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Categoricity and Mathematical Knowledge

Categoricity and Mathematical Knowledge

Fernando Ferreira, “Categoricity and Mathematical Knowledge,” Revista Portuguesa de Filosofia 73, no. 3–4 (2017): 1423–36, DOI 10.17990/RPF/2017_73_3_1423.

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Categoricity and Mathematical Knowledge

Type Journal Article
Author Fernando Ferreira
Rights © 2018 Aletheia - Associação Científica e Cultural | © 2018 Revista Portuguesa de Filosofia
Volume 73
Issue 3-4
Pages 1423-1436
Publication Revista Portuguesa de Filosofia
ISSN 0870-5283; 2183-461X
Date 2017
DOI 10.17990/RPF/2017_73_3_1423
Language English
Abstract We argue that the basic notions of mathematics (number, set, etcetera) can only be properly formulated in an informal way. Mathematical notions transcend formalizations and their study involves the consideration of other mathematical notions. We explain the fundamental role of categoricity theorems in making these studies possible. We arrive at the conclusion that the enterprise of mathematics is not infallible and that it ultimately relies on degrees of evidence.
Date Added 17/01/2018, 17:51:13
Modified 18/01/2018, 10:39:15


  • categoricity theorems,
  • epistemology of mathematics,
  • formal and informal reasoning,
  • infallibility,
  • Platonism


  • Benacerraf, Paul. ‘Mathematical truth’. In Philosophy of Mathematics (Selected Readings), edited by P. Benacerraf and H. Putnam, 403-420. Cambridge: Cambridge University Press, 1983.
    Bernays, Paul. ‘Sur le platonisme dans les mathématiques’, L’Enseignment Mathématique 34 (1935): 52-69. Translated into English by C. Parsons under the title ‘On platonism in mathematics’. In Philosophy of Mathematics (Selected Readings), edited by P. Benacerraf and H. Putnam, 258-271. Cambridge: Cambridge University Press, 1983.
    Button, Tim, and Walsh, Sean. ‘Structure and categoricity: determinacy of reference and truth value in the philosophy of mathematics’. Philosophia Mathematica 24, no. 2 (2016): 283-307. doi: 10.1093/philmat/nkw007.
    Dedekind, Richard. Was sind und was sollen die Zahlen? Braunschweig: Vieweg, 1888. Translated into English by W. W. Beman under the title The Nature and Meaning of Numbers in Essays on the Theory of Numbers (Mineola: Dover Publications, 1963).
    Enderton, Herbert. A Mathematical Introduction to Logic. San Diego: Academic Press, 2001.
    Feferman, Solomon. ‘What rests on what? The proof-theoretic analysis of mathematics’. In In the Light of Logic, 187-208. Oxford: Oxford University Press, 1998.
    Feferman, Solomon. ‘Why a little bit goes a long way: logical foundations of scientifically applicable mathematics’. In In the Light of Logic, 284-298. Oxford: Oxford University Press, 1998.
    Feferman, Solomon. ‘Why the programs for new axioms need to be questioned’. In the panel discussion ‘Does mathematics need new axioms?’. Bulletin of Symbolic Logic 6, no. 4 (2000): 401-413. doi: 10.2307/420965.
    Gödel, Kurt. ‘What is Cantor’s continuum problem?’. American Mathematical Monthly 54 (1947): 515-525. Revised and expanded version in Philosophy of Mathematics (Selected Readings), edited by P. Benacerraf and H. Putnam, 470-485. Cambridge: Cambridge University Press, 1983.
    Hrbacek, Karel and Jech, Thomas. Introduction to Set Theory. New York: Marcel-Dekker, 1999.
    Isaacson, Daniel. ‘The reality of mathematics and the case of set theory’. In Truth, Reference, and Realism, edited by Z. Novak and A. Simony, 1-75. Budapest: Central European University, 2011.
    Kreisel, Georg. ‘Informal rigour and completeness proofs’. In Problems in the Philosophy of Mathematics, edited by I. Lakatos, 138-157. Amsterdam: North-Holland, 1967.
    Leibniz. Principes de la Philosophie ou Monadologie, 1714. Translated into Portuguese by Luís Martins under the title Princípios de Filosofia ou Monadologia. Lisboa: Imprensa Nacional – Casa da Moeda, 1987.
    Parsons, Charles. ‘The uniqueness of the natural numbers’. Iyyun 39 (1990): 13-44. Stable URL:
    Peano, Giuseppe. Arithmetices principia, nova methodo exposita. Torino: Bocca, 1889. Partial translation into English by J. van Heijenoort under the title The principles of arithmetic, presented by a new method. In From Frege to Gödel, edited by J. van Heijenoort, 83-97. Cambridge, Massachusetts: Harvard University Press, 1967.
    Quine, Willard. Philosophy of Logic. Cambridge, Massachusetts: Harvard University Press, 1986.
    Quine, Willard. ‘Ontology and ideology’. Philosophical Studies 2 (1951): 11-15. doi: 10.1007/BF02198233.
    Uzquiano, Gabriel. ‘Models of second-order Zermelo set theory’. Bulletin of Symbolic Logic 5, no. 3 (1999): 289-302. doi: 10.2307/421182.
    Wang, Hao. ‘Eighty years of foundational studies’. dialectica 12 (1958): 466-497.
    Wang, Hao. From Mathematics to Philosophy. London: Routledge & Kegan Paul, 1974.
    Wang, Hao. A Logical Journay. From Gödel to Philosophy. Cambridge, Massachusetts: MIT Press, 1996. doi: 10.1111/j.1746-8361.1958.tb01476.x.
    Zermelo, Ernst. ‘Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre’, Fundamenta Mathematicae 16 (1930): 26-47. Translated into English by E. de Pellegrin under the title ‘On boundary numbers and domains of sets’. In Ernst Zermelo. Collected Works, vol. 1, edited by H.-D. Ebbinghaus and A. Kanamori, 401-431. Berlin: Springer-Verlag, 2010.


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