Paul Mansion, a young mathematician in an emerging Belgium

Mansion’s disciple Alphonse Demoulin (1869-1947), who in 1893 became his colleague in Ghent and was himself a reputed specialist in differential geometry, recorded many details about Paul Mansion’s childhood (Demoulin Reference Demoulin1929). Paul Mansion was born in 1844 in Marchin-lez-Huy in the province of Liège. He was the ninth child of a family of ten, and his father Paul-Joseph Mansion was a public official for the commune. Paul-Joseph was fifty-three years old when Paul was born, and died soon after. The family nevertheless remained relatively well off, as Paul-Joseph had acquired several properties and lands. Paul’s mother died when her son was only seventeen, and he passed into the care of his older brothers.

In 1844, Belgium was still quite a newly independent country. While formal independence from the Netherlands was achieved in 1830 after a short revolution and the ascension of Leopold I to the throne, it was not until 1839 that the king of Holland accepted the borders of the new state, as guaranteed by the Great Powers. The question of education arose early in Belgian political life, as there was a need to address the role played by the Catholic church in public education. In 1842, the liberal Prime Minister Jean-Baptiste Nothomb (1805-1881) reached a compromise with the Catholic party. Each commune would have a primary school for children, and this school could either be funded by the commune (subsidiée) or “adopted,” meaning that an existing private school would be officially chosen by the commune, which would also provide funding, and, if necessary, a building. The law stipulated that teachers must have a degree from a state-approved school. Religion was part of the syllabus and had to be taught by Catholic priests, so the Church still had some control over schools (Deprez Reference Deprez1970).

Mansion entered Marchin’s communal school when this regulation was in force. His teacher, Jean-Joseph Blaise, was one of the teachers with a degree from a state school. A brilliant pupil, Mansion pursued his studies at secondary school in Huy, and, in 1862, having decided to become a science teacher, was admitted to a program at the École Normale in Ghent designed to train teachers for secondary schools. In 1847, special training courses (cours normaux) were created at the state universities. The University of Liège was responsible for preparing students for an upper-level degree (agrégation) in teaching in the humanities, and Ghent University oversaw the scientific domains.Footnote ⁵ This division, however, created tension, as other departments, such as the Law department at Ghent University and the Medicine department at the University of Liège, feared that these new courses would result in a drastic decrease of the number of students in their programs. Thus, in the end, the planned division of educational programs was not carried out (for instance, the University of Liège continued to offer science courses). Even so, Ghent was the most natural place to study mathematics.

Mansion had two first-rate teachers during his schooling in Ghent: the mathematicians Félix Dauge (1829-1889) and Mathias Schaar (1817-1867). Mansion declared later that he never forgot Dauge’s lectures on mathematical methodology, about which Dauge published a celebrated book in 1883.Footnote ⁶ Schaar, a remarkable polymath, taught arithmetic and analytical mechanics. In 1865, Mansion graduated from the École Normale as an agrégé and, in October 1865, he began teaching at the preparatory school for civil engineering in Ghent where Schaar was the schools inspector. As Mansion probably, at this stage of his career, wished to obtain a university degree (which the agrégation was not), he also defended a doctorate in physics and mathematics on 13 August 1867. He may have been encouraged to do so by Schaar himself, as Schaar, who was seriously ill, had not been able to teach since January 1867. Schaar died very soon afterwards (in April), and so his position became vacant. Paul Mansion was appointed to the chair of Differential and Integral Calculus and Higher Analysis at the University of Ghent on 3 October 1867 at the rather young age of twenty-three. This appointment seems to have resulted from Dauge’s and Schaar’s efforts to ensure that their brilliant protégé would stay in Ghent.

A mathematician in Ghent

Mansion’s appointment in Ghent appears to have consolidated a local mathematical tradition that had been well-established since the beginning of the century. The founding figure of this development in Ghent mathematics had been the French mathematician Jean-Guillaume Garnier (1766-1840), who had arrived in 1817 following a rather unusual trajectory. At the end of the eighteenth century he had been an important actor of the French mathematical scene (it is for instance thanks to Garnier that Joseph Fourier was sent to Paris from his rather obscure position of mathematics teacher in Auxerre). But Laplace took a dislike to Garnier for reasons of competition with Poisson and Garnier resolved to seek his fortune elsewhere than in Paris (Droesbeke Reference Droesbeke2005). In an article devoted to Garnier, Mansion made the following comments, probably based on information obtained first-hand from his older colleagues, who had known Garnier well.

As Mansion noted, the outstanding importance of Garnier’s presence in Ghent can be seen through the list of his students, who formed the first generation of mathematicians in the independent Belgium, with Quetelet ranked first among them. Garnier supervised Quetelet’s thesis on conical sections, which was defended in July 1819 in Ghent. In 1825 the two men founded the first Belgian journal specializing in mathematics, entitled Correspondance mathématique et physique. Footnote ⁷

Mansion’s first paper, On the Problem of Points, published by the Royal Academy of Sciences in Brussels, was devoted to a probabilistic question (Mansion Reference Mansion1870).Footnote ⁸ It is not clear why Mansion chose this topic, but his professor Emmanuel-Joseph Boudin (1820-1893), who at the time taught the probability course at Ghent University, may have suggested it. In any case, the paper provided an opportunity for Mansion to become acquainted with the papers of another mathematician who would play an important role in his scientific life, a professor in Liège, the French-born Eugène Catalan (1814-1894).Footnote ⁹ Indeed, Catalan is one of the only references cited in Mansion’s Problem of Points, which, in addition to noting Poisson’s research on Pascal’s problem of points, also cites a paper by Catalan in the Journal de Liouville, in which the latter presents some combinatorics results (Catalan, Reference Catalan1842). Using Catalan’s results, Mansion proved that it was possible to obtain the solution to the general problem of points using only combinatorics, which was an alternative approach to what Laplace had done with integral calculus and what Poisson had done with a direct probabilistic approach.

Catalan had remained continuously interested in the discipline, mostly in connection with his focus on combinatorics. Moreover, as a student at the École Polytechnique during the 1830s, Catalan benefited from virtually the only high-level teaching in probability theory available at that time in France, although only a few students seem to have been drawn to this opportunity. Of these, Bravais and Catalan were more or less the only future mathematicians. Though Catalan’s main mathematical interest was not probability, he continued to devote some works to it here and there. Jongmans and Seneta (Reference Jongmans and Seneta1994), for instance, have investigated how in 1841 Catalan proposed a martingale model of drawing successive balls from an urn.

The letters exchanged between Catalan and Mansion shed light on their relationship (Fonds Mansion, KBR). They developed a real friendship, despite their difference in age (thirty years) and political views. In 1869, Mansion contacted Catalan to ask about the possibility of embarking on a PhD (doctorat spécial) in elliptic functions. In his detailed reply of 20 October 1869, Catalan explained that the Belgian system for PhDs remained quite obscure to him; but he was encouraging, and suggested that Mansion come and visit him in Liège. In 1870, with the help of Catalan, Mansion completed a PhD on the theory of multiplication of elliptic functions, subsequently defended in Liège. Catalan, a consummate networker, managed to get Mansion invited to Göttingen for a research stay with Alfred Clebsch (1833-1872) and Ernst Schering (1833-1872). Clebsch was Riemann’s successor at Göttingen and considered to be one of the top experts on elliptic functions of his time. He and Mansion soon became good friends, as testified by the collection of letters that Clebsch sent to Mansion over a short period of time.Footnote ¹⁰ The relationship ended very suddenly, as Clebsch died unexpectedly in 1872 after a brief illness.

In 1871, Mansion married Cécile Belpaire, daughter of the engineer Alphonse Belpaire (1817-1854), who had played an important technical role in the newly independent state, in particular in the organization of the railway system. For instance, he published a study devoted to the economic management of the particularly dense rail network in Belgium (Belpaire Reference Belpaire1847). The Belpaire family was wealthy and well-connected within Belgian political and intellectual circles, and this marriage certainly facilitated Mansion’s access to these circles and his role as an academic authority in Ghent, despite his young age.

In 1874, Catalan decided to resume publication of Garnier and Quetelet’s journal Correspondance mathématique under the fairly obvious title Nouvelle correspondance mathématique and asked Mansion to help him. Although the new journal was supposed to deal with mathematical topics taught in the upper classes of secondary schools and in engineering schools, Catalan, who from 1876 at 1880 was its primary motive force, gradually raised the level, so that the journal, which was less and less suitable for educational needs in Belgium, ultimately failed. In 1881, it was replaced by a new publication, Mathesis, founded by Mansion and Joseph Neuberg (1840-1926). The journal’s scope was the same as that originally intended for the Nouvelle correspondance, but Mansion and Neuberg had the wisdom to keep Mathesis at the level they had assigned to it, so that it could survive and develop (Demoulin, Reference Demoulin1929, 107). The journal continued to be published without interruption for thirty-five years until the end of 1915, when the war brought an end to the publication.

Mansion’s central research topic was analysis. He was a prolific author and published often, especially in the two journals that he edited (Nouvelle correspondance, Mathesis) and in the transactions of the Brussels Academy of Science. Mansion was also a regular contributor to the British journal Messenger of Mathematics. In 1873, he won a prize from the Belgian Academy of Science for his paper on the theory of second-order partial differential equations that was subsequently published as a book in Paris in 1875, and was distributed throughout Europe (Mansion Reference Mansion1875). Mansion also undertook the huge endeavor of publicizing Belgian mathematical activity, for instance in the Italian journal Bolletino, published in Rome by Balthasar Boncompagni. Since the very beginning of his career, Mansion also published many studies on the history of science and numerous obituaries for Belgian colleagues. In 1882, he was elected a corresponding member of the Sciences Class of the Royal Academy of Belgium, and then a full member in 1887. In 1892, he was officially put in charge of the course on probability theory at Ghent University, when he replaced Boudin as the chair of calculus of probability.

A Catholic scientist

One of the unusual characteristics that must be taken into consideration in order to understand Mansion’s scientific activities is his Catholicism. Mansion was a devout Catholic throughout his life, and a deeply committed militant for his faith. The University of Liège and Ghent University were the country’s two state universities, and Mansion’s first position was at Ghent University in 1867. During the almost fifty years that he spent at this non-religious, secular institution, Mansion openly displayed his commitment to Catholicism and perceived himself to be a defender of the faith. Mansion in fact engaged in numerous activities connected to his Catholic faith. He was a member of the congregation of Our Lady of the Seven Sorrows and a member of the Saint Vincent de Paul society in Ghent, and remained close to the Jesuits throughout his entire life. But, more importantly for the topic of this paper, very early in his scientific life Mansion sought a way to connect his faith with his professional life.

An opportunity arose in 1870, when he created the Cercles Cauchy with his friend the engineer Charles Lagasse de Locht (1845-1837).Footnote ¹¹ The idea was to hold regular meetings, bringing together several of the brightest students at Ghent to counter what they held to be atheistic propaganda based on allegedly scientific arguments. Lagasse had originally intended to name the Cercles after Leibniz, but Mansion insisted on replacing the protestant philosopher’s name with that of the Catholic mathematician Cauchy. Under Mansion and Lagasse’s impetus, other Cercles Cauchy appeared in Belgium. The Belgian Jesuit, mathematician, and teacher at Saint-Michel College in Brussels, Ignace Carbonnelle (1829-1889), instituted a Cercle Cauchy in Brussels, and some years later, in 1875, founded the Brussels Scientific Society (Société scientifique de Bruxelles).

The explicit motto of the Society was to oppose anti-religious positivism. Its aim was to help Catholic scholars promote the advancement and diffusion of science in order to combat rationalist and atheist doctrines and establish a barrier against emerging materialist and anti-religious movements. The Brussels Scientific Society was very active and published the proceedings of its meetings under the title Annales de la Société scientifique de Bruxelles. Mansion had been one of the Brussels Scientific Society’s founding members, and served as its president from 1889-1890. In his recent paper, Stoffel (Reference Stoffel2020) studies how Mansion’s program coincided with that of the Brussels Scientific Society.

In 1877, a scientific journal was launched and published in Louvain and in Paris under the title Revue des Questions Scientifiques. From the beginning, the journal—which still exists today—was conceived as a scientific journal that would offer a Catholic view of the advances in science, giving preference as authors to scholars who clearly belong to the Roman Church. Mansion made many contributions to the journal.

Nye (Reference Nye1976) provides a detailed study of the first years of the Brussels Scientific Society, and emphasizes how questions about determinism were of central concern to its members in those years. It is noteworthy that, alongside French members such as Hermite and Pasteur, the society included a Joseph Boussinesq (1842-1929), a Catholic mathematical physicist who was deeply involved with the study of randomness and probability. At the time Boussinesq was a professor in Lille, and he would later (in 1896) succeed Poincaré at the Sorbonne as chair of the calculus of probability and mathematical physics.

Mansion was close to another important prelate, and future cardinal, Désiré Mercier (1851-1926). According to Mercier’s biographer Laveille, while Mercier was a student of theology, his interest in science prompted him to attend various lectures delivered by Mansion in Ghent (Laveille Reference Laveille1928, 64.). Their shared commitment to Catholicism was certainly important to their relationship from the beginning. A letter that Mansion sent to Mercier some years later (in 1882) provides interesting insight into Mansion’s feelings about the Belgian academic situation with regard to Catholicism. At that time, one of Mercier’s cousins was a student in Ghent. Mansion wrote to Mercier:

Mercier was probably the most famous Belgian cleric of the nineteenth and twentieth centuries. Of course, today he is mostly remembered for his activity during World War I.Footnote ¹³ But Mercier had much earlier gained a singular position on the Belgian intellectual scene. As De Volder (Reference Volder2016, 13-14) expresses it, Mercier, who had an insightful mind and was a brilliant professor, was philosophically and scientifically progressive, while knowing how to maintain orthodoxy, which protected him from the setbacks suffered by other Catholic thinkers of the time. Mercier’s activity in the Belgian context is addressed in the collection of essays about Mercier written by Roger Aubert (Reference Aubert1994) and, more recently, in De Maeyer and Kenis’ article about the creation of a Catholic intelligentsia in Belgium in light of the “modernist crisis” (2017). Deeply interested in philosophy, natural sciences, and mathematics, Mercier was ordained as a priest in 1874, and defended a doctorate in theology in Louvain in 1877, at the precise moment that the reign of Pope Pie IX ended and his successor, Leo XIII, was elected (in February 1878).

Leo XIII had a strong desire to provide the Catholic Church with the tools needed to confront the problems posed by various recent scientific discoveries. Almost immediately after his election, the new pope, in his 1879 encyclical Aeterni Patris, chose to reintroduce Thomas Aquinas’s philosophy as a basis of reflection. This was not seen as a concession to liberalism, but as a means of emphasizing the importance of Aquinas’s teachings to prevent the errors made by modern philosophy (such as Kantianism).Footnote ¹⁴ Nevertheless, the encyclical clearly stated that philosophical work was the product of rational activity, and this made a scientific approach to the world possible for Catholic elites who had looked on the extremely rigid dogmatic answers given by the former pope with concern. This attitude, they thought, had increased the gap between the Church and academics, and Mercier shared this opinion. He was convinced that the rigidity of the old religious regime had been counterproductive to scientific progress, due primarily to the fact that most clerics and theologians were scientifically illiterate. Mercier, who had a strong personal relationship with the new pope, took this opportunity to implement his intellectual institutional agenda, the aim of which was to renew scholastic philosophy and create a resolutely positive attitude towards science within the Church, defending the idea that scientific activity must be guaranteed relative autonomy and freedom.

Mercier’s major achievement was the opening of the Higher Institute of Philosophy (Institut supérieur de philosophie) in Louvain in 1889. Pope Leo XIII was an enthusiastic supporter of his friend Mercier’s project. In the pre-project plan presented to Rome for approval in 1887, Mercier suggested that the institute offer lectures on philosophy in the morning and scientific talks in the afternoon (De Raeymaeker Reference De Raeymaeker1951, 534). The institute would soon become one of the main research centers on neo-Thomism and spiritual reflection on science, though initially there were many clashes with other elements within the University of Louvain, who viewed Mercier’s experiments with a critical eye.

The year 1894 saw the inauguration of the new building of the Institute of Philosophy, directed by Mercier, and its formal association with the University of Louvain, which Leon XIII required of the rector Abeloos. This resulted in a few years of open friction between Mercier and the university’s administrative board, who had been very critical from the start of Mercier’s plan to base the study of theology on scientific results. Nevertheless, Mercier succeeded in avoiding major threats to his Louvain institute, which was allowed to continue.Footnote ¹⁵

It is probable, however, based on the account of the beginning of the institute given by De Rayemaeker (Reference De Raeymaeker1951, 534–540), that this hostility from his colleagues prompted Mercier to look outside of Louvain for the teaching staff that he needed. He asked Mansion, his friend and former teacher in Ghent, to become his main expert on mathematics. Mansion gave a series of talks on the fundamental principles of mathematics during the first year of the institute (1890-1891) and another on the fundamental principles of mechanics during the institute’s second year (1891-92) (De Raeymaeker Reference De Raeymaeker1951, 538). He wrote several papers for Mercier’s journal Revue néo-scolastique, which was founded in 1894 as the official publication of the institute.Footnote ¹⁶

Both Mercier and Mansion were deeply interested in the role the church must play in the education of youth. In a letter dated 28 November 1906, Mansion wrote:

Mercier and Mansion had a good opportunity to present their views to a broader audience on the occasion of the third international congress of Catholic scientists, held in Brussels in September 1894. These congresses had been launched in 1886 at the instigation of another cleric, Maurice Le Sage d’Hauteroche d’Hulst (1841-1896).Footnote ¹⁸ In 1886, d’Hulst had organized an international congress of Catholic scientists that was to be held in Paris on Easter 1887. In his circular letter of 1 February 1886, published in the Annales de Philosophie Chrétienne,Footnote ¹⁹ d’Hulst mentions that the aim is to gather professors and writers who were known for adding real scientific value in the service of Christian convictions and to invite all Catholics who were interested in the development of science for the defense of the faith (Beretta Reference Beretta1996, 269). Mansion and Mercier were both members of the organizing committee. The proceedings of the congress testify to Mansion’s continual involvement in the preparation phase. He was a member of the preparatory commission (in fact, a vice president) and, as reported in the historical report, was probably the most active member in that he attracted almost all his Catholic colleagues from the university of Ghent (28 out of 30) and a considerable number of members from Ghent, Anvers and Western Flanders (Anon. 1895, 9). The seventh section of the proceedings, devoted to mathematical and natural sciences, includes Mansion’s presentation to the congress (Mansion Reference Mansion1895). It also contains a contribution by Charles Hermite, who attended the Congress, and was certainly considered the most valuable scientific endorsement of the meeting (Hermite Reference Hermite1895).

Belgium as fertile ground for probability

We have already mentioned how Quetelet played a central role in Belgian academic life and how, under his influence, probability theory became a central topic of study. Quetelet was probably the first to perceive that statistics must fundamentally rely on the calculus of probability (Droesbeke et Tassi 1997, 44). In a nice, recently published study, Donnelly (Reference Donnelly2016) undertakes an in-depth examination of the genesis of Quetelet’s interest in the mathematics of chance and their application to the study of the social domain. Quetelet’s initial decision to pursue a career as a writer in parallel with his mathematical studies (he defended a PhD on conical sections supervised by Garnier at Ghent University in 1819) sheds light on his taste for description, which he ultimately put to work in the service of science, after the decisive encounter with Laplace’s work during his stay in Paris in the 1820s. Following in his predecessor’s footsteps, Quetelet would establish his tutelage on the Belgian scientific scene in two sectors where descriptions play a particularly acute role, astronomy and probabilistic statistics, at the end of the 1820s. Quetelet’s taste for teaching made him a highly appreciated pedagogue throughout his entire life, celebrated for his clear and simple explanatory style.

Droesbeke (Reference Droesbeke2005) studied Quetelet’s continuous promotion of mathematical education in Belgium, with a special emphasis on popularization. Three important books published by Quetelet over three consecutive years, during his campaign for the creation of an observatory in Brussels, speak to his desire to disseminate scientific culture throughout the country. In order to make the calculus of probability more accessible to a wide audience, Quetelet sought to create a sort of arithmetic of probability based only on fractions and proportions. His former PhD adviser at Ghent University, Garnier, suggested the expression “arithmetic of probability” after reading Quetelet’s book with enthusiasm. Garnier commented that he “could not understand how it had not come before” (Droesbeke Reference Droesbeke2005, 15).

Though Quetelet’s frenetic activity had begun before the Belgian revolution, the state’s newly acquired independence clearly spurred him on further. Droesbeke (Reference Droesbeke2005) explains how Quetelet participated in, and often directed, the development of the Belgian educational system. In particular, he supported the creation of polytechnic schools, whose syllabi were required to include theoretical, as well as practical, scientific lectures, and he was also involved in the organization of the Royal Military School. Quetelet was, moreover, one of the developers of the Université Libre de Bruxelles, which was founded in 1834 in response to the establishment of the Catholic Institution of Higher Education in Malines, which was quickly transferred to Louvain in 1835.

The study of probability in Liège was promoted by the mathematician Anton Meyer.Footnote ²⁰ In her study of geodetics in the nineteenth century, Marie-France Jozeau describes how Meyer learned German least-square techniques when he visited Germany for a few weeks in 1846 in order to familiarize himself with new probabilistic methods for geodesy (Jozeau Reference Jozeau1997, 99–104). In 1838, probably with Quetelet’s support, Meyer had been offered the Chair of Higher Mathematics at the Université Libre de Bruxelles, and later at the University of Liège in 1849. Jozeau observes that Meyer proposed a new syllabus for the doctorate-level course in analysis, most likely inspired by Quetelet (ibid., 106). The syllabus now contained two explicitly separate parts: superior analysis, elliptic functions, and calculus of variations on the one hand, and probability and social arithmetic on the other hand. Between 1849 and 1857, the year of his sudden death, Meyer taught a course on probability that may have been the most structured course offered at a European university at the time, fulfilling Quetelet’s plan of connecting the theoretical approach to probability theory of the French mathematicians Laplace, Lacroix, and Poisson, with the Germans’ practical developments concerning the law of errors.

In 1856, Anton Meyer submitted a proof of the Central Limit Theorem in the special case of two-valued random variables to the Academy of Brussels. Meyer’s proof was not based on the usual procedure, which can be traced back to de Moivre and had also been detailed in Laplace (Reference Laplace1812). Instead, Meyer used an extension of Laplace’s generating functions.Footnote ²¹ Meyer’s article unfortunately appears to have been lost, but we know of its contents due to a brief report (Brasseur, Reference Brasseur1856) written by Jean-Baptiste Brasseur. The latter hoped that Meyer’s method would lead to a more exact discussion of how “terms of higher orders of smallness” could be neglected. Meyer’s paper was accepted for publication under the condition that the “convergence of the series” should be examined more thoroughly. Ultimately the paper was not published, as Meyer died the following year (Fischer Reference Fischer2010, 24-25).

A few months before his death, Meyer had been involved in a controversial argument with Liagre, a former student of Quetelet who taught probability theory in military academies, mostly from a practical point of view, and who in 1855 had presented a note to the Brussels Royal Academy in which he introduced the first elements for the so-called turning-point test.Footnote ²² Meyer published his reaction in 1857 as a note, in which he rather tactlessly criticized Liagre for his alleged sloppiness. This resulted in some controversy at the Royal Academy (as Liagre as an important member of the institution, while Meyer was only a corresponding member), illustrating the difficulty of maintaining a balance between a theoretical and a practical point of view in probabilistic questions.Footnote ²³

Around the time that he died, Meyer was completing the manuscript of a textbook on the calculus of probability. The manuscript remained untouched for more than fifteen years, but was eventually published in 1874 by the Société royale des sciences de Liège under the supervision of François Folie (1833-1905). Meyer’s book makes systematic use of Laplace’s method for the asymptotic development of integrals to present the law of errors. The method itself is presented in a very clear mathematical appendix (Meyer Reference Meyer1874, 421-435). Meyer’s course on probability was clearly designed for students with a quite extensive mathematical education. Thus, the book offers a rather remarkable continuation of Laplace’ Analytical Theory of Probability (1812), which Meyer had probably read with enthusiasm while he was in Paris.

In “On the Objective Significance of the Calculus of Probabilities” (Mansion, Reference Mansion1903, abbreviated henceforth as “Objective Significance”), Mansion wrote that Meyer’s 1874 textbook was the only genuine treatise on probability theory written in a European language between Poisson’s book in 1837 and Bertrand’s book in 1889. This statement may be somewhat exaggerated, and Hald (Reference Hald1998), for instance, seemed to feel that the value of Meyer’s book was due more to its clarity than its originality. The most essential fact regarding Meyer’s book was its subsequent translation into German in 1879 by the Prague mathematician Emanuel Czuber (1851-1925). We do not know how Czuber became aware of the existence of Meyer’s treatise, but his specialization in geodesy offers a reliable clue. In 1876, Czuber had indeed defended a professorial thesis in geodesy with the title Theorie und Praxis der Ausgleichsrechnung. He may also have begun to lecture on probability theory at the Deutsche Technische Hochshule as a privatdozent beginning in 1876 (Hyksova 2011, 112).

Czuber (Reference Czuber1878) clearly shows that, in addition to geodesy, Czuber was concerned with actuarial sciences in 1878. Czuber was probably attracted by the mathematical soundness of the approach used in Meyer’s textbook to base the law of errors on a precise mathematical structure. He added a preface to the translation of Meyer’s treatise that comments on its content. Czuber explained that, due to the recent development of actuarial science, he had felt the need to rewrite and fill out some parts of the book, in particular by extensively revising Chapter Eight, which was devoted to compensation of errors. The aim was to make this chapter useful for standard situations, but also to include the most recent theoretical developments on the subject. Czuber also completely revised Chapter Nine, which was devoted to insurance problems, as this topic was of major importance to his actuarial interests. In his 1901 entry on probability theory in the Encyklopädia des Mathematisches Wissenschaft, Czuber mentioned Meyer’s book as a fundamental reference textbook, along with Cournot (Reference Cournot1843) and Laurent (Reference Laurent1873). It is plausible that Mansion did not mention these treatises in “Objective Significance” (Mansion, Reference Mansion1903) because they were not specifically intended for students specializing in mathematics.

As previously mentioned, Meyer’s treatise contained the lectures on probability given in Liège. Eugène-Joseph Boudin (1820-1893), Mansion’s former teacher and Chair of Calculus of Probability prior to 1892, had taught a course in probability at the University of Ghent (the other state university) and the Ecole du Génie Civil for more than forty years. In 1913, Mansion wrote in his biographical note on Boudin for the Liber Memorialis of the University of Ghent:

In fact, it was a long time before Boudin’s treatise was properly published. Between 1865 and 1889, three rather confidential handwritten editions were published (Boudin, Reference Boudin1865, Reference Boudin1870, Reference Boudin1889). In 1869, the young Mansion provided a copy of Boudin’s first handwritten edition to Eugène Catalan, who replied:

The final edition of Boudin’s lessons, with considerable extensions and additions, would be published in a highly symbolic way much later, in 1916, while Ghent was under German occupation and the Germans were supporting the creation of a Flemish university to replace the university of Ghent. Mansion succeeded in having them printed in Ghent by the printing house Hoste, as a clandestine publication of the Paris academic publisher Gauthier-Villars. Unfortunately, I was not able to find any documents related to this matter in the Hoste archives in Ghent (Liberas archive). Mansion probably saw publication of the book, which was of course only released after the liberation of Belgium in 1918, as a patriotic act, and many subtle comments included in the book, emphasizing the role and presence of Belgium in the Concert of Nations, can be interpreted as such. He included his own “Objective Significance” (Mansion, Reference Mansion1903) as an appendix to the book.

Paul Mansion as a probabilist

Mansion’s publications on probability, though not very numerous, were spread out throughout his forty-five-year career, demonstrating his continuous interest in the discipline. Probability was, in particular, the topic of his first published study in 1868. Mansion also seems to have been interested in the technical aspects of probability (mostly limit theorems: the law of large numbers and central limit theorem), as well as the epistemological interpretation of probability’s role in contemporary science at the time. There is an obvious link between the two. As was the case for Poincaré in the last third of the nineteenth century, Mansion questioned whether a probabilistic approach could reveal a part of the objective reality. In other words, was this approach only a mathematical trick, or did it reflect some reality of the world?Footnote ²⁷

The first time that this question appears in Mansion’s work is a small note included by the philosopher Paul Janet in the second edition of his essay on final causes in 1882 (Janet, Reference Janet1882). In the preface to this edition, Janet commented that he had to modify his original text after having received many reactions to the first edition in 1876. As the book had been conceived as a polemical text, aimed at provoking comments on a topic rarely commented on in France, this had been quite natural for Janet. He mentioned that cross-examination of philosophical questions was a more common feature of Anglo-Saxon culture than of French culture. Janet (Reference Janet1876) questioned whether recent progress in science necessarily led to doubting the existence of God, and explored how classical approaches to the question needed to be adjusted to adapt to this change. In the second chapter of the book, Janet analyzed the so-called physical-theological proofs.

Among the comments he had received, Janet wrote, one of the most interesting was from Paul Mansion. Janet decided to reproduce Mansion’s letter as an appendix in his new edition (Janet Reference Janet1882, 720–725). Mansion’s note is entitled “The Epicurean argument and the calculus of probability.” In it, Mansion contests an assertion made by Janet in the book’s first edition regarding the Epicurean objection to the “final cause” proof of the existence of God. This proof states that the world is too well organized not to reflect the intelligent design of a creator. Epicureans object, stating that, if the world is an assembly of elementary particles randomly associated with one another, one must necessarily conclude that there exists a time in which the world will be distributed in any given arrangement (for instance the present state).

Janet asserted that the calculus of probability may be used to study the validity or invalidity of this argument: in particular because if the number of particles is infinite, the probability of assuming the present state of the world is zero, so that the Epicurean argument is dismissed. Mansion contested this use of probability, though, as he himself wrote, it seemed highly tempting at first glance. First, Mansion wrote, infinity is a mathematical abstraction, so it cannot be used in a proof about the real world. Second, even if the number of particles is finite, the conclusion that the world must assume all the possible states is valid only if one assumes that some internal forces guarantee that the atoms behave in such a way that the rules of the calculus of probability are applicable (in modern terms, that they are uniformly distributed over the various possibilities). This is an assumption we have no specific reason to make. Therefore, Mansion wrote, probability cannot be a useful approach to the Epicurean objection. He would later include a section along the same lines in Mansion (Reference Mansion1903) (section XI).

Mansion’s probabilistic technical research probably focused on the limit theorems because applying them was the only way to attribute an objective value to a probabilistic result. At the end of the nineteenth century, especially after Cournot’s studies, Laplace and Poisson’s works were revisited, and Mansion used several results of these works to improve the rate of convergence in the law of large numbers. In particular, Mansion was interested in the convergence rate of the probability of large deviations of the average number of successes from theoretical probability. With the exception of a small group of probabilists connected to Mansion and a few Russian mathematicians, these questions would not attract substantial attention until the arrival of a new generation of mathematicians after 1920. Another topic present in Mansion’s works is an extension of Laplace and Gauss’ least-squares method, extending Quetelet’s, Catalan’s, and Bouvier’s studies.Footnote ²⁸

Another aspect of Mansion’s interest in probability was his central position in a small and very active network of mathematicians linked in one way or another to the Belgian scientific scene. Mansion’s studies about the law of large numbers provide a good illustration of the publications from this group. In 1892, Mansion published a paper in the Annals of the Brussels Scientific Society (Mansion Reference Mansion1892). In 1895, it was followed by the paper published by the astronomer and meteorologist Edouard Goedseels (1857-1928) in the same journal (Goedseels Reference Goedseels1895). Then, two new papers by Mansion appeared in 1902 and 1904 (Mansion, Reference Mansion1902, Reference Mansion1904). In 1907, the Louvain mathematician Charles de La Vallée Poussin (1866-1962) extended Goedseels’ and Mansion’s results in two articles on Bernoulli’s theorem (Vallée Poussin Reference Vallée Poussin1907a, Reference Vallée Poussin1907b).Footnote ²⁹ Mansion and De la Vallée Poussin’s results were used in Montessus de Ballore’s Elementary Lessons in Probability Calculus (1908).

The French mathematician Robert Montessus de Ballore (1870-1937) was then a professor of mathematics at the Catholic University of Lille. Probability was a central topic in his communications with Mansion. In 1904, Mansion sent a copy of “Objective Significance” (Mansion Reference Mansion1903) to Montessus and suggested that he visit him in Ghent (Mansion to Montessus, 28 April 1904, in Fonds Montessus, SUA). In 1906, Montessus suggested that they publish a book together on probability theory, but Mansion declined, explaining that he was working on publishing his former colleague Boudin’s lecture notes (Mansion to Montessus, 25 December 1906, in Fonds Montessus, SUA). However, he expressed his interest in Montessus’ project, and when Montessus’ s Lessons (1908) was published in 1908, Mansion advised his colleague to send his book to the Jesuit father Fernand Willaert (1877-1953), an astronomer who was investigating the meaning of probability, in Louvain, so that he could write a review of the book for the Revue des Questions Scientifiques (Mansion to Montessus, 2 April 1908, in Fonds Montessus, SUA).

Willaert indeed wrote a review for the journal in which he praised the clarity and readability of the book (Willaert, Reference Willaert1908). He also devoted several pages to expressing his skepticism about Montessus’ assertion that Bernoulli’s theorem provided the calculus of probability with objective value. Willaert contended that there was in fact no genuine objective value to any mathematical abstraction: such a conclusion would be merely a psychological illusion, because mathematics is a powerful theoretical tool, and many of its results are close to what can be observed in reality. It is therefore somewhat impossible to determine the exact conditions under which an abstract model agrees with reality.

Even though he was slightly more positive about this possibility in his book, Montessus seems to have appreciated the review, and wrote to Willaert to express his gratitude. Willaert replied to thank Montessus for his kindness and to insist on the fact that the central problem regarding probability was indeed estimating the degree of agreement between the experimental result (for instance when one throws a die) and the theoretical result (Willaert to Montessus, 9 August 1908, in Fonds Montessus, SUA). This shows that, for Willaert, as well as for Mansion (and also Poincaré or Borel at that time), the question of the objectivity of calculus of probability results was under scrutiny.

It is significant to compare Montessus’ book with Borel’s Elements of Probability Theory (1909), which was published at the same time. While Montessus’ book mentioned elementary lessons on the calculus of probability, Borel evoked the theory of probability. Both authors declared in their introductions that their aim was to help disseminate a subject that was gaining more and more importance in the contemporary study of phenomena. But, while Montessus clearly announced that he was targeting a wide and non-specialist audience, Borel did not conceal his desire to inspire the curiosity of young mathematicians with his work. While Borel placed himself within the mathematical structure proposed by Poincaré, Montessus was clearly more on the side of the validity of practical application. Mansion and de la Vallée-Poussin’s approximating inequalities for Bernoulli’s theorem were part of this agenda.

Montessus’ book is probably also the first place where the results from Bachelier’s 1900 thesis on the theory of speculationFootnote ³⁰ were quoted, although I was unable to find any information about how Montessus became acquainted with Bachelier’s work. A reasonable hypothesis is that this information came from Poincaré: Montessus defended his PhD (on continuous fractions) in 1905 at the Sorbonne in front of a jury presided over by Poincaré; his second thesis (as usual, its topic had been chosen by the faculty, probably by Poincaré himself) was on the calculus of probability. For this occasion, Montessus needed to prepare a survey of recent works on probability, and necessarily took Bachelier’s thesis into account.

As mentioned above, Montessus was professor at the Catholic Institute in Lille – which was not a neutral fact in those years of tension between the church and the state in France. Although Montessus was certainly not as involved in the Catholic church as Mansion was, their common religion clearly played a role in their relationship. In his letter of 25 December 1906, Mansion congratulated his colleague for having received the great Paris Academy of Sciences prize for his thesis on continuous fractions, and wrote that he would share the news with the Brussels Scientific Society. He expressed his satisfaction because “it is always good for the public, and young people in particular, to learn that one of the world’s leading scientific bodies has rewarded the work of young Catholic scientists” (Mansion to Montessus, 25 December 1906, in Fonds Montessus, SUA).Footnote ³¹

Another member of Mansion’s “Belgian” probabilistic network was the Italian mathematician Ernesto Cesarò (1859-1906). After some disappointment in Italy during his secondary studies, the young Cesarò was sent to Liège (his brother was already a student there) to study engineering: he entered the mining school in 1874 and graduated in 1878. However, Cesarò decided to continue his studies at the University of Liège and began to publish mathematical works, receiving much encouragement from Catalan. Cesarò returned to Italy in 1883 and, after several complicated years, eventually obtained a chair in Palermo and then in Naples.

Though Cesarò is known today above all for his contributions to differential geometry and the theory of series, he was also interested in probability, as demonstrated by several publications, mostly in the Giornale di Battaglini or in Mansion’s journal Mathesis. Most of these publications were devoted to specific problems such as the “diamond cleavage” problem, a geometric probability problem in which the question is to decide which division in three pieces would be optimal for the jeweler (Cesarò, Reference Cesarò1886). However, Cesarò was also interested in broader aspects of probability theory. In his long two-part paper, “Considerations on the Concept of Probability” (Cesarò, Reference Cesarò1891), Cesarò studied how probability can be interpreted in cases where there are an infinite number of possibilities. For Cesarò, assigning a probability value as a limit was perfectly acceptable, and no more subjective than the attribution of a value to the sum of a non-absolutely convergent series. Mansion emphasized the soundness of Cesarò’s approach in “Objective Significance”, and regretted that the latter’s probabilistic work was not better known, due to its publication in an Italian journal that was not widely read (Mansion Reference Mansion1903, 59).

A reflection on evolution

The publication of Charles Darwin’s The Origin of Species in 1859 created a major crisis in the Roman Catholic world. The status of man in the natural world was at stake, and became the topic of numerous controversies about various possible interpretations of Darwin. The reception of Darwin’s book by Catholic theologians, and more generally by Catholic thinkers, immediately generated considerable discussion and argument. This crisis has been the subject of numerous historical studies. The present paper is obviously not the right place for such an overview, and we shall limit our comments to some elements that will enable us to place Mansion’s comments in context. Much of the information in this section is taken from Brundell (Reference Brundell2001), Blancke (Reference Blancke2013), and Artigas, Glick and Martinez (Reference Artigas, Glick and Martínez2006), which the reader can consult for more complete information.

The 1860s, which saw the end of Pope Pius IX’s rigid pontificate and the collapse of the Pontifical states under the pressure of Italian unification, was certainly not a suitable time for open discussion. The pope even called on Catholic intellectuals to assist him in his fight against modernity, including any non-fixist theories of the world, which were considered to be radically incompatible with biblical teaching. The situation began to change with his succession, in 1878, by Pope Leo XIII, who had already declared his wish for there to be a change in the dialogue between the Church and modern science. This wish primarily concerned questions about evolution, as Catholic intellectuals felt it was necessary to respond to theories about biological species when they were applied to humans. For many Catholic thinkers, leaving these questions to anti-clerical militants would have been simply negligent. Therefore, the last decades of the nineteenth century were a period of intense reflection on evolution, as Catholic theologians, philosophers, and scientists sought a version of Darwin’s theory that was compatible with the teachings of the Catholic Church.

Of special importance were the reflections by some English-educated converts from Anglicanism. Their responses, which were marked by considerable scholarly independence, were supported by a tradition of liberal thinking at various journals that offered them a wide audience. The most influential response was that of the biologist St. George Jackson Mivart, whose career path illustrates the immense difficulties that Roman Catholics experienced in responding to non-fixist theories. Mivart had been an enthusiastic Darwinist and a collaborator of Darwin and Huxley before he began to raise objections regarding Darwin’s concept of natural selection. In his widely disseminated book On the Genesis of Species, published in 1871, Mivart proposed a Lamarckian-type adaptation of Darwin’s theory: he accepted the principles of the evolution of species but, contrary to Darwin, who supported the idea of small, continuous variations, Mivart defended the idea of large, discontinuous variations (Artigas, Glick & Martinez Reference Artigas, Glick and Martínez2006, 238). Such discontinuities could be traced to a divine plan, which would have been more difficult to do for small variations arising by sheer chance. Mivart’s work received high praise from Catholic intellectuals, in particular the influential Reverend Newman, another convert from Anglicanism to whom we shall return in the next section.

Though Mivart was an avowed supporter of evolution, because he was an opponent of the most dangerous consequences of this theory as outlined by Thomas Huxley (“Darwin’s bulldog”), he was seen as an authentic supporter of the Catholic cause. Newman was also a declared evolutionist, as testified to by the following passage from a letter he sent to a friend:

In 1876, Pope Pius IX offered Mivart a Doctorate of Philosophy, and his successor Leo XIII, a close friend of Newman (whom he made Cardinal in 1879), also had a deep appreciation for Mivart. This did not prevent Mivart from getting into serious trouble later on. Between 1885 and 1887, Mivart became embroiled in a serious dispute with Reverend Murphy and Bishop Newport after the publication of articles in the journal Nineteenth Century, in which he resolutely defended the freedom of Catholics to question the scientific assertions in the Bible, and lamented the “harm that priestly ignorance of science causes Catholics” (Artigas, Glick & Martinez 2006, 244). In the Vatican, the small Jesuit group Civilità Cattolica, supporters of a rigid doctrinal approach to Church policy regarding evolution (resulting in a condemnation of evolutionary theory) and of strict control of scientific publications by theologians, had gained more and more power over the elderly pope, provoking a “U-turn in papal policy” (Brundell Reference Brundell2001, 93). Mivart was obviously suspect in their eyes, and in 1893 they succeeded in having him condemned by the Congregation of the Index (this was not directly for his writing on evolution, but rather for articles putting forward a non-orthodox concept of hell). Though Mivart was formally rehabilitated in 1894 after a retraction and a declaration of submission to the Church authorities, this contributed to his ultimate condemnation in 1900, shortly before his death.

In 1896, Mivart supported Reverend John Augustine Zahm’s book Evolution and Dogma, in which the University of Notre Dame academic enthusiastically promoted Mivart’s theory of evolution. The translation of the book into Italian provoked a violent reaction from the Holy Office. A complete explanation of the reasons behind his condemnation must also mention Mivart’s 1898 articles on the Dreyfus case in France, in which he violently attacked the hypocrisy and abominable nature of the ecclesiastical position on the subject (Root, Reference Root1985).

According to O’Brien (Reference O’Brien1931), the actions that the University of Louvain took for a rapprochement between evolution and the Scriptures was “one of the greatest, if not indeed the greatest, of all the universities conducted under the auspices of the Catholic Church” (115). Mivart made regular visits in Louvain, where he defended a Master’s thesis in medicine in 1884 and became a professor of Natural History the same year. He taught a course on the philosophy of natural history in Louvain between 1890 and 1893, but had to resign after his condemnation by the Index.

De Raeymaeker (Reference De Raeymaeker1951, 520) mentions that, as early as 1881, Mivart had been referenced in the course taught by Mercier in Malines, where he held the chair of philosophy. In his Louvain institute, Mercier prominently featured Mivart’s theory, writing of him in 1894: “Let us proudly quote the work of our colleague S. G. Mivart. … No one has done as much as he has to show the weaknesses of positivism and help to oppose it and restore honor, in England and America, to the fundamental tenets of the Scholastic” (Mercier Reference Mercier1894, 9). Also in 1894, worried about the rising tensions between the Institute and the other elements of the university, Mercier contacted the rector to confirm that, after his rehabilitation by the Holy Office, Mivart would still be able to teach at the Institute. He wrote in a letter to the rector:

In a section on transformism in “Objective Significance” (Mansion, Reference Mansion1903), Mansion explores how a probabilistic approach may help support Mivart’s views on transformism through discontinuous and large variations. In 1877, the Belgian philosopher and psychologist Joseph Delbœuf (1831-1887) from the University of Liège, who had a strong interest in mathematics, wrote a paper attempting to provide mathematical support for transformism (Delbœuf, Reference Delbœuf1877). This paper discussed how combinatorics demonstrates that evolution systematically favors the predominance of mutations. Delbœuf’s interest in evolutionary theory seems to have been more related to the fact that it was the burning scientific question of the time, rather than to its religious implications. Despite its oversimplifications, Mansion praised Delbœuf’s attempt, and further commented on some necessary adjustments that needed to be made to correct his false conclusion. The result was an argument in favor of Mivart’s vision of evolution.

Probabilism and faith

In relation to the excitement around the theory of evolution, reflections about the calculus of probability and its presence in scientific modeling were of particular interest to Mercier and Mansion. Probability, and its relationship to knowledge and certainty, had been continuously discussed at least since Hume. In the middle of the nineteenth century, the relationship between probability and religious faith was the main topic studied by one of the most influential theologians of the time, the Englishman John Henry Newman (1801-1890), a former Anglican minister who converted to Catholicism in 1845. In 1870, Newman published his major book Grammar of Ascent (Newman, Reference Newman1870), in which he asserted that, in everyday life, the accumulation of probabilities replaces formal logic. The latter is generally inoperative because it is too rigid to be applied to the continuously changing circumstances of life. For instance, I am certain that Britain is an island, although I have never verified this fact myself: it is the accumulation of probabilities in favor of the fact that Britain is an island that leads me to certainty. This concept was supported by many modernist theologians, especially Jesuits such as Henri Brémond (1865-1933), who published an overview of Newman’s main ideas (Bremond, 1905), which are scattered throughout Newman’s book in a somewhat confusing manner.

Newman’s central idea was that man has access to an “illative sense” allowing him to form a personal logic built on a cluster of probabilities, and to convert these probabilities into certainty. This “sense” is the central property of mankind that “helps him out” in confronting the questions of faith. While Newman’s probabilism was highly praised in Catholic circles, this was not true everywhere, even in the case of theologians (Anglican admittedly) such as Edwin Abbott (1838-1926), who were afraid that it would become an incitement to atheism by casting doubt on any belief. In their excellent paper, Smith, Berkove, and Baker (Reference Smith, Berkove and Baker1996) present Abbott’s famous parodic novel Flatland, a Romance of Many Dimensions as a response to Newman’s theories.

Like all Catholic intellectuals in the second half of the nineteenth century, Mercier was deeply interested in Newman. Boccaccini (Reference Boccaccini2017) presents a comparative study of Newman’s, Brentano’s, and Mercier’s approaches to knowledge. As he mentions, all three thinkers opposed the domination of Kantian idealism by developing a metaphysics based on gnoseological traditions with roots in medieval philosophy. Mercier was particularly interested in the new science of psychology. De Raeymaeker (Reference De Raeymaeker1951, 550) notes that, as early as 1891, a course of experimental psychology was taught at the Institute of Louvain by Ghent University professor of pharmacology Jean-François Heymans (1859-1932), and a laboratory was soon established. In their interesting—though slightly hagiographic—book, Misiak and Staudt (Reference Misiak and Staudt1954) present Mercier as the “Catholic pioneer of scientific psychology” (chapter 3, 34–52).

Mercier was engaged in reflection about probability. Louvain’s archive includes his lecture notes on probability theory, which were presented to students at the institute in 1891, among his personal papers. The content of the lectures was not particularly original, but two points deserve emphasis. First, Mercier’s aim was to teach the mathematical basis of probability theory. He presented his students with the principles governing probability (the principles of total probabilities and compound probabilities) and gave many examples of calculations using these principles. In keeping with his intellectual plan to provide his institute’s students with a strong scientific grounding, Mercier thought that a philosopher must begin by learning the techniques of a science before constructing a philosophical approach to it. Mercier most likely intuited that this was the only proper way to avoid condescension from scientists. Some philosophers shared Mercier’s sentiment. Léon Brunschvicg, for instance, wrote that “philosophical speculations that relate to the space of geometers without any other specification, whether a reality or a pure idea or a form of intuition, have lost touch with current science” (Brunchvicg 1912, 444).

A second observation is that Mercier does not mention the possibility of an objective probability, but only the concept of a subjective probability that mathematical probability is supposed to quantify and make calculable. Although he is not quoted in Mercier’s notes, this approach is in line with Newman’s use of probability simply as a tool to illuminate free will.

It was from Mansion’s lectures in Ghent that Mercier learned the basis of the calculus of probability, and he had asked Mansion to give lectures on mathematics, including its history and philosophy, at the institute since the very beginning. This formed part of the background of “Objective Significance” (Mansion, Reference Mansion1903), which presented his conception of the objective contents of probability a century after Laplace and Condorcet. As Mansion mentioned, in the second half of the nineteenth century several scientists, such as Bertrand and Poincaré, claimed that this objectivity was mostly an illusion. In his final edition of Boudin’s treatise (Boudin, Reference Boudin1916), in which he inserted “Objective Significance” (Mansion, Reference Mansion1903) as an appendix, Mansion stated his conclusion regarding the nature of probability:

At the end of “Objective Significance,” Mansion twists Laplace’s formula in a slightly provocative manner by asserting that for God, and only for him, “nothing is uncertain and the future, like the past, is present for Him” (Mansion Reference Mansion1903, 90). Perhaps Mansion would have praised what Georges Bernanos, another Christian militant, would write fifty years later: “what we call chance is perhaps after all only the logic of God” (Bernanos 1947, 1613).