Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T03:54:06.941Z Has data issue: false hasContentIssue false
  • English
  • Français

RELATIVE CATEGORICITY AND ABSTRACTION PRINCIPLES

Published online by Cambridge University Press:  26 February 2015

SEAN WALSH  and
SEAN EBELS-DUGGAN
SEAN WALSH*
Affiliation:
Department of Logic and Philosophy of Science, University of California, Irvine
SEAN EBELS-DUGGAN*
Affiliation:
Department of Philosophy, Northwestern University
*
*DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE 5100 SOCIAL SCIENCE PLAZA UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92697-5100, U.S.A. E-mail:[email protected] or [email protected]
†DEPARTMENT OF PHILOSOPHY NORTHWESTERN UNIVERSITY 1860 CAMPUS DRIVE, EVANSTON ILLINOIS 60208-2214, U.S.A. E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; Väänänen & Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright’s and Hale’s project of founding mathematics on abstraction principles (Hale & Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume’s Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991).

Type
Research Article
Information
The Review of Symbolic Logic , Volume 8 , Issue 3 , September 2015 , pp. 572 - 606
Copyright
Copyright © Association for Symbolic Logic 2015 

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

Antonelli, G. A. (2010a). The nature and purpose of numbers. The Journal of Philosophy, 107(4), 191–212.CrossRefGoogle Scholar
Antonelli, G. A. (2010b). Notions of invariance for abstraction principles. Philosophia Mathematica, 18(3), 276–292.Google Scholar
Antonelli, G. A. (2010c). Numerical abstraction via the Frege quantifier. Notre Dame Journal of Formal Logic, 51(2), 161–179.CrossRefGoogle Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21. Reprinted in Boolos (1998).Google Scholar
Boolos, G. (1990). The standard equality of numbers. In Boolos, G., editor. Meaning and Method: Essays in Honor of Hilary Putnam, pp. 261–277. Cambridge: Cambridge University Press. Reprinted in Boolos (1998), Demopoulos (1995).Google Scholar
Boolos, G. (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.Google Scholar
Burgess, J. P. (2005). Fixing Frege. Princeton Monographs in Philosophy. Princeton: Princeton University Press.CrossRefGoogle Scholar
Button, T., & Walsh, S. (2015). Ideas and results in model theory: Reference, realism, structure and categoricity. arXiv:1501.00472.Google Scholar
Cook, R. T., editor (2007). The Arché Papers on the Mathematics of Abstraction. Volume 71 of The Western Ontario Series in Philosophy of Science. Berlin: Springer.Google Scholar
Cook, R. T. (2012). Conservativeness, stability, and abstraction. British Journal for the Philosophy of Science, 63, 673–696.Google Scholar
Demopoulos, W., editor (1995). Frege’s Philosophy of Mathematics. Cambridge: Harvard University Press.Google Scholar
Ebbinghaus, H.-D. (1985). Extended logics: The general framework. In Barwise, J. and Feferman, S., editors. Model-Theoretic Logics, Perspectives in Mathematical Logic, pp. 25–76. New York: Springer.Google Scholar
Enderton, H. B. (2001). A Mathematical Introduction to Logic (second edition). Burlington: Harcourt.Google Scholar
Fine, K. (2002). The Limits of Abstraction. Oxford: The Clarendon Press.CrossRefGoogle Scholar
Fine, K. (2005). Précis. Philosophical Studies, 122(3), 305–313.Google Scholar
Fine, K. (2006). Our knowledge of mathematical objects. In Gendler, T. Z. and Hawthorne, J., editors. Oxford Studies in Epistemology, Vol. 1, pp. 89–109. Oxford: Clarendon Press.Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: Koebner.Google Scholar
Frege, G. (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number (second edition). Evanston: Northwestern University Press.Google Scholar
Hale, B. (1987). Abstract Objects. Oxford: Basil Blackwell.Google Scholar
Hale, B., & Wright, C. (2000). Implicit definition and the a priori. In Boghossian, P. and Peacocke, C., editors. New Essays on the A Priori, pp. 286–319. Oxford: Clarendon. Reprinted in Hale & Wright (2001).Google Scholar
Hale, B., & Wright, C. (2001). The Reason’s Proper Study. Oxford: Oxford University Press.Google Scholar
Heck, R. G. Jr. (1992). On the consistency of second-order contextual definitions. Noûs, 26(4), 491–494.Google Scholar
Hodes, H. (1984). Logicism and the ontological commitments of arithmetic. The Journal of Philosophy, 81(3), 123–149.CrossRefGoogle Scholar
Hodes, H. (1990). Where do the natural numbers come from? Synthese, 84(3), 347–407.CrossRefGoogle Scholar
Hodes, H. (1991). Where do sets come from? The Journal of Symbolic Logic, 56(1), 150–175.CrossRefGoogle Scholar
Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory (third edition). Volume 220 of Monographs and Textbooks in Pure and Applied Mathematics. New York: Dekker.Google Scholar
Jané, I., & Uzquiano, G. (2004). Well and non-well-founded Fregean extensions. Journal of Philosophical Logic, 33, 437–465.Google Scholar
Keenan, E. L., & Moss, L. S. (1985). Generalized quantifiers and the expressive power of natural language. In van Benthem, J. and ter Meulen, A., editors. Generalized Quantifiers in Natural Language, pp. 73–124. Dordrecht: Floris.Google Scholar
Kunen, K. (1980). Set Theory. Volume 102 of Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar
Kunen, K. (2011). Set Theory. London: College Publications.Google Scholar
Lavine, S. (1999). Skolem was wrong. Unpublished.Google Scholar
Linnebo, Ø. (2011). Higher-order logic. In Horsten, L. and Pettigrew, R., editors. The Continuum Companion to Philosophical Logic, pp. 105–127. London and New York: Continuum.Google Scholar
MacFarlane, J. (2002). Frege, Kant, and the logic in logicism. The Philosophical Review, 111(1), 25–65.CrossRefGoogle Scholar
Manzano, M. (1996). Extensions of First Order Logic. Volume 19 of Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.Google Scholar
Marker, D. (2002). Model Theory: An Introduction. Volume 217 of Graduate Texts in Mathematics. New York: Springer.Google Scholar
McGee, V. (1997). How we learn mathematical language. Philosophical Review, 106(1), 35–68.CrossRefGoogle Scholar
Parsons, C. (1990). The uniqueness of the natural numbers. Iyyun, 39(1), 13–44.Google Scholar
Parsons, C. (2008). Mathematical Thought and its Objects. Cambridge: Harvard University Press.Google Scholar
Peters, S., & Westerståhl, D. (2008). Quantifiers in Language and Logic. Oxford: Oxford University Press.Google Scholar
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic. Volume 17 of Oxford Logic Guides. New York: The Clarendon Press.Google Scholar
Shapiro, S. (2000). Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.Google Scholar
Shapiro, S., editor (2005). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press.Google Scholar
Shapiro, S., & Weir, A. (1999). New V, ZF and abstraction. Philosophia Mathematica, 7(3), 293–321.Google Scholar
Sher, G. (1991). The Bounds of Logic: A Generalized Viewpoint. A Bradford Book. Cambridge: MIT Press.Google Scholar
Simpson, S. G. (1999). Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Berlin: Springer.Google Scholar
Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7(2), 143–154.Google Scholar
Väänänen, J., & Wang, T. (2014). Internal categoricity in arithmetic and set theory. Notre Dame Journal of Formal Logic. To appear.Google Scholar
Walsh, S. (2012). Comparing Hume’s Principle, basic law V and Peano arithmetic. Annals of Pure and Applied Logic, 163, 1679–1709.Google Scholar
Walsh, S. (2014a). Fragments of Frege’s Grundgesetze and Gödel’s constructible universe. arXiv:1407.3861.Google Scholar
Walsh, S. (2014b). Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, 7(1), 84–119.Google Scholar
Wright, C. (1983). Frege’s Conception of Numbers as Objects. Volume 2 of Scots Philosophical Monographs. Aberdeen: Aberdeen University Press.Google Scholar
Wright, C. (1997). On the philosophical significance of Frege’s theorem. In Heck, R. G. Jr., editor. Language, Thought, and Logic: Essays in Honour of Michael Dummett, pp. 201–244. Oxford: Oxford University Press. Reprinted in Hale & Wright (2001).Google Scholar
Wright, C. (1998). On the harmless impredictavity of N = (Hume’s Principle). In Schirn, M., editor. Philosophy of Mathematics Today, pp. 393–368. Oxford: Clarendon Press. Reprinted in Hale & Wright (2001).Google Scholar
Wright, C. (1999). Is Hume’s Principle analytic? Notre Dame Journal of Formal Logic, 40(1), 6–30. Reprinted in Hale & Wright (2001).Google Scholar