## Periodic Reporting for period 1 - STRUCTURALISM (The Roots of Mathematical Structuralism)

**Reporting period:**2017-03-01

**to**2018-08-31

## Summary of the context and overall objectives of the project

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 thus describes the natural number structure, analysis the real number structure, geometry the structure of (Euclidian) space, and so on. Precise elaborations of that idea have been provided by a number of current 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 and early development of this philosophical position. The present project aims to fill this gap. It will give a first interdisciplinary investigation of the mathematical and philosophical roots of structuralism that combines historical, philosophical, and logical methods of analysis.

The present project has two principal objectives, one historical and the other systematic in spirit. The first aim is to provide a detailed reconstruction of these philosophical origins of mathematical structuralism and their immediate mathematical background. Specifically, the focus will be set on two historical developments in nineteenth century mathematics and early twentieth century philosophy of science: the first one concerns several conceptual changes in geometry between 1860 and 1900 that eventually led to a “structuralist turn” in the field. This includes the gradual implementation of model-theoretic techniques in geometrical reasoning, the unification of geometrical theories by algebraic methods--and, specifically, by the use of invariants--as well as the successive consolidation of formal axiomatics. The second development considered concerns the beginnings of 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 aim of the project will be to reconnect these early contributions with the contemporary debates on mathematical structuralism. Recent work on structuralism has focused mainly on systematic issues concerning the ontology of structural mathematics as well as on the consistent formulation of different structure theories. Nevertheless, several important research questions have not yet been sufficiently addressed in the literature. This concerns two issues in particular, namely (i) the compatibility between structuralism and mathematical practice and (ii) the relation between structuralism and other research programs in the philosophy of mathematics, most importantly neo-logicism. A central hypothesis of the present project is that these open research questions can be fruitfully addressed by reconsidering the formative contributions to structuralism by Carnap, Husserl, and Cassirer. Research on the project will thus not only establish a new understanding of the historical development of structuralism before 1960, but also yield new systematic insights relevant to contemporary philosophy of mathematics. Specifically, it will provide us with new ways to conceptualize key notions such as “mathematical structure”, “structure abstraction”, or “structural property” and to evaluate their significance in modern mathematical practice.

The present project has two principal objectives, one historical and the other systematic in spirit. The first aim is to provide a detailed reconstruction of these philosophical origins of mathematical structuralism and their immediate mathematical background. Specifically, the focus will be set on two historical developments in nineteenth century mathematics and early twentieth century philosophy of science: the first one concerns several conceptual changes in geometry between 1860 and 1900 that eventually led to a “structuralist turn” in the field. This includes the gradual implementation of model-theoretic techniques in geometrical reasoning, the unification of geometrical theories by algebraic methods--and, specifically, by the use of invariants--as well as the successive consolidation of formal axiomatics. The second development considered concerns the beginnings of 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 aim of the project will be to reconnect these early contributions with the contemporary debates on mathematical structuralism. Recent work on structuralism has focused mainly on systematic issues concerning the ontology of structural mathematics as well as on the consistent formulation of different structure theories. Nevertheless, several important research questions have not yet been sufficiently addressed in the literature. This concerns two issues in particular, namely (i) the compatibility between structuralism and mathematical practice and (ii) the relation between structuralism and other research programs in the philosophy of mathematics, most importantly neo-logicism. A central hypothesis of the present project is that these open research questions can be fruitfully addressed by reconsidering the formative contributions to structuralism by Carnap, Husserl, and Cassirer. Research on the project will thus not only establish a new understanding of the historical development of structuralism before 1960, but also yield new systematic insights relevant to contemporary philosophy of mathematics. Specifically, it will provide us with new ways to conceptualize key notions such as “mathematical structure”, “structure abstraction”, or “structural property” and to evaluate their significance in modern mathematical practice.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project was successfully implemented during the reporting period at the Department of Philosophy of University of Vienna in spring 2017. The present project group includes the PI, Georg Schiemer, a pre-doctoral scholar, Henning Heller, as well as two postdoctoral scholars, Francesca Biagioli and John Wigglesworth. Research on the project during the reporting period focused primarily on the historical topics related to subproject 1 “The geometrical roots of structuralism” and subproject 2 “Early mathematical structuralism”. Several results on more systematic topics in the philosophy of mathematics were also made concerning subproject 3 titled “Reconsidering mathematical structuralism”. Regarding the first subproject, the work on the team has focused on several methodological developments in nineteenth-century geometry and algebra, in particular on Felix Klein’s group-theoretical classification of geometries, on the history of algebraic representation theory, and on the introduction of modern formal axiomatics in the work of David Hilbert and others. Several research articles on these topics were written during the project phase by the team, including an article by Schiemer on the principle of duality and so-called transfer principles in nineteenth-century projective geometry. A second article currently under review is Biagioli’s paper on Klein’s structuralism underlying his work on non-Euclidean geometry.

Regarding the second subproject, research has focused on an analysis of early philosophical contributions to mathematical structuralism, in particular, by Rudolf Carnap and Ernst Cassirer. Concerning the last subproject, Wigglesworth and Schiemer have conducted research on the notions of structural abstraction and structural properties as well as on the application of structural mathematics in science. A joint article on the first topic has been published in The British Journal for the Philosophy of Science in 2018. During the reporting period, the team has organized three international workshops in Vienna and Groningen as well as one international conference titled “Varieties of Mathematical Abstraction” in Vienna.

Regarding the second subproject, research has focused on an analysis of early philosophical contributions to mathematical structuralism, in particular, by Rudolf Carnap and Ernst Cassirer. Concerning the last subproject, Wigglesworth and Schiemer have conducted research on the notions of structural abstraction and structural properties as well as on the application of structural mathematics in science. A joint article on the first topic has been published in The British Journal for the Philosophy of Science in 2018. During the reporting period, the team has organized three international workshops in Vienna and Groningen as well as one international conference titled “Varieties of Mathematical Abstraction” in Vienna.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In the first reporting period of the project, significant progress has been made on all three subprojects. In particular, concerning the mathematical roots of structuralism, research on the project has led to a refined understanding of various connections between developments in projective geometry in the nineteenth-century, Klein’s Erlanger Program, and modern axiomatics. Concerning the philosophical prehistory of structuralism, new insights were gained concerning both Carnap’s and Cassirer’s early structuralist views on mathematics. Concerning the logical and conceptual foundations of structuralism, the project has led to new proposals for the logical specification of the notions of structural abstraction and structural properties.

Research of the project until 2022 will likely have a significant impact on the history and philosophy of mathematics on two levels. First, it will change our present account of the historical development of a “structural turn” in mathematics in the nineteenth-century. Moreover, it will also lead to a novel and much refined understanding of how philosophy of mathematics evolved in the nineteenth and early twentieth century. Second, the project also has the potential to change our current understanding of mathematical structuralism. Three aspects are of particular importance here: first, the historical focus will challenge some of the traditional boundaries between positions in modern philosophy of mathematics that have thus far been conceived as non-related. This holds in particular for the possible connection between mathematical structuralism and neologicism with respect to the role of abstraction principles. Second, the focus on the mathematical and philosophical roots of structuralism will motivate an alternative treatment of several key notions in the contemporary debate (such as “(implicit) structural content”, “structure abstraction” or “structural property”); finally, it will allow a novel approach to several systematic issues currently under discussion in the debate on structuralism, for instance the proper understanding of the instantiation of structures by mathematical objects, the definition of inter-structural relations, as well as their significance in actual mathematical practice.

Research of the project until 2022 will likely have a significant impact on the history and philosophy of mathematics on two levels. First, it will change our present account of the historical development of a “structural turn” in mathematics in the nineteenth-century. Moreover, it will also lead to a novel and much refined understanding of how philosophy of mathematics evolved in the nineteenth and early twentieth century. Second, the project also has the potential to change our current understanding of mathematical structuralism. Three aspects are of particular importance here: first, the historical focus will challenge some of the traditional boundaries between positions in modern philosophy of mathematics that have thus far been conceived as non-related. This holds in particular for the possible connection between mathematical structuralism and neologicism with respect to the role of abstraction principles. Second, the focus on the mathematical and philosophical roots of structuralism will motivate an alternative treatment of several key notions in the contemporary debate (such as “(implicit) structural content”, “structure abstraction” or “structural property”); finally, it will allow a novel approach to several systematic issues currently under discussion in the debate on structuralism, for instance the proper understanding of the instantiation of structures by mathematical objects, the definition of inter-structural relations, as well as their significance in actual mathematical practice.