Abstract
The standard semantic analysis of sentences such as ‘The number of planets in the solar system is eight’ is that they are identity statements that identify certain mathematical objects, namely numbers. The analysis thereby facilitates arguments for a controversial philosophical position, namely realism about mathematical objects. Accordingly, whether or not this analysis is accurate should concern philosophers greatly. Recently, several authors have offered rival analyses of sentences such as these. In this paper, I will consider a wide range of linguistic evidence and show that all of these analyses, including the standard analysis, suffer significant drawbacks. I will then outline and present further evidence in favour of my own analysis, developed elsewhere, according to which such sentences are identity statements that identify certain kinds of facts. I also defend a novel and plausible approach to the semantics of interrogative clauses that corroborates my analysis. Finally, I discuss how realists about mathematical objects should proceed in light of the arguments presented in this paper.
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Notes
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Dolby (2009, p. 294).
References
Barwise, F., & Cooper, R. (1981). Generalised quantifiers and natural language. Linguistics and Philosophy, 4, 159–219.
Dolby, D. (2009). The reference principle: A defence. Analysis, 69, 286–296.
Felka, K. (2014). Number words and reference to numbers. Philosophical Studies, 168, 261–282.
Field, H. (1980). Science without numbers: A defence of nominalism. New Jersey: Princeton University Press.
Field, H. (1989). Realism, mathematics and modality. Oxford: Blackwell.
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. English edition: Frege, G. 1953. The foundations of arithmetic: A logico-mathematical enquiry into the concept of number (J. Austin, Trans). Oxford: Blackwell.
Ginzburg, J. (1995). Resolving questions II. Linguistics and Philosophy, 18, 567–609.
Groenendijk, J., & Stokhof, M. (1997). Questions. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 1055–1124). Cambridge: MIT Press.
Hofweber, T. (2005). Number determiners, numbers, and arithmetic. Philosophical Review, 114, 179–225.
Karttunen, L. (1977). Syntax and semantics of questions. Linguistics and Philosophy, 1, 3–44.
Keenen, L., & Westerståhl, D. (1997). Generalized quantifiers in linguistics and logic. In J. Benthem & A. Meulen (Eds.), Handbook of logic and language. Cambridge: MIT Press.
Knowles, R. (2015). What ‘the number of planets is eight’ means. Philosophical Studies, 172, 2757–2775.
Knowles, R. (Forthcoming). Heavy duty Platonism. Erkenntnis.
Künne, W. (2006). Properties in abundance. In P. F. Strawson (Ed.), Universals, concepts, and qualities: New essays on the meaning of predicates and abstract entities (pp. 249–300) Aldershot: Ashgate.
Mikkelsen, L. (2011). Copular clauses. In C. Maienborn, K. von Heusinger, & P. Portner (Eds.), Semantics: An international handbook of natural language meaning (Vol. 2, pp. 1805–1829). Berlin: Mouton de Gruyter.
Moltmann, F. (Forthcoming). The number of planets, a number-referring term? In P. Ebert & M. Rossberg (Eds.), Abstractionism. Oxford: Oxford University Press.
Moltmann, F. (2013a). Reference to numbers in natural language. Philosophical Studies, 162, 499–536.
Moltmann, F. (2013b). Abstract objects and the semantics of natural language. Oxford: Oxford University Press.
Montague, R. (1974). The proper treatment of quantification in ordinary english. In R. Thomason (Ed.), Formal philosophy. New Haven: Yale University Press.
Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44, 12–36.
Oliver, A. (2005). The reference principle. Analysis, 65, 177–187.
Prior, A. N. (1971). Objects of thought. Oxford: Clarendon Press.
Quine, W. V. (1948). On what there is. The Review of Metaphysics, 2, 21–38.
Rosefeldt, T. (2008). ‘That’-clauses and non-nominal quantification. Philosophical Studies, 137, 301–333.
Ross, J. (1972). Act. In D. Davidson & G. Harman (Eds.), Semantics of natural language. Dordrecht: D. Reidel and Company.
Schlenker, P. (2003). Clausal equations: A note on the connectivity problem. Natural Language and Linguistic Theory, 21, 157–214.
Sellars, W. (1960). Grammar and existence: A preface to ontology. Mind, 69, 499–533.
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Knowles, R. (2016). Semantic Assumptions in the Philosophy of Mathematics. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_4
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