Article ID: CBB001213914

The Ways of Hilbert's Axiomatics: Structural and Formal (2014)


Sieg, Wilfried (Author)

Perspectives on Science
Volume: 22, no. 1
Issue: 1
Pages: 133-157

Publication Date: 2014
Edition Details: Part of a special issue, “Hilbert's Axiomatics---Geometry, Physics, Logic”
Language: English

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as one direction for fascinating work.

Included in

Article Lupacchini, Rossella (2014) Hilbert's Axiomatics as “Symbolic Form”?. Perspectives on Science (pp. 1-34). unapi

Citation URI

Similar Citations

Article Abrusci, V. Michele; (2014)
On Hilbert's Axiomatics of Propositional Logic (/isis/citation/CBB001213913/)

Article Stillwell, John; (2014)
Ideal Elements in Hilbert's Geometry (/isis/citation/CBB001213909/)

Article Rédei, Miklós; (2014)
Hilbert's 6th Problem and Axiomatic Quantum Field Theory (/isis/citation/CBB001213911/)

Article Majer, Ulrich; (2014)
The “Axiomatic Method” and Its Constitutive Role in Physics (/isis/citation/CBB001213910/)

Book Gabriele Lolli; (2016)
Tavoli, sedie, boccali di birra: David Hilbert e la matematica del Novecento (/isis/citation/CBB100106060/)

Book Gauthier, Yvon; (2002)
Internal Logic: Foundations of Mathematics from Kronecker to Hilbert (/isis/citation/CBB000301820/)

Book Hilbert, David; Ewald, William; Sieg, Wilfried; Hallett, Michael; Majer, Ulrich; Schlimm, Dirk; (2013)
David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917--1933 (/isis/citation/CBB001551530/)

Article Lupacchini, Rossella; (2014)
Hilbert's Axiomatics as “Symbolic Form”? (/isis/citation/CBB001213908/)

Book Elena Anne Corie Marchisotto; Francisco Rodriguez-Consuegra; James T. Smith; (2021)
The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics (/isis/citation/CBB763111584/)

Book Gandon, Sébastien; (2012)
Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics (/isis/citation/CBB001201203/)

Article Willea, Matthias; (2011)
“Metamathematics” in Transition (/isis/citation/CBB001211006/)

Article Paseau, Alexander; (2011)
Mathematical Instrumentalism, Gödel's Theorem, and Inductive Evidence (/isis/citation/CBB001024149/)

Book William Boos; Florence S. Boos; (2018)
Metamathematics and the Philosophical Tradition (/isis/citation/CBB060593902/)

Chapter Mancosu, Paolo; Zach, Richard; Badesa, Calixto; (2009)
The Development of Mathematical Logic from Russell to Tarski, 1900--1935 (/isis/citation/CBB001210349/)

Article Pejlare, Johanna; (2007)
Torsten Brodén's Work on the Foundations of Euclidean Geometry (/isis/citation/CBB000773211/)

Chapter Hallett, Michael; (2010)
Frege and Hilbert (/isis/citation/CBB001211657/)

Authors & Contributors
Hallett, Michael
Majer, Ulrich
Gauthier, Yvon
Pejlare, Johanna
Corry, Leo
Cerroni, Cinzia
Perspectives on Science
Historia Mathematica
Revue d'Histoire des Mathématiques
Studies in History and Philosophy of Science
History and Philosophy of Logic
Kluwer Academic
Palgrave Macmillan
Cambridge University Press
de Gruyter
Raffaello Cortina Editore
Philosophy of mathematics
Formalization (philosophy)
Hilbert, David
Russell, Bertrand Arthur William
Gödel, Kurt
Kronecker, Leopold
Brodén, Torsten
Enriques, Federigo
Time Periods
20th century, early
19th century
20th century
20th century, late
Great Britain

Be the first to comment!

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

Log in or register to comment