Article ID: CBB069750857

Mathematical Formalization and Diagrammatic Reasoning: The Case Study of the Braid Group Between 1925 and 1950 (2019)


The standard historical narrative regarding formalism during the twentieth century indicates the 1920s as a highpoint in the mathematical formalization project. This was marked by Hilbert’s statement that the sign stood at the beginning of pure mathematics [‘Neubegründung der Mathematik. Erste Mitteilung’, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1 (1922), 157–177]. If one takes the braid group as a case study of research whose official goal was to symbolically formalize braids and weaving patterns, a reconsideration of this strict definition of formalism is nevertheless required. For example, does it reflect what actually occurred in practice in the mathematical research of this period? As this article shows, the research on the braid group between 1926 and 1950, led among others by Artin, Burau, Fröhlich and Bohnenblust, was characterized by a variety of practices and reasoning techniques. These were not only symbolic and deductive, but also diagrammatic and visual. Against the historical narrative of formalism as based on a well-defined chain of graphic signs that has freedom of interpretation, this article presents how these different ways of reasoning—which were not only sign based—functioned together within the research of the braid group; it will be shown how they are simultaneously necessary and complementary for each other.

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Authors & Contributors
Colyvan, Mark
Heijenoort, Jean van
Bernays, Paul
Cléro, Jean-Pierre
Formica, Giambattista
Schirn, Matthias
History and Philosophy of Logic
Journal Electronique d'Histoire des Probabilités et de la Statistique
Perspectives on Science
Science and Education
Cambridge University Press
Harvard University Press
J. Vrin
Pensa Multimedia
Philosophy of mathematics
Philosophy of science
Probability and statistics
Frege, Gottlob
Gödel, Kurt
Quine, Willard Van Orman
Wittgenstein, Ludwig
Carnap, Rudolf
Russell, Bertrand Arthur William
Time Periods
20th century
19th century
21st century
18th century

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