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WEYL REEXAMINED: “DAS KONTINUUM” 100 YEARS LATER

Part of: History of mathematics and mathematicians Philosophical aspects of logic and foundations General logic

Published online by Cambridge University Press:  10 June 2020

ARNON AVRON
ARNON AVRON*
Affiliation:
SCHOOL OF COMPUTER SCIENCE TEL AVIV UNIVERSITYTEL AVIV6997801, ISRAEL E-mail: aa@cs.tau.ac.il
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Abstract

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Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918, he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book, he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’.

In this paper, we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal.

Keywords

WeylDas Kontinuumpredicative mathematicsfoundations of mathematics

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) 01A60: 20th century
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 1 , March 2020 , pp. 26 - 79
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Association for Symbolic Logic

References

REFERENCES

Abiteboul, S., Hull, R., and Vianu, V., Foundations of Databases , Addison-Wesley, Boston, 1995.Google Scholar
Adams, R. and Luo, Z., Weyl’s predicative classical mathematics as a logic enriched type theory , Types for Proofs and Programs: International Workshop, Types 2006 (T. Altenkirch and C. McBride, editors), Lecture Notes in Computer Science, vol. 4502,Springer, New York, 2007, pp. 1–17.Google Scholar
Adams, R. and Luo, Z., Classical predicative logic-enriched type theories . Annals of Pure and Applied Logic , vol. 161 (2010), pp. 1315–1345.CrossRefGoogle Scholar
Adams, R. and Luo, Z., Weyl’s predicative classical mathematics as a logic-enriched type theory . ACM Transactions on Computational Logic , vol. 11 (2010), pp. 1–29.CrossRefGoogle Scholar
Avron, A., Transitive closure and the mechanization of mathematics , Thirty Five Years of Automating Mathematics, (F. D. Kamareddine, editor), Applied Logic Series, vol. 28, Springer, New York, 2003, pp. 149–171.CrossRefGoogle Scholar
Avron, A., Safety signatures for first-order languages and their applications , First-Order Logic Revisited (V. Hendricks, F. Neuhaus, U. Scheffler, S. A. Pedersen, and H. Wansing, editors), Logos Verlag Berlin, Germany, 2004, pp. 37–58.Google Scholar
Avron, A., Constructibility and decidability versus domain independence and absoluteness . Theoretical Computer Science , vol. 394 (2008), no. 3, pp. 144–158.CrossRefGoogle Scholar
Avron, A., A new approach to predicative set theory , Ways of Proof Theory (R. Schindler, editor), Ontos Series in Mathematical Logic, vol. 2, De Gruyter, Berlin, 2010, pp. 31–63.Google Scholar
Avron, A., Why predicative sets? , Fields of Logic and Computation III (Gurevich’ Festschrift) , Lectures Notes in Computer Science, vol. 12810, Springer, New York, 2020, pp. 30–45.CrossRefGoogle Scholar
Avron, A. and Cohen, L., Formalizing scientifically applicable mathematics in a definitional framework . Journal of Formalized Reasoning , vol. 9 (2016), no. 1, pp. 53–70.Google Scholar
Avron, A., Lev, S., and Levi, N., Safety, absoluteness, and computability, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018) (D. Ghica and A. Jung, editors), Leibniz International Proceedings in Informatics (LIPIcs), vol. 119, Schloss Dagstuhl–Leibniz-Zentrum füer Informatik, Germany, 2018, pp. 8:1–8:17.Google Scholar
Bernard, J., Notes on the first chapter of ‘Das Kontinuum’: Intention, extension and arithmetism . Philosophia Scientiae , vol. 13 (2009), pp. 155–176.CrossRefGoogle Scholar
Cohen, S. and Avron, A., The middle ground–ancestral logic . Synthese , vol. 196 (2019), pp. 2671–2693.CrossRefGoogle Scholar
Crosilla, L., Predicativity and Feferman, Feferman and Foundations , Outstanding Contributions to Logic, Springer, New York, 2018, pp. 423–447.Google Scholar
Ewald, W., From Kant to Hilbert , Clarendon Press, Oxford, 1996.Google Scholar
Feferman, S., Systems of predicative analysis . The Journal of Symbolic Logic , vol. 29 (1964), no. 1, pp. 1–30.CrossRefGoogle Scholar
Feferman, S., Weyl vindicated: Das Kontinuum 70 years later , Remi e Prospettive Della Logica e Della Scienza Contemporanee , vol. I (1988), pp. 59–93, Reprinted in [18].Google Scholar
Feferman, S., In the Light of Logic . Logic and Computation in Philosophy, Oxford University Press, Oxford, 1998.Google Scholar
Feferman, S., The significance of Hermann Weyl’s ‘Das Kontinuum’, Proof Theory—Historical and Philosophical Significance (V. Hendricks, S. A. Pedersen, and K. F. Jørgensen, editors), Synthese Library, vol. 292, Kluwer Academic Publishers, Berlin, 2000, pp. 179–194.Google Scholar
Feferman, S., The development of programs for the foundations of mathematics in the first third of the 20th century , Storia Della Scienza, vol. 8 (S. Petruccioli, editor), Istituto Della Enciclopedia Italiana, Italy, 2004, pp. 112–121, Appeared in translation as ‘Le scuole di Filosofia della Matematica’.Google Scholar
Feferman, S., Predicativity, The Oxford Handbook of the Philosophy of Mathematics and Logic (S. Shapiro, editor), Oxford University Press, Oxford, 2005, pp. 590–624.CrossRefGoogle Scholar
Hatcher, W. S., The Logical Foundation of Mathematics , Pergamon Press, Oxford, 1982.Google Scholar
Hölder, O., Der angebliche circulus vitiosus und die sogennante Grundlagenkrise in der Analysis . Sitzungsberichte der Leipziger Akademie , vol. 78 (1926), pp. 243–250, English translation in [27, pp. 143–148].Google Scholar
Kunen, K., Set Theory , Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
Landau, E., Foundations of Analysis , Chelsea Publishing Company, New York, 1951, Translated from German ‘Grundlagen der Analysis’ by F. Steinhardt.Google Scholar
Longo, G., Interfaces of incompleteness , Systemics of Incompleteness and Quasi-Systems (G. Minati, M. Abram, and E. Pessa, editors), Springer, Cham, 2019, pp. 3–55.CrossRefGoogle Scholar
Mancosu, P., From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in 1920s , Oxford University Press, Oxford, 1998.Google Scholar
Mancosu, P., The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900–1940 , Oxford University Press, Oxford, 2010.CrossRefGoogle Scholar
Martin, R. M., A homogeneous system for formal logic . The Journal of Symbolic Logic , vol. 8 (1943), no. 1, pp. 1–23.CrossRefGoogle Scholar
Myhill, J., A derivation of number theory from ancestral theory . The Journal of Symbolic Logic , vol. 17 (1952), no. 3, pp. 192–197.CrossRefGoogle Scholar
Parsons, C., Realism and the debate on impredicativity, 1917–1944, Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman (W. Sieg, R. Sommer, and C. Talcott, editors), Lecture Notes in Logic, vol. 15, Association for Symbolic Logic, A. K. Peters Ltd., Mansfield, 2002, reprinted in [32, pp. 372–389].Google Scholar
Parsons, C., Philosophy of Mathematics in the Twentieth Century , Harvard University Press, Cambridge, MA, 2014.CrossRefGoogle Scholar
Poincaré, H., Les mathématiques et la logique, II, III . Revue de Métaphysique et Morale , vol. 14 (1906), pp. 17–34, 294–317, translated in [15].Google Scholar
Poincaré, H., La logique de l’infi . Revue de Métaphysique et Morale , vol. 17 (1909), pp. 461–482.Google Scholar
Poincaré, H., Dernière pensées, Flammarion, 1913 , translated by J. Bolduc as Mathematics and Science: Last Essays , Dover Press, New York, 1963.Google Scholar
Pólya, G., Eine erinnerung an Hermann Weyl . Mathematische Zeitschrift , vol. 126 (1972), pp. 296–298.CrossRefGoogle Scholar
Schütte, K., Proof Theory , Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
Shapiro, S., Foundations without Foundationalism: A Case for Second-Order Logic , Oxford University Press, Oxford, 1991.Google Scholar
Shoenfield, J. R., Mathematical Logic , Addison-Wesley, Boston, 1967.Google Scholar
Simpson, S. G., Subsystems of Second-Order Arithmetic , Springer-Verlag, Berlin, 1999.CrossRefGoogle Scholar
Ullman, J. D., Principles of Database and Knowledge-Base Systems , Computer Science Press, New York, 1988.Google Scholar
Weaver, N., Predicativity beyond $\varGamma_{0}$ , 2009, arXiv:math/0509244v3 [math.LO].Google Scholar
Weyl, H., Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis , Veit, Leipzig, 1918, Translated into English in [48].Google Scholar
Weyl, H., Der circulus vitiosus in der heutigen Begründung der Analysis . Jahresbericht der Deutschen Mathematiker-Vereinigung , vol. 28 (1919), pp. 85–93, English translation in [48].Google Scholar
Weyl, H., Über der neue Grundslagenkrise der Mathematik . Mathematische Zeitschrift , vol. 10 (1921), pp. 37–79, English translation in [27, pp. 86–118].CrossRefGoogle Scholar
Weyl, H., Mathematics and logic: A brief survey serving as a preface to a review of the philosophy of Bertrand Russell . American Mathematical Monthly , vol. 53 (1946), pp. 2–79.Google Scholar
Weyl, H., Philosophy of mathematics and natural science , Princeton University Press, 1949, An expanded English version of Philosophie der Mathematik und Naturwissenschaft , Leibniz Verlag, München, 1927.Google Scholar
Weyl, H., The Continuum: A Critical Examination of the Foundation of Analysis , Thomas Jefferson University Press, Kirksville, Missouri, 1987, translated by S. Pollard and T. Bole.Google Scholar
Whitehead, A. N. and Russell, B., Principia Mathematica, 3 vols. , second ed., Cambridge University Press, Cambridge, 1910–1932, pp. 1925–1927.Google Scholar