Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-29T11:25:09.947Z Has data issue: false hasContentIssue false
  • English
  • Français

FREGE’S THEORY OF REAL NUMBERS: A CONSISTENT RENDERING

Part of: General and miscellaneous specific topics Proof theory and constructive mathematics Mathematical Logic and Foundations General logic Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  22 February 2021

FRANCESCA BOCCUNI  and
MARCO PANZA
FRANCESCA BOCCUNI
Affiliation:
VITA-SALUTE SAN RAFFAELE UNIVERSITY MILAN, ITALY E-mail: [email protected]
MARCO PANZA
Affiliation:
IHPST (CNRS AND UNIVERSITÉ DE PARIS 1, PANTHÉONE SORBONNE) PARIS, FRANCE E-mail: [email protected] CHAPMAN UNIVERSITY ORANGE, CA, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Frege’s definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege’s ill-fated Basic Law V. We restate Frege’s definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege’s own indications is possible at all.

Keywords

FregeFrege’s theory of real numbersfoundation of real analysisLogicismEuclid’s Principle

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical 03-03: Historical (must also be assigned at least one classification number from Section 01) 03B30: Foundations of classical theories (including reverse mathematics) 03F35: Second- and higher-order arithmetic and fragments
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 624 - 667
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Adeleke, S. A., Dummett, M. A. E, & Neumann, P. M. (1987). On a question of Frege’s about right-ordered groups. Bulletin of the London Mathematical Society, 19, 513–521.CrossRefGoogle Scholar
Antonelli, A. & May, R. (2005). Frege’s other program. Notre Dame Journal of Formal Logic, 46(1), 1–17.CrossRefGoogle Scholar
Batitsky, V. (2002). Some measurement-theoretic concerns about Hale’s ‘reals by abstraction’. Philosophia Mathematica, 10(3), 286–303.CrossRefGoogle Scholar
Blass, A. (1984). Existence of bases implies the axiom of choice. In Baumgartner, J. E., Martin, D. A., & Shelah, S., editors. Axiomatic Set Theory. Vol. 31 of Contemporary Mathematics. Providence, RI: AMS, pp. 31–33.CrossRefGoogle Scholar
Błaszczyk, P. (2013). Nota o rozprawie Otto Höldera “Die Axiome der Quantität und die Lehre vom Mass”. Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, 5, 129–144.Google Scholar
Boolos, G. (1998a). Iteration again, in [8], pp. 88–104.Google Scholar
Boolos, G. (1998b). Saving Frege from contradiction, in [8], pp. 171–182.Google Scholar
Boolos, G. (1998c). Logic, Logic, and Logic. Burgess, J., & Jeffrey, R., editors. Cambridge, MA: Harvard University Press.Google Scholar
Bourbaki, N. (n.d.) Elements of Mathematics, Algebra I. Berlin, Heidelberg: Springer.Google Scholar
Conti, L. (2019). Paradossi di Russell e programmi astrazionisti. Spiegazioni e soluzioni a confronto, PhD thesis. FINO, Northwestern Philosophy Consortium, University of Pavia.Google Scholar
de Bruijn, N. G. (1957). Embedding theorems for infinite groups. Nederlandse Akademie van Wetenschappen. Proceedings, Series A, 60 (=Indagationes Mathematicae), 19(5), 560–570.Google Scholar
de Bruijn, N. G. (1964). Addendum to ‘Embedding Theorems for Infinite Groups’. Nederlandse Akademie van Wetenschappen. Proceedings , Series A, 67 (=Indagationes Mathematicae), 26(5), 594–595.Google Scholar
Dummett, M. (1991). Frege: Philosophy of Mathematics. London: Duckworth.Google Scholar
Euclid (1908). The Thirteen books of Euclid’s Elements. London: Duckworth. Translated with Introduction and Commentary by T. L. Heath, Cambridge University Press, Cambridge (3 vols.).Google Scholar
Frege, G. (1884). Grundlagen der Arithmetik, W. Breslau: Köbner.Google Scholar
Frege, G. (1893–1903). Die Grundgesetze der Arithmetik. 2 Vols. Jena: Verlag von H. Pohle.Google Scholar
Frege, G. (1953). The Foundations of Arithmetic (translated by Austin, J. L.). Oxford: Blackwell.Google Scholar
Frege, G. (2013). Basic Laws of Arithmetic (translated by Ebert, P. A. & Rossberg, M.). Oxford: Oxford University Press.Google Scholar
Gauss, C. F. (1831). Theoria residuorum biquadraticorum. Commentatio secunda. Göttingische gelehrete Anzeigen, I(64), 624–638.Google Scholar
Gauss, C. F. (1863–1933). Werke. Leipzig-Berlin: Königlichen Gesellshaft der Wissenshaften zu Göttingen (12 vols).Google Scholar
Hale, B. (2000). Reals by abstraction, Philosophia Mathematica, 8, 100–123.CrossRefGoogle Scholar
Hale, B. (2002). Real numbers, quantities, and measurement. Philosophia Mathematica, 10(3), 304–323.CrossRefGoogle Scholar
Herrlich, H. (2006). Axiom of Choice. Berlin-Heidelberg: Springer.Google Scholar
Hölder, O. (1901). Die axiome der quantität und die lehre vom mass. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physischen Classe, 53, 1–64.Google Scholar
Karrass, A., & Solitar, D. (1956). Some remarks on the infinite symmetric groups, Mathemathische Zeitschrift, 66, 64–69.CrossRefGoogle Scholar
Mycielski, J., & Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice . Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 10, 1–3.Google Scholar
Panza, M. (2015). From Lagrange to Frege: Functions and expressions. In Benis-Sinaceur, H., Panza, M., & Sandu, G., editors. Functions and Generality of Logic. Reflections on Frege’s and Dedekind’s Logicisms. Cham, Heidelberg, New York, Dordrecht, London: Springer, pp. 59–95.Google Scholar
Panza, M. (2016). Abstraction and epistemic economy. In Costreie, S., editor. Early Analytic Philosophy. New Perspectives on the Tradition. Berlin: Springer, pp. 387–428.CrossRefGoogle Scholar
Panza, M. (2018). Was Frege a logicist for arithmetic? In Coliva, A., Leonardi, P., & Moruzzi, S., editors. Eva Picardi on Language, Analysis and History . Cham, Switzerland: Palgrave MacMillan, Springer Nature, pp. 87–112.CrossRefGoogle Scholar
Panza, M. (FC1) Anzahlen et Werthverläufe: quelques remarques sur les §§I.34-40 des Grundgesetze de Frege, forthcoming in L’épistémologie du dedans. Mélanges en l’honneur d’Hourya Benis-Sinaceur, Garnier, Paris.Google Scholar
Panza, M. (FC2) Was Frege a Logicist? Forthcoming (This is a much extended version of Panza (2018), still unpublished).CrossRefGoogle Scholar
Panza, M. (FC3) Reals by Abstraction. An enquiry about Epistemic Economy in Mathematics. Forthcoming monograph.Google Scholar
Panza, M., & Sereni, A. (2019). Frege’s constraint and the nature of Frege’s foundational program, The Review of Symbolic Logic, 12(1), 97–143.CrossRefGoogle Scholar
Payne, J. (2013). Abstraction relations need not be reflexive. Thought: A Journal of Philosophy, 2(2), pp. 137–147.CrossRefGoogle Scholar
Schirn, M. (2013). Frege’s approach to the foundations of analysis (1874–1903). History and Philosophy of Logic, 34(3), 266–292.CrossRefGoogle Scholar
Scott, D. (1958–1959). A General Theory of Magnitudes, Unpublished typewritten document drawn from a series of lectures on geometry given at the University of Chicago in 1958–59.Google Scholar
Sereni, A. (2019). On the philosophical significance of Frege’s constraint. Philosophia Mathematica, 27(2), 244–275.CrossRefGoogle Scholar
Shapiro, S., & Snyder, E. (2020). Frege on the real numbers. In Rossberg, M., & Ebert, P., editors. Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press, pp. 343–383.Google Scholar
Shapiro, S., & Weir, A. (2000). ‘Neo-Logicist’ logic is not epistemically innocent. Philosophia Mathematica, 8, 160–189.CrossRefGoogle Scholar
Simons, P. (1987). Frege’s theory of real numbers. History and Philosophy of Logic, 8, 25–44.CrossRefGoogle Scholar
Wright, C. (2000). Neo-Fregean foundations for real analysis: Some reflections on Frege’s constraint. Notre Dame Journal of Formal Logic, 41(4), 317–334.CrossRefGoogle Scholar