Hilbert, David [addendum]
Bernays's entry on Hilbert still reads, after forty years, as a wonderful account of the essential contributions by Hilbert to the foundations of geometry and proof theory. However, recent developments have substantially increased the understanding of Hilbert's original investigations and pushed these investigations further. The following bibliography will help the reader navigate among the most important recent contributions. It is divided into four parts: (1) contributions to Hilbert's biography and mathematical work emerging from Hilbert's famous list of problems given in Paris in 1900; (2) historical work related to Hilbert's foundational views; (3) logico-foundational and philosophical developments related to Hilbert's program; and (4) ongoing work of publication of Hilbert's and Bernays's work.
Mathematics, Foundations Of.
Bibliography
Biography and Mathematical Developments Emerging from Hilbert's Problems
The only book-length biography available is Constance Reid's Hilbert (New York: Springer, 1970). For mathematical developments arising from Hilbert's list of 23 problems in Paris, see Jeremy Gray's The Hilbert Challenge (Oxford: Oxford University Press, 2000) and Benjamin H. Yandell's The Honors Class: Hilbert's Problems and Their Solvers (Natick, MA: A. K. Peters, 2002).
Historical Studies of Hilbert's Foundational Work
There has been a substantial amount of work in the study of the development of Hilbert's foundational views. A novelty in this area is the study of his work on the foundations of physics. Most of this scholarship is characterized by the extensive use of the Hilbert Nachlaß in Göttingen. The work has been divided below by periods.
Foundations of Geometry (1891–1899)
Toepell, Michael-Markus. Über die Entstehung von Hilberts "Grundlagen der Geometrie." Göttingen, Germany: Vandenhoeck und Ruprecht, 1986.
Axiomatizations of Physical Theories (1895–1917)
Corry, Leo. "David Hilbert and the Axiomatization of Physics." Archive for History of Exact Sciences 51 (1997): 83–198.
Corry, Leo. "Hilbert and Physics (1900–1915)." In The Symbolic Universe: Geometry and Physics (1890–1930), edited by Jeremy Gray, 145–187. New York, Oxford University Press, 1999.
Majer, Ulrich. "The Axiomatic Method and the Foundations of Science: Historical Roots of Mathematical Physics in Göttingen (1900–1930)." In John von Neumann and the Foundations of Quantum Physics, edited by Miklos Redi and Michael Stoeltzner, 11–33. Dordrecht, Netherlands: Kluwer, 2001.
Sauer, Tilman. "The Relativity of Discovery: Hilbert's First Note on the Foundations of Physics." Archive for History of Exact Sciences 53 (1999): 529–575.
From Axiomatics to Proof Theory (1900–1922)
Peckhaus, Volker. Hilbertprogramm und kritische Philosophie. Göttingen, Germany: Vandenhoeck und Ruprecht, 1990.
Mancosu, Paolo. "Between Russell and Hilbert: Behmann on the Foundations of Mathematics." The Bulletin of Symbolic Logic 5 (3) (1999): 303–330.
Sieg, Wilfried. "Hilbert's Programs: 1917–1922." The Bulletin of Symbolic Logic 5 (1999): 1–44.
Mancosu, Paolo. "The Russellian Influence on Hilbert and His School." Synthese 137 (2003): 59–101.
Development of Logic in Hilbert's School (1905–1928)
Hallett, Michael. "Hilbert and Logic." In Québec Studies in the Philosophy of Science, pt. 1, vol. 177, edited by Mathieu Marion and Robert Cohen, 135–187. Dordrecht, Netherlands: Kluwer, 1995.
Peckhaus, Volker. "Hilberts Logik. Von der Axiomatik zur Beweistheorie." Intern. Zs. f. Gesch. u. Ethik der Naturwiss. Tech. u. Med. 3 (1995): 65–86.
Zach, Richard. "Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic." The Bulletin of Symbolic Logic 5 (3) (1999): 331–366.
Moore, Gregory "Hilbert and the Emergence of Modern Mathematical Logic." Theoria 12 (1) (1997): 65–90.
Avigad, Jeremy, and Richard Zach. "The Epsilon Calculus." Stanford Encyclopedia of Philosophy. Available at http://plato.stanford.edu/entries/epsil on-calculus/.
Proof Theory and Finitism Until Gödel's Theorems (1922–1931)
Majer, Ulrich. "Hilberts Methode der idealen Elemente und Kants regulativer Gebrauch der Ideen." Kant-Studien 84 (1993): 51–77.
Mancosu, Paolo. "Hilbert and Bernays on Metamathematics." In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, pp. 149–188. New York: Oxford University Press, 1988.
Zach, Richard. 2003 "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program." Synthese 137 (2003): 211–259.
Logico-Foundational and Philosophical Contributions (1970–2003)
Logico-Foundational Developments
The best starting point are the articles from "A Symposium on Hilbert's Program":
Feferman, Solomon. "Hilbert's Program Relativized: Proof-Theoretical and Foundational Reductions." Journal of Symbolic Logic 53 (1988): 364–383.
Simpson, Steven. "Partial Realizations of Hilbert's Program." Journal of Symbolic Logic 53 (1988): 349–3631.
Sieg, Wilfried. "Hilbert's Program Sixty Years Later." Journal of Symbolic Logic 53 (1988): 338–348.
For further references see Richard Zach's "Hilbert's Program" in the Stanford Encyclopedia of Philosophy. Available at http://plato.stanford.edu/entries/hilbe rt-program/.
Philosophical Contributions
Kitcher, Philip. "Hilbert's Epistemology." Philosophy of Science 43 (1976): 99–115.
Tait, William. "Finitism." Journal of Philosophy 78 (1981): 524–546.
Detlefsen, Michael. Hilbert's Programme: An Essay on Mathematical Instrumentalism. Dordrecht, Netherlands: Reidel, 1986.
Hallett, Michael. "Physicalism, Reductionism and Hilbert." In Physicalism in Mathematics, edited by Andrew D. Irvine, 183–257. Dordrecht, Netherlands: Kluwer, 1990.
Parsons, Charles. "Finitism and Intuitive Knowledge." In The Philosophy of Mathematics Today, edited by Matthias Schirn, 249–270. Oxford: Oxford University Press, 1998.
Translations and Editions of Hilbert and Bernays
Van Hejenoort 1967, Ewald 1996, and Mancosu 1998 provide an extensive coverage of translations into English of most of Hilbert's and Bernays's published articles on the foundations of mathematics for the period 1900–1931:
Ewald, William. From Kant to Hilbert: Readings in the Philosophy of Mathematics. Oxford: Oxford University Press, 1996.
Mancosu, Paolo., ed. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. New York: Oxford University Press, 1998.
van Heijenoort, Jean. 1967 From Frege to Gödel. Cambridge, MA: Harvard University Press, 1967.
In addition there are two scholarly editions in the making. The Hibert Edition (six volumes, Springer Verlag) includes a selection of the original unpublished lecture notes (in German) preserved at the University of Göttingen. The first volume, edited by Michael Hallett, appeared in 2004. There is also in preparation an edition of Bernays's foundational writings in English, a description of which is available at http://www.phil.cmu.edu/projects/bernay s/. This edition is scheduled to appear some time after 2004 for Open Court.
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