Book contents
- Mathematics and Its Logics
- Mathematics and Its Logics
- Copyright page
- Contents
- Acknowledgements
- Introduction
- Part I Structuralism, Extendability, and Nominalism
- 1 Structuralism without Structures
- 2 What Is Categorical Structuralism?
- 3 On the Significance of the Burali-Forti Paradox
- 4 Extending the Iterative Conception of Set: A Height-Potentialist Perspective
- 5 On Nominalism
- 6 Maoist Mathematics?
- Part II Predicative Mathematics and Beyond
- Part III Logics of Mathematics
- Index
- References
1 - Structuralism without Structures
from Part I - Structuralism, Extendability, and Nominalism
Published online by Cambridge University Press: 26 January 2021
Book contents
- Mathematics and Its Logics
- Mathematics and Its Logics
- Copyright page
- Contents
- Acknowledgements
- Introduction
- Part I Structuralism, Extendability, and Nominalism
- 1 Structuralism without Structures
- 2 What Is Categorical Structuralism?
- 3 On the Significance of the Burali-Forti Paradox
- 4 Extending the Iterative Conception of Set: A Height-Potentialist Perspective
- 5 On Nominalism
- 6 Maoist Mathematics?
- Part II Predicative Mathematics and Beyond
- Part III Logics of Mathematics
- Index
- References
Summary
As with many “isms,” “structuralism” is rooted in some intuitive views or theses which are capable of being explicated and developed in a variety of distinct and apparently conflicting ways. One such way, the modal-structuralist approach, was partially articulated in Hellman [1989] (hereinafter MWON). That account, however, was incomplete in certain important respects bearing on the overall structuralist enterprise. In particular, it was left open how to treat generally some of the most important structures or spaces in mathematics, for example, metric spaces, topological spaces, differentiable manifolds, and so forth. This may have left the impression that such structures would have to be conceived as embedded in models of set theory, whose modal-structural interpretation depends on a rather bold conjecture, for example, the logical possibility of full models of the second-order ZF axioms.
- Type
- Chapter
- Information
- Mathematics and Its LogicsPhilosophical Essays, pp. 19 - 42Publisher: Cambridge University PressPrint publication year: 2021
References
Brown, D. K. and Simpson, S. G. [1986] “Which set-existence axioms are needed to prove the separable Hahn-Banach theorem?,” Annals of Pure and Applied Logic 31: 123–144.CrossRefGoogle Scholar
Burgess, J. [1983] “Why I am not a nominalist,” Notre Dame Journal of Formal Logic 24(1): 93–105.CrossRefGoogle Scholar
Burgess, J., Hazen, A., and Lewis, D. [1991] “Appendix,” in Lewis, D., Parts of Classes (Oxford: Blackwell), pp. 121–149.Google Scholar
Chihara, C. [1973] Ontology and the Vicious Circle Principle (Ithaca, NY: Cornell University Press).Google Scholar
Dedekind, R. [1888] “The nature and meaning of numbers,” reprinted in Beman, W. W., ed., Essays on the Theory of Numbers (New York: Dover, 1963), pp. 31–115, translated from the German original, Was sind und was sollen die Zahlen? (Brunswick: Vieweg, 1888).Google Scholar
Feferman, S. [1964] “Systems of predicative analysis,” Journal of Symbolic Logic 29: 1–30.CrossRefGoogle Scholar
Feferman, S. [1968] “Autonomous transfinite progressions and the extent of predicative mathematics,” in Logic, Methodology, and Philosophy of Science III (Amsterdam: North Holland), pp. 121–135.CrossRefGoogle Scholar
Feferman, S. [1977] “Theories of finite type related to mathematical practice,” in Barwise, J., ed., Handbook of Mathematical Logic (Amsterdam: North Holland), pp. 913–971.CrossRefGoogle Scholar
Feferman, S. [1988] “Weyl vindicated: Das Kontinuum 70 years later,” in Temi e Prospettive della Logica e della Filosofia della Scienza Contemporanee (Bologna: CLUEB), pp. 59–93.Google Scholar
Feferman, S. [1992] “Why a little bit goes a long way: logical foundations of scientifically applicable mathematics,” in Hull, D., Forbes, M., and Okruhlik, K., eds., Philosophy of Science Association 1992, Volume 2 (East Lansing, MI: Philosophy of Science Association), pp. 442–455.Google Scholar
Feferman, S. and Hellman, G. [1995] “Predicative foundations of arithmetic,” Journal of Philosophical Logic 24: 1–17.Google Scholar
Field, H. [1992] “A nominalist proof of the conservativeness of set theory,” Journal of Philosophical Logic 21: 111–123.CrossRefGoogle Scholar
Hellman, G. [1989] Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press).Google Scholar
Hellman, G. [1990] “Modal-structural mathematics,” in Irvine, A. D., ed., Physicalism in Mathematics (Dordrecht: Kluwer), pp. 307–330.CrossRefGoogle Scholar
Hellman, G. [1994] “Real analysis without classes,” Philosophia Mathematica 2(3): 228–250.Google Scholar
Hellman, G. [1999] “Some ins and outs of indispensability,” in Cantini, A., et al., eds., Logic and Foundations of Mathematics (Dordrecht: Kluwer), pp. 25–39.CrossRefGoogle Scholar
Mayberry, J. [1994] “What is required of a foundation for mathematics,” Philosophia Mathematica 2(1): 16–35.CrossRefGoogle Scholar
O’Neill, B. [1983] Semi-Riemannian Geometry with Applications to General Relativity (Orlando, FL: Academic Press).Google Scholar
Parsons, C. [1990] “The structuralist view of mathematical objects,” Synthese 84: 303–346.CrossRefGoogle Scholar
Resnik, M. [1981] “Mathematics as a science of patterns: ontology and reference,” Noûs 15: 529–550.CrossRefGoogle Scholar
Shapiro, S. [1989] “Structure and ontology,” Philosophical Topics 17: 145–171.CrossRefGoogle Scholar
Shapiro, S. [1991] Foundations without Foundationalism: A Case for Second-Order Logic (Oxford: Oxford University Press).Google Scholar
Shapiro, S. [1997] Philosophy of Mathematics: Structure and Ontology (New York: Oxford University Press).Google Scholar
Smorynski, C. [1982] “The varieties of arboreal experience,” Mathematical Intelligencer 4: 182–189.CrossRefGoogle Scholar
Tait, W. W. [1986] “Critical notice: Charles Parsons,” Mathematics in Philosophy, Philosophy of Science 53: 588–606.CrossRefGoogle Scholar