Article ID: CBB001211483

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

unapi

Smadja, Ivahn (Author)


Synthese
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.

...More
Included in

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

Citation URI
http://data.isiscb.org/isis/citation/CBB001211483/

Similar Citations

Article Schirrmacher, Arne; (2003)
Planting in his Neighbor's Garden: David Hilbert and Early Göttingen Quantum Physics (/isis/citation/CBB000641699/) unapi

Article Rowe, David E.; (2001)
Einstein Meets Hilbert: At the Crossroads of Physics and Mathematics (/isis/citation/CBB000102532/) unapi

Article Dunlop, Katherine; (2012)
The Mathematical Form of Measurement and the Argument for Proposition I in Newton's Principia (/isis/citation/CBB001211480/) unapi

Article Wright, Aaron Sidney; (2014)
The Advantages of Bringing Infinity to a Finite Place: Penrose Diagrams as Objects of Intuition (/isis/citation/CBB001201044/) unapi

Article Valente, Giovanni; (2008)
John Von Neumann's Mathematical “Utopia” in Quantum Theory (/isis/citation/CBB000933675/) unapi

Book John von Neumann; R. Lupacchini; G. Gottardi; (2018)
Metamatematica hilbertiana e fondamenti della meccanica quantistica (/isis/citation/CBB497992538/) unapi

Article Yan, Chen-guang; Deng, Ming-li; (2009)
Riemann's Idea of Geometry and its Impact on the Theory of Relativity (/isis/citation/CBB000952285/) unapi

Article Maanen, Jan van; (2006)
Diagrams and Mathematical Reasoning: Some Points, Lines, and Figures (/isis/citation/CBB000850085/) unapi

Article Nascimento, Carlos Arthur Ribeiro do; (2008)
Avicena e as Ciências Mistas (/isis/citation/CBB000950536/) unapi

Chapter Celeyrette, Jean; (2008)
Bradwardine's Rule: A Mathematical Law? (/isis/citation/CBB000760472/) unapi

Article González Redondo, Francisco A.; (2003)
La contribución de Leonard Euler a la matematización de las magnitudes y las leyes de la mecánica, 1736--1765 (/isis/citation/CBB000530003/) unapi

Article Davey, Kevin; (2003)
Is Mathematical Rigor Necessary in Physics? (/isis/citation/CBB000410734/) unapi

Article Raynaud, Dominique; (2014)
Building the Stemma Codicum from Geometric Diagrams (/isis/citation/CBB001321046/) unapi

Book Petkov, Vesselin; (2010)
Minkowski Spacetime: A Hundred Years Later (/isis/citation/CBB001032084/) unapi

Book Gray, Jeremy; Parshall, Karen Hunger; (2007)
Episodes in the History of Modern Algebra (1800--1950) (/isis/citation/CBB000774194/) unapi

Authors & Contributors
Rowe, David E.
Davey, Kevin
González Redondo, Francisco A.
Corry, Leo
Schirrmacher, Arne
Celeyrette, Jean
Journals
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
Publishers
Kluwer Academic
American Mathematical Society
Springer
Bibliopolis
Concepts
Physics
Mathematics
Mathematics and its relationship to science
Mathematical physics
Diagrams
Geometry
People
Hilbert, David
Minkowski, Hermann
Einstein, Albert
Bradwardine, Thomas
Von Neumann, John
Levi-Civita, Tullio
Time Periods
20th century, early
19th century
Medieval
17th century
11th century
20th century
Places
Netherlands
China
Institutions
Universität Göttingen
University of Chicago
Comments

Be the first to comment!

{{ comment.created_by.username }} on {{ comment.created_on | date:'medium' }}

Log in or register to comment