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Catherine Goldstein: The Question of Long-Term Histories and Mathematical Identity

Jenny Boucard
Jemma Lorenat

Communicated by Notices Associate Editor Laura Turner

1. Introduction: Configurations in the History of Mathematics

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“Mathematics is the art of giving the same name to different things.” With this quote from Henri Poincaré’s The Future of Mathematics (1908), Catherine Goldstein opened the plenary conference on the history of mathematics for the 2018 International Congress of Mathematicians.⁠Footnote1 It underlines two important points. First, at the beginning of the twentieth century, this question of absorbing mathematical objects under the same name—and the same general concept—was central to the gradual establishment of a theory of structures. Second, this point of view also played a important part in the construction of a retrospective history of mathematics, by looking to the (sometimes very distant) past for traces of the objects of current mathematics Gol19a. Goldstein is one of several historians who, particularly since the 1980s, have been working on a new history of mathematics, integrating developments in the history of science and technology while taking serious account of the specific nature of mathematics. Two interrelated motifs run through Goldstein’s research: the identity of mathematics over time and the implications of long-term histories.

1

Both authors are former doctoral students of Catherine Goldstein. We would like to thank Catherine for the opportunity to write this article and her support in its research. Thank you also to Laura Turner for suggesting this piece and for her editorial guidance.

Currently research director (directrice de recherche) at the Institut de mathématiques de Jussieu-Paris Rive Gauche in Paris, Goldstein began her research career in pure mathematics. She was one of the first French students to complete a thesis in number theory under the supervision of John Coates at Université Paris-Sud.⁠Footnote2 She defended her doctoral thesis on -adic -functions in 1981, just as she obtained a tenured position as a researcher in the Centre national de la recherche scientifique at the Mathematics Laboratory of the Université Paris-Sud. For historians of science, the 1980s “crackled with debate”—as historian Lorraine Daston recounts—invigorated by ethnographic and sociological approaches, feminist theory, and political movements Das09, 803. Interested in the humanities and social sciences, Goldstein seized the opportunity to attend a series of lectures on the history, sociology, and philosophy of science organized by the philosopher Michel Serres, along with other academics such as Bernadette Bensaude-Vincent, Bruno Latour, and Isabelle Stengers. This seminar became a collective book, A History of Scientific Thought: Elements of a History of Science (published in 1989 in French, translated into English in 1995), for which Goldstein wrote two chapters. This experience was a turning point for her.

2

Coates, who had studied at the École Normale Supérieure, held positions in the United States and Australia prior to joining the faculty at Paris-Sud.

The scholars involved in this collective project desired to go beyond the so-called internalist (“that of concepts and results” Gol19b, 1) versus externalist (“that of institutions or scientific politics”) approach to history by showing the inextricable interweaving of “internal contents which are exclusively scientific and external conditions which are exclusively social” Ser95, 14 in the history of science and technology. Goldstein, along with other historians of mathematics, was convinced that the specific nature of mathematics (as compared to the natural sciences) called for the use of appropriate historical methods. The aim of this paper is to present some of the methods that Goldstein developed to obtain new results in the history of mathematics.

In her introduction to the second volume of Ernest Coumet’s Collected Works, Goldstein recalls:

For many of us, it seemed that it was time to move on [from the internalist/externalist debates]: that it was appropriate to approach the texts of the past as closely as possible, because it was in their technical details, in their organisation, both material and intellectual, at the bend of an unexpected word or the unexpected meaning of a word, an imported notation, an approximation of classifications or formulations, that the traces of a wider culture, of a hitherto unsuspected collective, could be caught; and, in the opposite direction, that it was necessary to examine how these traces were imprinted in symbolism, in a particular type of exercise, in priorities selecting the direction of the development of mathematics. Everything here remained to be done. Gol19a, p. 11, to verif.

This program is already apparent in the two chapters written by Goldstein for Elements of a History of Science. The first, titled “Stories of the Circle,” reveals the unexpected heterogeneity of seemingly uncontroversial pieces of mathematics. Here, the plural of “stories” is essential: it underlines the multiplicity of historical lines, as opposed to the idea that there is one single, linear history—the true history—of mathematical concepts. In this chapter, Goldstein proceeds from Mesopotamia and Egypt to Hilbert’s Foundations of Geometry identifying multiple precise ways that the circle has manifested within mathematics. The circle in the Śulvasūtra (first millennium BCE, India) is introduced with the constructive and numerical procedure of converting a square into a circle of equal area, the circle in Euclid’s Elements (third century BCE, Greece) is a plane figure like the straight line without numerical values, while the algebraic approach to the circle (for instance, in the work of Descartes in the seventeenth century) identifies it “with the conics, whose equations are of the same type” Gol95a, pp. 182–184. Such reorganization has profound consequences. To take one among many, the circle as a conic section leads to the introduction of circular points at infinity.

The second chapter deals with the question of “Working with Numbers in the Seventeenth and Nineteenth Centuries,” going from what Goldstein calls Homo ludens to Homo faber. Starting from the “same” mathematical statement in the seventeenth and nineteenth centuries—what is now known as Fermat’s last theorem, namely the impossibility of the equation in nonzero integers—Goldstein shows how different were the epistemological, mathematical, and social contexts involved. Number theory is a particularly good case for historical investigation:

The choice of mathematics and more particularly number theory as a field of study is in this respect more interesting for being an extreme case: among the sciences, mathematics, and the theory of numbers in particular, retains, rightly or wrongly, a reputation for splendid and unchanging isolation which discourages over-hasty explanation of its professional development. Gol95c, p.345

Number theory, both as an “extreme case” and as a familiar field, has been Goldstein’s topic of historical investigation par excellence since the 1980s, even as her research has also encompassed other themes in the history of mathematics. The two chapters published in Ser95, investigations of a mathematical object and a mathematical domain over a long period of time, also reflect two central questions in Goldstein’s work: how to construct long-term histories of mathematics and to how to analyze the identity of mathematical objects. These issues are also fundamental in her keynote address for the 2018 International Congress of Mathematicians, titled “Long-Term History and Ephemeral Configurations.” Goldstein motivates her choice of the history of Hermitian forms in the second half of the nineteenth century as “a minimal example: the concrete case of a rather technical and apparently stable concept” Gol19b, 493.⁠Footnote3 She based her presentation on the sociological concept of configuration, which she first borrowed and adapted from Norbert Elias in her early work in the 1990s. Elias introduced the concept of configurations in the early twentieth century to study interdependent networks between individuals. Goldstein first adapted configurations of “knowledge, objectives, and priorities” to inform a social history of texts Gol95b, p. 9. She describes a configuration as the interaction of texts and persons in which “mathematics weaves together objects, techniques, signs of various kinds, justifications, professional lifestyles, epistemic ideas” Gol19b, 491. Marking these configurations as ephemeral does not restrict the history of mathematics to local studies but rather is a call to investigate “the various ways in which they [local studies] are, or not, connected” Gol19b, 511 thereby forming long-term histories of mathematics.

3

Goldstein defines Hermitian forms in this address as “simply an expression of the type , with coefficients in , such that (here, the bar designates the complex conjugation); in particular, the diagonal coefficients are real numbers.”

While the details are crucial to fully appreciating the value of this scholarship, here we focus on three examples of Goldstein’s research to provide a sense of how methodological frameworks can address questions of long-term histories and identity through methods and concepts—such as configurations—adapted from the human and social sciences. For a deeper investigation, the reader is recommended to read the original texts as cited in the bibliography.

2. Mathematical Versus Historical Readings: One Theorem and Its Multiple Identities

In the conclusion to Un théorème de Fermat et ses lecteurs (A Fermat Theorem and Its Readers), Goldstein traces the history of her book back to 1988 and “a question from a colleague, Norbert Schappacher, about the identity of Fermat’s and Frenicle’s proofs” Gol95b, 177. The theorem in question—that the area of a right triangle with rational sides is not a square—was proved by Pierre de Fermat and Frenicle de Bessy at around the same time in the seventeenth century. At first Goldstein supposed that the proofs were “similar”: Fermat and Frenicle had similar number-theoretic knowledge, they wrote without algebraic symbolism, used the method now known as infinite descent. But then she began to have doubts, not just about the texts, but about the stability of similarity.

To address this notion of stability and answer the central question of her book—what a “historical description of a mathematical theorem” can be—Goldstein identifies two categories of readings: “mathematical readings aim to find problems to solve, methods to use, in brief, a source of inspiration to produce new mathematics”; whereas historical readings (perhaps by the same people!) “search in the text for information about the activity of past mathematics and its development in order to produce historical reflections and commentaries” Gol95b, 6–7. She is then able to “observe at close hand the concrete reading and working practices” of mathematicians and historians Gol95b, 16. In presenting many mathematical and historical readings of the famous theorem, Goldstein demonstrates how perceptions of mathematics are shaped by prior mathematical knowledge. She directs attention to the slippery question of identity (which becomes a problem to be studied rather than a self-evident given), the plurality of contexts, the inadequacy of historiographical dichotomies.

In spite of its geometrical appearance and apparent specificity, the theorem, considered by Fermat and Frenicle as about numbers, can be seen as a “cornerstone (…) of the study of quadratic forms (…and) of the arithmetic of elliptic curves” Gol95b, 5. This position has important effects for both types of reading. For example, Goldstein analyzes how the rewriting of Fermat’s and Frenicle’s proofs in algebraic symbolism erases specifities of each text. An algebraic rewriting requires the clarification of initially ambiguous points, such as the relationships between elements introduced as a reasoning progresses, and standardizes initially differentiated terminologies Gol95b, 88–89.

Goldstein compares the two texts “point by point” Gol95b, p.76 and shows how a nonalgebraic reading reveals the differences in Fermat’s and Frenicle’s approaches. Both proofs rely on a method of infinite descent, but each author had different conceptions of the process. Frenicle considered the decrease in size of the triangles, the next is smaller than the last “diminishing always” until forced to stop at 1. Fermat, by contrast, first asserted that there can only be a finite number of whole numbers smaller than a given number without reference to the triangles.⁠Footnote4 For the study of right triangles, Frenicle’s approach is “more direct, shorter, more economical” while that of Fermat is “richer in varied connections with other problems” Gol95b, 80.

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As a further example of multiplicities of readings, Goldstein notes that Fermat’s use of infinite descent has been read as anticipating the modern height function Gol95b, 11.

Goldstein also discusses a historical reading of Fermat’s theorem based on elliptic curves, showing how this approach “gives a unified meaning to a corpus of apparently heterogeneous statements” Gol95b, 100. Goldstein notes that this second kind of reading is more controversial because the concepts involved are a priori more distant than the content of the original texts. But she highlights that all readings she studied are “strictly speaking anachronistic” Gol95b, 106. More subtle instances of anachronism underscore the fragility of identity of mathematics. As a case in point, neither Frenicle nor later historical readings provide any justification that “if a product of two prime numbers together is a square, each of the two numbers must be so.” Does this assertion point to the mathematical knowledge of the seventeenth century or is it a feature of the style of his Traité des Triangles Rectangles en Nombres? Goldstein asserts that ignoring the lack of justification or replacing the result would be to tacitly replace “the mathematical knowledge of the original readership by another” Gol95b, 59.

Evaluating a text depends on the configuration in which it is read. That is, “the similarity between two mathematical texts, whether contemporary or distant in time, has no absolute significance” Gol95b, p.178. These configurations may articulate “a general conception of the history of mathematics and of the subject to be dealt with (…); technical knowledge; a vision of the state of the mathematical field concerned, in the past and in the present, in particular problems or texts whose connection with the one studied may be relevant; an individual reading of the texts; a chronology” Gol95b, 108. In particular, the construction of a long-term historical narrative depends on the situation of the historian and their choices accordingly. It is in this sense that “the so-called long-term histories of mathematics are local histories” Gol95b, 111.

3. Men, Letters, and Numbers: A Micro-Social Analysis to Solve Fermat’s Paradox

You sent me 360, whose aliquots⁠Footnote5 are the same number as 9 to 4, and I’m sending you 2016, which has the same property. —Letter from Fermat to Mersenne, February 20, 1639

5

The aliquot parts of a number are the divisors of that number, except itself. In the case of Fermat’s example here, the aliquot parts of 360 are: 1, 2, 180, 3, 120, 4, 90, 5, 72, 6, 60, 8, 45, 9, 40, 10, 36, 12, 30, 15, 24, 18, 20. Their sum is 810. We then have: .

In his history of mathematics “from Ancient to Modern Times,” Morris Kline states that “Fermat’s work in the theory of numbers determined the direction of the work in this area until Gauss made his contributions.” He then claims that Fermat “had great intuition, but it is unlikely that he had proofs for all of his affirmations” Kli72, 274–275. More generally, contrasting biographical accounts on Fermat bring to light an apparent paradox: sometimes presented as a theorist, his correspondence on number theory reveals a number of particular statements, without proofs and where the methods are very rarely made explicit. This combination led one of his biographers, Michael S. Mahoney, to describe him as a “problem-solver.”⁠Footnote6 The quotation in the epigraph to this section is in fact a typical example of Fermat’s exchanges on number theory in his letters.

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See Gol09, 25–27. Goldstein introduces her paper with a paradoxical fictional biogaphy of Fermat constructed from pieces taken from a dozen biographies and writings by the mathematician to underline this apparent paradoxal aspect of Fermat’s historical persona, among others. Note that what follows concerns Fermat’s number theory, but many of these results are also valid for his work in geometry and probability.

For Goldstein, two pitfalls explain the paradox that seems to punctuate Fermat’s biography: a too narrow focus on Fermat and neglect of the fact that certain characteristics of his work are not specific to him but are, on the contrary, characteristic of the environment in which he evolved. Our current knowledge of Fermat’s work on number theory is based on the version annotated by Fermat of Bachet’s edition of Diophantus’s Arithmetica, published by Fermat’s son in 1670 as well as a collection of letters, handwritten and partially published between the seventeenth and twentieth centuries, and mostly exchanged in the context of Marin Mersenne’s network. Fermat was then a wealthy magistrate who worked in Toulouse and Castres. He was in contact with the learned circles of Bordeaux and Paris, and corresponded with Mersenne, René Descartes, Bernard Frenicle de Bessy, Blaise, Pascal, and Gilles Personne de Roberval, among others.⁠Footnote7 Here again, studying the configuration in which Fermat’s texts were produced allows the resolution of these apparent biographical contradictions. Goldstein engages in a micro-analysis of Mersenne’s correspondence as a “specific social space for mathematics” Gol09, 41, studying letters as “both social links and texts” Gol13, 254 and thereby underlying the interdependance between mathematical content and “paratextual and social features.”

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Mersenne organized, between 1617 and his death in 1648, a network of mathematicians, first with meetings in Paris, then in the form of a network of correspondents linking mathematical amateurs in French and European cities. At least 2,000 letters were exchanged between 180 correspondents between 1635 and 1648 in this “all-mathematical academy.” At a time when mathematical journals did not yet exist, this network was fundamental to mathematical communication, especially for those who did not live in Paris or near other practitioners in capital cities.

Several important results are obtained by this socio-historical approach of mathematical texts. The specific form of exchanges on number theory in Mersenne’s correspondence—where sharing challenges and particular problems prevail—can be explained: first, the different participants in letter networks do not usually know each other and this allows the exchange of mathematics without disclosing methods before a relationship of trust is established. It also has to be recalled that most of the correspondents were amateurs in mathematics, in the sense that they earn their living from another profession, which left them little time to work on questions of numbers—numerous correspondents, including Fermat, Descartes, and De Beaune, explicitly complained about a lack of time. Secondly, this form enabled assessing the difficulty of a question and the value of the solution method without making it explicit. A lexicographical analysis of the network of letters (in which the word “method” appears very frequently—“several dozen occurrences in Fermat’s correspondence” Gol95b, 135) demonstrates the strategic intention in the statement of problems, which might appear disparate at first glance.

A global study of Mersenne’s correspondence does not only make it possible to catch the collective rules in this epistolary academy; it also makes it possible to distinguish between what is specific to an individual and what comes under the norms and values constituting a collective. First, the fact that Fermat exchanged particular number problems confirms that he had a thorough understanding of the collective norms of Mersenne’s correspondence. Then, Fermat occupies a singular place. He was the only one who mastered algebra and arithmetic and he was very familiar with the workings of the Mersenne academy. He transformed his statements so that they were in the desired form and adapted his questions to his readership: if he sent a problem to an algebraist like Descartes, for example, he made sure that the question could not be solved by algebra, whereas his questions for an arithmetician like Frenicle could only be solved by algebra. This gave him a mathematical superiority even if he was in a marginal geographical position and it allowed him to prove (to others but also to himself) that his methods were more efficient than algebra or the arithmetical methods used by Frenicle. This kind of social history of mathematics therefore shows Fermat “occupying a special position in a social configuration of knowledge practices, where he dominated the operation and specific features” Gol09, 55 and at the same time resolves apparent contradictions in historical images of Fermat.

4. From Great Men to Large Numbers? Capturing the Collective in Nineteenth-Century Number Theory through Citation Networks

Our main goal in this part has been to reshape the global representation of the history of the D[isquisitiones] A[rithmeticae] […]. We were not satisfied with the usual summary—a period of latency and awe, then a succession of a small number of brillant contributors, one or two per generation, who were deeply involved with the book, and finally the blossoming of algebraic number theory at the turn of the twentieth century. What we wanted was to pay more attention to the relations among mathematicians (and among their results), and to the actual mechanisms of knowledge transfer, in particular from one generation to another, that is, to understand some part of the dynamics in the changing role of the D. A. and of number theory. GS07b, 97–98

This quotation appears in the conclusion of the first two long chapters by Goldstein and Schappacher introducing the collective book The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae (2007). In understanding the dynamics of number theory publications in the nineteenth century, Goldstein pursues a double ambition: to go beyond the “great number-theorists,” mostly German, making up most of the stories about number theory⁠Footnote8 and to develop and test suitable methods for the case under study, namely, identifying and analyzing a corpus of nineteenth-century number-theoretic texts. From the systematic exploration of research journals containing mathematics—which were booming throughout the period—and several review journals created in the second half of the nineteenth century such as the Jahrbuch über die Fortschritte der Mathematik, she has studied number theory over the long nineteenth century from several corpora of texts: first, notes in the weekly proceedings of the Paris Academy of Sciences Gol94, then 3,500 articles and books reviewed in the Jahrbuch, Gol99, and finally references to Gauss’s Disquisitiones arithmeticae in major journals and complete works of identified mathematicians published from 1801 to 1914 GS07aGS07b.

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From this point of view, the description given by André Weil and taken up by Goldstein Gol99, 189 as an example of this kind of heroic story is particularly representative: “The great number-theorists of the last century are a small and select group of men. The names of Gauß, Jacobi, Dirichlet, Kummer, Hermite, Eisenstein, Kronecker, Dedekind, Minkowski, Hilbert spring to mind at once. To these one may add a few more, such as the universal Cauchy, H. Smith, H. Weber, Frobenius, Hurwitz” Wei75, p. 1.

In purely quantitative terms, the initial results of these surveys already contrast strikingly with the standard image mentioned above: a handful of German mathematicians are replaced by dozens of authors of various nationalities working on a wide variety of number-theoretic subjects. However, despite an interest in quantitative methods for the study of collective phenomena, Goldstein underlines the extent to which their application, relatively common in social history, poses particular problems for nineteenth-century mathematics, for example due to the difficulty of “gathering (…) homogeneous series of sufficient size” Gol99, 195. Questions arise at every stage of such a historical investigation. How do we select the relevant texts, or in other words, how do we decide what is and isn’t number theory? The “actors’ point of view” is heterogeneous over the period under consideration, with classification categories depending on the journals considered and evolving over time. Another example: purely quantitative approaches put the texts in the corpus under study on the same level, whether they are extensive academic works, a brief research note (the classic format of the weekly proceedings of the Paris Academy of Sciences) or a few lines of response to a question posed in a journal aimed at mathematics amateurs and teachers (such as the Educational Times). All these categories of text provide relevant information, provided that their form and links with other texts are taken into account. To create a social analysis of texts, Goldstein applied a form of network analysis to these corpora. However, the heterogeneity of the texts identified and the disparate citation practices in the nineteenth century preclude any automated processing of such a corpus. On the contrary, it is necessary to work on the basis of qualified links (i.e. differentiating between quotations of a result or a method, of homage or opposition, cf. Gol99, 203–204), taking into account explicit and implicit quotations (such as the vague mention of a result). It is then possible to identify clusters of texts, “characterized by strong intercitation” Gol99, 205, by combining computerized sorting and search capabilities with manual groupings.

This approach, based on the analysis of qualified quotations, renews our understanding of “the shaping of arithmetic.” First of all, the status of number theory, its objects and methods, are far from stable. Goldstein and Schappacher distinguish two main periods in the reception of Gauss’s Disquisitiones arithmeticae. The very early reception of the work was very timid and rather French. Attempting to expand algebra (in the sense of equation theory), mathematicians mobilized the arithmetic tools of Gauss’s section VII, which presents an algebraic method for solving binomial equations, also providing the conditions of constructibility of regular polygons with ruler and compass. In the 1820s, new readers of Gauss’s work became familiar with Disquisitiones arithmeticae as part of their mathematical training, including Jacobi, Dirichlet, Galois, and Libri. The numerous research projects undertaken at the time can be grouped under a single, international research field in the sense of the sociologist Pierre Bourdieu GS07a, 52, that Goldstein and Schappacher called Arithmetic Algebraic Analysis. Contemporary research in analysis, for example, on complex numbers, Fourier analysis, and elliptic functions, is used to address a number of arithmetical questions, such as those developed by Gauss in his work on residues, reciprocity laws, and the theory of quadratic forms. Here, the field of research is not organized around a mathematical object per se, but is delimited by a set of tools and methods shared by the various players. However, their practices are not uniform, and their objectives are often different. Note that many of them develop a common discourse on the unity of mathematics, which gives legitimacy to number theory: the unity here comes from the possibility of being able to build links between different mathematical objects and different proofs, of being able to transfer methods from one domain to another.

Then, for the period 1870–1914, three main text clusters are identified, none of them linked directly to algebraic number theory. As for the previous period, they do not fall within strictly national frameworks. The first cluster, named “L–G cluster” (Legendre and Gauss), attracted authors from a wide range of social positions (academics, teachers, engineers, military personnel, amateurs, etc.), mostly British, French, and Italian. The texts referred to both Legendre and Gauss, and avoided any recourse to analysis. This network enabled nonacademic authors to participate and constituted a gateway into circles where advanced number theory had no place. Nevertheless, most of the authors had no students, and the number of papers of articles in this cluster declined rapidly. At the end of the nineteenth century, they turned to media not taken into account in the network analysis conducted so far, such as the congress reports of the Association française pour l’avancement des sciences, whose aim was to promote useful and entertaining science. The authors and arithmetic practices identified in this cluster have been the subject of a number of recent studies in the history of teaching and the history of amateurs in sciences. A good example is the highschool teacher Édouard Lucas (1842–1891), who developed a “fabric geometry” that applies number theory to textile issues, around the figure of the chessboard. Along with other mathematicians, teachers, and engineers in particular, he promoted mathematical recreations and a visual number theory, not well represented in academic circles, but adapted to pedagogical issues and the desire to popularize mathematics at the time Déc07. The second main cluster, the “D-cluster” (Dirichlet), groups together texts whose authors adopted a “complex-analytic approach, inherited from Riemann and centering around Dirichlet series” GS07b, 74. Ernesto Cesáro, Pafnuti Tchebycheff, and Rudolf Lipschitz are among the regular authors of this cluster. The third, “H–K cluster” (Hermite–Kronecker), focuses on the arithmetic theory of forms, to which authors such as Émile Picard and Luigi Bianchi contribute. In addition to these three main clusters, there are several smaller ones, one of which includes the texts that engaged with Kummer’s number ideals, Dedekind’s ideals and algebraic number theory in the form promoted by Hilbert at the turn of the twentieth century. This cluster is surprisingly small, as measured by the number of texts, compared with its important place in the standard historiography of number theory.

Indeed, the authors from the H–K cluster who studied decomposable forms considered them as an alternative to ideals. Some, such as the Germans Eduard Selling and Paul Bachmann, even began by generalizing Kummer’s work on ideal numbers before turning to the theory of ternary forms. A systematic study of number-theoretical publications then shows the existence of “alternative paths of development” GS07b, 76, quantitatively and qualitatively important, to algebraic number theory and these diverse groupings are a far cry from an exclusively German number theory centered on algebraic number theory.

5. Conclusion

In a conference given in a seminar on science policy, Goldstein remarked that her family had no connection whatsoever with academia. When she entered the mathematical community, she was struck by the contradiction between, on the one hand, her mathematician colleagues’ sense of belonging to a strong organization with collective values of integrity, rigor, care for others, etc., and, on the other, an individualistic presentation of self, embedded in a history spanning several centuries built around a few “great” white Western men Gol21. One of her goals upon entering the history of mathematics, was to analyze this contradiction by studying the craft or the art of mathematicans. The “various ways” in which local configurations are, or are not, connected is a way of understanding how mathematics actually works, which is also important for understanding the nature of the work of mathematicians today. Construed broadly, mathematical practices are at once technical and political, as the examples presented above illustrate. Goldstein’s individual and collective research contributions extend well beyond the themes treated in this paper.⁠Footnote9 Two recent projects highlight other historical configurations that circumscribe the politics of mathematics.

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See http://webusers.imj-prg.fr/~catherine.goldstein for a more complete documentation.

First, politics can occupy center stage, as in the recent collaborative project around the history of mathematics in World War I. Until recently, this war was understood primarily as a site of loss in which mathematics was disrupted. The War of Guns and Mathematics shows how the practices and social position of mathematics were also impacted. Pulling together international expertise and archival access, the work draws on individual experiences of mathematicians “in specific places through the war” demonstrating disparate experiences among similarly situated actors as well as symmetries across national boundaries AG14, p. 43. Second, Goldstein’s own political activity has included leadership in the formation and advancement of Femmes et mathématiques.⁠Footnote10 Goldstein served on the council until 2002, was elected president in 1991, and has continually brought her historical expertise to bear in connected scholarly contributions, such as interrogating the assumptions that underlie the popular, but poorly defined question of whether there exists a particularly feminine kind of mathematics Gol94. In her historical work on gender and science, Goldstein again draws attention to the pitfalls in assuming a “univocal, long-term historical process,” in particular, that of “increasing estrangement between domestic women and public science” Gol00, p. 3. Such research corrects contemporary assumptions around women as eternally domestic that ignores both “subtler mechanisms of exclusion” and, in the other direction, opportunities, ideals, positive aspirations, and the actual possibilities of life Gol00, pp. 8, 27.

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The association began in 1987 with the objective of increasing the participation of girls and women in mathematical fields, from creating early educational programming to maintaining awareness of gender equality among scientific professionals.

Moreover, in her position at the Institut de Mathématiques de Jussieu, Goldstein has supervised doctoral research in the history of mathematics, from seventeenth-century controversies over indivisibles to the geometry of numbers in the twentieth century. This breadth of time and mathematical discipline is undergirded by sustained attention to and documentation of specific methodological approaches, such as network analysis, social history of texts, and microhistory. Like in mathematical research, Goldstein has trained her students—and exemplified through her talks and papers—the value of clearly defining methodological choices, what counts as evidence, and the process of securing precise historical results.

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[Das09]
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Catherine Goldstein, Neither public nor private: mathematics in early modern france, “The Work of Science. Gender in the Coordinates of Profession, Family and Discipline 1700–2000” (Berlin, June 2000), 2000.
[Gol09]
Catherine Goldstein, L’arithmétique de Pierre Fermat dans le contexte de la correspondance de Mersenne: une approche microsociale, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. Fascicule Spécial, 25–57. MR2561374,
Show rawAMSref \bib{Goldstein2009b}{article}{ label={Gol09}, author={Goldstein, Catherine}, title={L'arithm\'{e}tique de Pierre Fermat dans le contexte de la correspondance de Mersenne: une approche microsociale}, journal={Ann. Fac. Sci. Toulouse Math. (6)}, volume={18}, date={2009}, number={Fascicule Sp\'{e}cial}, pages={25--57}, issn={0240-2963}, review={\MR {2561374}}, }
[Gol13]
Catherine Goldstein, Routine controversies: mathematical challenges in Mersenne’s correspondence (English, with English and French summaries), Rev. Hist. Sci. 66 (2013), no. 2, 249–273, DOI 10.3917/rhs.662.0249. MR3184408,
Show rawAMSref \bib{Goldstein2013}{article}{ label={Gol13}, author={Goldstein, Catherine}, title={Routine controversies: mathematical challenges in Mersenne's correspondence}, language={English, with English and French summaries}, journal={Rev. Hist. Sci.}, volume={66}, date={2013}, number={2}, pages={249--273}, issn={0151-4105}, review={\MR {3184408}}, doi={10.3917/rhs.662.0249}, }
[Gol19a]
Ernest Coumet, Œuvres d’Ernest Coumet. Tome 2, Sciences: Concepts et Problèmes. [Sciences: Concepts and Problems], Presses Universitaires de Franche-Comté, Besançon, 2019. Edited by Catherine Goldstein. MR3967054,
Show rawAMSref \bib{Goldstein2019}{collection}{ label={Gol19a}, author={Coumet, Ernest}, title={\OE uvres d'Ernest Coumet. Tome 2}, series={Sciences: Concepts et Probl\`emes. [Sciences: Concepts and Problems]}, note={Edited by Catherine Goldstein}, publisher={Presses Universitaires de Franche-Comt\'{e}, Besan\c {c}on}, date={2019}, pages={459}, isbn={978-2-84867-662-3}, review={\MR {3967054}}, }
[Gol19b]
Catherine Goldstein, Long-term history and ephemeral configurations, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 487–522. MR3966737,
Show rawAMSref \bib{Goldstein2019b}{article}{ label={Gol19b}, author={Goldstein, Catherine}, title={Long-term history and ephemeral configurations}, conference={ title={Proceedings of the International Congress of Mathematicians---Rio de Janeiro 2018. Vol. I. Plenary lectures}, }, book={ publisher={World Sci. Publ., Hackensack, NJ}, }, date={2018}, pages={487--522}, review={\MR {3966737}}, }
[Gol21]
Catherine Goldstein, Et les autres? Histoires et imaginaires, Séminaire “Politique des sciences”, Institut Henri Poincaré, Paris, 2021.
[GS07a]
Catherine Goldstein and Norbert Schappacher, A book in search of a discipline (1801–1860), The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae, Springer, Berlin, 2007, pp. 3–65, DOI 10.1007/978-3-540-34720-0. MR2308277,
Show rawAMSref \bib{Goldstein2007a}{article}{ label={GS07a}, author={Goldstein, Catherine}, author={Schappacher, Norbert}, title={A book in search of a discipline (1801--1860)}, conference={ title={The shaping of arithmetic after C. F. Gauss's \textit {Disquisitiones arithmeticae}}, }, book={ publisher={Springer, Berlin}, }, date={2007}, pages={3--65}, review={\MR {2308277}}, doi={10.1007/978-3-540-34720-0}, }
[GS07b]
Catherine Goldstein and Norbert Schappacher, Several disciplines and a book (1860–1901), The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae, Springer, Berlin, 2007, pp. 67–103, DOI 10.1007/978-3-540-34720-0. MR2308278,
Show rawAMSref \bib{Goldstein2007b}{article}{ label={GS07b}, author={Goldstein, Catherine}, author={Schappacher, Norbert}, title={Several disciplines and a book (1860--1901)}, conference={ title={The shaping of arithmetic after C. F. Gauss's \textit {Disquisitiones arithmeticae}}, }, book={ publisher={Springer, Berlin}, }, date={2007}, pages={67--103}, review={\MR {2308278}}, doi={10.1007/978-3-540-34720-0}, }
[Kli72]
Morris Kline, Mathematical thought from ancient to modern times, Oxford University Press, New York, 1972. MR472307,
Show rawAMSref \bib{Kline1972}{book}{ label={Kli72}, author={Kline, Morris}, title={Mathematical thought from ancient to modern times}, publisher={Oxford University Press, New York}, date={1972}, pages={xvii+1238}, review={\MR {472307}}, }
[Ser95]
Michel Serres, Introduction, A History of Scientific Thought: Elements of a History of Science (Michel Serres, ed.), Blackwell, Cambridge, 1995, pp. 1–16.
[Wei75]
André Weil, Introduction to Ernst Eduard Kummer’s Collected papers: Volume I: Contributions to number theory, Springer-Verlag, Berlin-New York, 1975. MR465760,
Show rawAMSref \bib{Weil1975}{book}{ label={Wei75}, author={Weil, Andr\'{e}}, title={Introduction to Ernst Eduard Kummer's Collected papers: Volume I: Contributions to number theory}, publisher={Springer-Verlag, Berlin-New York}, date={1975}, review={\MR {465760}}, }

Credits

The opening photo is courtesy of ICM2018.

The sketches of the authors are courtesy of Jemma Lorenat.