable of feeding imaginations and for the most recent ones, images stirred up in a frightening manner. In the conception of the two philosophers, there is a dialectical relationship between repression and the power of the specter, if not between the power of the specter and the intensity of its repression. Here, there is no repression, only the presence of hypothetical substances that would float in a universe of abstract extrapolations for lack of control based on reasoned criticism. Starting from the notion in line with its etymological sense, we find ourselves at the end of the investigation in a world of specters. Products of scholarly or profane imaginations prey to hypostases of abstract processes whose robustness - from a historical point of view - comes from the very long duration of these processes or the vast extent of their formation and their uses, all kinds of diversities amalgamated in the inebriation of abstraction. Such specters do not float in the air. They are carried and maintained by those who are familiar with them or who, conversely, fear them. We are now on the field of a sociology of knowledge where we know how to consider such abstractions held as substances. Two author names come to mind: those of Thomas Kuhn (1922–1996) and Maurice Halbwachs (1877–1945). The former indeed analyzed the regimes of scientists’ adherence to the presuppositions of their work.28 And the latter developed a theory of collective memory capable of accounting for, as we will see, the cohabitation of the specters that we identify, and the way in which specialists and laypeople engage with them.29

26

Roudinesco, 2006. Alunni, 2018. 28 Kuhn, 1962, 1977. 29 See Brian, 2008, 2013 and Brian et al. (dir.), 2011. 27

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The survivals of old presuppositions – for example those associated with ratios or averages – fall under what Sir Francis Bacon called the idol of the theater adopted afterwards by specialists of this or that calculation in the manner of postures played without critical reflection in accordance with the traditions attached to them. However, it is clear that they are maintained by communities that recognize themselves in this act of preservation and consider it a guarantee of their specialty. This is then the very principle of collective memory of what – in Durkheimian vocabulary – Maurice Halbwachs called a group; such as for example “the group of musicians”, in the sense of performers who perpetuate performance traditions. Here it is about, for example in the case of the proportion of sexes at birth, what he would have called “the group of demographers” and regarding the know-how from the 19th century, for example that of psychotechnicians attached to the method of the intelligence quotient, sealed in the acronym IQ, or that of dieticians attached, them, to the body mass index, the BMI. As for M/100F, IQ and BMI, these are all emblems of the collective memory, respectively, of demographers, psychotechnicians and dieticians.30 The analytical scheme proposed by Halbwachs, theorist of collective memory, offers the interest, from a sociological and historical point of view, of highlighting where lies the power of each of these specters when criticisms are formulated: in the consistency of the concerned communities, in other words in the corporatism they shelter. The purpose of this book will undoubtedly not fail to offend them. In the Halbwachsian theory of memory, it is important to distinguish two concepts often confused when it is only superficially invoked: collective memory on the one hand and social memory on the other. Indeed, for this sociologist, there is no collective memory without a particular community that maintains it in the manner of faithful or specialists whose social existence as a community would be jeopardized if this memory weakened. Social memory, it is not the property of a particular community, but it applies to society understood in the broad sense, that is to say where many communities and specialties coexist. An extrapolated example from his study on the memory of musicians will make feel the difference between the two Halbwachsian concepts of memory. The Boléro by Maurice Ravel (1875–1937), composed and created in 1928, is undoubtedly the most performed musical work of the 20th century and broadcast today. This is to say that this piece is a pinnacle of the social memory of the music of the last century. However, each instrumentalist, in order to interpret it, relies on a corpus of gestures passed down among their peers. Therefore, there are as many collective memories specific to distinct musical communities as there are specialized performers engaged in the execution of the piece. The relationship of these specialized collective memories to the social memory of the piece is like that of accomplished skills to profane familiarities. Confusing the two would be to gloss over the former.31

30

See the publications indicated in the previous note. Halbwachs’ article on the memory of musicians was published on the eve of the Second World War, and for reasons of topicality, he had taken as a case study the Ride of the Valkyries by Richard 31

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For each of the two generating skills that have been studied, we observe today both states of memories, these are the theater idols to return to the vocabulary of Sir Francis Bacon and a state of social memory that deserves attention. Thus, the idea that the numerical ratio between the two sexes would be a constant ratio is the principle of frequent institutional or political reasoning about parity. This reasoning pattern is not strictly demographic but logical: two categories, this would imply 50–50%, while by the very movement that led to the awareness of discrimination between men and women, more complex conceptions of genders emerge that could ruin the logical balance, or substitute it with more complex equations, if not, to escape new complexities, revert by reaction to some unknown simplism that would favor one or the other gender. The presupposition of a constant ratio of the numbers of the two sexes thus stirs two specters today: one in the collective memory of demographers and the other in the necessarily eclectic social memory of a principle balance. One can immediately guess that attempts to intervene on the two types of memories, collective or social, to be effective, must take different forms. In the case of remnants of a state of specialized collective memory, since their discussion involves specialized skills, interventions should be placed in the arena of these specialties precisely. For our part, this was the principle of our already cited works on the indices of proportion of the two sexes at birth.32 Does this mean that as soon as a fallacy is demonstrated in some reasoning accepted within a scholarly community, it would renounce its fallen idol. To envisage this, from the point of view of a sociology of knowledge, would be to confuse the individual rationality of scientists faced with the demonstration of the fallacy and the collective rationality of a corporation. However, this collective rationality, in such a case, for reasons of collective self-defense of the community will go towards the maintenance of the idol of the specialty, its a totem. The conditions under which a discipline can reflect on itself, renounce contradicted premises, and so to speak refound itself, was indeed the subject of the two main works of Thomas Kuhn.33 At this point, it is clear that the issue of scientific revolutions pertains to the collective memory of a discipline and not to its social memory. There are authors who miss their targets as they aim at a collective memory by attacking the social memory of a fragment of competence. The characteristic of such gestures is a prophetic style and the fact that they are expressed in works classified by publishers under the heading “general public” when it is not through video recordings full of clichés. Thus in the case of presuppositions induced by reasoning built on the average, a former academic has loudly proclaimed: “the average harms you”.34

Wagner (1813–1883) whose performance then implemented the collective memories of the communities of interpreters; even as its social memory was annexed by the Nazis for their propaganda. In the case of Ravel’s Boléro, such political influence is absent. 32 Brian and Jaisson, 2007. 33 Kuhn, 1962, 1977. 34 Rose, 2016, 2018a, b.

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Through these inappropriate gestures in the scientific world and his mistake in choosing his target, the author has undoubtedly exposed himself to the benefit of public notoriety but not to that of his scholarly authority. To go further, it is appropriate to consider the works of three well-known authors: the historian Lucien Febvre (1878–1956), the philosopher and historian of science Alexandre Koyré (1892–1964), and the philosopher and sociologist of science Thomas Kuhn (1922–1996). The first expressed bluntly to mid-20th century students that he had been trained up to his higher studies in such a way that he had held, around 1900, the state of previously acquired sciences as a peak of human knowledge. But, as soon as he entered the École normale supérieure, a few years later, he was deeply shaken by the eruption of Einsteinian physics that passionate classmates made him aware of. He then became aware that the recourse to sciences as he had learned it before this shock had been an illusion of tranquility, and that science itself was at work as soon as the shaken scholars, driven by worry (inquiétude), assumed that they were both custodians of the legacy of previously formed sciences and engaged in the exploration of possibilities to which this worry could lead. One of the founders of the historiographical renewal of the 20th century, he was also the director of an encyclopedic enterprise carried out in Paris during which he matured and expressed a deep conception of the historicity of sciences. It governed his requests for contributions to the Encyclopédie française addressed to scholars from all fields.35 “The Encyclopédie, sets itself the task of analyzing and explaining the creations and manifestations of contemporary humanity. Explaining , is essentially showing in our current conceptions a temporary balance between what we knew, conceived and achieved yesterday and what we will know, conceive or achieve tomorrow”.36

This passage is taken from a typewritten document addressed to the authors approached by the director of the Encyclopédie and now preserved in the archive of one of them, it has not been published before 2023, except that all of the historian’s interventions in this collective work from the 1930s to the 1950s or elsewhere during the same period on the question of sciences decline this theme.37 Alexandre Koyré was four years younger than Febvre, which means they shared the same scientific background considered accomplished at the end of the 19th century and the same experience of a upheaval in the face of physics for which the years 1905 and 1915 were decisive. Should we therefore be surprised that the 16th-century historian and the philosopher both approached the scientific renewals subsequent to the articles of Albert Einstein (1879–1955) published at the beginning of the 20th century by holding the modern theorist as the counterpart of a Nicolas Copernicus (1473–1543), the promoter of heliocentrism in the case of

35

The collection Febvre, 2023, gathers and analyzes the texts that manifest his intellectual journey in this regard between the years 1910 and 1950. 36 Id., ibid., paragraph [4§3], p. 49. 37 See our analysis in the postface to the cited work: “Position of the Febvrian problem”, Id., ibid., p. 179–189.

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Febvre: or of an Isaac Newton (1642–1727), the designer of universal gravitation in that of Koyré.38 It is through entirely different paths that Thomas Kuhn directly confronted the problem of a shift in the scientific horizon from one era to another, first in his lectures, then in a project for a book intended for a contemporary encyclopedic enterprise.39 This was the starting point for his great book on the structure of scientific revolutions.40 Which still feeds today the reflection of philosophers and historians of science.41 This work has helped to promote the notion of paradigm to characterize a coherent regime of exercise of a scientific discipline devoid of internal contradiction. It is thus clear that the three authors, although from different disciplines, wanted to respond to the same intellectual conjuncture occasioned by the radically different conditions of practicing physics in the nineteenth and then the 20th century. That’s why, if we return to the question of numbers and abstraction we would be inclined to think that there would have been a paradigm of “ratios”, another of “averages”, finally a third that we will no doubt call “networks”. We have then shown two important things: as far as they were paradigms, it is necessary to note: firstly that each of these systems of know-how, of presuppositions and hypostasized notions has endured and that today for calculators, ratios, averages and networks are like the elements of a toolbox which they do not seek to know if they are compatible or not. Yet the contradictions they present could be resolved in the manner of the “resorptions” indicated by Kuhn.42 But these ways of resorption, if we suggest them here are not even explored today. Here is therefore our second conclusion, in terms of abstraction, what could be considered as incoherent paradigms do coexist. We thus live, as scientists, in a world of systems each time coherent and often outdated, sometimes contradictory with each other. In other words, the world of abstraction is populated by specters in the simple sense of the word that we have chosen to retain. No doubt, what metaphysics owed, from a historical point of view, to religious dogmas provided the 19th century with the illusion of a unified or coherent general scientific horizon constantly reaffirmed until science itself witnessed its dissipation at the beginning of the following century. The great philosophical and historical works subsequent to this collective denial were like a collective work of mourning. We want to believe that works like this one, oriented towards the history of abstraction, can contribute to the accomplishment of this mourning. A question then comes to mind: what remains, under these necessary conditions of the unity of the sciences? It cannot reside, one guesses, in the disparate collection of pseudosubstances forged on obsolete know-how. It will reside in the very action of the scientists, in their gestures and in the forms of control they will give themselves: in a

38

As for the philosopher, see Koyré, 1957, p. 169 in the 1968 reprint; and for the historian see the cited work, Febvre, 2023. 39 Neurath et al., 1938. 40 Kuhn, 1962. 41 Blum et al., 2016. 42 Kuhn, 1962, p. 173–189.

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word, in the action of “the studious man, the man tense in his study,”43 in the terms of Gaston Bachelard, that is in reason in action. Despite his dazzling demonstration, Thomas Kuhn, by firmly placing himself under the aegis of the sociology of science, jeopardized his analysis. Indeed, the sociology of science of that time, that of Talcott Parsons (1902–1979) or Robert King Merton (1910–2003) mainly, is “structural-functionalist”. The social world was considered from above and the sociologist described mechanisms with functional characteristics. This is the meaning of the word “structure” in the title of the master work of 1962. It is also the reason for its plan which consists of examining science almost as a naturalist if not as a physiologist doctor. Two criticisms have been made against such approaches. Firstly, it was: where then would the sociological point of view stand in such a case?44 A question that can be retranslated as: does this sociological posture not consist in establishing the sociology of science, or even Thomas Kuhn’s theory, as a new general metaphysics? Another salvo of criticism came from the science studies of the 1980s and 1990s: at this altitude, do we really grasp what scientists are actually doing? It was essentially the criticism of a “Bottom up” approach. Thomas Kuhn’s second reference work consisted in responding to these criticisms often expressed shortly after the publication of the first, those oriented towards the ambiguities of the notion of paradigm, or those aimed at the approach itself.45 Scientific activity is then no longer characterized by normal or critical regimes but by an “essential tension” that works between scholarly traditions on the one hand and the potential for innovations on the other. Kuhn could not have known the outcome of historian Lucien Febvre’s reflection on the historicity of science. However, it is clear that the essential tension in question in Kuhn is nothing other than the “provisional balance” that Febvre asked the authors envisaged for his encyclopedia to explain in their articles.46 Thus, the truly phenomenological upheaval induced by the physical and even biological advances of the very beginning of the 20th century led their eminent commentators to break away from the metaphysical perspective resulting from the philosophical tradition formed from Aristotle to Kant. Febvre thus anticipated the philosophers’ break away from this traditional point of view. Doesn’t he expect from the contributors to the Encyclopédie française that they “show in the scholar [...] not a possessor of the absolute sitting at the center of an immutable universe, but a perpetual investigator in constant defiance against the conceptions of others or his own.”47 In the 1970s–1990s, sociology in various languages took up the investigation of scientific activities in the hope of breaking with previous philosophical expectations. But in doing so, the problem clearly conceived by the historian Lucien Febvre and stated as well by the philosophers Gaston Bachelard and Alexandre

43

Bachelard, 1951, reprised in id. 1972, see the epigraph of this chapter. This is the critique of objectivism notably in Bourdieu et al., 1967, p. 34. 45 Kuhn, 1977. 46 Febvre, 2023, paragraph [4§3], p. 49 (document written in 1934). 47 Id., ibid., paragraph [4§6], p. 50. 44

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Koyré was lost sight of. Many factors have combined in this loss.48 Two authors have taken stock of these decades: they have highlighted that these works, themselves, have not freed themselves from the unifying perspective of the metaphysics of previous centuries: this moment of the sociology of science, according to Terry Shinn and Pascal Ragouet have drawn a picture of the sciences where they appeared undifferentiated: the traits that would characterize one among them would be common to the others. The two authors then argue for a sociology of the differentiation of sciences.49 The question then becomes, how do the sciences distinguish themselves from each other, given that no one will doubt the observations of differentiated claims between the various scientific specialties? The answer comes immediately to mind, through the corpus of skills and methodological and technical baggage taught during their respective learning, or even through the routine or critical maintenance of particular genealogies. In the terms of historian Lucien Febvre, it is to say “by the legacy” of each specialty that the new entrants are required to assume. In those of sociologist Maurice Halbwachs, it would be by specialized “collective memory“. Thus we have a twofold conceptualization that has the advantage of accounting for the maintenance of the distinction between scholarly skills (between disciplines or specialties as you will): a historical and sociological conceptualization (or even praxeological if we think about what Febvre meant by the explanation of scholarly work). Such an approach responds to the weakness of the sociological literature of the late 20th century, since it is opposed to an analysis of the consolidation of the scientific disciplines thus differentiated.50 These legacies specific to a scholarly competence are most often founded, but in the contrary case, Sir Francis Bacon saw them as theatre idols. For Thomas Kuhn, it was about prevailing scientific traditions: one of the poles of the tension he finally considered essential. For us, these are disciplinary spectres often helpful but sometimes guardians of unfounded orthodoxies. The sociology of collective memory developed by Maurice Halbwachs - which is a general sociology and not a sociology primarily oriented towards the sciences - shows that living as a scholar in the world of spectres of previous knowledge, those founded or obsolete, is in principle hardly different from ordinary life in a world saturated with forms of collective memories maintained or not and often even in an agonistic way. As for figures, numbers and data, the spectres of their abstraction, omnipotent, generate among their commentators of all kinds discourses that cannot resist once subjected to a reasoned critique. Among the less informed, it is the illusion of omnipotence that is maintained by the promoters of new technologies eager to be flattered by it; by vain denouncers to the point of imagining themselves thus reaching the heights of criticism51; or even by superstitious laymen. Remains the always

48

Brian, 2023. Shinn and Ragouet, 2005. 50 Brian, 2013. 51 On this principle of magnification in the act of denunciation, see Boltanski et al., 1984. 49

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current task of the philosopher of sciences, that designated by Bachelard. It consists in this case in developing this rational critique and, about data, in providing the means to fight against perverse amalgams. Historian interlocutors have asked this question: would it mean that these handwritten figures, these printed statistics, these recorded data should remain dead letters, and that historical research should abandon the project of reusing them? Far from it. As long as they are subjected, as can be subsequent methods, to a reasoned critique, they can feed constructions developed experimentally. In response to the crisis of economic and social history founded, particularly in Fernand Braudel (1902–1985) on an anachronistic and uninterrogated recourse to conceptions of historical time of economic exchanges specific to the mid-20th century while such works were on earlier periods during which these anachronistic conceptions were not shared.52 The historian, his successor as the director of the journal Les Annales, Bernard Lepetit (1948–1996) proposed an experimental conception of empirical research in history.53 This proposal stemmed from a perspective opened by JeanClaude Perrot (1928–2021), the critic of Braudel and the mentor of Lepetit, for whom extracting the history of the economy from the routine history of its ideas and providing it with the support of a history of the conditions of formation, circulation and reception of ancient explorations of economic and social phenomena should lead intellectual history towards a renewal of the agenda of economic and social history.54 This program can lead to tangible results. It was implemented, at the invitation of Lepetit and at the cost of significant work to reconstruct numerical knowledge about the population at the end of the 18th century, with the aim of reconstructing the state of the French population on the eve of the Revolution.55 In short, the critical and rationalist historicism that has governed our investigation into data and their older avatars leads to an experimental and constructive program of reusing old traces of abstract operations. Febvre set the course to follow in investigations into ancient intellectual activities four years before the publication of his major book on Rabelais (1942) where he implemented them: “The task of tomorrow: first to inventory in detail, then recompose for the studied period, the mental material that the men of this period had at their disposal; by a powerful effort of scholarship, but also of imagination, to reconstruct the universe, the entire physical, intellectual, moral universe, in the midst of which each of the generations that preceded them moved; to get a clear and assured sense of what, on the one hand, the insufficiency of factual notions on this or that point, on the other hand, the nature of the technical material in use at such a date in the society that is being studied, necessarily engendered gaps and distortions in the representations that [such a historical community] was forging of the world, of life, of religion”.56

52

Perrot, 1981. Lepetit et al., 1989. 54 On this subject, see: Brian, 2021. 55 Brian, 2001; for which developments are underway. 56 Febvre, 1938, p. 8●12–7, col. 1, or, id., 2023, p. 199. 53

References

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But, what counts this invitation to the austerity of such a powerful effort of scholarship against the charms of the inebriation of abstraction, and the school conveniences that consist in sticking to reiterate the intoxicating overviews? Most often little, and it is the lot of the scrupulous worker to constantly confront the accepted facilities. Except that by putting in tension the legacy of the past and the potential of new works (Febvre); established traditions and innovations (Kuhn); various states of collective scientific memories (Halbwachs); or to put it in other terms by making confront specters of old or contemporary states of the sciences we work for the science itself.

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Duthé, G., Meslé, F., Vallin, J., Badurashvili, I., & Kuyumjyan, K. (2012). High sex ratios at birth in the caucasus: Modern technology to satisfy old desires. Population and Development Review, 38, 487–501. Febvre, L. (1938). Psychologie et histoire. Encyclopédie française, t. VIII. Société de gestion de l’Encyclopédie française, p. 8●12–3. Febvre, L. (1942). Le Problème de l’Incroyance au XVIe siècle. La Religion de Rabelais. Albin Michel. Febvre, L. (2023). Histoire et Sciences. éd. EHESS. Gouraud, C. (1848). Histoire du calcul des probabilités depuis ses origines jusqu’à, avec une thèse sur la légitimité des principes et des applications de cette analyse. Durand. Graunt, J. (1662). Natural and political observations made upon the bills of mortality. T. Raycroft. (reprint: New York, Arno Press, 1975; trans. fr. by É. Vilquin, Observations naturelles et politiques faites sur les bulletins de mortalité. Paris, éd. INED, 1977). Hacking, I. (1990). The taming of chance. Cambridge University Press. Halbwachs, M. (1912). La Théorie de l’homme moyen. Essai sur Quetelet et la statistique morale. Alcan. (20102, Chilly-Mazarin, éd. Sens). Halbwachs, M. (1933). Recherches statistiques sur la détermination du sexe a la naissance. Journal de la Société de statistique de Paris, 74(5), 164. (reproduced in id. et al., 2005, Le Point de vue du nombre (1936). Paris, éd. INED, p. 381; first pub. in Encyclopédie française, t. VII. Paris, Comité de l’Encyclopédie Francaise, 1936, p. 7●76.01–7●94.04). Koyré, A. (1957). From the closed world to the infinite universe. Johns Hopkins Press. Kuhn, T. S. (1962). The structure of scientific revolutions. University of Chicago Press. Kuhn, T. S. (1977). The essential tension. Selected studies in scientific tradition and change. University of Chicago Press. Lepetit, B., et al. (1989). Tentons l’expérience. Annales. Économies, Sociétés, Civilisations, 44(6), 1317. Lottin, J. (1912). Quetelet statisticien et sociologue. Louvain. Mandelbrojt, Szolem, 2020, « Proposition de création d’une chaire d’Analyse générale et calcul des probabilités (1946) », Revue de Synthèse, 141, n° 3–4, p. 371. Neurath, O., et al. (eds.). (1938). International encyclopedia of unified science. University of Chicago Press. Perrot, J.-C. (1981). Le présent et la durée dans l’œuvre de Fernand Braudel (note critique). Annales. Économies, Sociétés, Civilisations, 36(1), 3–15. Perrot, J.-C. (1992). Une histoire intellectuelle de l’économie politique XVIIe-XVIIIe siècles. éd. EHESS. Perrot, J.-C, 2021, « Histoire des sciences, histoire concrète de l’abstraction (1998) », Revue de Synthèse, 142, n°3–4, p.492. Poisson, S.-D. 1830. Mémoire sur la proportion des naissances des filles et des garçons. Mémoires de l’Académie royale des sciences de l’Institut de France. Paris, Didot, IX, p. 239. Shinn, T., & Ragouet, P. (2005). Controverses sur la science. Pour une sociologie transversaliste de l’activité scientifique. Raisons d’agir éd. Rose, T. (2016). The end of average: How we succeed in a world that values sameness. HarperCollins. Rose, T. (2018a). La Tyrannie de la norme. Pocket. Rose, T. (2018b). The end of average: Harvard’s todd rose on why individuality is the key to the future (on line on et Youtube: https://youtu.be/-34ASwa_Ztk). Roudinesco, É. (2006). Jacques Derrida: spectres de Marx, spectres de Freud. In Bibliothèque publique d’information, Un jour Derrida. Actes du colloque [. . .]. Centre Pompidou. Simondon, G. (1958). Du mode d’existence des objets techniques. Aubier.

Index

A Abstraction, 1–3, 5–7, 10, 12, 15, 26, 36, 54, 64, 75, 132, 145–147, 151, 153, 157, 159, 161 Académie française, 127 Accounts keeping, 23 Algebra, 5, 19, 26, 127 Algorithms, 27, 151, 153 Allard, A., 24 Alunni, Ch., 153 Amado, É., 32 Amalgamation, 5, 15, 130, 146 Anachronism, 5, 23, 61, 146, 147 Analytical Theory of Heat, 84 Analytical theory of probabilities, 43, 83, 104, 127 Anthropology, 121 Anthropology (physical), 121 Anthropometry, 110, 111, 121, 130, 131 Antiquity, 3, 8, 20, 24, 25, 27, 48, 54, 63, 147 Anxiety, 121 Apian, P., 40 Arabic figures, 24, 27 Archibald, R.-C., 24, 29 Aristotle, 30, 31, 40–42, 158 Arithmetic learning, 28 Arithmetics, 3, 4, 9, 24–33, 35, 36, 38, 40, 42, 48, 50, 53, 55, 58, 59, 64, 80, 85, 97, 119, 147, 148, 151 Armatte, M., 79 Ars conjectandi, 49 Art, 2, 13, 27, 32, 34, 48, 49, 56, 71, 77, 101, 110 Art of thinking, 29

Astronomy, 81, 83, 85, 94 Auget de Monthyon J.-B., 72 Augustine of Hippo, 132 Average, 75, 78, 118, 119, 121, 131, 133, 137 Average distance (in astronomy), 76 Average man, 111, 129, 132

B Bachelard, G., 59, 87, 145, 153, 158, 160 Bacon, F., 37, 64, 137, 138, 147, 152, 154, 155, 159 Badurashvili, I., 150 Bailly, J.-S., 62 Baltic sea, 21, 64 Baras, M.-M.-A., 61–63 Barbut, M., 91, 136 Barrême, F., 23 Barrow, I., 35, 36 Bartolozzi, M., 24 Bayes, T., 86, 126 Beginnings (of modern probability), 44 Behar, C., 48 Belgium, 73, 74, 76, 92, 93, 114 Bell curve, 81, 87, 96, 104, 106, 119–123, 125, 128, 130, 131, 133–137 Benoît, P., 19, 24 Berger, P.-L., 7 Berlin, 46, 108, 113, 114, 128 Bernoulli, J., 41, 42, 48–50, 60, 85, 89, 92, 126 Bernoulli, N., 48–51, 54 Bertillon, A., 121, 130, 131 Bertillon, J., 117, 118, 121, 123, 130, 148, 152 Bertillon, L.-A., 121, 123, 130, 148

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 É. Brian, Are Statistics Only Made of Data?, Methodos Series 20, https://doi.org/10.1007/978-3-031-51254-4

163

164 Big data, 83 Binomial, 88, 96, 97, 106, 111 Birth ratio, 43, 44, 148 Blayo, Y., vi Body mass index (BMI), 109, 151, 154 Bompaire, M., 24 Borel, É., 126, 127, 132 Bottin, J., 23 Bourdeau, M., 8 Bourdieu, P., 158 Bourguet, M.-N., 84 Braudel, F., 160 Bret, P., 83 Brian, É., 19, 21, 23, 27, 42, 43, 46, 48, 57, 58, 63, 72, 74, 81, 87, 100, 109, 114, 118, 123, 134, 146, 148, 150, 153, 159, 160 Broido, A.-D., 136 Bru, B., 55, 57, 84 Bru, M.-F., 126 Brunschwig, J., 40 Budapest, 113, 128 Bureau des longitudes, 76 Busse Berger, A.-M., 39

C Cadoux, G., 132, 133 Cahen, F., 37, 55 Calculability, 8 Capitalism, 46, 48, 51 Caselli, G., 149 Cause-effect relationships, 86 Causes, 77, 84, 86, 89, 94, 96, 98, 101, 103, 105, 126, 128, 133, 136, 137 Caveing, M., 25, 30 Ceccarelli, G., 44 Celestial mechanics, 76 Central-limit theorem, 49 Chabrol de Volvic, G., 86, 87 Chamboredon, J.-C., 158 Chemla, K., 24 Chenu, J., 20 Cifoletti, G., 19, 20, 29, 32, 34 Circulation, 1, 2, 13–15, 23, 40, 48, 74, 75, 122, 137, 146, 150, 151, 160 CIS, 108, 112, 114, 118, 121 Clauset, A., 136 CNN, 120 Cole, J., 89 Collective memory, 83, 153–155, 159 Combinatorial analysis, 41 Common knowledge, 120 Comparability, 115 Compensation of highs and lows, 82

Index Condorcet, M.-J.-A.-N., 4, 6, 19, 28, 34, 41–43, 46–48, 51, 54, 56–63, 71, 72, 77–80, 82, 83, 86–88, 94, 95, 98, 106, 127, 128 Conic sections, 90 Coquard, J.-M., 19, 29, 31 Coquery, N., 22 Corroboration, 63–64, 137–138 Couffignal, L., 7 Coumet, E., 41, 44, 50 Courses, 6, 9, 14, 23, 34, 41, 44, 46, 48, 49, 56, 71, 72, 81, 82, 89, 91, 94, 95, 112, 133, 134, 160 Cousin, V., 73, 128 COVID pandemic, 120 Coyer, M., 98 Crampe-Casnabet, M., 4 Crépel, P., 58 Cultural history, 51 Cybernetics, 7–10, 12

D D’Alembert, J., 4, 13, 36, 42, 56, 127 Dactyloscopy, 121 Daston, L., 51, 52 Data, 1–15, 19–24, 27, 36, 45, 47, 54, 61, 64, 71–138, 147, 150–152, 159, 160 Demography, 37, 55, 59, 121, 130, 133, 148 Deparcieux, A., 48 De Roover, R., 24 Derrida, J., 153 Desrosières, A., 89 De usu artis conjectandi in jure, 49 See also Bernoulli, N. Deviation, 80, 81, 86, 89, 94, 102, 104, 106, 107, 117, 124, 133, 135–137, 150 Diagrams, 118 Didactics, 118 Diderot, D., 4, 36, 56 Digitization, 2, 114 Digits, 32 Dijksterhuis, E.-J., 38 Dispersion scheme, 119 Djebbar, A., 24 Double entries, 20, 23 Drobisch, M.-W., 128 Droesbeke, J.-J., 82 Dual monarchy (Austria and Hungary), 113 Dupâquier, J., 55 Duration, 26, 29, 36, 64, 76, 106, 115, 148, 150, 153 Durkheim, É., 89, 129 Duthé, G., 150 Duvillard de Durand, E.-É., 48, 62, 76, 82

Index E Econometry, 125, 133 Economy, 46, 73, 77, 127, 160 Edinburgh, 92, 98, 107 Edinburgh Medical and Surgical Journal, 106 Effects, 7, 8, 15, 26, 27, 53, 78, 84, 86, 89, 91, 94, 101, 103, 105, 110, 115, 126, 128, 133, 136, 146, 151 Egypt, 19 Egyptian arithmetics, 25 Egyptian Expedition, 83 Ehrhardt, C., vii 80/20 law, 136 Encyclopédie (18th century), 156 Encyclopédie française (20th century), 156, 158 Epidemiology, 133 Errors, 25, 27, 35, 77–79, 81, 83, 86, 87, 89, 90, 94, 98–100, 103, 104, 107, 109, 116, 127, 147 Estimation, 51, 52, 61 Euclid, 3–12, 24, 26, 28–37, 39–41, 48, 59, 147 Eugenics, 130 Euler, L., 83, 95 Exponential, 81, 82, 87, 93, 94, 106, 121, 135 Extent, 36, 55, 56, 59, 64, 96, 104, 117, 118, 137, 148, 153 External moral constraints, 128, 129 Extrapolation, 58, 64, 132, 136, 137, 145, 153

F Febvre, L., 14, 28, 37, 38, 45, 46, 64, 82, 112, 137, 147, 156–160 Fermat, P. de, 20, 41 Fibonacci, L., 24, 25, 27, 31, 39, 48 See also Leonardo of Pisa Figures, 19, 25, 28, 32, 45, 48, 57, 60, 64, 74, 80, 97–101, 103, 106, 108, 109, 115, 116, 121, 128, 129, 137, 146, 147, 149, 150, 159, 160 Finance, 21, 23, 46, 48, 57, 136 Forcadel, P., 30–32 Foucault, M., 55, 56 Fourier, J., 72, 73, 77, 78, 82–87, 90–92, 94, 104, 116, 119, 128, 133, 135 Fourier’s series, 83 Foville, A. de, 123, 125 Fractional notation, 20 Fractional numbers, 25, 26 Franci, R., 24 Franklin, J., 41 Fréchet M., 132

165 Frederick II (Holy Roman Emperor), 24, 26, 63 Frederick II Hohenstaufen, 24 Free will, 101, 126, 128, 129, 132, 133 Freud, S., 153

G Gaignat de L’Aulnais, C.-F., 28 Galton G., 130 Garber, D., 38 Garnier, J.-G., 90, 91 Gauss, C.-F., 81, 86, 117 Gaussian distribution, 135 Gaussian noise, 133 Gauvin, J.-F., 38, 39 Gavroglu, K., 157 GCD, 24 Geeks, 146 General sociology, 159 Geometry, 3, 24, 25, 28, 32, 36, 41, 127 Gerschenkron, A., 73 Ghys, É., 133 Gladiator, 97, 98, 110 Global history, 64 Glock, H.-J., 10 Goldstein, C., 19, 20 Goonatilake, S., 25, 27 Gouraud, C., 101, 126–128, 131, 151 Graphic, 118 Graphs, 120 Graunt, J., 148, 149, 153 Gravitation, 76 Grotius, H., 50–52, 54 Guilbaud, G.-Th., 39, 49, 136

H Habit (mutual), 30, 32–34, 36 Habit (skill), 25, 36 Hacking, I., 6, 12, 41, 44, 51, 75, 146 Halbwachs, M., 79, 89, 129, 133, 149, 151, 153, 154, 159, 161 Hald, A., 46, 48, 51 Heat theory, 87 Hecht, J., 55, 61 Henrion, D., 3, 32, 33 Hérigone, P., 34, 35 Herrnstein, R., 125 Herschel, J.-F.-W., 92, 93, 95, 105, 110, 115, 116 Heuschling, X., 74, 116 Homogeneity, 85 See also Variability

166 Hoock, J., 19, 21 Horvath, R., 74 Høyrup, J., 24 Huygens C., 41 Hypostasis, 37, 44, 137, 151

I IBM 370, 109 IBM-PC, 109 Idol of the theater, 37, 121, 154 Idols, 138, 145–161 Inclination to crime, 111 India, 36, 64 Indian numbers, 24 Indian Ocean, 19 INED, 56 Insurance primes, 45, 53 Intellectual history, 82, 160 Intelligence Quotient (IQ), 125, 151, 154 International Congress of Statistics, 108, 111, 118, 137 See also CIS International Institute of Statistics, 74, 118 Irson, C., 22, 23, 28, 54 Israel, G., 38, 62

J Jadart, H., 20 Jaisson, M., 43, 100, 148–150, 153, 155 Jeannin, P., 19, 21 Joas, C., 157 Jorland, G., 42 Journal de la Société de Statistique de Paris, 121 Journals, 2, 15, 44, 74, 91, 114, 160 Jurisprudence, 22, 45, 52, 62, 82, 127 Justinian I., 52, 54, 63

K Kahane, J.-P., 80, 84 Kaiser, W., 19, 21 Kant, I., 4, 158 Keleti, C., 113 Kiaer, A., 117 Knapp, G.-F., 128–130, 133 Knorr-Cetina, K., 7 Know-how, 35, 131, 137 Kolmogorov A., 43 Koyre, A., 37–39, 156–158 Kramp, C., 82, 84, 88, 93, 104

Index Kuhn, T.-S., 153, 155–159, 161 Kuyumjyan, K., 150

L La Chapelle, J.-B., 36 Lacoarret, M., 24, 29 Lacroix, S.-F., 82, 83, 90, 94, 102, 116, 128 Lalande J., 62 La Michodière, J.-B.-F., 21, 57–61, 63 Language, 1, 3, 5, 14, 23, 24, 28, 29, 32, 34–36, 55, 56, 74, 102, 114, 123, 158 Laplace, P.-S., 42–44, 49, 55, 58, 60, 61, 63, 72, 73, 76–78, 80–89, 92, 94, 100, 102– 105, 108, 109, 116, 125–128, 130, 132, 135, 148 Lavoisier, A.-L., 42, 62 Law, 19, 129 Law of large numbers, 88–90, 92, 94, 126 See also Central-limit theorem LCM, 24 Lebart, L., 6 Lebesgue H., 132 Le Courtois, O., 125, 137 Le Gonidec, A., 82 Leibniz G.W., 34, 49 Lemarchand, Y., 20, 28 Leonardo of Pisa, 24–29, 31, 32 Leonardo of Vinci, 39 Lepetit, B., 160 Levasseur, É, 117 Le Roux, R., 7 Lévy, C., 14 Lévy, P., 43, 55, 61, 132, 134, 135 Libraries, 2, 15, 63, 114, 116, 118, 119, 131 Loccenius, J., 51 Logic, 19 London, 43, 54, 60, 114, 116, 149 Longitudes, 87 Lottin, J., 77, 79, 80, 83, 91, 96, 98, 99, 101, 111, 132, 133, 151 Luckmann, T., 7

M Maillefer, J., 20, 55 Maïstrov, L.-E., 43, 45–47, 51, 131 Mandelbrojt, Sz., 145 Manuscripts, 2, 5, 6, 20, 21, 24, 75, 109, 146, 147, 150 Maps, 118 Maritime insurances, 44–54 Marshall, H., 98

Index Martin, F., 47, 51 Marx, K., 7, 45, 153 Mazars, G., 24 McWatters, C., 28 Mediterranean, 64 Mediterranean Sea, 54 Melville, H., 27 Merchants, 15, 19–21, 23, 24, 26, 28, 34, 48, 55, 63, 75, 146, 149 Meslé, F., 150 Messance, L., 21 Meteorology, 85, 97 Meusnier, N., 41, 48–50 Michelet J., 46 Middle Ages, 38, 44 Ministère des travaux publics, de l’Agriculture et du Commerce, vi Model, 133 Moheau, J.-B., 21, 59–61, 63, 72, 78 Moivre, A. de, 126 Monogenism (vs Polygenism), 121 Montucla, J.-É., 46–48 Moral sciences, 101, 127, 148 Moral statistics, 124, 128–130, 133 Morineau, A., 6 Mosconi, J., 8 Mukherjee, R.S., 153 Multiplier of births, 59, 60, 63 Murray, C., 125 Mutual Habit, 35 Mystique, 133

N Napoleon Bonaparte, 83 Natality (crude rate of), 59 Neo-Thomism, 132 Nero, 97 Networks, 7, 15, 62, 73, 74, 79, 113, 118, 136, 150, 151, 153, 157 Neumann-Spallart, F.-X., 74 Neurath, O., 157 Neveu, J., 134 Newman, M., 136 Newton I., 76 Nixon, J.-W., 74 Normal law, 81 Numbers, 6, 12, 19, 20, 24–33, 35–39, 41, 43, 46–48, 50, 55–61, 63, 64, 75, 78, 80, 82, 84, 85, 88, 92, 94, 96, 98, 99, 101, 102, 104, 105, 111, 112, 114, 116, 117, 119, 127, 129, 130, 132, 134, 135, 137, 146, 147, 149–151, 155, 157, 159

167 O Objectivism, 158 Observatory, 73, 76 Oceans, 2, 19, 20, 52, 84, 118 Office, 84, 121 Oki, S., 43 Olivi, P., 41

P Pan-normality, 86, 90, 92, 96, 115, 119, 133, 135 Panofsky, E., 13, 14 Pareto, V., 136 Paris, 20, 30, 34, 39, 42, 43, 46, 48, 57, 59–63, 71, 73, 76, 82, 83, 85, 88, 90, 95, 96, 108, 109, 112, 113, 117, 118, 126, 130, 148, 156 Pascal B., 41 Passeron, J.-C., 158 Paul the Apostle, 132 Pearson, K., 117 Peletier, J., 32, 36 Péquignot, J., 2 Periodization (in the history of probability), 40, 43, 46, 47 Perrot, J.-C., 1, 3, 6, 14, 19, 21, 24, 27, 55, 56, 72, 82, 146, 160 Peyrard, F., 30, 36 Piron, S., 44 Poincaré, H., 89, 90, 92, 126, 127, 133 Poisson, S.-D., 85, 88, 126, 128, 133, 135 Political arithmetics, 12, 13, 21, 43, 55–64, 71, 76, 78, 134, 148, 152 Political economy, 7, 14, 61, 82 Political sciences, 56, 77, 78, 95, 96, 102, 114 Population, 55–63, 71, 79, 84–86, 91, 100, 121, 122, 136, 160 Portet, P., 24 Postal costs, 114 Pothier, R.-J., 52–54 Potomac, 110 Potts, R., 36 Power laws, 135 Practice, 25 Practice of numbers, 25 Pradier, P.-C., 46 Predictability (illusion of), 121 Premiums, 44–46, 48, 52–54 Presupposition, 64, 75, 79, 118, 119, 131, 137 Prints, 6, 28, 102 Probabilities, 6, 40–64, 71–138, 148, 150

168 Production, 7, 13, 23, 29, 45, 75, 112, 115, 129, 130, 137, 146 Proportionality of effects to causes, 133 See also Cause-effect relationships Proportions, 23, 24, 26–30, 32, 33, 37, 39, 40, 59, 60, 88, 92, 100, 110, 148–150, 154, 155

Q Quetelet, A., 10, 44, 73, 75–80, 83, 85, 86, 90–97, 99–112, 114–119, 122, 125, 126, 128–130, 132, 133, 135, 137, 150, 151, 153 Queteletism, 133

R Rabelais, F., 14, 38, 160 Ragouet, P., 159 Rate, 27, 36, 45, 47, 50, 51, 53, 54, 59, 89, 115, 116 Ratio, 4, 64, 78, 89, 104, 119, 138, 147–150, 153–155, 157 Relation, 6, 29, 36, 37, 40, 45, 55, 56, 76, 90, 97, 112, 125, 129 Reliablity, 72 Renn, J., 157 Residual (in statistical models), 125 Rich, N., 113 Risk-taking, 53 Rommevaux, S., 24 Rose, T., 155 Roudinesco, É., 153 Roussel, F.-G., 2 Roux, S., 38 Royer-Collard, P.-P., 127 Rüdiger, C., 44

S Saint Petersburg, 116 Satire, 119 Schiltz, M.-A., 159 Schützenberger, M.-P., 7 Schweber, L., 130 Scientific law, 77 Scientific police, 130 Scottish soldiers, 110 Sealed (ratio), 64 Seasonality, 120 Seasonal phenomena, 121 Second-order probability, 88 Seine (département), 84–86, 94 Seneta, E., 101

Index Sepkowitz, K., 120 Sex ratio (primary), 43, 60, 88 Shafer, G., 6, 49 Shalizi, C.-R., 136 Shinn, T., 159 Sigler, L., 25 Simondon, G., 152 Simplism, 14, 64, 133, 137, 155 Siobhan, R., 120 Slave Trade, 28 Social mathematics, 77 Social memory, 154, 155 Social Physics, 116 Sociology, 7, 9, 89, 129, 130, 132, 133, 153, 155, 158, 159 Softwares, 125 Spectres, 159 State, 2–6, 29, 41, 46, 48, 57, 60, 72, 79, 94, 96, 101, 105, 106, 110–113, 133, 146, 147, 152, 155, 156, 161 Statistical Society, 123 Statistics, 1–3, 6, 12, 15, 45, 47, 52, 72–76, 79, 84, 85, 87–91, 94, 96, 98, 104–106, 112–119, 121, 123, 126, 128–130, 132– 134, 136, 137, 148, 152, 160 Statistique, 92, 148 Stevin, S., 31, 33 Stigler, S.-M., 46, 87 Stockholm, 119 Superstitions, 54, 106, 146, 151

T Taillefer, 28 Thales, 4, 32 The New York Times, 120 Theory of numbers, 19 Théré, C., 21 Tilly, C., 108, 109 Translations, 10, 12, 24–36, 40, 43, 49–51, 93, 148 Turgot, A.-R., 42, 48, 57 Turquan, V., 47

U Universal language, 5, 34 Uses, 5, 23, 29, 30, 32, 36, 55, 64, 126, 153

V Vallin, J., 149, 150 Van Steenberghen, F., 132 Variability, 44, 58, 59, 63, 82, 86, 94, 100, 117, 118, 125, 130, 149

Index

169

Variance, 136 Vienna, 114 Vital statistics, 130 Vitrac, B., 29, 30, 35 Von Mayr, G., 89, 117

Waves, 120, 121, 123 Wiener, N., 3–12 Wikipedia, 3, 6, 44 Wittgenstein, L., 9, 13 Wunsch, G., 149

W Walter, C., 125, 137 Warwick, K.-M., 6

Z Zajdenweber, D., 136