Periodic Reporting for period 4 - STRUCTURALISM (The Roots of Mathematical Structuralism)
Reporting period: 2021-09-01 to 2022-02-28
Mathematical structuralism is a dominant position in contemporary philosophy of mathematics. Roughly put, it is the view that mathematical theories describe abstract structures: Peano arithmetic describes the natural number structure, geometry the structure of space, and so on. Precise elaborations of this idea have been given by a number of philosophers of mathematics, resulting in a number of different versions of structuralism. It is striking that in spite of its present relevance, little scholarly attention has so far been dedicated to the historical emergence of this philosophical position. The project “The Roots of Mathematical Structuralism” aimed to fill this gap. It gave a first interdisciplinary investigation of the mathematical and philosophical roots of structuralism that combined historical, philosophical, and logical methods of analysis. The project had two principal objectives, one historical and the other systematic in spirit. The first aim was to reconstruct the philosophical origins of mathematical structuralism and their immediate mathematical background. Specifically, the focus was set on two historical developments in nineteenth century mathematics and early twentieth century philosophy: the first one concerns several conceptual changes in geometry between 1860 and 1900 that eventually led to a “structuralist turn” in the field. Research focused on the gradual implementation of model-theoretic techniques in geometrical reasoning, the unification of geometrical theories by algebraic methods as well as the successive consolidation of structural axiomatics. The second development considered in the project concerns the philosophical reflection on these mathematical transformations between 1900 and 1940. This includes different attempts by thinkers such as Rudolf Carnap, Edmund Husserl, and Ernst Cassirer to spell out the philosophical implications of the new structuralist methodologies at work in modern geometry. The second central aim of the project was to reconnect these early contributions with contemporary debates on mathematical structuralism. Research on the project thus not only aimed to established a refined understanding of the historical development of structuralism before 1960, but also yielded new systematic insights relevant to contemporary philosophy of mathematics. Specifically, it led to new ways to conceptualize several of the key notions discussed in structuralism and to evaluate their relevance in modern mathematics.
The project was successfully implemented during the final reporting period at the Department of Philosophy of the University of Vienna. The research team working on the “Roots of Mathematical Structuralism” project consisted of the PI (Georg Schiemer), two predoctoral scholars (Henning Heller and Inger Bakken Pedersen), as well as three postdoctoral scholars (John Wigglesworth, Francesca Biagioli, and Günther Eder). Research on the project has focused on several historical topics related to subproject 1 “The geometrical roots of structuralism” and to subproject 2 “Early mathematical structuralism”. Moreover, systematic results on the philosophical and logical foundations of structuralism were established relating to subproject 3 “Reconsidering mathematical structuralism”. Regarding the first subproject, the team has investigated several methodological developments in nineteenth-century geometry and algebra, in particular, related to Felix Klein’s group-theoretical classification of geometries, to the history of representation theory, as well as to the development of formal axiomatics in work by David Hilbert and others. Several research publications on these topics were written by team members during the project period, including on the structuralism underlying Klein’s Erlangen Program (by Henning, Schiemer, Biagioli) and on the method of implicit definition. Regarding the second subproject, research has mainly focused on the analysis of different contributions to mathematical structuralism in early twentieth-century philosophy, in particular, by Rudolf Carnap, Edmund Husserl, and Ernst Cassirer. A number of research articles on the philosophical roots of modern structuralism are meanwhile published or currently prepared for submission (by Biagioli, Eder, Schiemer, Pedersen). Concerning the last subproject, Wigglesworth and Schiemer have worked on the notions of “structural abstraction” and “structural properties” as well as on a modal account of structural information in mathematics. Several articles on these topics are now published, e.g. in The British Journal for the Philosophy of Science, Philosophia Mathematica, Erkenntnis, among other journals. The team has organized eight scientific events related to the project, including the international conferences “Varieties of Mathematical Abstraction” (2018), “Structuralist Foundations” (2021), and “Modern Geometry and its Foundations” (2022) at the University of Vienna.
In the four reporting periods of the project, significant progress has been made on all three subprojects of the “Roots of Mathematical Structuralism” project. In particular, concerning the mathematical origins of structuralism, research on the project has led to a refined understanding of various connections between the developments in projective geometry, Klein’s Erlangen Program, and modern axiomatics. Regarding the philosophical prehistory of structuralism, new insights were gained concerning early structuralist views by Rudolf Carnap, Ernst Cassirer, and Edmund Husserl. Concerning the logical foundations of mathematical structuralism, the project has led to new proposals for the logical explication of the notions of structural abstraction, structural properties, and implicit structural content. Research on the project will likely have a significant impact both on the history and philosophy of mathematics on at least two levels. First, it led to a much more detailed understanding of the historical development of the “structural turn” in mathematics in the nineteenth-century. Moreover, it will also lead to a novel account of how philosophy of mathematics evolved in the late nineteenth and early twentieth century. Second, the research results of the project have the potential to modify our present understanding of mathematical structuralism as a dominant position in the philosophy of mathematics. Three aspects are of importance here: first, the historical focus of the project has challenged some of the traditional boundaries between different views in philosophy of mathematics. This concerns, in particular, of the connection between mathematical structuralism and (neo-)logicism with respect to the role of abstraction principles. Second, the focus on the roots of structuralism motivated a new treatment of several key notions in the current debate, including the notions of “(implicit) structural content”, “structure abstraction” or “structural property”; finally, the project has developed novel approaches to several systematic issues currently debated in structuralism, in particular, on the proper understanding of structural content, on the explication of structural properties, as well as the significance of a structuralist style of reasoning in mathematical practice.