BERND BULDT, VOLKER HALBACH and REINHARD KAHLE
REFLECTIONS ON FREGE AND HILBERT
The papers in this special issue of Synthese revolve around Gottlob Frege and David Hilbert, who are arguably the most prominent figures to have shaped the modern philosophy of mathematics. It is well-known that Frege’s system in the Basic Laws of Arithmetic suffers from inconsistency not only because of its notorious Axiom V, but also because in it Frege embraces a principle of full comprehension regarding concepts. Although Frege’s logicist program failed in its original form, there have been attempts to breath new life into it. Such attempts must, however, either introduce restrictions to Axiom V or to the comprehension principle for concepts in order to make Frege’s system consistent. Following a suggestion made by Michael Dummett (one of the early promoters of reviving logicism) Richard Heck was able to show that Frege’s system becomes consistent, if ramified predicative second-order logic is adopted. This result prompted further research and many authors tried particularly to establish the consistency of larger fragments as well. Fernando Ferreira, in his contribution below, proves that extending Heck’s fragment of Frege’s system with the axiom of reducibility for concepts, which are true only of a finite number of individuals, still produces a consistent system; a system that, as he was also able to show, can be used to interpret Peano’s elementary arithmetic. These consistent fragments of Frege’s system are sufficient for carrying out considerable portions of Frege’s attempted logicist reduction and much is known especially about the reconstruction of arithmetic within this fragment. Bob Hale, in his article, explores a route to advance these results by extending the logicist reduction to analysis—without taking the detour via set theory. Kai Wehmeier and Peter Schroeder-Heister discuss the so-called (and still controversial) “Permutation Argument” that Frege introduced to show that an arbitrary value-range may be identified with the True, any other with the False. They consider the two leading interpretations of that argument, the metalogical and the mathematical interpretations, and come to the result that the former is invalid, while the latter is not. The remaining four papers focus on the origin, the legacy, and the broader context of Hilbert’s philosophy of mathematics. The first of the four papers, by David McCarty, puts Hilbert’s well-know optimism (“We must know. We will know.”) in a new perspective by construing Hilbert’s Program as a response to Emil du Bois-Reymond’s skeptical doctrines Synthese (2005) 147: 1–2 DOI 10.1007/s11229-004-6203-9
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concerning the sciences, which were extended to mathematics by du BoisReymond’s brother Paul. Michael Rathjen investigates a certain extension of Hilbert’s original program: He investigates to what extent constructive consistency proofs can be given for systems of classical (infinitistic) mathematics. More precisely, he takes the constructive standpoint to be captured by Martin-Löf’s type theory and it is then asked what subsystems of second-order arithmetic can be reduced to Martin-Löf’s type theory. Dirk Schlimm and Wilfried Sieg have devoted their entire article to reconstructing the development of Dedekind’s analysis of natural numbers. It is the first in a series of publications which should show why and how Dedekind was a precursor to and a main source of inspiration for views Hilbert held on both mathematics and the foundations of mathematics. The last paper deals with intuitionism. In it Schlimm critically examines and tries to refute certain views put forward by Bill Tait in order to clarify Brouwer’s original ideas about what intuition is and the role it is supposed to play in science and mathematics. We would like to thank the authors of this issue and are indebted to the many referees who saw their papers through all the revisions. Most of the papers are based on talks given at the workshop Philosophy of Mathematics: A Centennary of Hilbert’s ‘Problem Address’, which was part of the conference GAP.4 organized by the “German Society for Analytical Philosophy” in Bielefeld in 2000. We gratefully acknowledge the help of the conference organizers and the financial support we received from the from research group Logic in Philosophy, supported by the DFG, at Konstanz and Tübingen. It was a pleasure to do it all; we hope you, the reader, will enjoy this issue as much as we enjoyed putting it together. Bernd Buldt Universität Konstanz Fachgruppe Philosophie Postfach 5560 D 21 78434 Konstanz, Germany E-mail:
[email protected] Volker Halbach New College Oxford, OX1 3BN, U.K. Reinhard Kahle Universidade Nova de Lisboa Departamento de Informática 2829-516 Caparica, Portugal E-mail:
[email protected] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
