Abstract
The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benacerraf P. and Putnam H. (1983). Philosophy of Mathematics. Cambridge University Press, Cambridge
Bishop E. (1967). Foundations of Constructive Analysis. McGraw-Hill, New York
Bolzano, B.: 1837, Theory of Science, translated by R. George, Berkeley, University of California Press, 1972.
Boolos, G.: 1995, Introductory Note to Gödel (1951), in Gödel (1995), pp. 290–304.
Brouwer, L. E. J.: 1948, ‘Consciousness, Philosophy and Mathematics’, Philosophy of Mathematics, in Benacerraf and Putnam (1983), 90–96.
Burgess J. and Rosen G. (1997). A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press, Oxford
Chihara C. (1990). Constructibility and Mathematical Existence. Oxford University Press, Oxford
Coffa A. (1986). ‘From Geometry to Tolerance: Sources of Conventionalism in Nineteenth-Century Geometry” in From Quarks to Quasars: Philosophical Problems of Modern Physics, University of Pittsburgh Series. Pittsburgh University Press, Pittsburgh, 3–70
Divers J. and Miller A. (1999). ‘Arithmetical Platonism: Reliability and Judgement-Dependence”. Philosophical Studies 95: 277–310
Dummett M. (1963). ‘The Philosophical Significance of Gödel’s Theorem’. Ratio 5: 140–155
Dummett, M.: 1973, “The Philosophical basis of Intuitionistic Logic’, in Dummett (1978), pp. 215–247; reprinted in Benacerraf and Putnam (1983), pp. 97–129.
Dummett M. (1978). Truth and Other Enigmas. Harvard University Press, Cambridge, Massachusetts
Field H. (1980). Science Without Numbers. Princeton University Press, Princeton
Field H. (1989). Realism, Mathematics and Modality. Blackwell, Oxford
Frege, G.: 1884, Die Grundlagen der Arithmetik, Breslau, Koebner; The Foundations of Arithmetic, translated by J. Austin, 2nd edn, Harper, New York, 1960.
Gödel, K.: 1951, ‘Some Basic Theorems on the Foundations of Mathematics and their Implications’, in Gödel (1995), pp. 304–323.
Gödel K. (1995). Collected Works III. Oxford University Press, Oxford
Hellman G. (1989). Mathematics Without Numbers. Oxford University Press, Oxford
Heyting, A.: 1931, ‘The Intuitionistic Foundations of Mathematics’, in Benacerraf and Putnam (1983), pp. 52–61.
Heyting A. (1956). Intuitionism, an Introduction. North Holland, Amsterdam
Hilbert, D.: 1900, ‘Mathematische Probleme’, translated in Bulletin of the American Mathematical Society 8 (1902), 437–479.
Lewis D. (1997). ‘Finkish Dispositions’. Philosophical Quarterly 47: 143–158
Mancosu, Paolo, and J. Hafner 2005, ‘The Varieties of Mathematical Explanation’, in K. Jørgensen, P. Mancosu et al., (eds.), Visualization, Explanation and Reasoning Styles in Mathematics, Kluwer.
Mancosu, Paolo, Klaus Jørgensen, and Stig Pedersen (eds.) 2005, Visualization, Explanation and Reasoning Styles in Mathematics, Synthese Library 327, Springer, Dordrecht.
McCarty D. (2005). ‘Problems and Riddles: Hilbert and the Du Bois-Reymonds. Synthese 147: 63–97
Menzies P. (1998). ‘Possibility and Conceivability: A Response-Dependent Account of their Connections’. European Review of Philosophy 3: 255–277
Penrose, R.: 1996, ‘Beyond the Doubting of a Shadow: A Reply to Commentaries on Shadows of the mind’, Psyche 2(23) ( http://psyche.cs.monash.edu.au/ v2/psyche-2–23-penrose.html).
Resnik M. (1997). Mathematics as a Science of Patterns. Oxford University Press, Oxford
Shapiro S. (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press, Oxford
Shapiro S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press, New York
Shapiro, S.: 2000, ‘The Status of Logic’, in P. Boghossian and C. Peacocke (eds.), New Essays on the a Priori, Oxford University Press, pp. 333–366.
Shapiro S. (2001). ‘Why Anti-Realists and Classical Mathematicians Cannot Get Along’. Topoi 20: 53–63
Shapiro S. and Taschek W. (1996). ‘Intuitionism, Pluralism and Cognitive Command’. Journal of Philosophy 93: 74–88
Steiner M. (1978). ‘Mathematical Explanation and Scientific Knowledge’. Noûs 12: 17–28
Steiner M. (1980). ‘Mathematical Explanation’. Philosophical Studies 34: 135–152
Tennant N. (1987). Anti-Realism and Logic. Oxford University Press, Oxford
Tennant N. (1997). The Taming of the True. Oxford University Press, Oxford
Wang H. (1974). From Mathematics to Philosophy. Routledge and Kegan Paul, London
Wang H. (1987). Reflections on Kurt Gödel. The MIT Press, Cambridge, Massachusetts
Wright C. (1992). Truth and Objectivity. Harvard University Press, Cambridge, Massachusetts
Wright C. (2001). ‘On Being in a Quandary: Relativism, Vagueness, Logical Revisionism’. Mind 110: 45–98
Yablo, S.: 2005, ‘The Myth of the seven’, in Mark Eli Fictionalist Approaches to Metaphysics, Oxford University Press, Kalderon, Oxford, pp. 88–115.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Shapiro, S. The Objectivity of Mathematics. Synthese 156, 337–381 (2007). https://doi.org/10.1007/s11229-005-5298-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-005-5298-y