Many of the most notable features of modern mathematics, including the role of abstraction and the relationship between pure and applied subjects, trace to the German university town of Göttingen and the two mathematical titans who presided there at the turn of the twentieth century, David Hilbert and Felix Klein (Rowe 1989). Their respective mathematical programs, in their own ways, proved both mathematically and institutionally ambitious, driven by a universalism that tied together theories and research efforts through foundational reasoning that cut across conceptual, professional, and national boundaries. The ferment associated with Göttingen underwrote a new structural approach to mathematical theory, especially in the field of Algebra, with profound consequences for the practice and philosophy of modern mathematics (Corry 2004).
Around the end of the Great War, Göttingen’s internationally-oriented mathematics community became the hub for a new publishing venture spearheaded by Ferdinand Springer, Jr., grandson of the founder of the Springer publishing house (Munroe 2007). Adding mathematical publications to his catalogue, Springer, Jr. lay claim to what continues to this day as a profitable niche catering to mathematical researchers. This effort gave rise to a profound new development in mathematical bibliography with the 1931 launch of the Zentralblatt für Mathematik und ihre Grenzgebiete, based editorially in Göttingen. Siegmund-Schultze (1994) argues that the Zentralblatt marked a radical departure from previous bibliographical enterprises, especially the then-dominant, Berlin-based, half-century-old Jahrbuch über die Fortschritte der Mathematik. Where the Jahrbuch’s publishers aimed to produce a comprehensive (albeit German-centered) view of annual developments in the discipline to serve as a durable reference at the cost of long publication delays, the Zentralblatt editors released a rolling record of new mathematics as rapidly as they could produce it. Instead of relying like older bibliographical projects had on subsidies from scientific societies, Springer found that such a guide to the current literature justified its cost as an authoritative advertisement for the press’s own publications (cf. Fyfe et al. 2017).
The Zentralblatt quickly proved that the kind of rapid, wide-ranging, international bibliographic apparatus for which Rudio called in 1897 could become an indispensable and self-sustaining matrix joining a dispersed variety of publications into a unified mathematical literature. While the reviewing journal relied on an international pool of editors and the even broader disciplinary connections they maintained, it did not require the kind of top-down international coordination Rudio presumed necessary. By publishing authors from across Europe and by advertising their publications in a common forum and format that entered them into ongoing research conversations, Springer and the Zentralblatt’s editors helped to sediment a common market and a common research community, reinforcing international activity by making it mutually available to participating scholars.
At the same time Springer launched the Zentralblatt, the publisher released a monograph that Corry (2004) identifies as a watershed in the rise of mathematical structures, Göttingen-trained Bartel van der Waerden’s two-volume (1930–1931) Moderne Algebra. The first volume was not reviewed in the Zentralblatt, but the second drew an extended notice on pages 8–10 of the first issue of the review journal’s second volume. For Corry, van der Waerden’s book marked a milestone for “the reflexive character of mathematics,” using mathematical methods to make claims about the discipline of mathematics as a whole. New mathematical theories of structures from what Corry calls the “body of mathematics” defined and developed new philosophies of mathematical structuralism for what Corry terms the “image of mathematics.” Mathematicians powerfully asserted that the domains and concepts of mathematics themselves obeyed structural principles susceptible to mathematical analysis, and correspondingly that studying these structures constituted the essence of mathematics.
Analyzing the life and worldview of the Zentralblatt’s founding managing editor, Otto Neugebauer, Siegmund-Schultze (2016) observes the biographical and institutional convergence in Göttingen of modern international theoretical programs and modern international reviewing infrastructures. This, he notes, came despite the apparent difference between backward-looking bibliographic labor and forward-looking original scholarship. However, seen another way, the modernization associated with mathematical structures and structuralism was very much of a piece with the modernization in mathematical infrastructure associated with Springer, the Zentralblatt, and further financial and institutional developments in and around Göttingen. This link between mathematical structures and structuralisms and their supporting infrastructures suggests a corresponding reflexive relationship between the material, personal, and institutional bases of mathematics and the discipline’s image and body.
New mathematical infrastructures supported and required new means of practicing mathematics and new conceptions of what mathematics was. At each level—basis, body, and image—the operative object is a kind of abstract relation. In structuralist images of mathematics, abstract relations connect principles, fields, and kinds. In the structural body of mathematics, such relations tie together ideal concepts and entities. Post-1930 mathematical infrastructures, meanwhile, joined mathematicians through relations mediated by the production and circulation of bibliographic abstracts in the mold of entries from the Zentralblatt. Just as mathematicians applied theoretical developments from the body of mathematics to the image of mathematics and vice versa, I propose, their infrastructural relations emphasized and facilitated distinct kinds of theoretical abstraction.