The Prehistory of Mathematical Structuralism

This is an open access title available under the terms of a CC BY-NC-ND 4.0 licence. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations. Recently, debates about mathematical structuralism have picked up steam again within the philosophy of mathematics, probing ontological and epistemological issues in novel ways. These debates build on discussions of structuralism which began in the 1960s in the work of philosophers such as Paul Benacerraf and Hilary Putnam; going further than these previous thinkers, however, these new debates also recognize that the motivation for structuralist views should be tied to methodological developments within mathematics. In fact, practically all relevant ideas and methods have roots in the structuralist transformation that modern mathematics underwent in the 19th and early 20th centuries. This edited volume of new essays by top scholars in the philosophy of mathematics explores this previously overlooked 'pre-history' of mathematical structuralism. The contributors explore this historical background along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics, such as Dedekind, Hilbert, and Bourbaki, who are responsible for the introduction of new number systems, algebras, and geometries that transformed the landscape of mathematics. Second, they reexamine a range of philosophical reflections by mathematically inclined philosophers, like Russell, Cassirer, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysical aspects of structuralism. Overall, the essays in this volume show not only that the pre-history of mathematical structuralism is much richer than commonly appreciated, but also that it is crucial to take into account this broader intellectual history for enriching current debates in the philosophy of mathematics. The insights included in this volume will interest scholars and students in the philosophy of mathematics, the philosophy of science, and the history of philosophy.

Erich H. Reck is Professor of Philosophy at the University of California at Riverside. He is the author of a number of articles in the philosophy of mathematics, the history and philosophy of logic, and the history of 19th/20th-century philosophy. In addition, he is the editor, or co-editor, of several related collections of essays: From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy (OUP, 2002); Frege's Lectures on Logic: Carnap's Student Notes, 1910-1914 (Open Court, 2004); Gottlob Frege: Critical Assessments of Leading Philosophers, Vols. I-IV (Routledge, 2005); The Historical Turn in Analytic Philosophy (Palgrave Macmillan, 2013); and Logic, Philosophy of Mathematics, and their History: Essays in Honor of W.W. Tait (College Publications, 2018). Currently he is working on a book on the mathematician and philosopher of mathematics Richard Dedekind. Georg Schiemer is Assistant Professor at the Department of Philosophy at the University of Vienna as well as external fellow at the Munich Center for Mathematical Philosophy at LMU Munich. He is currently principal investigator of the project "The Roots of Mathematical Structuralism" funded by an ERC Starting Grant. His research focuses on the history and philosophy of mathematics and early analytic philosophy. He is also interested in the history and philosophy of logic and formal philosophy of science.

1. Erich Reck & Georg Schiemer: The Prehistory of Mathematical Structuralism: Introduction and Overview Part I: Mathematical Developments 2. Paola Cantù: Grassmann's Concept Structuralism 3. José Ferreirós & Erich Reck: Dedekind's Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions 4. Dirk Schlimm: Pasch's Empiricism as Methodological Structuralism 5. Georg Schiemer: Transfer Principles, Klein's Erlangen Program, and Methodological Structuralism 6. Wilfried Sieg: The Ways of Hilbert's Axiomatics: Structural and Formal 7. Audrey Yap: Noether as Mathematical Structuralist 8. Gerhard Heinzmann & Jean Petitot: The Functional Role of Structures in Bourbaki 9. Colin McLarty: Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures Part II: Logical and Philosophical Reflections 10. Jessica Carter: Logic of Relations and Diagrammatic Reasoning: Structuralist Elements in the Work of Charles Sanders Peirce 11. Janet Folina: Poincaré and the Pre-History of Mathematical Structuralism 12. Jeremy Heis: 'If Numbers Are To Be Anything At All, They Must Be Intrinsically Something': Bertrand Russell and Mathematical Structuralism 13. Erich Reck: Cassirer's Reception of Dedekind and the Structuralist Transformation of Mathematics 14. Wilfried Sieg: Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory 15. Georg Schiemer: Carnap's Structuralist Thesis 16. Sean Morris: Explication as Elimination: W.V. Quine and Mathematical Structuralism