Periodic Reporting for period 1 - FORMALISM (The Formal Turn - The Emergence of Formalism in Twentieth-Century Thought)
Periodo di rendicontazione: 2022-09-01 al 2025-02-28
The philosophy of science and logic underwent a wide-ranging formal turn in the first half of the twentieth century. This shift is characterized by a new emphasis on formal methods and the adoption of a general formalist viewpoint towards the disciplines in question. Roughly put, formalism presented a novel methodological and epistemological stance: it is characterized by the use of non-representational forms of reasoning as well as the focus on the purely formal features of a field of inquiry, studied in abstraction from its traditional subject matter and from considerations about meaning or interpretation. As such, the position was often linked to a new understanding of the nature of scientific objectivity and rigor. Moreover, formalist positions not only dominated research at the time, but also led to fundamentally revised conceptions of logic, mathematics, and the philosophy of the exact sciences, determining the shape of these disciplines today.
The project will give an interdisciplinary study of early contributions to formalism in twentieth-century thought. This general objective will be addressed in terms of three interrelated projects. The first subproject is historical in character and aims at a historical reconstruction of the emergence of formalist thinking in nineteenth-century mathematics and neo-Kantian philosophy of science. On the mathematical side, the project will retrace several developments in projective geometry, symbolic algebra, and number theory (Peacock, Hankel, von Staudt) in the nineteenth century that contributed to a formalist conception of these fields. Regarding its philosophical roots, the focus will be on contributions to a formalist notion of scientific objectivity in the work of the Marburg school of neo-Kantianism (Ernst Cassirer, Paul Natorp). The second project will give a comparative study of different formal turns in logic, the foundations of mathematics, and philosophy of logical empiricism between 1920 and 1940. This includes research on the foundations of mathematics and the formality of logic by members of the Göttingen school (David Hilbert, Paul Bernays) as well as central contributions to a scientific formalism in logical empiricism (Rudolf Carnap, Moritz Schlick). The third project will analyze the philosophical implications of the formal turn for the subsequent shaping of these fields. The emergence of formalist thinking was closely related to the development of new theories of formal semantics in logic and philosophy. The aim will therefore be to connect the early contributions to formalism with contemporary debates on the philosophy of model theory and inferentialism as well as with the logical study of scientific theories.
The project will give an interdisciplinary study of early contributions to formalism in twentieth-century thought. This general objective will be addressed in terms of three interrelated projects. The first subproject is historical in character and aims at a historical reconstruction of the emergence of formalist thinking in nineteenth-century mathematics and neo-Kantian philosophy of science. On the mathematical side, the project will retrace several developments in projective geometry, symbolic algebra, and number theory (Peacock, Hankel, von Staudt) in the nineteenth century that contributed to a formalist conception of these fields. Regarding its philosophical roots, the focus will be on contributions to a formalist notion of scientific objectivity in the work of the Marburg school of neo-Kantianism (Ernst Cassirer, Paul Natorp). The second project will give a comparative study of different formal turns in logic, the foundations of mathematics, and philosophy of logical empiricism between 1920 and 1940. This includes research on the foundations of mathematics and the formality of logic by members of the Göttingen school (David Hilbert, Paul Bernays) as well as central contributions to a scientific formalism in logical empiricism (Rudolf Carnap, Moritz Schlick). The third project will analyze the philosophical implications of the formal turn for the subsequent shaping of these fields. The emergence of formalist thinking was closely related to the development of new theories of formal semantics in logic and philosophy. The aim will therefore be to connect the early contributions to formalism with contemporary debates on the philosophy of model theory and inferentialism as well as with the logical study of scientific theories.
Research on the project will be structered in terms of three interrelated work packages. The first work package will analyze several central strands in nineteenth-century mathematics as well as in neo-Kantian philosophy that had a formative impact on the subsequent formal turn in the philosophy of logical empiricism and in logic. The period investigated will range from 1830 until 1910. Research on the present subproject will be guided by the following two core hypotheses: (a) Nineteenth-century research on symbolic algebra, number theory, and geometry is characterized by the invention of several formalist methods that led to a novel conception of the nature and content of mathematical theories. (b) Neo-Kantian philosophy of the Marburg school was strongly influenced by these mathematical developments and contains a first formulation of a formalist epistemology of science. Work package (2) will give a comparative study of formalist approaches in mathematical logic and the philosophy of science in the period between 1920 until 1950. Research on this work package will be guided by the following two core hypotheses: (c) One can identify several ways to characterize the formal content of deductive theories in the different research traditions in twentieth-century logic and in the “formalist” foundation of mathematics. (d) Theories of scientific formalism related to logical empiricism were directly rooted in the mathematical and philosophical developments investigated in work package (1). Work package (3) will study the wide-ranging philosophical implications of the formal turn for the philosophy of logic and mathematics as well as for formal philosophy of science. Research on the present subproject will be guided by two core hypotheses: (e) The theoretical developments surveyed in subprojects 1 and 2 suggest two general ways to think about formality or formal content, namely in terms of invariance and definability conditions. (f) The formality thesis is historically and conceptually tied to the rise of new theories of semantics, namely model theory and inferentialism.
The project will have a significant impact on scholarship in each of the fields investigated and, more generally, on the intellectual history of the twentieth century. Three key points should be mentioned here. First, the project will provide the first systematic study of the development and implementation of formalist methods in nineteenth-century mathematics and in neo-Kantian philosophy. As such, it will change our present account of the historical development of these disciplines. Second, it will lead to a novel and much refined understanding of how formal philosophy of science, the foundations of mathematics, and mathematical logic evolved in the first decades of the twentieth century. In particular, the project will show in detail how these disciplines interacted in the period studied. Moreover, research on the project aims to establish formalism as a central strand or paradigm in philosophical and logical thought in the twentieth century. Finally, the project has the potential to change our current understanding of formalism in the philosophy of logic, mathematics and the sciences. In particular, it will likely develop new theoretical contributions to systematic issues currently under discussion in these fields, for instance, on the adequate explication of key notions such as the formal content of statements in mathematics and logic or criteria of formal equivalence of mathematical or scientific theories.