Abstract
The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts there are.
Similar content being viewed by others
References
Boolos, G. (1987). The consistency of Frege’s foundations of arithmetic. In J. Thomson (Ed.), On beings and sayings: Essays in honor of Richard Cartwright (pp. 3–20). Cambridge: MIT Press, Reprinted in Boolos [1998].
Boolos, G. (1990). The standard of equality of numbers. In G. Boolos (Ed.), Meaning and method: Essays in honor of Hilary Putnam. Cambridge: Harvard University Press, Reprinted in Boolos [1998].
Boolos, G. (1997). Is Hume’s principle analytic? In R. Heck (Ed.), Logic, language, and thought. Oxford: Oxford University Press, Reprinted in Boolos [1998].
Boolos G. (1998). Logic, logic, and logic. Cambridge, Harvard University Press
Burgess J.P. (2005). Fixing Frege. Princeton, Princeton University Press
Cook R., Ebert P. (2005). Abstraction and identity. Dialectica 59(2): 121–139
Eklund, M. (2008). Bad company and neo-Fregean philosophy. Synthese, doi: 10.1007/s11229-007-9262-x.
Fine K. (2002). The limits of abstraction. Oxford, Oxford University Press
Fine K. (2005a). Class and membership. Journal of Philosophy 102(11): 547–572
Fine K. (2005b). Our knowledge of mathematical objects. In: Gendler T.S., Hawthorne J. (eds). Oxford studies in epistemology (Vol. 1). Oxford, Oxford University Press, pp. 89–109
Frege G. (1953). Foundations of arithmetic (trans.: Austin, J. L.). Oxford, Blackwell
Frege G. (1964). Basic laws of arithmetic (Ed. and trans.: Montgomery Furth). University of California Press, Berkeley and Los Angeles
Hale B., Wright C. (2001). Reason’s proper study. Oxford, Clarendon
Heck R.G. (1996). The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik. History and Philosophy of Logic 17, 209–220
Hodes H. (1984). On modal logics which enrich first-order S5. Journal of Philosophical Logic 13, 423–454
Leitgeb H. (2005). What truth depends on. Journal of Philosophical Logic 34, 155–192
Linnebo Ø. (2004). Frege’s proof of referentiality. Notre Dame Journal of Formal Logic 45(2): 73–98
Linnebo Ø. (2006). Sets, properties, and unrestricted quantification. In Rayo A., Uzquiano G. (eds). Absolute generality. Oxford, Oxford University Press, pp. 149–178
Linnebo, Ø. (2008). Introduction. Synthese, doi: 10.1007/s11229-007-9267-5.
Parsons, C. (1983). Sets and modality. Mathematics in philosophy (pp. 298–341). Cornell: Cornell University Press.
Shapiro S. (2000). Frege meets Dedekind: A neologicist treatment of real analysis. Notre Dame Journal of Formal Logic 41(4): 335–364
Uzquiano, G. (2008). Bad company generalized. Synthese, doi: 10.1007/s11229-007-9266-6.
Wright C. (1999). Is Hume’s principle analytic?. Notre Dame Journal of Formal Logic 40(1): 6–30 Reprinted in Hale and Wright [2001]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Linnebo, Ø. Bad company tamed. Synthese 170, 371–391 (2009). https://doi.org/10.1007/s11229-007-9265-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-007-9265-7