Abstract
In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong D. (1979). A theory of universals. Cambridge, Cambridge University Press
Bealer G. (1982). Quality and concept. Oxford, Oxford University Press
Boolos, G. (1990). The standard of equality of numbers. In G. Boolos (Ed.), Meaning and method: Essays in honor of Hilary Putnam. Cambridge: Cambridge University Press. Pages references to the reprint in Boolos (1998).
Boolos G. (1998). Logic, logic and logic. Cambridge, MA, Harvard University Press
Bostock D. (1974). Logic and arithmetic: Natural numbers. Oxford, Clarendon Press
Bostock D. (1979). Logic and arithmetic: Rational and irrational numbers. Oxford, Clarendon Press
Cook R. (2001). The state of the economy: Neo-logicism and inflation. Philosophia Mathematica 10(3): 43–66
Dummett, M. (1967). Frege, Gottlob. In P. Edwards (Ed.), The encyclopaedia of philosophy. London: Macmillan 1967. Reprinted as ‘Frege’s Philosophy’ in Dummett’s Truth and other enigmas. London: Duckworth, 1978.
Dummett M. (1991). Frege philosophy of mathematics. London, Duckworth
Dummett M. (1998). Neo-Fregeans in Bad Company. In: Schirn M. (ed) Philosophy of mathematics today. Oxford, Clarendon Press
Ebert, P. (2005). The context principle and implicit definitions. University of St. Andrews, Doctoral Dissertation.
Evans G. (1982). The varieties of reference. Oxford, Clarendon Press
Field H. (1980). Science without numbers. Oxford, Blackwell
Franklin, L. R. (2001). ‘Plato’s Joints’, available online at http://www.columbia.edu/~lrf2001/PlatosJoints.doc
Frege, G. (1879). Begriffsschrift. Halle: L. Nebert; part translated in P. Geach & M. Black, 1952. Translations from the philosophical writings of Gottlob Frege. Oxford: Blackwell.
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: W. Koebner. Reprinted with English translation by J. L. Austin as The foundations of arithmetic. Oxford: Blackwell, 1950.
Hale B. (1994). Dummett’s critique of Wright’s attempt to resuscitate Frege. Philosophia Mathematica 2(3): 122–147. Reprinted in Hale and Wright 2001.
Hale B. (2000a). Reals by abstraction. Philosophia Mathematica 8(3): 100–123
Hale B. (2000b). Abstraction and set theory. Notre Dame Journal of Formal Logic 41(4): 379–398
Hale, B. & Wright, C. (2000). Implicit definition and the a priori. In B. Paul, & P. Christopher (Eds.), New essays on the a priori. Oxford: Clarendon Press. Page references are to the reprint in Hale & Wright (2001).
Hale B., Wright C. (2001a). The reason’s proper study: Essays towards a neo-Fregean philosophy of mathematics. Oxford, Clarendon Press
Hale, B., & Wright, C. (2001b). ‘To bury Caesar ...’. In Hale & Wright (2001a), pp. 335–396.
Hale B., Wright C. (2003). Responses to commentators. Philosophical Books 44(3): 245–263
Hale, B., & Wright, C. (forthcoming a). ‘Abstraction and Additional Nature’ forthcoming in Philosophia Mathematica.
Hale, B., & Wright, C. (forthcoming b). ‘The Metaontology of Abstraction’ forthcoming in David Chalmers et al. (Eds.), Metametaphysics. Oxford University Press.
Heck R. (1997). Finitude and Hume’s principle. Journal of Philosophical Logic 26: 589–561
Horwich P. (1997). Implicit definition, analytic truth and apriori knowledge. Noûs 31: 423–440
Horwich P. (1998). Meaning. Oxford, Clarendon Press
King J.C. (1999). Are complex ‘That’ phrases devices of direct reference? Noûs 33(2): 155–182
King, J. C. (2001) Complex demonstratives: A quantificational account. MIT Press.
Lewis D. (1986). On the plurality of worlds. Oxford, Basil Blackwell
Linnebo Ø. (2004). Predicative fragments of frege arithmetic. Bulletin of Symbolic Logic 10: 153–174
Mellor D.H., Oliver A. (1997). Introduction In Properties. Oxford, Oxford University Press (pp. 1–33)
Rumfitt I. (2003). Singular terms and arithmetical logicism. Philosophical Books 44(3): 193–219
Russell, B. (1905). On denoting. Mind, 14, 479–493. Page references are to the reprint in R. C. Marsh (Ed.), Logic and Knowledge. London: George Allen & Unwin, 1956.
Shapiro S., Weir A. (2000). Neo-Logicist logic is not epistemically innocent. Philosophia Mathematica, (3) 8(2): 160–189
Stirton W. (2003). Caesar invictus. Philosophia Mathematica 11: 285–304
Sullivan P., Potter M. (1997). Hale on Caesar. Philosophia Mathematica 5: 135–152
Swoyer C. (1996). Theories of properties: From plenitude to paucity. Philosophical Perspectives 10: 243–264
Weir A. (2003). Neo-Fregeanism: An embarrassment of riches. Notre Dame Journal of Formal logic 44: 13–48
Williamson, T. (2003). Understanding and inference. Proceedings of the Aristotelian Society, Supp., LXXVII, 249–293.
Williamson, T. (2006). Conceptual truth. Proceedings of the Aristotelian Society Supp., LXXX, 1–41.
Wright C. (1983). Frege’s conception of numbers as objects. Aberdeen, Aberdeen University Press
Wright, C. (1997). On the philosophical significance of Frege’s Theorem. In H. Richard, Jr. (Ed.), Language, thought and logic: Essays in honour of Michael Dummett. Oxford: Clarendon Press. Page references are to the reprint in Hale & Wright (2001).
Wright C. (1998a). On the (Harmless) impredicativity of Hume’s Principle. In: Schirn M. (ed) Philosophy of mathematics today. Oxford, Clarendon Press, Reprinted in Hale & Wright (2001).
Wright C. (1998b). Response to Dummett. In: Matthias S. (ed) Philosophy of mathematics today. Oxford, Clarendon Press. Reprinted in Hale & Wright (2001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hale, B., Wright, C. Focus restored: Comments on John MacFarlane. Synthese 170, 457–482 (2009). https://doi.org/10.1007/s11229-007-9261-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-007-9261-y