Article ID: CBB001211483

Local Axioms in Disguise: Hilbert on Minkowski Diagrams (2012)


Smadja, Ivahn (Author)

Volume: 186, no. 1
Issue: 1
Pages: 315-370

Publication Date: 2012
Edition Details: Part of a special issue, “Diagrams in Mathematics: History and Philosophy”
Language: English

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski's Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating rigorously with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as drawn formulas, and formulas as written diagrams, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski's diagrammatic methods in number theory prompted Hilbert's axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert's assessment of Minkowski's diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.

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Article Mumma, John; Panza, Marco (2012) Diagrams in Mathematics: History and Philosophy. Synthese (pp. 1-5). unapi

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Authors & Contributors
Rowe, David E.
Davey, Kevin
González Redondo, Francisco A.
Corry, Leo
Schirrmacher, Arne
Celeyrette, Jean
Physics in Perspective
British Journal for the Philosophy of Science
Llull: Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas
Historia Mathematica
British Society for the History of Mathematics Bulletin
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
Kluwer Academic
American Mathematical Society
Mathematics and its relationship to science
Mathematical physics
Hilbert, David
Minkowski, Hermann
Einstein, Albert
Bradwardine, Thomas
Von Neumann, John
Levi-Civita, Tullio
Time Periods
20th century, early
19th century
17th century
11th century
20th century
Universität Göttingen
University of Chicago

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