Mathematical Formalization and Diagrammatic Reasoning: The Case Study of the Braid Group Between 1925 and 1950 (2019)
Friedman, Michael (Author)
British Journal for the History of Mathematics
Volume: 34
Issue: 1
Pages: 43-59
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.
...MoreSimilar Citations
Article
Kraal, Anders;
(2013)
The Emergence of Logical Formalization in the Philosophy of Religion: Genesis, Crisis, and Rehabilitation
(/isis/citation/CBB001213917/)
Article
Peter Olen;
(2016)
From Formalism to Psychology: Metaphilosophical Shifts in Wilfrid Sellars’s Early Works
(/isis/citation/CBB228770461/)
Article
Yemima Ben-Menahem;
(2016)
Poincaré’s Impact on Twentieth-Century Philosophy of Science
(/isis/citation/CBB415413095/)
Book
Mario Castellana;
(2019)
Federigo Enriques e la “Nuova Epistemologia”
(/isis/citation/CBB605748137/)
Article
Manzano, Maria;
Alonso, Enrique;
(2014)
Completeness: From Gödel to Henkin
(/isis/citation/CBB001213927/)
Chapter
Carlo Casolo;
(2018)
La Biblioteca Universale
(/isis/citation/CBB604701261/)
Essay Review
Klement, Kevin C.;
(2007--2008)
A New Century in the Life of a Paradox
(/isis/citation/CBB001567033/)
Book
Colyvan, Mark;
(2001)
Indispensability of Mathematics
(/isis/citation/CBB000102198/)
Article
Lavers, Gregory;
(2013)
Frege, Carnap, and Explication: “Our Concern Here Is to Arrive at a Concept of Number Usable for the Purpose of Science”
(/isis/citation/CBB001212885/)
Article
Eberhardt, Frederick;
(2011)
Reliability via Synthetic a priori: Reichenbach's Doctoral Thesis on Probability
(/isis/citation/CBB001211469/)
Article
Formica, Giambattista;
(2010)
Von Neumann's Methodology of Science: From Incompleteness Theorems to Later Foundational Reflections
(/isis/citation/CBB001034599/)
Chapter
Weiner, Joan;
(2010)
Understanding Frege's Project
(/isis/citation/CBB001211648/)
Article
Marco Buzzoni;
(2021)
A Neglected Chapter in the History of Philosophy of Mathematical Thought Experiments: Insights from Jean Piaget’s Reception of Edmond Goblot
(/isis/citation/CBB742523599/)
Article
Jan Halák;
(2022)
Mathematics embodied: Merleau-Ponty on geometry and algebra as fields of motor enaction
(/isis/citation/CBB074170500/)
Book
Heijenoort, Jean van;
(2002)
From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931
(/isis/citation/CBB000201415/)
Article
Marquis, Jean-Pierre;
(2012)
Mario Bunge's Philosophy of Mathematics: An Appraisal
(/isis/citation/CBB001252236/)
Book
Bernays, Paul;
(2003)
Philosophie des mathématiques
(/isis/citation/CBB000501923/)
Article
Christine Chambris;
Jana Visnovska;
(2022)
On the history of units in French elementary school arithmetic: The case of proportionality
(/isis/citation/CBB935801822/)
Book
Juliette Kennedy;
(2014)
Interpreting Godel: Critical Essays
(/isis/citation/CBB427455296/)
Article
Iulian D. Toader;
(2021)
Why Did Weyl Think That Emmy Noether Made Algebra the Eldorado of Axiomatics?
(/isis/citation/CBB095893246/)
Be the first to comment!