Philosophical Studies

, Volume 162, Issue 3, pp 499–536 | Cite as

Reference to numbers in natural language

  • Friederike MoltmannEmail author


A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.


Numbers Abstract objects Tropes Frege Referential terms Adjectival Strategy Abstraction 



I would like to thank the audiences of presentations of earlier versions of this paper at the University of St Andrews, the University of Geneva, Hong Kong University, the IHPST (Paris), Kyoto University, Oxford University, the University of Venice, and the University of St Petersburg for very stimulating discussions. The paper has also greatly benefited from comments by Kit Fine, Matti Eklund, Thomas Hofweber, and Richard Kayne. This research was partly supported by the Chaire d’Excellence Semantic Structure and Ontological Structure (Agence Nationale de la Recherche ANR-06-EXC-012-0).


  1. Aristotle. Metaphysics. Google Scholar
  2. Armstrong, D. M. (1978). A theory of universals. Cambridge: Cambridge University Press.Google Scholar
  3. Bacon, J. (1995). Universals and property instances—the alphabet of being. Oxford: Blackwell.Google Scholar
  4. Bigelow, J. (1988). The reality of numbers. Oxford: Clarendon Press.Google Scholar
  5. Booles, G. (1984). To be is to be value of a variable (or to be the values of some variables). Journal of Philosophy, 81, 430–449.CrossRefGoogle Scholar
  6. Bostock, D. (1974). Logic and arithmetic 1: Natural numbers. Oxford: Oxford University Press.Google Scholar
  7. Brogaard, B. (2007). Number words and ontological commitment. Philosophical Quarterly, 57(226), 1–20.CrossRefGoogle Scholar
  8. Campbell, K. (1990). Abstract particulars. Oxford: Blackwell.Google Scholar
  9. Carlson, G. (1977). A unified analysis of the English bare plural. Linguistics and Philosophy, 1, 413–457.CrossRefGoogle Scholar
  10. Den Dikken, M., Meinunger, A., & Wilder, C. (2000). Pseudoclefts and ellipsis. Studia Linguistica, 54, 41–89.CrossRefGoogle Scholar
  11. Dummett, M. (1973). Frege: Philosophy of language. Cambridge: Harvard University Press.Google Scholar
  12. Dummett, M. (1995). Frege’s philosophy of mathematics. London: Duckworth.Google Scholar
  13. Fine, K. (1998). Cantorian abstraction: A reconstruction and defence. Journal of Philosophy, 55(12), 599–634.CrossRefGoogle Scholar
  14. Frege, G. (1884). Die Grundlagen der Arithmetik. Translated as Foundations of Arithmatics by J. L. Austin, 1974. Oxford: Basil Blackwell.Google Scholar
  15. Gottlieb, D. (1980). Ontological economy: Substitutional quantification and mathematics. Oxford: Oxford University Press.Google Scholar
  16. Grimshaw, J. (1997). Complement selection and the lexicon. Linguistic Inquiry, 10, 279–326.Google Scholar
  17. Hale, B. (1987). Abstract objects. New York: Blackwell.Google Scholar
  18. Heycock, C., & Kroch, A. (1999). Pseudocleft connectedness: Implications for the LF interface levels. Linguistic Inquiry, 30, 365–397.CrossRefGoogle Scholar
  19. Higgins, F. R. (1973). The pseudo-cleft construction in English. PhD dissertation, MIT, Cambridge, published in 1979 by Indiana University Linguistics Club.Google Scholar
  20. Hodes, H. (1984). The ontological commitment of arithmetics. Journal of Philosophy, 81, 123–149.CrossRefGoogle Scholar
  21. Hodes, H. (1990). Where do natural numbers come from? Synthese, 84, 347–407.CrossRefGoogle Scholar
  22. Hofweber, T. (2005a). Number determiners, numbers, and arithmetic. Philosophical Review, 114(2), 179–225.CrossRefGoogle Scholar
  23. Hofweber, T. (2005b). A puzzle about ontology. Nous, 39(2), 256–283.CrossRefGoogle Scholar
  24. Jacobson, P. (1994). Binding connectivity in copula sentences. In M. Harvey & L. Santelmann (eds.), Proceedings of SALT IV (pp. 161–178). Ithaca, NY: Cornell University.Google Scholar
  25. Kayne, R. (2007). Several, few, many. Lingua, 117(5), 832–858.CrossRefGoogle Scholar
  26. Kripke, S. (1973). John Locke lectures. Oxford: Oxford University Press.Google Scholar
  27. Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377.CrossRefGoogle Scholar
  28. Lowe, J. (2006). The four-category ontology. A metaphysics foundation for natural science. Oxford: Oxford University Press.Google Scholar
  29. Mayberry, J. R. (2000). The foundations of mathematics in the theory of sets. Cambridge: Cambridge University Press.Google Scholar
  30. Mikkelsen, L. (2004). Specifying who: On the structure, meaning, and use of specificational sentences. PhD dissertation, University of California, Santa Cruz.Google Scholar
  31. Moltmann, F. (1997). Intensional verbs and quantifiers. Natural Language Semantics, 5(1), 1–52.CrossRefGoogle Scholar
  32. Moltmann, F. (2003a). Nominalizing quantifiers. Journal of Philosophical Logic, 32, 445–481.CrossRefGoogle Scholar
  33. Moltmann, F. (2003b). Propositional attitudes without propositions. Synthese, 135, 70–118.CrossRefGoogle Scholar
  34. Moltmann, F. (2007). Events, tropes and truthmaking. Philosophical Studies, 134, 363–403.CrossRefGoogle Scholar
  35. Moltmann, F. (2008). Intensional verbs and their intentional objects. Natural Language Semantics, 16(3), 239–270.CrossRefGoogle Scholar
  36. Moltmann, F. (2009). Degree structure as trope structure: A trope-based analysis of comparative and positive adjectives. Linguistics and Philosophy, 32, 51–94.CrossRefGoogle Scholar
  37. Montague, R. (1973). The proper treatment of quantification in ordinary English. In J. Hintikka et al. (Eds.), Approaches to natural language. Dordrecht: Reidel (reprinted in R. Thomason (Ed.), Formal philosophy. Selected papers by Richard Montague (pp. 247–270), New Haven: Yale University Press.Google Scholar
  38. Oliver, A., & Smiley, T. (2004). Multigrade predicates. Mind, 113, 609–680.CrossRefGoogle Scholar
  39. Pustejovsky, J. (1991). The generative lexicon. Cambridge, MA: MIT Press.Google Scholar
  40. Romero, M. (2005). Concealed questions and specificational subjects. Linguistics and Philosophy, 25, 687–737.CrossRefGoogle Scholar
  41. Schiffer, S. (1996). Language-created and language-independent entities. Philosophical Topics, 24(1), 149–167.CrossRefGoogle Scholar
  42. Schlenker, P. (2003). Clausal equations (a note on the connectivity problem). Natural Language & Linguistic Theory, 21, 157–214.CrossRefGoogle Scholar
  43. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition: Number concepts underlying the development of analysis in 17–19th century France and Germany. New York: Springer.Google Scholar
  44. Searle, J. (1979). Expression and meaning. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  45. Sharvit, Y. (1999). Connectivity in specificational sentences. Natural Language Semantics, 7, 299–304.CrossRefGoogle Scholar
  46. Van Inwagen, P. (2000). Quantification and fictional discourse. In A. Everett & T. Hofweber (Eds.), Empty names, fiction, and the puzzles of non-existence (pp. 235–248). Stanford: CSLI Publications.Google Scholar
  47. Williams, D. C. (1953). On the elements of being. Review of Metaphysics, 7, 3–18.Google Scholar
  48. Woltersdorff, N. (1970). On universals. Chicago: Chicago University Press.Google Scholar
  49. Wright, C. (1983). Frege’s conception of numbers as objects. Cambridge: Cambridge University Press.Google Scholar
  50. Yi, B.-Y. (1998). Numbers and relations. Erkenntnis, 49, 93–113.CrossRefGoogle Scholar
  51. Yi, B.-Y. (1999). Is two a property? The Journal of Philosophy, 96(4), 163–190.CrossRefGoogle Scholar
  52. Yi, B.-Y. (2005). The logic and meaning of plurals. Part I. Journal of Philosophical Logic, 34, 459–506.CrossRefGoogle Scholar
  53. Yi, B.-Y. (2006). The logic and meaning of plurals. Part II. Journal of Philosophical Logic, 35, 239–288.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.IHPST (Paris1/ENS/CNRS)ParisFrance

Personalised recommendations