Bibliography Panel from Barcelona

MB INTRODUCTION

JA Bibliographies and Power [TALK TEXT ONLY]

After Patrick Wilson’s Two Kinds of Power in 1968 [@wilson68:twokinds], we had all come to accept that the results of bibliographical control were caused by power struggles external to the bibliographical systems. A few years after The Order of Things in 1966, Wilson published his thought about how knowledge, particularly practical knowledge in how to do things, has always been a form of power. As a philosopher and librarian he conceived of the activity of “bibliographical control” as allocating finite resources to the problem of the vast unread, so that which is of value in the unread can be found. It seemed to him after the currency of a text has past, most would only be read to a few more times. Since we cannot predict which things those will be, we cannot decide on what to describe in detail, what to describe briefly, what to dispose of, and what to retain. He considers the demands of scholars unreliable and self-serving: Alluding specifically to demands from historians of science alongside those who study extra-sensory perception, he thinks of this question as falling onto the power struggle between the disciplines.

Said succinctly, if the only reason to develop a bibliography of materials is because it is useful to me, then the whole argument rests on who me is. If I have power, then me is enough. If I do not have power, then me is not enough.

Putting aside Wilson’s apocalyptic claim of the vast unread–it was well into the Cold War after all and apocalypse was in the air–serious scholars of bibliography have still not carefully examined his claims. We have accepted a sort of Machiavellian history of ideas, where the changes in how knowledge is organised reflect larger external struggles. Shifting from Wilson’s Cold-War American studies to global museology, we look to the ascending powers and political undergirding to determine what the history of bibliographies could be.

The disturbingly frequent mis-reading of D.F. McKenzie’s sociology of texts as ending the flawed positivist project of the New Bibliography and heralding an age of humanist archival studies, draws on this same stance: it perversely argues we ought to abandon the critical investigation of sources because this critical investigation is not sufficiently critical. As Johanna Drucker has recently pointed out in her essay on bibliographies for Information: a new companion [@blair_information:_2021], forensic approaches have come to seem un-critical. This, of course, is absurd.

Among other things we’re doing in this project, we’re investigating a mode of historiography which takes the year-by-year changes in mathematical and scientific bibliographies seriously. As McKenzie actually did, we combine extrinsic analyses of power with the intrinsic systems of knowledge.

Said succinctly again, we consider paperwork history not as a mere result of biopower, but as an autonomous historical domain worthy of study. I’ll demonstrate some of the sort of analysis I’ve been thinking about.

The Zentralblatt für Mathematik was begun by mathematicians Richard Courant, Otto Neugebauer, and Harald Bohr in 1931 as an effort to provide a central listing of current work in mathematics. The work was partially funded by a grant from the Rockefeller Foundation’s International Education Board made in 19 November 1926 to make Göttingen a centre for natural sciences.^[“University of Gottingen” report from History-2 - Projects, Volumes 1-3, 1910-1971; Rockefeller Foundation records; Rockefeller Foundation records, Administration, Program and Policy, RG 3; Rockefeller Foundation records, administration, program and policy, SG 3.1 and SG 3.2; Administration, Program and Policy, Subgrp 1; General Program and Policy, Series 900; Rockefeller Archive Center] Building on its strategic origins, and continuing until the present day, it has by-and-large served the purpose of listing current work in mathematics, although its structures and patterns have changed over the years in response to historical events. Today, it is widely used in the award-winning online portal zbMath, which includes material from 1931 to the present day, various other sources, and applies a modern classification scheme to it. One can even download datasets containing historical mathematics and use those to study the history of mathematics with distant reading, or quantitative humanities, techniques. Yet, the portal is a result of repeated remediation and changes of context. Things are missing and things are added, which means that one can not use this database responsibly without understanding its history. Rather than going backward, let’s start from the beginning of the journal’s bibliographical history.

The 1930s numbered issues came at the end of a long tradition of excerpting at central locations. From Callimachus to Pliny and Gessner, humanists kept commonplace books and studied the techniques of excerpting, the ars excerpendi, to serve their personal meditations.[@vine19:miscel; @blair17:gessner] A reader would collect excerpts in a florilegium, commonplace, or even slips of paper. This reader might be a professional managing books or a virtoso studying the philosophy of nature. The shift from personal to published was commonplace, Chinese encyclopedists had long practised the art before Europe, but the 17th to 18th century saw a shift to shared excerpting projects along with new information methods. The centralising shift is generally thought to come about partially because of technical advancements in writing technology and partially in response to Francis Bacon’s explicit call for such a shift.[@BaconNewOrganonReading] The 19th century and early 20th century further improved writing technology, and the Zentralblatt falls securely into that tradition.[@carvalho04:forty; @mitchell37:inks; @mitchell04:inks; and @waters40:inks describe the situation as it appeared over the first third of the twentieth century. The works are particularly interested in the idea of permanence and stability, which is echoed in the aims of these bibliographies.]

Zentralblatt covers in the 1930s:

The Zentralblatt thus constructed an image as a shared excerpting project that aims to provide the benefit of the private meditation on new mathematics, but for a broad, public audience. The constructed image can be seen in the covers where the names of editors appeared identifying a social scene where these editors invited materials from the world, allocated them to special reviewers, who produced short summaries. The actual practice articulates to this imagined construction, but differs in specific details in interesting ways. The judgement of the various staff were obviously involved in routing and assessing journals. Traces of this individual judgement can be seen on the coloured marks on the cards used to organise the activity of the journal. The use of coloured pencils in office settings had been firmly established by the 1930s and indicate both multiple contributors and multiple kinds of information. On this card, the red appears to be a judgement about the topic of the article drawn from the title whereas the blue is a clarification from the entry as to where it should fall in an index.

Example of annotated index card:

Using these cards, which had printed slips for the items, each issue would then classify the topic of the materials reviewed and at the end of a volume, not quite a year usually, they would produce an index of the topics. These topics were–at first–arranged by convenient topic, but structures, hierarchies, and patterns of mathematics were identified and established by the editors. In establishing these particular categories, and not others, they developed a structural image of mathematics that placed work into context.[@barany21:abstr] Even if the editors viewed their work as a neutral description of how mathematics actually exists, it was based on their social experience, shaped by material conditions, and thus reproduced a particular sense of both the possibilities and actualities of mathematics for the time.

After indexing, the individually numbered issues are dated, so we know that the index of topics came out after the next volume had begun. This was, in part, because the process of producing the index from the separately indexed numbered issues took some time: concepts had to be reconciled and arranged following the social process above. It is helpful too to note that the printed proofs must have provided the raw material for the card index of what was reviewed, by whom, where it was published, and how it was indexed.

Thus every few years from the 1930s, the Zentralblatt reconsidered the basic structures of mathematics. At the same time abstract algebra was beginning to reconsider the basic structures of mathematics. The bibliographical efforts here are especially important in the history of bibliography broadly because of the profound self-consciousness about abstract classification that the mathematicians studying formal systems of classification brought to the practical classification they faced. During this period the journal was printed, so the basic structure of the imagined site of submission and evaluation was similar. Different editors and different political regimes in Germany changed the purposes and imagined communities of the journal, but it remained a sort of print review journal with expert editors who would decide what was good, real, and bad mathematics. Indeed, this role of authority is precisely what about the Zentralblatt appealed to politicians who wished to use this authority in strategic ways to further their aims.^[TK cite something by Reinhard Siegmund-Schultze and Nazis]

Advertisement for the Zentralblatt on CD-ROM:

While the printed period has been partially examined, in the 1990s, the Zentralblatt had begun to appear in digital form, on BBSs, through WAIS, and on CD-ROM. The process proceeded in the typical way that digital conversion projects of the era did. They first hired a team of keyboardists to fill-in field-based records that meant to model the content of the textual entries and as the project progressed they began using Optical Character Recognition (OCR) to digitise more rapidly.^[TK I think this is buried in the MR archives, I need to hunt; a similar phenomenon is documented by Saztec when they prepared the CD-ROMs for the British Library Catalogue. CITE SATZEC video] The scene promoted by the CD-ROM is like the one in the illustrated advertisement: complete, longest running, comprehensive, and associated with the best European organisations. Rather than imagining itself as a review of contemporary mathematics, the new position was one drawing on its history–a distinctive selling point. This reframing of longest-running meant little without some sort of access to the past and in addition to experimenting with reprinting projects.^[WAITING ON cite the reprint: ask Linda Hall Library for a picture.]

The Fachinformationszentrum Karlsruhe had undertaken comprehensive retrospective digitisation to fit with this new comprehensively retrospective vision for the Zentralblatt and by 1997 the project to keyboard all the old issues to CD-ROMs was complete. The CD-ROM versions, like the earlier But, resources are not unlimited, so the early efforts at classification along with a host of other aspects were left out. This was fine, because one can certainly have recourse to the early issues, if needed. It’s the 1990s and every library has a periodical room.

The media archeology of the CD-ROMs and the database is fascinating, but that could be a whole other talk on it’s own.

Moving further forward to the online portal, it ingested the content from the 1990s, which ingested the content from the 1930s, so we can only see those moments through that chain of remediation. Each step, of course, removed context.

One small example is the nominal date of the numbered issues. It is not present in the zbMath database, so I built a second database to add that context back in for 1931–1937. There’s a lot of other useful information we can get from this, but to show that it has meaning, I’ll share an obvious result.

I conjecture it took longer to review dissertations than articles.

The average number of years between publishing a work in mathematics and it being reviewed is less than one year, about five months. However, dissertations took on average fourteen months to be reviewed.

Said another way, of the nearly 25,000 items from 1931 to 1937 only four took three or more years to review. Of the 136 dissertations, 98 took a year or more to be reviewed, 13 took two or more years, and 2 took three or more years. The only review that took more than four years is a dissertation.

Furthermore, the dissertations from central Europe were mostly reviewed within the year. It is the dissertations from outside of central Europe which took longer.

This is not a particularly surprising result, but it follows Andrew Piper’s model of quantitative humanities. We have a conjecture, we have some data, we test it, it is confirmed.

Critically, however, we could not do this kind of work without understanding the step-by-step bibliographical history of the Zentralblatt.

Slide 3: RS

This builds on our broader work thinking about the clerical offices that produced bibliographies. [… fill the remaining time, if any]

JV on classification

As with other literary practices in mathematics, historically, centralised bibliographic styles arise from the consolidation of epistemic communities. The substance of their centralisation consists not only in the institutionalisation of their editing and printing, but essentially also in their epistemologically programmatic core, embedded deeply within classificatory systems on one hand and the mindsets of reviewers and editors on the other. Leaving their complexities aside for a moment, we could say that the interconnectedness between social and epistemological factors constitutes the context through which major historical periodical bibliographies achieved their centralising effect. As such they were natural outgrowth of the literary and programmatic aspects of major institutionalised centres or centralised communities of mathematicians rather than the other way around, while both the literary and epistemological requirements underlined the necessity either founding or contending against established centralised bibliographies. Centralised bibliographical practices, with their established classifications and standards of referencing, were but one type of representatives of epicentres of a modern (re)mapping of centres and margines of international mathematics. Comprehensive bibliographic projects kickstarted in the major Western European centres like Berlin, London, and Paris were set to not only to list but to redraw the inner anatomy of mathematical science but concurrently also externalised its structural image, connecting it with points on maps and lists of renowned mathematicians associated with the academic localities which these points represented.

Major periodical bibliographies, most prominently the Berlin-centred Jahrbuch über die Fortschritte der Mathematik (JFM), began publication as early as 1868. Its founders Carl Ohrtmann and Felix Müller explicitly modelled the JFM on existing “annual reports on the advancements” of the natural sciences, particularly the Fortschritte der Physik (1847), and they drew on classificatory ideas developed by Ampère and others (Fraser 2022, pp. 7–8). The resulting system arranged mathematics into a quasi-natural taxonomy, beginning with history and philosophy, then algebra, number theory, series, and analysis, followed by synthetic and analytic geometry, and concluding with applied sciences, including mechanics, physics, optics, astronomy, and geodesy (Teschke 2021, p. 63). In this way the Jahrbuch’s classificatory scheme presented mathematics as a structural image resembling a hierarchical tree of fields and subfields, closely aligned with the classificatory practices of the physical sciences. The principal objective of the Jahrbuch was to list, and later to review, the latest advances (Fortschritte) across the entirety of mathematics, conceived as the growth of a unified, tree-like organism. This image was widely shared among mathematicians of the late nineteenth century, and most memorably articulated by Felix Klein (1895, Vol. 2, p. 232; 1932, p. 12).

By the closing decades of the nineteenth century, at roughly the same time that the International Congress of Mathematicians first convened, referencing practices and bibliographical enterprises had assumed an increasingly universalist character, with comprehensive undertakings emerging in France, the United Kingdom, Germany, and beyond. In France, the Répertoire bibliographique des sciences mathématiques (RBSM), conceived in 1885 under the auspices of the Société Mathématique de France, was designed above all to catalogue the mathematical literature in a systematic fashion; between 1894 and 1912 it issued around 20,000 entries drawn from dozens of journals (Rollet and Nabonnand 2002, pp. 11–25; Nabonnand and Rollet 2003, pp. 9–14). The classification scheme for this early French attempt at a comprehensive mathematical bibliography was devised and supervised by a small international commission under the presidency of Henri Poincaré (Rollet and Nabonnand 2002, pp. 18–20; Nabonnand amd Rollet 2003, pp. 10–12). Like the Jahrbuch, the RBSM sought to impose order upon the expanding body of mathematical knowledge by means of a methodical subject classification; yet it differed in both purpose and format – for it was a catalogue of references rather than a reviewing journal – and thus its organisation and use diverged markedly from that of the Jahrbuch (Nabonnand and Rollet 2003, pp. 11–13).

The classification scheme of RBSM conceived in 1885 was articulated semi-hierarchically into three parallel principal domains – analysis, geometry, and applied mathematics – subdivided into 24 top-level classes, labelled A to X, and further into subclasses, divisions, sections, and subsections, thereby yielding a five-tier taxonomy of notable complexity (Rollet and Nabonnand 2002, pp. 18–20). A published example shows how a code such as “L 1 4c” denoted Geometry: class L (conic sections and quadric surfaces), subclass 1 (conics), division 4 (tangents), section c (tangents under specified conditions) (Rollet and Nabonnand 2002, p. 19). This structure was first set out in the Projet de classification circulated in 1888, which articulated the positivist aspiration to map the mathematical sciences by content rather than by author (Nabonnand and Rollet 2003, pp. 10–11). Poincaré’s Commission defended this systematic ordering as a way of reducing informational overload and furnishing scholars with a navigable “map” of the discipline (Rollet and Nabonnand 2012, pp. 64–66). Compared with the German Jahrbuch, which also used subject divisions but retained distinctions of method (for instance, between synthetic and analytic geometry), the RBSM classified works purely by subject matter, thus enacting a different epistemological conception of mathematical order (Nabonnand and Rollet 2003, pp. 12–13). Importantly, the classification scheme of RBSM grew to dominate the French mathematical periodicals and found a unique use for its codes for classifying articles – and even unsolved problems – across major mathematical periodicals in the country. In 1893, the French classification scheme became the “structural spine” of the newly established French-Dutch Revue semestrielle des publications mathématiques, a periodical bibliography that paralleled the German Jahrbuch.

Before the establishment of Zentralblatt in the early 1930s, Jahrbuch and Revue semestrielle were arguably the two most influential periodical reviewing bibliographies in the world. Their influence extended beyond the mere listing, exclusion, or re-evaluation of mathematical literature. More significantly, by classifying and organising mathematical knowledge, they provided one of the most immediate materialisations of structural images of mathematics. These structural images, moreover, were produced from within distinct epistemic communities, shaped by specific programmatic trends, disciplinary emphases, and institutional geographies. Shifts in the centres that controlled the mechanisms of classification and evaluation accordingly brought about structural changes of their own. Yet, as many mathematicians soon recognised, both forms of classification also exhibited inherent limitations.

The initial classification systems, while groundbreaking, struggled to accommodate new fields or new conceptions of mathematics that emerged in the 20th century. For instance, early Jahrbuch volumes had traditional categories (e.g. arithmetic, algebra, geometry), but no place for novel areas like topology or abstract algebra until those fields matured. Mathematicians and librarians noted the difficulty of classifying certain cross-disciplinary works or fundamentally new ideas. By 1905 the Jahrbuch editors introduced significant adjustments – for example, splitting “Equations” and adding categories for group theory, etc. – indicating that they themselves saw the need to overhaul the taxonomy as mathematics progressed (Corry 2007). These changes can be read as admission of the earlier scheme’s inherent limitations. Leo Corry’s analysis of the Jahrbuch’s algebra entries shows that only as the abstract, structural approach took hold (circa 1930) did the reviewing classification fully catch up by creating new headings like “Universal Algebra,” whereas previously many algebraic innovations were pigeonholed under old headings or scattered. Contemporary mathematicians who used these bibliographies were aware of such shortcomings. In 1931, when Neugebauer and colleagues launched Zentralblatt, they explicitly promised to treat new and applied areas “as seriously as pure mathematics” (Neugebauer 1931), implicitly criticizing the old Jahrbuch for not doing so. Likewise, commentators in the 1930s (e.g. in Italy, as noted by Guerraggio and Nastasi 2006, p. 214) described Jahrbuch and the Revue semestrielle as “now outdated because of the development of new” directions in mathematics."

The problem of rigidity in classification systems, however, is not merely a matter of editors’ [alleged inability]{.mark} to respond to new developments that drastically reshaped the anatomy of fields and subfields. In fact, albeit with an average delay of a few years, the editors of the Jahrbuch made numerous attempts to restructure their classification system quite radically in order to [accommodate new trends and emerging fields.]{.mark} By contrast, the Revue semestrielle undertook almost no such restructuring. Importantly, as indicated above, the central problem lay in how to account both for changes within subfields and for the emergence of new ones, which were often driven by the distinctly “modern” syntheses of different areas. A direct answer may be obtained by looking more closely at how these classifications were applied in practice in the most problematic cases. For this purpose, we shall refer to the classification systems of the Jahrbuch and the Revue (i.e. the RBSM) as vertical and horizontal, respectively.

By vertical classification we mean a tree-like hierarchical structure, in which the major subfields are categorically separated from one another and each instance of mathematical work can consequently be sorted as belonging only to a specific branch of the tree. In a vertical ordering of mathematical knowledge, every emerging subfield and even minor changes between individual branches of the tree necessitate a comprehensive restructuring of the entire system (see Figure 1). By contrast, a horizontal system of classification, as exemplified by the RBSM, enumerates the major building blocks or categories of mathematical knowledge in parallel, thereby avoiding the impossible task of extending the principal disciplinary divisions (algebra, arithmetic, geometry) into a strict hierarchical order. The alphanumeric system of the RBSM is a good illustration of how the major “ingredients” of mathematics can be arranged in a long list. Such parallel listing, while avoiding explicit statements about certain aspects of structuredness of mathematical knowledge – of the kind keenly pursued by, for instance, the German modernists – nonetheless presents a very clear epistemic image, with concrete [practical determinacies]{.mark}. In this case, major new developments, such as the emergence of an entire subfield, could simply be managed by “pigeonholing” a new category under a new letter or by “neutrally” extending existing lettered categories with additional numbers. In practice, however, the inventors of the classification in the late 1880s had already foreseen what must have seemed a much simpler solution (see [Cayley’s 1888 letter to Poincar]{.mark}é) – one that eventually became a standard practice, stubbornly maintained by the editors of the Revue in dealing with all kinds of novelty. The parallel (semi-hierarchical) listing of the RBSM made it possible to combine alphabetical and numerical signifiers, a practice that preserved epistemic rigidity of the founders while at the same time efficiently recording the more subtle structural transformations associated with the modernist integration and overlapping of major mathematical subfields – for instance, the algebraisation and arithmetisation of geometry, and so forth. Unsurprisingly, from the early 1920s onwards, combined signifiers, involving between two and three individual categories, became predominant in the entries of the Revue.

Emerging fields such as topology can be used to illustrate the practical differences between the two aforementioned types of classification. Since Klein’s Erlangen Programme of 1872, analysis situs—the nominal precursor of the unified field of topology as we know it today—was treated as one of the sub-branches of geometry. As a category, it was included in the classification system of the Jahrbuch in the same year that Klein’s unification of the field was announced. However, since the early volumes were published with less delay than Klein’s programme, the classification was probably amended within a year or two after 1872. From the 1871–2 volume onwards, the then relatively narrow subfield of analysis situs was designated as VIII 2, with class VIII denoting “Pure, Elementary, and Synthetical Geometry,” parallel to “Analytic Geometry.” By the 1905 volume—published in 1908—the classification already accommodated a special subcategory for set and group theories. Set theory was pigeonholed alongside philosophy and pedagogy, while group theory represented one of the main branches of algebra. The next major change came in the combined volumes for 1916–18, which were compiled and published half a decade after the end of the First World War (1922–3). In this revised system, set theory became an independent class (III), while analysis situs was reclassified as a subcategory of a unified class of geometry (V), entitled “Continuity-related and Set Theoretical Observations.” This retrospective reclassification of analysis situs already incorporated the theoretical patterns of what was then also known as “set theoretical topology.” It drew a sharp line between topology and set theory, the latter now liberated from the status of a mere philosophical side-branch of mathematical knowledge. The final significant change in classification occurred in the early 1930s, applied retrospectively to the year 1925. In this last version of the Jahrbuch’s system – before publication came under the control of the National Socialist movement – the name “topology” appeared for the first time, replacing the verbose description used in 1922.

By contrast, the classification system of the Revue, and thereby also of the French RBSM, changed very little in the three decades following its establishment. From the outset, analysis situs was placed under the ostensibly marginal category Q, situated immediately before the applied fields and comprising “various geometries; general on n-dimensional geometry; non-Euclidean geometry.” A separate category P was created for “geometrical transformations,” while set and group theories were assigned to the broad and agglomerative category J, covering “combinatorial analysis; probabilistic calculus; calculus of variations; general theory of groups of transformations,” and so forth. Within this scheme, the most common signifiers for set and group theories were J4a–d and J5a–d, respectively. Unlike the Jahrbuch, the Revue did not respond to emerging fields by constructing new classifications in which, for instance, set-theoretical topology or combinatorial topology would appear as discrete domains. Instead, retaining its horizontal enumerative system, the editors and reviewers codified new developments through combinations of existing signifiers. By the late 1920s, when the field of “topology” was already nominally recognised in European mathematics, though not yet fully systematised, the most common markers for it were combinations such as Q3(x) with J5(y), supplemented by various others, including D for functions with real and complex variables.

The comparison of the two classificatory schemes, however, does not explain the striking discrepancies between how the Jahrbuch and the Revue recorded works on analysis situs or topology throughout their existence. For example, in 1908/9 the Jahrbuch recorded a total of 44 publications in the field of analysis situs, while the Revue counted only 11. By 1928 the disparity had grown: the Jahrbuch listed 133 publications in topology, whereas the Revue recorded only 25 under analysis situs. Strikingly, even after the two publications merged between 1932 and 1936, the same pattern of discrepancy persisted, despite classifications being printed side by side and the Jahrbuch allowing for the assignment of multiple categories to a single publication. If not the result of different classificatory schemes, what else could explain such enduring discrepancies?

Aside from recording a considerably smaller number of publications, the classificatory practice of the Revue tended to identify as analysis situs only works produced in a handful of European centres of topology at the time—Amsterdam, Budapest, Greifswald, and so forth—and almost never those of the emerging American “topology.” The Jahrbuch, by contrast, consistently counted as analysis situs or topology not only earlier works in point-set geometry but also a greater number of papers explicitly related to analysis situs. It thus presented a livelier and more coherent picture of the field, encompassing a broad array of authors from both Europe and the United States. Importantly, the works listed as topology in the Jahrbuch were not absent from the Revue. The latter simply classified the same works under different headings. A partial explanation can be found in the joint volumes covering the years 1932–36. There, one sees that the reviewers of the Revue often classified works published in the Polish Fundamenta Mathematicae as set theory, while the reviewers of the Jahrbuch insisted that the same works were undoubted examples of pure topology (signifier V2). From 1932 onwards, the Revue did not make the same “mistake” in the case of the major topological works of the 1930s, authored by figures such as Lefschetz, Reidemeister, Hopf, Alexandroff, and others, which paved the way for a unified field of topology. Yet, if published in journals such as Fundamenta Mathematicae, works by these same mathematicians would still have been classified by the Revue as set theory or as belonging to another subcategory of geometry or algebra.

Even though, retrospectively, this may appear as a straightforward failure on the part of the Revue to describe something obvious, the situation looked far less clear-cut within American and European mathematics at the time. Even in the mid-1930s, topology was not yet a unified field—neither in theory nor in terms of its major centres or cohesion within the international community of topologists. The unequal geographical distribution of leading figures, together with the still predominantly national organisation of teaching and research, meant that awareness of the field was concentrated in particular national contexts. The gradual formation of an international network of topologists began only in the late 1920s. The organisation of the first international conferences and major courses in a more unified version of topology—which grew out of these early networks—bore significant fruit only in the early to mid-1930s. Before these developments, the uneven treatment of the emerging field across different bibliographies was not accidental. It reflected the proximity between the epistemic communities in which new knowledge was being developed and those responsible for defining and applying classificatory schemes. In other words, classification practices mirrored local intellectual alignments rather than the state of mathematics as a whole.

By contrast with the Revue, the Jahrbuch was, despite its slower pace, more directly connected to the small but vibrant communities of topologists that had begun to appear across the former Habsburg Empire and the German states of the interwar period. Yet, as a publication whose founders and editors were rooted in the academic establishment of Berlin—above all the Prussian Academy of Sciences—it also embodied the outlook of a conservative epistemic community, often in contest with newer centres of modern mathematics, most notably Göttingen. Cooperation between the Jahrbuch and the Göttingen-dominated German Mathematical Society (Deutscher Mathematiker-Verein, DMV) existed from the outset, but began to falter around 1910, owing largely to political tensions between Felix Müller and the DMV leadership under Felix Klein and others. A strong connection with the DMV was crucial for ensuring access both to information on trends in modern mathematics and to lists of the most pertinent reviewers and advisers. Indeed, one of Müller’s principal arguments in his political campaigns against foreign bibliographies—such as that of the British Royal Society in the early 1900s—or against rival German initiatives was that the editors of the Jahrbuch commanded the most exhaustive lists of international mathematicians and mathematical periodicals. The relationship with the DMV improved only after Müller’s death in 1928, when Georg Feigl, a topologist close to the DMV leadership, became one of the Jahrbuch’s chief editors. From 1925 onwards, under the presidencies of mathematicians such as Schoenflies and Meger, Austrian and German topologists had assumed a more central position within the German mathematical community. In this way, the reorganisation of 1928 introduced renewed perspectives on the scope and development of topology, enabling the editors to adapt their classifications and to provide more comprehensive coverage of the literature.

By the late 1920s, however, the Jahrbuch was already mired in organisational difficulties. Some of these were a direct consequence of the post-war economy and the concentration of financial power in the Notgemeinschaft der Deutschen Wissenschaft, whose mathematical commission was overseen by Felix Klein. With Göttingen’s revival in 1928 as the ascendant centre of modern mathematics—sustained by both domestic and international funding—an alternative soon emerged in the form of the world-famous Zentralblatt für Mathematik und Ihre Grenzgebiete. When, in 1931, Neugebauer announced from his specially designated office at the Göttingen Mathematical Institute that the Zentralblatt would cover new developments with greater takt and precision, his pledge was not underpinned by a fundamentally more advanced classificatory system. In fact, the early classification of the Zentralblatt was little more than a synthesis of those employed by the Jahrbuch and the Revue. In this case, too, the true innovation lay not in structure but in position: classification and reviewing would now be overseen from the very epicentre of modern mathematics, one of the very birthplaces of new epistemologies. Thus, the centrality in the name of “Zentralblatt” (German for “central journal”) was not achieved by virtue of an improved structural image of mathematics, but its institutional power, embodied in the centrality of names listed in its directory of advisors and reviewers, and its dense network of connections with leading centres across the globe. While in practice, the Zentralblatt grew out of the lessons and experiences of Jahrbuch and the Revue, the above is what constituted the international centrality of Göttingen, through which the publication became viewed as “an international coordinating and synthesizing influence in mathematics [rather] than a mere mechanical bibliographical aid.” ([Warren Weaver diaries 1939,]{.mark} p. 38)

[[Tentative and lightly weak synthesis – I do not like the way it is written, so it ought to be replaced or amended after team discussion – doing some “strong programming” here (I know this is a very ambiguous way to put it)]]{.mark}

The major reviewing and abstracting periodical bibliographies were not mere mechanical aids, tasked with sorting mathematical literature into neutrally arranged pigeonholes or producing balanced reviews and concise abstracts. From their very conception, the classifications they employed functioned as representations of mathematical epistemologies. In the allegedly “exhaustive” bibliographies of the era, these epistemologies acquired clear-cut structural images which no longer served merely as referential frameworks for thinking about foundational issues in mathematical science, but as concrete instruments of “coordinating and synthesising” practice. Moreover, the structural images reproduced through classification and reviewing were objects of an inherently social nature. Emerging from shifting concentrations of contending national and nascent international epistemic communities, these structural images were advanced from the institutionalised centres of mathematical life. For this reason—particularly in cases of novel trends and developments—this relation can often better account for the discrepancies and inadequacies of classification and reviewing. As we have emphasised above, these phenomena reveal that the “institutional images” of mathematics, together with their corresponding lists and maps, played an equal role in meeting the needs of the emerging “international mathematical community” and its “internationalised mathematics.”

As historical documents, instruments, or objects, these bibliographies were highly contextual entities, often in friction not only with one another but also with emerging peripheral structures and their socio-epistemic contexts. Viewed from a linear progressivist perspective, the internationalisation of modern mathematics would appear to entail the decontextualisation of the structural image of mathematics from its original epistemic and social epicentres, and the creation of a new image capable of more equally encompassing a broader institutional and epistemic geography of mathematical practice. Yet paradoxically, the examples above suggest that the consolidation of a bibliographic practice of structuring mathematics required a structural homeomorphism analogous to that of its systems of classification: its social and epistemic centres needed to overlap. Only this type of coherence could generate a veritable Zentralblatt. Thus, the alleged contextual broadening intended to accommodate the international mathematical community and the total image of mathematical knowledge was in practice achieved through a new round of centralisation—one that involved the proper directory of expert mathematicians. This new directory did indeed transcend earlier national boundaries and made significant advances in overcoming the linguistic divides in which “world mathematics” had been written. However, as history shows, this merely produced a new form of centralised practice, complete with its own geographical, social, and epistemic constraints, and with mechanisms that efficiently excluded or pigeonholed the provincial.

________

[[I am not mentioning Soviet mathematics as a relevant example… Probably a bit unrelated here. Also, I think that I am not really getting the message across – should be revised.]]{.mark}