Abstract
According to Hume’s principle, a sentence of the form ⌜The number of Fs = the number of Gs⌝ is true if and only if the Fs are bijectively correlatable to the Gs. Neo-Fregeans maintain that this principle provides an implicit definition of the notion of cardinal number that vindicates a platonist construal of such numerical equations. Based on a clarification of the explanatory status of Hume’s principle, I will provide an argument in favour of a nominalist construal of numerical equations. The neo-Fregean objections to such a construal will be examined and rejected. And the implications of the nominalist construal for the use of numerals and for the understanding of ontological questions for the existence of numbers will be spelled out.
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Notes
Quinean quasi-quotation will be employed wherever necessary (cf. Quine 1981, ch. 6). ‘F’ and ‘G’ will be used as predicate variables, ‘F’ and ‘G’ as the corresponding object variables. Accordingly, phrases like ‘the Fs’, ‘a F’, etc. are short for ‘the object to which F applies’, ‘an object to which F applies’, etc.
cf. Rundle (1979, p. 257).
Suppose, for example, that we are to apply the procedure to given sets of cups and spoons. The assumption in question, then, tells us that it does not matter which spoon is paired with which cup, nor how they are paired (whether, for instance, a spoon is put to the left of a cup or the other way around). Either they will pair off or they won’t.
In the quoted passage, Wright is concerned with the relation between statements that claim that two lines are parallel and the corresponding statements about the identity of the lines’ directions. But his considerations are supposed to be transferable to the case of statements that claim the bijective correlatability of two concepts and the corresponding numerical equations.
Again, the practice of transitive counting determines a consistent use of number ascriptions only on the true, though non-trivial assumption that the procedure’s outcome does not depend on which F is counted with which numeral.
In this sense, even a connective such as ‘and’ could be said to refer to something, viz. a truth-function.
It can be granted that there is no trivial for proof for this intuitive principle. Still, the Neo-Fregean’s would certainly accept the principle and classify it as a logical truth.
References
Alston, W. P. (1958). Ontological commitments. Philosophical Studies, 9(1–2), 8–17.
Balaguer, M. (1998). Platonism and anti-platonism in the philosophy of mathematics. New York: Oxford University Press.
Burgess, J. P., & Rosen, G. (2005). Nominalism reconsidered. In S. Shapiro (Ed.), The Oxford handbook of the philosophy of mathematics and logic (pp. 515–535). Oxford: Oxford University Press.
Contessa, G. (2016). It Ain’t easy: Fictionalism, deflationism, and easy arguments in ontology. Mind, 125(499), 763–773.
Dummett, M. (1991). Frege: Philosophy of mathematics. London: Duckworth.
Eklund, M. (2006). Neo-Fregean ontology. Philosophical Perspectives, 20, 95–121.
Fine, K. (1994). Essence and modality. Philosophical Perspectives, 8, 1–16.
Frege, G. (1950). The foundations of arithmetic. Oxford: Basil Blackwell.
Frege, G. (1972). Review of Dr. E. Husserl’s philosophy of arithmetic. Mind, New Series, 81 (323), transl. E.W. Kluge, 321–337.
Glock, H.-J. (2002). Does ontology exist. Philosophy, 77(300), 235–260.
Greimann, D. (2003). What is Frege’s Julius Caesar problem? Dialectica, 57(3), 261–278.
Hale, B. (1994). Is platonism epistemologically bankrupt? The Philosophical Review, 103(2), 299–325.
Hale, B. (2010). The bearable lightness of being. Axiomathes, 20, 399–422.
Hale, B., & Wright, C. (2001). The reason’s proper study: Essays towards a neo-Fregean philosophy of mathematics. Oxford: Clarendon Press.
Hale, B., & Wright, C. (2005). Logicism in the 21st-century. In S. Shapiro (Ed.), The Oxford handbook of the philosophy of mathematics and logic (pp. 166–202). Oxford: Oxford University Press.
Hale, B., & Wright, C. (2009). The metaontology of abstraction. In D. Chalmers, R. Wasserman, & D. Manley (Eds.), Metametaphysics: New essays on the foundations of ontology (pp. 178–212). Oxford: Clarendon Press.
Heck, R. G., Jr. (2000). Cardinality, counting, and equinumerosity. Notre Dame Journal of Formal Logic, 41(3), 187–209.
Hofweber, T. (2016). Ontology and the ambitions of metaphysics. Oxford: Oxford University Press.
Kim, J. (2011). A strengthening of the Caesar problem. Erkenntnis, 75(1), 123–136.
MacBride, F. (2003). Speaking with shadows: A study of neo-logicism. The British Journal for the Philosophy of Science, 54, 103–163.
MacBride, F. (2006). The Julius Caesar objection: More problematic than ever. In F. MacBride (Ed.), Identity and modality (pp. 174–202). Oxford: Oxford University Press.
Pederson, N. J. (2009). Solving the Caesar problem without categorical sortals. Erkenntnis, 71(2), 141–155.
Putnam, H. (1972) [2010]. Philosophy of logic. Abingdon: Routledge.
Quine, W. V. O. (1981). Mathematical logic. Cambridge, MA: Harvard University Press.
Roeper, P. (2015). A vindication of logicism. Philosofia Mathematica, 24(3), 360–378.
Rundle, B. (1979). Grammar in philosophy. Oxford: Clarendon Press.
Russell, B. (1937). The principles of mathematics. London: Allen & Unwin.
Schaffer, J. (2009). On what grounds what. In D. Chalmers, R. Wasserman, & D. Manley (Eds.), Metametaphysics: New essays on the foundations of ontology (pp. 347–383). Oxford: Clarendon Press.
Schiffer, S. (1994). A paradox of meaning. Nous, 28, 279–324.
Schiffer, S. (2003). The things we mean. Oxford: Oxford University Press.
Sider, T. (2011). Writing the book of the world. Oxford: Oxford University Press.
Tennant, N. (2009). Natural logicism via the logic of orderly pairing. In S. Lindström, E. Palmgren, K. Segerberg, & V. Stoltenberg-Hansen (Eds.), Logicism, intuitionism, formalism: What has become of them? (pp. 91–125). New York: Springer.
Thomasson, A. L. (2007). Ordinary objects. New York: Oxford University Press.
Thomasson, A. L. (2013). Fictionalism versus deflationism. Mind, 122(488), 1023–1051.
Thomasson, A. L. (2015). Ontology made easy. Oxford: Oxford University Press.
Thomasson, A. L. (2017). Why we should still take it easy. Mind, 126(503), 769–779.
Wittgenstein, L. (1974). Philosophical Grammar. Anthony Kenny (tr.); Rush Rhees (ed.). Oxford: Blackwell.
Wittgenstein, L. (1975). In C. Diamond (Ed.), Wittgenstein’s lectures on the foundations of mathematics, Cambridge 1939, From the notes of R. G. Bosanquet, N. Malcolm, R. Rhees and Y. Smithies. New York: Cornell University Press.
Wright, C. (1983). On Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.
Wright, C. (1992). Truth and objectivity. Cambridge: Harvard University Press.
Yablo, S. (2000). A paradox of existence. In A. Everett & T. Hofweber (Eds.), Empty names, fiction, and the puzzles of non-existence (pp. 197–228). Palo Alto: CSLI Publications.
Zalta, E. N. (1999). Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege’s Grundgesetze in object theory. Journal of Philosophical Logic, 28(6), 619–660.
Acknowledgements
I would like to thank David Dolby, Severin Schroeder and an anonymous reviewer for their comments on previous drafts. Further to this, I received helpful suggestions from the audience of the 5th conference of the Latin American Association for Analytic Philosophy (ALFAn V) at which I presented an earlier version of this paper.
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Büttner, K.M. Hume’s principle: a plea for austerity. Synthese 198, 3759–3781 (2021). https://doi.org/10.1007/s11229-019-02309-3
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DOI: https://doi.org/10.1007/s11229-019-02309-3