Gwendoline Malogne-Fer

Protestant Churches and Same-Sex Marriage in France: "Theological" Criteria and Sociological Approaches

Introduction

The reflection presented here was born from my personal questions about the academic study of religion, including the place of non-believer scientists in the field of the French sociology of religion and the potential influence of researchers' religious beliefs and belonging on sociological analyses. My own research on Polynesian Protestantism questions the relevance of the classical academic boundary between sociology and anthropology. The grand partage of the earth into distinct "cultural areas" [START_REF] Lenclud | Le grand partage ou la tentation ethnographique[END_REF] still prompts researchers studying nonwestern societies to adopt an anthropological approach. Anthropology has historically focused on non-Christian religions, with Christianity assigned to sociology, at least until the more recent development of an anthropology of Christianity [START_REF] Coleman | The anthropology of global Pentecos-talism and evangelicalism[END_REF][START_REF] Robbins | Becoming sinners: Christianity and moral torment in a Papua New Guinea society[END_REF]). Yet for me and for many other researchers of my generation who received no religious education, Protestantism is as "exotic" as a non-Christian religion.

The advice given me by my first thesis supervisor, an anthropologist, was to "look beneath the Christian varnish" to discover the true Polynesian culture and beliefs; to me this was difficult to apply, as it implied a "true Christianity" with which I lacked familiarity. In addition, such an unveiling would be difficult in Tahiti and its neighbouring islands, as Christianity has been present since the end of the 18th century. I turned to the sociology of religion, but ended up facing new difficulties related to the eurocentric nature of many paradigmatic concepts that have Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, contributed to the distinctive identity of this discipline in France since the 1960-70s, notably, the theory of secularisation and the concept of religious modernity and its various reformulations, such as ultra-modernity, advanced modernity, or late modernity (for a critical analysis, see [START_REF] Obadia | Paradoxes, utopies et cécités du modernisme en religion: le cas du bouddhisme de France[END_REF]. I also had methodological and epistemological issues with the use of concepts with Christian origins, which can be understood in both a secular and a religious sense and owe part of their success to their polysemy.

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My argument here is that the sociology of religion in France is not simply a branch of sociology, but a discipline whose specificity is continually reinforced by academics who advocate the irreducible distinctiveness of the "religious" object. Even if the diversity of theoretical approaches represents an intellectual richness conducive to debate, the risk is that the alleged specificity of the religious object ultimately means that an irreducible part of religion will always elude the scope of sociological analysis. This chapter draws on the confrontational debates triggered in 2012-2013 by the "marriage for all" bill (same-sex marriage) and considers the way these debates were analysed from a Protestant and a sociological point of view. The debates revealed tensions within French society, as well as within the sociology of religion, more specifically, the sociology of Christianity. Although I focus on two articles by Jean-Paul Willaime, the following reflection should not be read as a critique of the work of this eminent sociologist of Protestantism; in fact, his insights furthered my own doctoral research on women's accession to the pastorate and the professionalisation of this ministry. Rather, my objective is to draw on a case I previously studied1 to highlight the divergent approaches coexisting in the sociology of religion.

To understand what was at stake in the debate on same-sex marriage and the commitments of actors in both academic and religious fields, we need to review the Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, chronology of events and the stances of the various Christian churches on the issue. The notion of "field" elaborated by Bourdieu is particularly relevant to understand the relationships of power and the strategies involved in the positions adopted by religious, political, and academic actors (Obadia, 2014: 299-300; also [START_REF] Bourdieu | Genèse et structure du champ religieux[END_REF].

The debate on the "marriage for all" bill highlighted a convergence of views on this kind of issue by Protestant and Catholic churches. At the beginning of the 2000s, Willaime noted the significant convergence of Protestants and Catholics in matters of sexual ethics [START_REF] Willaime | Entre éthique de responsabilité et éthique de conviction: l'éthique sexuelle en débat dans le protestantisme français des années 1970-2000[END_REF] during the debate on same-sex civil union (civil solidarity pact, pacs). This convergence contrasted sharply with the context of the 1970s, when Catholic and Protestant churches showed strong divergences. There were also clear divergences at the time within the various Protestant faiths: the Lutheran-Reformed churches, dominant in the French Protestant Federation, argued in favour of contraception, decriminalisation of abortion, and recognition of divorce, while Evangelicals 1 In a presentation with V. Aubourg at the Congress of the French Sociology Association (2013) and in a recent publication [START_REF] Malogne-Fer | Les sciences sociales au secours des Églises protes-tantes: l'exemple du 'mariage pour tous[END_REF], I explained the use of social sciences by Catholic and Protestant churches in the debate on samesex marriage.

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were more conservative. Despite the more recent move towards convergence, many in religious and academic fields still see this as an unbridgeable gap. This chapter explains the evolution of Protestant churches' positions on same-sex marriage and seeks to shed light on the convergence of views, on the one hand, between Catholic and Protestant churches and, on the other hand, between the different Protestant streams. It questions the relevance of a strict distinction between Lutheran-Reformed Protestants and Evangelicals as a sociological key to analysing Protestant attitudes towards same-sex marriage. It shows why intra-Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, Protestant "theological" criteria do not provide a sufficient sociological explanation: there is a risk of reproducing theological criteria as they are elaborated in the Protestant field without examining the social and institutional stakes of these distinctions. Finally, the chapter turns to a controversy between two major sociologists of religion during the debate on same-sex marriage to illustrate the tensions generated by an incomplete process of secularisation of the sociology of religion in France.

Protestant Stances on the "Marriage for All" Bill

French Protestant churches have two different federations. The French Protestant Federation (FPF), established in 1905, includes churches from all denominational backgrounds, but its leaders are mainly Reformed and Lutheran ministers. Its mission is to represent French Protestantism and Protestant ac-tors to state authorities. Since 2010, the FPF has faced competition from the Conseil National des Évangéliques de France (Cnef), which has a doctrinal basis and attracts approximately 70 percent of Evangelicals in France.2 Both federations took stances on the bill; even though their arguments differed, their official statements show both federations -as well as numerous Protestant churches -were against samesex marriage.

On 13 September 2012, the Cnef and the Assemblies of God (the main Pentecostal denomination within the Cnef) released two quite different statements on same-sex marriage. The Cnef's statement begins with a reminder that its position rests on biblical ethics but goes on to make a different type of argument: it notes the need for a large and democratic debate and cites the difference between men and women which "has always been a fundamental fact of anthropology," with marriage considered a "foundation that structures society." Not least, it is in the best interests of the child to be brought up by Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, 2 The distinction between these federations is less clear at the church level, as nine church unions are simultaneously affiliated with both.

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a father and a mother (Cnef, 2012). In contrast, the Assemblies of God formulate their arguments in biblical terms:

The biological difference between men and women has existed since creation. Male and female sexual identities are the timeless foundation of the couple, which makes marriage, procreation and family possible.... By their doctrine, the Assemblies of God remain strongly committed to respect for ethical biblical values, as expressed in the first point of their theology, where the Bible is presented as the unerring rule of faith and behaviour for the assembly, as well as for each Christian.3 (addfrance.org/ communique-de-presse.html) Pastor Thierry Le Gall, in charge of the communications of both the Assemblies of God and the Cnef, differentiates the statements as follows:

The Assemblies' statement is a little bit timeless and outside society, to say: this is what we believe and will always believe; the discourse is more spiritual, and that's what we wanted to emphasise. The Cnef discourse is societal; it's a stance taken within the society.

Interview, pastor Thierry le Gall, 28 August 2013, côtes-d'armor

The Cnef's argumentation strives to appropriate a "discourse under constraints" [START_REF] Fassin | La voix de l'expertise et les silences de la science dans le débat démocratique[END_REF] and borrows expressions from the vocabulary of the social sciences to gain political legitimacy. Le Gall explains:

When we started thinking about this statement, the specification was to translate Evangelical thought in terms acceptable by civil society. It was made for media first, then for political leaders and also for Evangelicals; this can explain its form and its low profile.... And this statement has been a gateway for us, to be heard and Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, received in the institutional spheres.... Even if this statement was not perfect, it brought us visibility for the first time; ... it enabled us to be heard in Parliamentary settings, to express our position and get better identified as an established body that has something to say and represents about 600,000 voters today in France.

Interview, pastor Thierry le Gall, 28 August 2013, côtes-d'armor Opinions on same-sex marriage are usually diverse amongst the Lutheran-Reformed churches and the leaders of the FPF, but public stances do not 3 Unless otherwise indicated, the translations are my own.

[page 39] reflect this diversity. On 27 August 2012, Laurent Schlumberger, President of the French Reformed Church, said in internal correspondence that a process of collective reflection was still going on within the church. Therefore, he asked those ministers supporting the blessing of same-sex couples to remain silent until the conclusions of the working group led by theologian Isabelle Grellier were released.4 In September 2012, a pastor working at the FPF head offices explained to colleagues that the FPF would not take any official stance on the issue; the diversity of opinion amongst churches within the FPF was too strong to reach a common position. But on 2 October 2012, Claude Baty, President of the FPF, declared to Le Monde that his federation opposed the government's project, not for theological reasons, but for anthropological ones: "'Marriage for all' suggests a false equality. Heterosexual and homosexual couples, this is not the same thing" (http://religion.blog.lemonde.fr/2012/10/02/le-gouvernement-ne-veut-pas-de-debatsur-le-mariage-pour-tous).

Ten days later, on 13 October 2012, the council of the FPF endorsed the position of its President in a new statement:

It is not a question of putting homosexuality and same-sex marriage at the core of Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, theological debates; the issue is fundamentally social and collective. It's about the way a society perceives and builds itself, and the symbols that structure the field of its identity. In this perspective, we need to say clearly that distinctions made between homosexuality and heterosexuality don't reflect an outdated moralism, but rather a deeply-felt need of the society. Society requires to be structured, both symbolically and concretely, by the representation and the acceptance of an original and fundamental difference that shapes our deepest interiority, in bodies and ways of being.... Marriage is not a celebration of love, a display of feelings, but rather a social organisation that contributes to structure relations by symbolising the difference between generations, sexes, between marriageable and non-marriageable people.... The FPF thinks that the current project of "marriage for all" brings confusion into the social symbolical order and doesn't help structuring families. This is not a question of morality but of anthropology and symbol.

FPF, 2012

Addressed to the presidents of local church councils, it said: "The Church is not committed to the blessing of same-sex couples.... A reflection is still going on. If choices should be made and the position of the Church should evolve, this would be as a last resort the prerogative of the national synod to take such decision."

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The statement had not been subject to long discussions. In fact, it borrows most of its arguments from a text on homosexuality entitled "A few elements to reflect on homosexuality," published by the ethical commission of the FPF in 1994. This suggests that for the FPF, the terms of the debate on same-sex marriage had not significantly changed over 20 years. Its reflections remained strongly influenced by the national context and did not mention the positions adopted by the Protestant churches of Northern Europe in favour of same-sex marriage or the blessing of same-sex couples (in Denmark, the Netherlands, Norway, and Sweden). Rather, the FPF appropriated the so-called "anthropological" arguments developed by the Catholic Church.

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, Reasons for Using "Anthropological" rather than "Biblical" Arguments How can we explain the disappearance of all or part of the biblical arguments and the emphasis placed on allegedly "anthropological" arguments? What kind of resources were selected from the social sciences to support the Protestant stances on same-sex marriage?

Euphemising biblical arguments while emphasising "anthropological" ones can be understood as a way to comply with the constraints of political commitment in a secular state and as a strategy to legitimate the churches' participation in the public debate, in a partly de-Christianised French society. Thus, for Catholic and Protestant institutions, the use of non-biblical arguments allowed a kind of generalisation facilitating the mobilisation of a larger circle than committed Christians, so that they did not appear simply as groups defending specific interests and convictions. This non-confessional vocabulary also enabled individuals to be recognised not only as church members but also as concerned citizens engaged in major social debates.

Fassin remarks that the "anthropology" to which religious actors referred during the debate on civil union was quite different from the academic discipline, because it aimed to re-establish a transcendent truth:

In fact, anthropology is requested to say simply but with authority that there are men and women -the order of heterosexuality, based on the or-der of gender differences.... This is what speaking about an anthropological transcendence means: just as religion could once stand above politics, today in a "democratic and secular" society, some people would like to place anthropology above politics. Fassin, 2001: 96

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Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, The mobilisation of these "anthropological" arguments was part of an adjustment of religious convictions to the social context. To speak with legitimacy, religious authorities needed to elaborate a discourse that took into account, on the one hand, the legal prohibition of homophobic language [START_REF] Fassin | La voix de l'expertise et les silences de la science dans le débat démocratique[END_REF], and, on the other hand, the French regime of laïcité, not to mention a double process of diversification and individualisation of moral norms.

The Catholic Church had been working on developing such a discourse since the 1970s and updated it in the 1990-2000s during the debate on civil union. In her study of Catholic opposition to the "theory of gender" -initiated by the Vatican in 1995 and appropriated by the French episcopate in 2005 -Beraud shows how the Catholic hierarchy turned to "human sciences, and particularly to psychoanalysis and law, in a quest of invariants, of universal and intangible norms" (Beraud, 2011: 234-235). An interdisciplinary working group, made up mostly of lay peoples, was established:

The request was explicit. "Today, how does one say that homosexuality is a tragedy?" wondered Cardinal Philippe Barbarin, Archbishop of Lyon. "Bring us some "biscuits" [good arguments] on this," he said to Jacques Arènes, a Catholic lay person and psychoanalyst. The program was clearly stated: "We have to highlight a discourse on differences as a component of human being and selfrealisation through a vocabulary drawn from anthropology and human sciences, which is the most acceptable today." Beraud, 2011: 234-235 In other words, the use of psychoanalysis and anthropology was officially promoted as part of a strategy to dress up an unchanged Catholic doctrine in new rhetorical clothes.

Protestant actors drew on this rhetoric for purposes of intra-Protestant competition and distinction. The stance of the Fpf underlines the influence of this Catholic Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, discourse and the effect of the reshaping of the Protestant institutional fieldespecially with the creation of the Cnef -on the federation. Interestingly, the Fpf's position against same-sex marriage does not reflect its internal diversity and is actually quite similar to the Cnef's position. The use of "anthropological" arguments was intended to distance the FPF from Evangelicals and their biblical arguments. But these "anthropological" arguments do not draw on any anthropological or sociological work; the anthropology mentioned by the FPF is not an academic discipline but a concept referring, in a normative and nondescriptive way, to human beings as a whole.

The working file "Church and Homosexuality" (subtitled "Symbolic significance of publicly established things") issued by the Lutheran-Reformed

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Communion in 2002, to which the FPF still refers today, proposes three kinds of approaches to the issue of same-sex marriage: a biblical approach, a psychoanalytical approach, and a psycho-sociological approach. The authors from the human and social sciences mentioned in the document are Freud, Lacan, theologian and Pastor Jean Ansaldi (1987), and Pastor Antoine Nouis. The third part, "To take the psycho-sociological approach some steps further" includes excerpts from the writings of Michel Johner (a professor of ethics at the Evangelical Theological Faculty of Aix-en-Provence who opposes the blessing of same-sex couples) and Tony Anatrella, a Catholic priest and psychoanalyst whose virulent and homophobic stances contrast with the tone of the statements made by Protestant churches and their federation. Rather than being a reflection based on marriage as intrinsically linked with procreation, the argumentation of the FPF in this working file establishes a hierarchy of sexualities in which homosexuality is described as inferior, because of its "essential impossibility to procreate."5 To avoid any contradiction that might undermine the relevance of this theory, the leaders of the FPF do not refer to the social sciences' studies on homosexual Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, parenting or to the social sciences more generally. In fact, the main intellectual influence appears to have been the conservative discourse of the Catholic Church.

In addition to their distinctive "anthropological" arguments, the leaders of the FPF presented their position on same-sex marriage as an illustration of the democratic functioning and the inclusiveness of their federation, thus differentiating it from the Cnef by casting itself the only federation to comprise the whole diversity of Protestantism. Claude Baty explains:

What could be feared on such a sensitive issue was a confrontation be-tween Evangelicals and Lutheran-Reformed Protestants. This text [the statement issued by the FPF] has been voted without hesitation by both of them, underlining the existence of a fruitful dialogue within the Protestant federation. With all the diversity of its members, the federation took a stance against the "marriage for all" not for religious reasons but for matters of social, symbolical and anthropological structuring.

Baty, 2012

Theological Differences and Scientific Categories

In summary, the two national Protestant federations took stances against same-sex marriage and advanced "anthropological" arguments inspired by 5 Quoted in L'homosexualité: éléments de réflexion, Ethical Commission of the fpf (1994).

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Catholic rhetoric. The Cnef, seeking political recognition, adopted this position to make an Evangelical voice heard in the public space, while preserving the possibility for its members to emphasize biblical arguments. For the FPF, still the only federation recognised by political authorities, the use of "anthropological"

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, arguments served two purposes: first, to distance itself from the biblical argumentation usually put forward by Evangelicals by using more consensual arguments; second, to secure its status as the exclusive representative of Protestantism in the political arena, by emphasising its democratic and inclusive functioning.

The distinction between the two federations and more generally between (liberal) Lutheran-Reformed Protestants and (conservative) Evangelicals on issues of sexual morality needs to be carefully specified. The statements re-leased by these federations during the debate on same-sex marriage took place in a particular political and historical context. Recent debates on the blessing of same-sex couples, and the reactions that followed the 2015 decision of the United Protestant Church to allow such blessing under some conditions suggest that while there are liberals amongst Reformed Protestants, conservatives are present in all Protestant streams: Reformed, Lutheran, and Evangelical.6 Therefore, a relevant sociological framework of analysis cannot be based on intra-Protestant theological distinctions alone.

Additional sociological data show that opinions on the topic vary according to age and gender. Women as a whole are more in favour of "marriage for all" than men.7 In addition, intergenerational gaps are important in such issues, regardless of religious affiliation, and Lutheran-Reformed church members in France constitute an ageing population. Quantitative analyses also reveal significant differences between pastors and ordinary church members: when asked if "same-sex couples should be able to get blessed by churches," 36 per-cent of Protestants answered positively in 2010, compared to only 26 percent of Protestant pastors. To the question "Should the right to abortion be advo-cated?" 72 percent of Protestants answered positively, compared to 54 percent of their pastors. These differences suggest personal opinions are not uniquely . 6 In June 2014, the union of Protestant churches in Alsace and Lorraine decided, as no consen-sus could be reached, to make no decision on the blessing of same-sex couples and refused to examine this question again in the following three years. The decision of the national synod of the United Protestant Church in May 2015 generated a significant movement of internal opposition, called the "attestants" (testifying Protestants).

. 7 ifop survey, 2012 (93 and f.). To the question "Do you think homosexual couples in France should have the right to marry?" 63 percent of all respondents and 43 percent of Protestants said "yes." Amongst Protestant respondents, 53 percent of the women said "yes" and 38 per-cent of the men; 51 percent of the respondents under 35 supported same-sex marriage, while the over-65s strongly opposed it, with only 35 percent answering "yes."

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determined by religious identities; social positions also play a role. Within a single church, pastors are ex officio in charge of reminding parishioners of sexual ethical norms, while church members who have to cope with these norms tend to be less normative. From the perspective of the sociology of social movements, the debate on same-sex marriage also underlines the strong mobilising capacity of the Catholic Church, in contrast to the small impact of Protestants, who represent approximately two percent of the French population.8

How can we explain the prominent role of intra-Protestant distinctions, specifically between Lutheran-Reformed Protestants and Evangelicals, in French academic analyses of the debate on same-sex marriage? Part of the answer lies in the relationships between sociologists of Protestantism and Protestant-ism itself. This sensitive issue is rarely discussed in publications, workshops, or conferences. In his conclusion to the proceedings of a symposium of the French Association for the Sociology of Religion (AFSR) on "The Religion of Sociologists," Bréchon remarks that the challenge consists in asking sociologists to publically expose the links that may exist between their convictions and their theories.... It is clear that readers will not find in this volume any sensational confession. I have to say that those who spoke about the influence of their personal beliefs or ideological convictions mostly mentioned the years of their youth.

bréchon , 1997: 245-246 With one exception, "no one has really taken the risk to conduct a self-analysis for the present days" (Bréchon, 1997: 245-246).

In fact, the point is not so much the beliefs of the sociologists of religion, but their interests in the religious field (Bourdieu, 2010: 2). Thus, emphasising the difference between Lutheran-Reformed Protestants and Evangelicals as the key to understanding Protestant stances on same-sex marriage tends to transform an alleged difference into a scientific and "objective" truth. Such an analytical framework serves not only scientific interests but also, objectively, the interests of the FPF's leaders and their strategy of distinction.

In November 2013, in an article entitled "Protestant diversity and homosexuality in France," Willaime presented the positions of the FPF and the Cnef on same-sex marriage as an illustration of this "difference," understood as an opposition between moderate and conservative Protestants: "On the marriage 8 The President of the FPF did not call for demonstrations, and many Evangelicals were re-luctant to join Catholics in these demonstrations, presented by most media as Catholic initiatives.

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for all, the FPF adopted... a position which can be described as moderate. In the Evangelical Protestant world..., a marked opposition to same-sex marriage is dominant" (Willaime, 2013: 82-83). In the article, Willaime mentions two elements to demonstrate the moderation of the FPF's stance: the kind of argumentation used and the decision-making process. First, he points to the FPF's defence of the institution of marriage "not from a moral point of view, but from the perspective of a Christian anthropology sensitive to the symbolical and structuring dimensions of social fundamental institutions" (Willaime, 2013: 82-83). The notion of "Christian anthropology" does not refer to the academic field but to the definition promoted by the Catholic Church, and the diffusion of this notion since the 1990s has caused sociologists and anthropologists to criticise a Catholic misappropriation of the social sciences [START_REF] Fassin | La voix de l'expertise et les silences de la science dans le débat démocratique[END_REF]. For example, sociologist of religion Hervieu-Léger points out that the "anthropology" to which the Catholic Church refers should be understood not as a discipline of the social sciences "that explores the diversity of family and kinship patterns in space and time" but as a family model built by the Catholic Church itself [START_REF] Hervieu-Léger | Mariage pour tous: le combat perdu de l'Eglise[END_REF]. Moreover, the debate on "marriage for all" has produced, as Rochefort remarks, a paradoxical scheme, as "conservative religious actors use secular discourse, while dissent theologies innovate through reinterpretations of religious heritage" (Rochefort, 2014: 229).

Second, Willaime points to the decision-making processes of the fpf. The stances taken were, according to him, preceded by extensive consultation of local churches. But the statement made by the President of the FPF in October 2012 was not preceded by any consultation; the consultation he mentions was, in fact, conducted in 2002, when the working file "Church and Homosexuality" issued by the Lutheran-Reformed Communion was distributed in local churches.

This selective interpretation can be understood as a reprise in the academic field of the arguments posited by the FPF leaders in terms that legitimated and reinforced the dominant position of this federation within the Protestant field. In the intra-Protestant competition for representativeness, an academic discourse emphasising the opposition between a conservative Evangelicalism (represented by the Cnef) and a moderate, inclusive Protestantism (represented by the FPF and Lutheran-Reformed Protestants) can hardly be seen as neutral. Of course, Evangelicals are more conservative than Lutheran-Reformed Protestants, but the increasing conservatism of Protestant churches affiliated with the FPF should be included in the picture. Finally, the Catholic use of "anthropological" rather than biblical arguments is a rhetorical strategy that aims to defend a body of doctrine. The use of this same "anthropological" argument within the academic field seems more problematic as it introduces -and tends

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to legitimate -religious categories into the sociological analysis of religious facts.

The controversy between Willaime and Hervieu-Léger during the debate on samesex marriage sheds light on these epistemological issues and underlines the tensions generated by the incomplete secularisation of the sociology of religion in Francean essential part of the history of this academic discipline.

From Religious Sciences to the Sociology of Religion (and Vice Versa)

The new academic discipline called "religious sciences" was officially recognised in 1886 with the establishment of the fifth section of the Ecole pratique des hautes études (ephe). It took place in a specific historical context and raised epistemological and methodological issues that French historians of Catholicism [START_REF] Langlois | Aux origines des sciences religieuses en France: la laïcisation du savoir (1810-1886[END_REF][START_REF] Poulat | Le développement institutionnel des sciences religieuses en France[END_REF]) and Protestantism (Cabanel) have analysed in detail. As Langlois explains, the education policy of the third Republic and the secularisation of teaching programs and staff dominated the context. In 1880, the government re-established the state monopoly on university education and supressed funding for the faculties of Catholic theology. Two options were possible: the creation of chairs in history of religion in the main French universities or the establishment of a few chairs concentrated in one prestigious Parisian institution. The second option was preferred. That same year, a university chair in history of religions was created at the Collège de France, with the aim of delivering generalist and comparative teaching [START_REF] Langlois | Aux origines des sciences religieuses en France: la laïcisation du savoir (1810-1886[END_REF]. The first holder of this chair was Albert Réville, a liberal Prot-estant. According to Langlois, this choice brought to the study of religious facts "the most indisputable scientific legitimacy (Collège de France/ephe) but un-der a model that avoided human sciences, developed in the universities, where the scientific study of religions also had its place" [START_REF] Langlois | Aux origines des sciences religieuses en France: la laïcisation du savoir (1810-1886[END_REF].

What exactly were these "religious sciences" elaborated in France at the end of the 19th century? For one thing, the new discipline distanced itself from the-ology by opting for a scientific methodology: the historical approach. A second distinctive feature was the strong presence of liberal Protestants amongst the teachers and researchers:

For a quarter of a century, we find a very strong core of Protestants, even pastors or former pastors, most of them closely linked with the new Parisian faculty of Protestant theology, which appears to be the main recruitment pool of the masters and disciples of the young science.

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Albert Réville (a former pastor) was amongst them, as were his son Jean Réville and Maurice Vernes (also a pastor's son). The latter finally left the faculty after having elaborated an agnostic and extremely liberal approach. In effect, he contributed to the first criticism of religious scientists; in his view, they should restrict themselves to collecting and classifying data. To this, Jean Réville responded: "Dissecting documents [requires] knowing by experience what a religious feeling or a religious thought is. Put in other words, even the most erudite person in the world would never offer anything else than a mummy instead of a living organism" (Reville, 1886).

As Cabanel remarks, religious feeling is "at the core of the French-Dutch -or we may say Protestant -conception of religious sciences" (Cabanel, 1994: 60). This implies a vigorous opposition to any hostile, ironic or condescending approaches to religion. This idea that "religious feeling" and the personal religious dispositions of the researcher will determine his/her ability to understand re-ligious facts led Durkheimian sociologists to suspect these religious sciences of being based on nonscientific premises. Durkheim aimed to study religious facts rather than religion and criticised the promoters of religious sciences of "wanting to express straightaway the content of religious life" (Cabanel, 1994: 61) instead of approaching this content through a careful study of the external forms of religious phenomena such as services, rituals and practices. In Durkheim's view, "the socalled religious phenomena consist in compulsory beliefs, linked with well-defined practices oriented towards objects given by these beliefs" (Cabanel, 1994: 61).

Similarly, Mauss strongly criticised Tiele's introduction to religious sciences as turning away from scientific methods in favour of general considerations: "When you try to reach them [the deeper and more intimate conditions of religious life] straightaway, through a simple introspection, your own prejudices, personal and subjective impressions take the place of things" (quoted by Cabanel, 1994: 62).

Simply stated, the sociological school advocated an analysis of religious phenomena as natural and objective things with social functions and refused to consider the philosophical, psychological or theological approaches of the religious sciences as relevant. This led Albert Réville to denounce the "exaggerations of the sociologists" who gave a determinant role to the social context in their study of individuals (Cabanel, 1994: 63).

Langlois argues that the choice to establish a fifth section at the ephe dedicated to the religious sciences instead of several chairs in French universi-ties contributed to the development of "a model of avoidance of human sciences" and the lasting isolation of this discipline [START_REF] Langlois | Aux origines des sciences religieuses en France: la laïcisation du savoir (1810-1886[END_REF]. He notes that the creation, in 1956, of the Groupe de sociologie des religions (GSR) by the CNRS followed a similar pattern. The objective was to develop the sociology of religion within "prestigious" institutions, at the margins of universities;

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill,.

[page 48]

as a consequence, the sociology of religion did not become a branch of general sociology. This means that sociologists of religion preferred to base the coherence of their discipline on its particular -"religious" -object, rather than on a sociological methodology shared with other branches of sociology. By so doing, they reinforced in the sociological field the impression that studying "religion" is a specific task, as if the object studied should rub off on the methodology to be used.9

This isolation of "religious sciences," only taught at the faculty of Protestant theology in Strasbourg and at the EPHE in Paris, was further reinforced in 1975 when the fourth section "Economic and social sciences" left the EPHE to form a new, independent institution: Ecole des hautes études en sciences sociales (ehess), which now comprises historians, sociologists, anthropologists and political scientists. Sociology of religion is today a declining discipline, mainly taught at the EPHE and the ehess [START_REF] Lassave | Les sociologues des religions et leur objet[END_REF]. In 1998, the GSR was split into two research centres: Groupe sociétés, religions, laïcités (cnrs-ephe) and Centre interdisciplinaire des faits religieux (cnrs-ehess).

The incomplete process of secularisation of the discipline tends to resurface in this context of decline and isolation, with the need for religious erudition considered an "admission ticket" by the defenders of the specificity of this academic subfield.

And the criticism Willaime (as director of studies in religious sciences at the EPHE and sociologist of Protestantism) levels at Hervieu-Léger (director of studies at the Ehess and sociologist of Catholicism) should be resituated in this field configuration. Without getting into all the details of their biographies and the reasons for their involvement in the sociology of religion, both sociologists have been part, in one way or another, of the churches they study. Williaime, who was a holder of a Ph.D. in religious sciences from the faculty of Protestant theology in Strasbourg, chose to write his thesis on "The Profession of Pastor," "as if to

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, distance himself from the profession for which his previous training in theology destined him" [START_REF] Lassave | Les sociologues des religions et leur objet[END_REF]. Mean-while, Hervieu-Léger conducted her first fieldwork on Catholic student missions while she was herself part of this protest movement in the university setting [START_REF] Lassave | Les sociologues des religions et leur objet[END_REF].

During the demonstrations against the same-sex marriage bill, Hervieu-Léger published a text in the newspaper Le Monde, in which she expresses her surprise that the Catholic Church uses "anthropological" rather than biblical arguments to oppose same-sex marriage: "In this debate on marriage for all, it comes as no surprise that the Catholic Church makes its voice heard. 9 There is an exception in this process of isolation; since the 1990s, the history of religion has found a place in the context of the democratisation of university education.

[page 49]

How the Church carefully avoids any reference to a religious interdiction is more surprising" (Hervieu-Léger, 2013). Then she underlines the weakness of the arguments put forward by the Catholic Church:

The arguments used by the Church -the end of civilisation, the loss of foundational landmarks of human beings, the threat of the familial cell's dislocation, the confusion of sexes, etc. -are the same as those once used to criticise the professional involvement of women outsides the domes-tic household or to fight against the establishment of divorce by mutual consent.

hervieu-léger, 2013

Finally, Hervieu-Léger remarks that same-sex marriage is one of many examples of the diversity of contemporary familial patterns characterised by the demand for equality and the dissociation of marriage and religious affiliation. Without denying the legitimacy of the Catholic Church to speak on the bill, she regards its arguments as unconvincing and concludes that "on these various issues, a word which places emphasis on freedom is expected" [START_REF] Hervieu-Léger | Mariage pour tous: le combat perdu de l'Eglise[END_REF].

This critical approach to the Catholic institution is in line with her previous work on the "exculturation" of Catholicism, defined as a loosening grip of Catholicism on French society [START_REF] Hervieu-Léger | Catholicisme, la fin d'un monde[END_REF]. The process of exculturation implies the Catholic Church is facing disaffection with its traditional activities; it also means the Church must now elaborate values and norms that make sense in a society where an increasing proportion of individuals have no Catholic culture at all. The use of "anthropological" arguments by the Catholic Church could, in such a context, be seen as a way to cope with exculturation. But Hervieu-Léger gives little consideration to these efforts as she questions "the capacity of the Church to nurture and elaborate... collectively shared representations whose religious origin might remain identifiable in the French society today" (Hervieu-Léger, 2003: 131).

In 2014, Willaime published a text in Transversalité, a publication of the Catholic Institute of Paris. In this text, he develops his criticism of the stance taken by Hervieu-Léger and provides his ongoing reflections on the definition of "religion" and the specific methodology of the sociology of religion. He ends with a vibrant call for better public recognition of the role of religion in society. The text uses two repertoires; while employing an ostensibly sociological approach, it nonetheless advocates the political involvement of churches in society. In several passages, it echoes the controversies that opposed, more than a century before, sociologists and promoters of "religious sciences." In other

[page 50]

words, these debates remain relevant today, and Willaime's definition of religion as a particular object of study that requires a specific approach seems in line with the heritage of the religious sciences. His warning about the risk of "reductionist tendencies that marked the history of sociology" (Willaime, 2014: 114) can be read as an implicit criticism of the Durkheimian sociological school. He is especially critical of "the approaches that tend to limit themselves to the study of the non- religious factors of religion... and reduce the sociological study of religion to the analysis of the social activities in which they are manifested, for example the various forms of ritual practices" (Willaime, 2014: 114). The definition of religion he proposes makes religion an object of study that is not reducible to the sociological approach:

I consider religion as a symbolic activity which has its own consistency; that is to say that while religion is socially determined -in many ways -, it remains relatively independent from these determinations.... This could be summarised by saying that, in its own way, religion makes society (by creating links), makes truth (by building legitimacy) and that historical processes are always evolving. What matters to me is reflecting the singularity of this social activity, while staying within a sociological framework. This entails recognising that this activity can't be reduced to any other forms of activities.

willaime, 2014: 115-116

Defining "religion" also means drawing the boundaries of a discipline -the sociology of religion. And as religion is defined here more by its essence than its social expressions, the risk is that this object constructed by and for the sociology of religion might end up being similar to the "culture" studied by classical anthropology. This implies the same limits as those underlined by Bensa in La fin de l'exotisme, where he shows the epistemological consequences of defining culture as an encompassing entity that exists beyond and outside individuals, and as a radical otherness that anthropologists have the task of defining. To find their disciplinary object in social reality, these anthropologists tend to exaggerate the homogeneity of culture, to minimise the temporal dimension of the social facts they study, and to consider actors as individuals governed by an encompassing reality:

But it is clear that the discourse [of the anthropologist] which takes on the air of "I was there" rarely produces a chronicle of what really happened "there." This kind of dodge probably comes from the dualist scheme which supposes the existence, as

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, the saying goes, of a "good god

[page 51]

behind the wardrobe," that is to say an elsewhere more significant that the social relations implemented and observable in the frame of a given experience. Bensa, 2006: 10 The definition proposed by Willaime also echoes debates in France in the 1960-70s, in the context of the institutionalisation of the sociology of Catholicism. Thus, Henri Desroche, a former Catholic priest who left the Dominican order in 1951, says religion should not escape -even partially -the sociological analysis:

I tend to think that no area of the religious domain should be closed to the sociological investigation. The history of gods is part of the history of men.... Put in other words, there is not a part of religion accessible to the sociological approach and another set apart for the sole religious approach. Desroche, 1968: 131 To this, Desroche adds: "It would be a mistake to interpret Durkheim as a pure-ly and simply 'reducist' .... Society and religion function in a complementary manner to form a total act" (Desroche, 1968: 61-62).

In addition, the sociology of religion as defined by Willaime requires re-searchers studying religion to have specific "dispositions" following Bourdieu's understanding of a disposition as something larger than belief (2010). He doesn't criticise Hervieu-Léger for not being a Christian believer (she was once involved in the Catholic Church) but for not believing enough in the Catholic institution. From his perspective, she is disrespectful when she says the Church has lost the battle against "marriage for all" or when she states "same-sex marriage obviously does not mean the end of civilisation," thus contradicting the official discourse of the Catholic Church. Rather, by publicly and explicitly contradicting the arguments of (even if, like anyone else, she owns the right to defend a militant point of view based on her political and religious convictions). The question I asked myself was whether a sociologist should put the religious group she studies on trial like this and deliver such a prediction. Personally, this is not how I see the position of the sociological analysis.

Willaime, 2014: 128, emphasis added

The conflicting views expressed by these two sociologists who both have a Christian background are probably partly based on the gendered dimension of the debate on same-sex marriage. They may also be explained by the different ability of the Catholic Church and the Lutheran-Reformed churches to manage internal diversity and critics. But beyond this difference of opinion, the kind of arguments put forward should elicit our attention. When Willaime regrets that a sociologist criticises the church she studies, he means sociologists can-not properly study a religion without having a kind of sympathy with it. The lack of loyalty to the institution studied (to which, in this case, the sociologist is supposed to belong) appears to him to be a methodological bias. The risk of such a perspective is that it may imply no critical approach to religion can be legitimate within the sociology of religion, especially when non-believing or non-practising sociologists develop it.

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill, More generally, according to Willaime,

The [Marxist-inspired] scheme has strongly influenced a part of sociology which is still dominated, in the name of "liberation," by the criticism of religion, and considers religions as traditions on the way to becoming totally obsolete. Such an approach is unable to overcome a disqualification of its object while intending to study it, and this obviously creates a bias in the analysis. Willaime, 2014: 115 By equating a critical perspective on religion with a militant approach that "disqualifies" the object of study, he minimises the inherently critical dimension of sociology. By redefining the sociology of religion, he tends, in fact, to disqualify sociologists who don't believe enough in "their" church -the church they study.

The debate on the beliefs of the sociologists of religion is an old one, as illustrated in the opposition between Durkheim and Réville at the start of the 20th century and its regular reappearance throughout the century (Desroche, 1968: 20;[START_REF] Wach | La sociologie et la religion[END_REF].

In 1980, a presentation given by Bourdieu at a conference of the French Association for Social Sciences of Religion revived this debate, despite meeting with strong resistance:

[page 53]

The question is not to know, as we often pretend, whether those who practise sociology of religion have faith or not, nor even whether they belong to the Church or not. Leaving aside the problem of faith in God, in the Church, and in all that the Church teaches and guarantees, it is about posing the problem of investment in the object, of adherence linked to a form of belonging, and of knowing what belief, understood in this sense, contributes to determine the relationship to the scientific object, to determine the investments in this object and the choice of this object. definition of axiological neutrality he defends implies sociologists should, at least in part, share the illusio of the religious institutions they study. In this perspective, the essential critical approach comes not from sociological methods but, more importantly, from religious faith: "Religions, by the very fact that they refer to an elsewhere, to one form or another of the beyond, invite us to decentre our gaze, to keep a critical distance from the world, the secular" (Willaime, 2014: 120).

Willaime goes on to defend a very positive view on the role of religion in society:

Religions today constitute sub-cultures, groups to identify with, providing landmarks and perspectives to the individual. They are meaningful and structuring minorities which offer fraternal communion and re-sources for hope in a global society dominated by instrumental and market relationships, and marked by concerns about the more or less radical relativisation of the foundational models of modern society.

Willaime, 2014: 126

The purpose becomes the rehabilitation of religion in the public space, with this expression of religious faith seen as a "chance for democracy... at a time when the democratic life is dominated by opinion and the sensational... and the political life faces globalisation and the emotional democracy of opinion" (Willaime, 2014: 126).

Willaime doesn't really specify in this text which particular religious expressions might be included in this vision of democratic life, even though he makes it clear that "all religious expressions are not equal" (Willaime, 2014: 124). Nevertheless, Christianity as a whole seems to be the main implicit reference underlying this defence of the legitimacy of religious public and political expression in ultramodern, postsecular and disenchanted societies.

[page 54]

This positive view of religion goes with a pessimistic view of social relationships (to be re-enchanted by religion) and democratic regimes (which religion needs to protect from their own excesses): "Against totalitarian drifts and human rights violations, Christianity, even if it has been slow to adopt the logic of the defence of human rights, also represents a resource of awareness and ethical mobilisation" (Willaime, 2014: 127).

While religions are, at least in part, defined as social organisations, it is as if they are located outside social and political life and, therefore, preserved from the ills affecting ultramodern societies. It is difficult to understand how religions presented as preserved -or even disconnected -from social dynamics could have an influence on these same social dynamics. Is it because religion is defined as "a symbolic activity which has its own consistency" (Willaime, 2014: 115-116) that it is protected from the harmful effects of social dynamics? The legitimacy of religious intervention in the public space is built on a strong bipolarisation of religion and society, making it difficult to under-stand how churches could become involved in society without losing their specificity.

It is not clear what kind of fieldwork and empirical data these general reflections draw on. But the perspective outlined by Willaime is evident: on the one hand, he makes a vibrant call for the greater visibility of and respect for a religious conservative stance in the public space and in politics, and on the other, he defends a sociology of religion as posited by sociologists who believe in "their" churchthe one to which they attend and/or study. Willaime also sees the reference to "an elsewhere" as the core of religion, and his definition of the social role of religion focuses on the ontological distance which separates it from the secular world. In such conditions, it seems quite complicated for non-believing sociologists who base their research work on empirical observation in this "secular world" to find the "good god behind the wardrobe," as Bensa writes (2006: 10) -an elsewhere supposedly more meaningful than the social relations observed through fieldwork.

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill,.

Conclusion

The debate on "marriage for all" and the stances adopted by Protestant churches and sociologists examined in this chapter show the kind of difficulties the sociology of religion faces today. The use of "anthropological" rather than biblical arguments in the religious field and the uncritical reprise of religious categories in the academic field comprise a double movement which jeopardises the sociology of religion's autonomy vis-à-vis religious discourses. The reprise

[page 55]

of Catholic arguments promotes an approach to social facts in religious terms, even as it reinforces the churches' strategy of presenting religious convictions as scientific truths. The fact that "anthropological" arguments mask biblical arguments reintroduces religious logics into a discipline, which is tempted to remain anchored in a restrictive definition of "religious sciences," thus distancing it from the critical debates of the more general social sciences. Invoking "anthropological evidence" can be interpreted as an attempt to escape the "rules of systematic discussion and general criticism" that should organise scientific exchanges [START_REF] Bourdieu | Sociologists of belief and beliefs of sociologists," translated by Véronique Altglas and Matthew Wood[END_REF].

The conclusion of Willaime's article in Transversalités can thus be read as an attempt to cut short any scientific discussion on same-sex marriage, by reducing the contribution of anthropology to a form of cultural relativism, while asserting the legitimacy of a moral religious norm:

Why should the fact of "being in conflict with all that anthropologists describe of the variability of familial and kinship patterns in time and space" be regarded as a problem? ... Because other models exist, it doesn't mean that they must be considered as valuable. And because a great diversity exists in time and space, it doesn't necessarily mean, unless we deliberately choose to support relativism, that we have to celebrate this diversity from a normative point of view.

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill,. Willaime, 2014: 132 In an article where he questions the sociological relevance of the "religious" object, Michel remarks that even in widely secularised societies, religion retains a specific status. More than other objects, religion inspires attitudes of reverence and conformism, most notably because, as Proust wrote in Swann's Way, "The facts of life do not penetrate to the sphere in which our beliefs are cherished; as it was not them that engendered those beliefs, so they are powerless to destroy them; they can aim at them continual blows of contradiction and disproof without weakening them." Michel, 2003: 161 As sociology values the objective observation of facts, sociologists are necessarily confronted by the effects of beliefs when they turn their attention to religious facts. This should inspire them to elaborate a critical analytical frame-work for the study of subjective religious experiences and to carefully reflect on their own relationship with religious truth.

[page 56] This well-known difficulty associated with the scientific study of religion has been increased by new interactions between the scientific, religious and political fields, as implied in the debate on the concept of post-secular society (Stavo-Debauge and Roca I Escoda, 2016). The idea that the symbolic resources offered by religions make a specific contribution to public debate paves the way for new kinds of religious mobilisations and places the social sciences of religion in an awkward situation. While they can have an interest in the social revaluation of their object of study in public debate, such sociologists can't uncritically accept the strategies of religious actors who seek the recognition of a particular legitimacy based on a transcendent truth.

Gwendoline Malogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study of Religion,Leiden,Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », in Véronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill, the Church, she gives the impression of personally breaking with her object of study, so that her point of view remains radically external and sociological. Paradoxically, this exteriority is precisely what Willaime considers at odds with the epistemological and methodological requirements of sociology:According to Danièle Hervieu-Léger, the opposition of the Catholic Church to same-sex marriage represents -and this is not purely hypothetical... -"the path towards the end of Catholicism in France." Nothing less! I was struck by the predictive character of my colleague's stance

	[page 52]

  It seems Willaime interprets the criticism levelled by Hervieu-Léger against the Catholic Church as a lack of "adherence linked to a form of belonging," while the GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,

	Bourdieu, 2010: 2-3

  GwendolineMalogne-Fer, 2018. « Protestant Churches and same sex Marriage : "theological" criteria and sociological approach », inVéronique ALTGLAS et Matthew WOOD (dir.), Bringing Back the Social Into the Study ofReligion, Leiden, Brill,


Alexandre Reiffers-Masson

Thierry Chonavel

Yezekael Hayel

Estimating Fiedler value on large networks based on random walk observations

Keywords: Fiedler value, stochastic approximation, random walk based observations

In this paper, we describe an iterative scheme which is able to estimate the Fiedler value of a network when the topology is initially unknown. The only available information is the one obtained through a random walk process over the network. Our algorithm is based on the Rayleigh quotient optimization problem and the theory of stochastic approximation. We explain the different tools used to construct our algorithm and we describe our iterative scheme. Finally, we illustrate its performance through a numerical study.

I. INTRODUCTION

Understanding the connectivity of large scale networks is essential for the study of their performance. Especially, in the context of overlay networks [START_REF] Gkantsidis | Towards topology aware networks[END_REF], [START_REF] Di | Distributed estimation and control of algebraic connectivity over random graphs[END_REF], ad-hoc wireless networks [START_REF] Barbarossa | Distributed detection and estimation in wireless sensor networks[END_REF] and consensus/swarming algorithms for distributed networks [START_REF] Di | Bio-inspired decentralized radio access based on swarming mechanisms over adaptive networks[END_REF], such understanding is critical. In overlay networks, connectivity between the nodes is crucial for a better load balancing and in general for a better quality of service. A classical example of such networks are P2P networks used for file sharing and content distribution application [START_REF] Gkantsidis | Towards topology aware networks[END_REF], [START_REF] Di | Distributed estimation and control of algebraic connectivity over random graphs[END_REF]. In distributed systems (distributed ledgers for instance) it is well known that the connectivity of the network is crucial to speed-up the convergence of distributed algorithms running over the network (see [START_REF] Di | Bio-inspired decentralized radio access based on swarming mechanisms over adaptive networks[END_REF] for an application to consensus algorithms). Spectral graph theory [START_REF] Rk | Spectral graph theory[END_REF] is a classical approach to study the connectivity of a network using graph analysis. The eigenvalue decomposition of a Laplacian matrix provides powerful heuristics for solving the RatioCut problem [START_REF] Von | A tutorial on spectral clustering[END_REF] and therefore can be also used to solve clustering problem in networks [START_REF] Von | A tutorial on spectral clustering[END_REF], for critical links identifications [START_REF] Gkantsidis | Towards topology aware networks[END_REF], [START_REF] Bertrand | Seeing the bigger picture: How nodes can learn their place within a complex ad hoc network topology[END_REF] and to design strongly connected networks [START_REF] Ghosh | Growing well-connected graphs[END_REF].

Set-up: One of the main quantities of interested in such problems is the Fiedler value [START_REF] Fiedler | Algebraic connectivity of graphs[END_REF] (and the Fielder vector associated to it). The Fiedler value is defined as the non-trivial second smallest eigenvalue of the Laplacian matrix and the Fiedler vector is the associated eigenvector. In this paper, we answer the following question about this connectivity metric: Is it possible to learn efficiently the Fiedler value and the Fielder vector of a large network by only using the information gathered by a random walk evolving on the graph? In our setup, we assume that it is not possible to have an initial full observation of the global network, and the only available information is the one obtained through a random walk process over the graph. Random walk is a standard scheme used to sample nodes and edges in a network (see [START_REF] Das Sarma | Distributed random walks[END_REF] and the references therein). Moreover, we would like to derive efficient algorithms, which are able to manage efficiently the memory/storage of the devices used to run our algorithms.

Contributions: The main contribution of the paper is the design of an efficient iterative scheme which converges to the Fiedler vector. The information used to learn the Fiedler vector is obtained using a random walk, which discovers, step-by-step the topology of the network. The convergence of the algorithm can be deduced from recent results in stochastic approximation [START_REF] Suhail | Stochastic approximation on riemannian manifolds[END_REF]. To be able to use the theory of stochastic approximation we show the equivalence between our problem and the problem of computing the leading eigenvector of a particular subset of matrices.

Related works: Our problem belongs to the class of works related to online/streaming computation of eigenvectors in a large-scale setting. First, solving large-scale eigenvalue problems using deterministic methods is a well studied problem, and a summary of the different methods can be found in [START_REF] Arbenz | Lecture notes on solving large scale eigenvalue problems[END_REF]. In these lecture notes, the streaming context with stochastic feedbacks is not discussed. Then the different works on the online leading eigenvector problem [START_REF] Garber | Online learning of eigenvectors[END_REF] and online/streaming principal component analysis [START_REF] Mitliagkas | Memory limited, streaming pca[END_REF], [START_REF] Jain | Streaming pca: Matching matrix bernstein and near-optimal finite sample guarantees for oja's algorithm[END_REF] are also close to our set-up. These algorithms are more oriented towards Principal Component Analysis than the computation of Fiedler vectors. Also, our problem can find some similarities with the online distributed computation of leading eigenvectors [START_REF] Barbarossa | Distributed detection and estimation in wireless sensor networks[END_REF], [START_REF] Bertrand | Distributed computation of the fiedler vector with application to topology inference in ad hoc networks[END_REF], [START_REF] Avrachenkov | Distributed spectral decomposition in networks by complex diffusion and quantum random walk[END_REF]. In these papers, the authors propose distributed algorithms to track the Fiedler value of a stochastic network based on an extension of the power iteration algorithm. A recent survey of these different methods can be found in [START_REF] Avrachenkov | Distributed spectral decomposition in networks by complex diffusion and quantum random walk[END_REF]. Another approach, similar to ours, is the one proposed in [START_REF] Vivek S Borkar | Asynchronous gossip for averaging and spectral ranking[END_REF], [START_REF] Borkar | Oja's algorithm for graph clustering, markov spectral decomposition, and risk sensitive control[END_REF] based on stochastic approximation theory to compute particular eigenvectors. In [START_REF] Vivek S Borkar | Asynchronous gossip for averaging and spectral ranking[END_REF], the authors derive a stochastic approximation variant of the gossip algorithm, applied to the computation of the leading eigenvector of a nonnegative matrix. Along the same lines, the authors of [START_REF] Borkar | Oja's algorithm for graph clustering, markov spectral decomposition, and risk sensitive control[END_REF] proposed a stochastic approximation algorithms inspired from Oja's algorithm. Different extensions are suggested, especially related to Spectral graph clustering. The main difference between our work and the papers cited above is the fact that we observe the network through a random walk exploration process. This kind of observations cannot be handled in the same way as the ones described in the previous works, therefore, forced us to come up with a different approach.

Finally, to the best of our knowledge, we have only identified two papers [START_REF] Doshi | Fiedler vector approximation via interacting random walks[END_REF], [START_REF] Lucińska | Graph clustering using early-stopped random walks[END_REF] that use random walks to compute the Fiedler vector. In [START_REF] Doshi | Fiedler vector approximation via interacting random walks[END_REF], a stochastic process that converges to the Fiedler vector is suggested. Its construction is based on a set of interacting random walks. We also mention that this paper contains, from the best of our knowledge, an up to date survey of the different algorithms that compute the Fiedler value and the Fiedler vector. Their approach is different than ours, in their case they use 2n competitive random walks, where n is the number of nodes. In our case we only need information gathered by one random walk. In [START_REF] Lucińska | Graph clustering using early-stopped random walks[END_REF], an algorithm is derived from a random walk and the authors proved that the algorithm converges to a local approximation of the Fiedler vector. In our case, we converge exactly to the Fiedler vector.

II. PROBLEM DEFINITION

In this paper, we are interested in the following question: Is it possible, using only random walk based observations of a graph topology, to estimate the Fiedler value and the associated eigenvectors of an undirected graph? In this section, we will first provide basic definitions and then we will describe our observation model. Finally, we will formulate the estimation problem that we want to solve.

Let G := (V, E) be an undirected unweighted graph, where V := {1, . . . , n} is the set of nodes and E is the set of edges. The adjacency matrix of G is the matrix A := [a ij ] i,j=1:n , where the ij-entry of A is given by a ij := 1 (i,j)∈E . The neighborhood of a node i is the set N (i) := {j ∈ V | a ij = 1}. We assume that for all node i, a ii = 0. The degree of a node i is given by deg(i) := n i =1 a ii and the degree matrix is equal to D := [δ ij deg(i)] i,j=1:n where δ •• is the Kronecker delta operator. We also define the Laplacian matrix as L := D -A.

The spectrum of the Laplacian matrix will be used to measure the connectivity of the graph. More precisely, we are interested in the estimation of the second smallest eigenvalue of L denoted by λ * . We assume that the graph is connected and therefore λ * is strictly positive [START_REF] Van Mieghem | Graph spectra for complex networks[END_REF]. λ * is called the algebraic connectivity (or Fiedler value) and we denote by v * the associated eigenvector. In this work, a discrete time random walk X(k) is used to explore the graph step-by-step. At each time step k, the stochastic process X is in vertex i k ∈ V and the neighboring nodes N (i k ) are observable. Then, after the k-th step of the random walk process, the σ-algebra H(k) is given by H(k) := {i 1 , N (i 1 ), . . . , i k , N (i k )} where for all k ≥ 0 the node i k is sampled following the law π 0 P k with P = D -1 A and with π 0 being the initial distribution of the random walk process. The goal of this work is to derive, for every k ≥ 0, a consistent estimator of λ * (resp. v * ) denoted by λ(H(k)) (resp. v(H(k))).

III. COMPUTATION OF THE FIELDER VALUE AS A LEADING EIGENVECTOR PROBLEM

To be able to derive an efficient estimator of the Fiedler value and the Fiedler vector, we consider a dynamical system approach. We first recall that the leading eigenvalue and the associated eigenvector is the solution of an optimization problem over the unit sphere. Then we describe the ordinary differential equations which converges to the solution of this optimization problem. We will next show that the computation of the Fiedler vector can be reformulated as a a leading eigenvector problem. Finally, we describe a stochastic approximation which is tracking the O.D.E. and therefore converges to the Fiedler vector.

A. Rayleigh quotient optimization problem

For any matrix W , the Rayleigh quotient r W : R n → R is the smooth function given by:

r W (x) = x T W x x -2 .
(1)

Let U n-1 := {x ∈ R n | x = 1}
be the unit sphere in R n . Then by considering the restriction of the Rayleigh quotient on the unit sphere U n-1 , the eigenvalues of a real symmetric matrix are characterized using the Rayleigh Quotient [START_REF] Golub | Matrix Computations[END_REF] and in particular:

λ min (W ) = min x∈U n-1 r W (x). Therefore, the largest eigenvalue of W can be obtained by solving an optimization problem on the manifold U n-1 . One approach to solve this optimization problem is to apply the gradient flow in order to search for the dominant eigenvector of W . This leads to the well known Rayleight Quotient Gradient Flow [START_REF] Helmke | Optimization and dynamical systems[END_REF] defined as the following ordinary differential equations:

ẋ = -(W -r W (x)I n )x, x(0) = x 0 ∈ U n-1 , . (2)
It can be proved that under specific conditions (λ min < λ 2 , W is a symmetric matrix) that x converges to the eigenvector associated to the eigenvalue λ min (see [START_REF] Helmke | Optimization and dynamical systems[END_REF]).

B. Problem transformation

In this section we explain how to transform the problem of tracking the second eigenvector (in increasing order of the eigenvalues) of the Laplacian matrix L into the one of tracking the first eigenvector of a modified positive matrix.

Let {λ i } i=1:n represent the eigenvalues of L, with λ 1 < λ 2 < λ 3 ≤ . . . ≤ λ n , and v 1 , . . . , v n the corresponding unit norm eigenvectors. Recall that λ 1 = 0 and v 1 = 1 √ n 1I, with 1I = [1, . . . , 1] T . Then, tracking v 2 can be done equivalently by tracking the eigenvector associated with the largest eigenvalue of the matrix I -1 n 1I1I T -L for < λ -1 n . This method is called deflation and it is proposed in [START_REF] Barbarossa | Distributed detection and estimation in wireless sensor networks[END_REF]. When λ n is large this could lead to numerical issues. Alternatively, we propose to work with the restriction of L to the space orthogonal to vector 1I. Clearly a basis of this subspace is given by the columns of the matrix M ∈ R n×n-1 with M ii = 1, M i+1,i = -1 and other entries equal to 0. Proposition 1. Let denote by λ 1 < λ 2 < λ 3 ≤ . . . ≤ λ n the eigenvalues of a Laplacian matrix L and v 1 , . . . , v n corresponding unit norm eigenvectors. Then, we consider the following matrix:

S = Q T LQ, with Q = [q 1 , . . . , q n-1 ] ∈ R n×n-1 and q k = 1 k(k + 1) [1, . . . , 1, -k, 0, . . . , 0] T , (k entries are equal to k(k + 1) -1
). The eigenvalues of matrix S are λ 2 < λ 3 ≤ . . . ≤ λ n and corresponding unit norm eigenvectors

Q T v 2 , . . . , Q T v n .
Proof The columns of Q are unit norm vectors, orthogonal to 1I and to each other: q i ⊥ q j for i = j. Let L = V ΛV T denote the EVD of S, where Λ = diag(λ 1 , λ 2 . . . λ n ). Then, we have

Q T V = [0, Q T v 2 , . . . , Q T v n ].

Since in addition the

matrix Q = [1I | Q] is orthogonal, for 1 < i, j ≤ n we have (Q T v i ) T (Q T v j ) = ( QT v i ) T ( QT v j ) = v T i Q QT v j = δ i,j , (3)
where δ i,j = 1 if i = j and 0 otherwise. Then, the EVD of

S is S = k=2:p λ k (Q T v k )(Q T v k ) T .
In the next section, based on this space reduction, we propose a method to track efficiently the leading eigenvector of matrix S when first the global Laplacian matrix is known and second, when observing one node at a time through a random walk process. The particular choice that we made for matrix transform Q is based on algorithmic complexity considerations, which we will discussed at the end of the next section.

C. Tracking the leading eigenvector 1) Known Laplacian case: The leading eigenvector Q T v 2 of matrix S can be computed iteratively with a gradient descent approach applied to the Rayleigh quotient function given by Equation (1), or equivalently to the minimum eigenvalue approximation x T Sx with x constrained to be in the unit length vector manifold. Up to second order terms both approaches are equivalent and yield to the following update for x:

x k+1 = x k -k (S -r S (x k ))x k , x k+1 = x k+1 x k+1 ,
where k is a decreasing step-size, such that k k = +∞ and k

2 k < +∞ (typically, k = 1 k+1
). This latter method assumes complete knowledge of matrix L which implies the computation of matrix S following proposition 1. When the network is large or for particular reasons, it may not be possible to have such information. In the next section, we propose an iterative method when we have a partial observation of the matrix based on a random walk exploration of the network.

2) Random walk case: At each instant k, the random walk X(k) is located at vertex i k and the neighboring nodes N (i k ) are observable. At instants k, line i k of L is observed and denoted by l i k . Then matrix L is replaced by the following instantaneous matrix L k = (0 l i k 0)

T , where only line i k of L k is nonzero. Then, the instantaneous version of matrix S is

S k = Q T L k Q.
It should be noted that the probability to observe the nodes i k when moving on the graph according to a random walk is not uniform. Thus E[S k ] = S and this should be accounted to avoid asymptotically biased estimates. Indeed,

E[S k ] = E[E[S k | i k ]] = i=1:n π i,k S k = 1 n S
where π k = [π 1,k , . . . , π n,k ] T is the instantaneous distribution of the Markov process X. Unfortunately, as the topology is unknown, neither instantaneous nor asymptotic distributions of this Markov process are known. Then instantaneous estimate p i,k of π i,k is updated at instant k in our procedure as we explore the network by the random walk. Let p 0 ∈ R n + , with 1I T p 0 = 1, y 0 ∈ U n-1 and λ 0 ∈ R + denote initial values. Then, at time k + 1 (k ≥ 0) the random walk X visits node i k+1 and updates are defined as follows:

p k+1 = p k + p,k (e i k+1 -p k ), x k+1 = Qy k , y k+1 = y k -k [Q T ] i k+1 l i k+1 x k+1 n p i k+1 ,k+1 -λ k y k , y k+1 = y k+1 -1 y k+1 , λ k+1 = λ k + λ,k [x k+1 ] i k+1 l i k+1 x k+1 n p i k+1 ,k+1 -λ k ,
where the step-size sequences { p,k }, { k } and { λ,k } should be such that

k p,k = k k = k λ,k = +∞, k 2 p,k + 2 k + 2 λ,k < +∞,
and satisfy also the condition: k p,k → 0, k λ,k → 0. These conditions are satisfied considering k = 1 1+k log(1+k) and λ,k = p,k = 1 k+1 . The latest conditions on the step-size sequences will ensure that the evolution of the sequence y k is much slower that the evolution of the sequences p k and λ k . Therefore it implies that: (1) p k have converged to the stationary distribution of the Markov process X; (2) the iterative procedure can be viewed as a noisy discretization of the following limiting system of a two-time scale ordinary differential equations (see ch.6 in [START_REF] Vivek | Stochastic approximation: a dynamical systems viewpoint[END_REF]):

ẏ = -(Q T LQy -λy) and λ = y T Q T LQy.
This O.D.E. is equivalent to the following Rayleight Quotient Flow (2):

ẏ = -(Q T LQ -y T Q T LQy)y = -(S -y T Sy)y.
This fact will ensure the convergence of our scheme using the theory of stochastic approximation on manifold [START_REF] Suhail | Stochastic approximation on riemannian manifolds[END_REF] to the leading eigenvector w 2 corresponding to the highest eigenvalue λ 2 of the matrix S. Then, the Fiedler vector v 2 is obtained as

v 2 = Q T w 2 .
Remark about the complexity per iteration: Simplifications in expressions involving S k = Q T L i k Q where L i k only has one nonzero line and lead to linear complexity. In addition, although computations of the form Qy are involved in the algorithm, this does not yields high numerical burden due to the particular structure of matrix Q. Indeed, Q only has two distinct nonzero terms in each of its columns. Thus computation of Qy only involves about 2n product terms, hence linear calculation complexity.

IV. NUMERICAL ILLUSTRATIONS

In our simulations, for known L of size n = 50 we choose k = 10 2 /(k log(k +1)+1). The ratio of the smallest eigenvalues of S is about 0.93 and after 300 iterations norm error is about 10 -11 .

For the same L, when it is unknown and we use a random walk, stepsizes are set to k = 10 2 /(k log(k + 1) + 1), p,k = 1/(k + 1) and λ,k = 10 -2 /(k + 1). After 10 6 iterations the MSE E[ v 2 -x 2 ], computed on the last half of estimates, is about 10 -2 .

One reason for the increase of the number of iterations compared to the case where L is known at each instant is because only a fraction 1/n of the matrix L is observed at each time in the case of a random walk on the graph. From earlier discussion it also appears that computational complexity of loops is also roughly reduced by a factor proportional to n in the case of line observations.

V. CONCLUSIONS AND PERSPECTIVES

In this paper, we have described an iterative scheme that converges to the Fiedler value and the Fiedler vector of a network using information gathered by a random walk. The follow-up of this work will address (i) the characterization of the exact rate of convergence of this algorithm, (ii) the extension of this algorithm to a set-up where we suppose that the number of nodes is not known in advance and (iii) the extension of this work to a set-up where the network is evolving over time.

Fig. 1 .

 1 Fig. 1. Tracking with known L.

Fig. 2 .

 2 Fig. 2. Tracking with random walk exploration.