Book contents
- Mathematics and Its Logics
- Mathematics and Its Logics
- Copyright page
- Contents
- Acknowledgements
- Introduction
- Part I Structuralism, Extendability, and Nominalism
- Part II Predicative Mathematics and Beyond
- 7 Predicative Foundations of Arithmetic
- 8 Challenges to Predicative Foundations of Arithmetic
- 9 Predicativism as a Philosophical Position
- 10 On the Gödel–Friedman Program
- Part III Logics of Mathematics
- Index
- References
8 - Challenges to Predicative Foundations of Arithmetic
from Part II - Predicative Mathematics and Beyond
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
- Part II Predicative Mathematics and Beyond
- 7 Predicative Foundations of Arithmetic
- 8 Challenges to Predicative Foundations of Arithmetic
- 9 Predicativism as a Philosophical Position
- 10 On the Gödel–Friedman Program
- Part III Logics of Mathematics
- Index
- References
Summary
This is a sequel to our article “Predicative foundations of arithmetic” (Feferman and Hellman [1995], reproduced as Chapter 7 in this volume), referred to in the following as PFA; here we review and clarify what was accomplished in PFA, present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by PFA was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons [1992]. Another critique is due to Daniel Isaacson [1987]. Most recently, Alexander George and Daniel Velleman [1998] have examined PFA closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic.
- Type
- Chapter
- Information
- Mathematics and Its LogicsPhilosophical Essays, pp. 117 - 138Publisher: Cambridge University PressPrint publication year: 2021
References
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
Dummett, M. [1978] “The philosophical significance of Gödel’s theorem,” in Truth and Other Enigmas (London: Duckworth), pp. 186–201 (first published in 1963).Google Scholar
Feferman, S. [1964] “Systems of predicative analysis,” Journal of Symbolic Logic 29: 1–30.Google 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. [1985] “Intensionality in mathematics,” Journal of Philosophical Logic 14: 41–55.CrossRefGoogle Scholar
Feferman, S. [1996] “Gödel’s program for new axioms: why, where, how and what?” in Hájek, P., ed., Gödel ‘96, Lecture Notes in Logic 6 (Berlin: Springer), pp. 3–22.CrossRefGoogle Scholar
Feferman, S., and Hellman, G. [1995] “Predicative foundations of arithmetic,” Journal of Philosophical Logic 24: 1–17.CrossRefGoogle Scholar
George, A., and Velleman, D. [1998] “Two conceptions of natural number,” in Dales, G. and Olivieri, G., eds., Truth in Mathematics (Oxford: Oxford University Press), pp. 311–327.CrossRefGoogle Scholar
Hallett, M. [1990] “Physicalism, reductionism, and Hilbert,” in Irvine, A. D., ed., Physicalism in Mathematics (Dordrecht: Kluwer), pp. 183–257.CrossRefGoogle Scholar
Hellman, G. [1989] Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press).Google Scholar
Hellman, G. [1996] “Structuralism without structures,” Philosophia Mathematica 4: 100–123.CrossRefGoogle Scholar
Hintikka, J. [1956] “Identity, variables, and impredicative definitions,” Journal of Symbolic Logic 21: 225–245.Google Scholar
Isaacson, D. [1987] “Arithmetical truth and hidden higher-order concepts,” in Paris Logic Group, eds., Logic Colloquium ‘85 (Amsterdam: North-Holland), pp. 147–169.Google Scholar
Parsons, C. [1990] “The structuralist view of mathematical objects,” Synthese 84: 303–346.CrossRefGoogle Scholar
Parsons, C. [1992] “The impredicativity of induction,” in Detlefsen, M., ed., Proof, Logic and Formalization (London: Routledge), pp. 139–161 (revised and expanded version of a 1983 paper).Google Scholar
Quine, W. V. O. [1961] “A basis for number theory in finite classes,” Bulletin of the American Mathematical Society 67: 391–392.CrossRefGoogle Scholar
Schütte, K. [1965] “Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik,” Archiv für Mathematische Logik und Grundlagen-forschung 7: 45–60.CrossRefGoogle Scholar
Simpson, S. [1985] “Friedman’s research on subsystems of second-order arithmetic,” in Harrington, L. A., Morley, M., Scedrov, A., and Simpson, S. G., eds., Harvey Friedman’s Research on the Foundations of Mathematics (Amsterdam: North-Holland), pp. 137–159.Google Scholar
Wang, H. [1963] “Eighty years of foundational studies,” reprinted in A Survey of Mathematical Logic (Amsterdam: North-Holland), pp. 34–56.Google Scholar