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Reference to numbers in natural language

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Abstract

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.

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Notes

  1. This is a consequence of Frege’s context principle, on one reading of it (Wright 1983; Hale 1987). If an expression has a meaning (‘Bedeutung’) only in the context of a sentence and the contribution of a singular terms to the truth conditions of a sentence is that of reference to an object, then there is no further question that if an expression is in fact a singular term, it refers to an object.

  2. The term ‘trope’ is due to Williams (1953). Other recent references on tropes are Bacon (1995), Campbell (1990), Lowe (2006), and Woltersdorff (1970). Tropes already played an important role as ‘accidents’ or ‘modes’ in ancient philosophy (Aristotle) as well medieval Aristotelian philosophy. They also play an important role in early modern philosophy (Locke, Berkeley, Hume).

  3. For the notion of a ‘natural property’ or a ‘sparse’ conception of properties, see Armstrong (1978) and Lewis (1983).

  4. For the notion of a quantitative trope see Campbell (1990), who calls such tropes ‘instances of quantities’. On the topic of reference to quantitative tropes in natural language in general see Moltmann (2009).

  5. See Kayne (2007) for a recent discussion of phrases like a large number of people. Kayne in fact assumes that the plural determiners several, few, and many are modifiers of an unpronounced noun number (of course on the amount-specification reading).

  6. The use of the number of as a numeral replacement is also indicated by the possibility of plural agreement rather than singular agreement in English:

    1. (i)

      A great number of women were arrested.

  7. Of course John noticed the number fifty is possible on a non-perceptual reading of notice or else when the object of the noticing is in fact an inscription of the numeral. Unlike that sentence, though, John noticed the number of women can naturally describe a perceptual situation of John looking at the women and counting them.

  8. This is despite the fact that it is a standard assumption in the recent philosophical literature on tropes that tropes come with two fundamental relations: similarity and co-location (Williams 1953; Campbell 1990).

  9. Another pair of examples making the same point is the following:

    1. (i)

      a. The number of bathrooms in this house is average.

      b. ?? Three is average.

  10. Some evaluative predicates seem to receive a more intensional interpretation than the trope analysis would predict. If in a particular context, the women are in fact the students, then (ia) and (ib) can still have distinct truth conditions:

    1. (i)

      a. The number of women is unusually high.

      b. The number of students is unusually high.

    Even though on the present account the number of women would refer to the same trope as the number of students, the difference between (ia) and (ib) appears to be independent of the view of what the subject terms stand for. Relative adjectives such as high generally may depend for their evaluation on the description used, not just the referent. Thus the same contrast appears in (ii), supposing that the gymnast and the basketball player in question are in fact the same person:

    1. (ii)

      a. This gymnast is unusually tall.

      b. This basketball player is unusually tall.

  11. There are some predicates that can be attributed to tropes, but cannot apply with the same reading to the corresponding plurality. An example is large. Large when applied to a plurality has a very different reading than when applied to a number trope:

    1. (i)

      a. The women are large.

      b. The number of women is large.

  12. The distinction is reflected in some languages, for example German. In German, Zahl is used relationally or non-relationally, whereas Anzahl is used only relationally (though, in fact, non-relational uses such as die Anzahlen ‘the numbers’—which to my ears are entirely deviant—are found in Frege (1884)). This means in particular that Anzahl can occur only in the the number of-term construction and does not allow for the plural:

    (i):

    a. Die Zahl/Anzahl der Planeten ist acht.

       ‘The number of the planets is eight’.

    b. die Zahl/* Anzahl acht

       ‘the number/number eight’

    c. diese Zahl/* diese Anzahl

       ‘this number/number’

    d. Er addierte die Zahlen/* die Anzahlen.

       ‘He added the numbers/the numbers’.

  13. Bigelow (1988) takes numerals to express relations of distinctness. Thus two expresses the relation that holds of two entities x and y in case x and y are distinct. The most plausible version of such an approach would take numerals to be multigrade predicates (since a numeral like two can also take three distinct arguments of which it would then be false). The multigrade predicate view as such is discussed at length in Oliver and Smiley (2004). In general there are two criteria for taking a predicate to be multigrade rather than a plural predicate (a predicate that can be true of several individuals at one). If the order of the arguments matters and if an individual can occur as an argument more than once, then the predicate must be multigrade rather than a plural predicate. This is obviously not the case for numerals, and thus numerals should better be regarded plural predicates. For further discussion of the two views and a defence of the plural-predicate view of numerals, see Yi (1998).

  14. See for example Booles (1984) and Yi (2005, 2006) for discussion.

  15. Functional uses of NPs are not generally clearly distinguishable from referential uses. Functional and referential uses can be involved in antecedent-anaphora relations, as in (i), and one and the same term appears to be used both referentially and functionally in cases of co-predication, as in (ii):

    (i):

    a. The number of students is high. It had increased a lot over the last years.

    b. The president of the US is a democrat. He is not always a democrat of course.

    (ii):

    a. The number of students, which has increased a lot in the last few years, is very high.

    b. The president of the US, who is elected every four years, is currently a democrat.

    In general, neither anaphora nor co-predication requires strict identity of an object, as is familiar from theories of discourse and lexical semantics (see, for example, Pustejovsky 1991).

  16. Montague (1973) took the temperature in the temperature is rising to stand for a function from times to numbers (which on the present view would rather be temperature tropes).

  17. I have become aware that the intuitions discussed in this section do not hold in the same way for speakers accustomed to the way the number of is used in parts of mathematics. As an anonymous referee has pointed out, in a number of areas of mathematics, such as elementary cominatorics, ‘the number of Xs’, is explicitly defined as a pure number, thus not as a number trope. It is likely that that use influences the way the data in this section are judged.

  18. (46) presupposes that the bearers of two number tropes to which addition applies are non-overlapping pluralities. This may not seem entirely adequate since it seems not impossible to add ‘the number of students’ to ‘the number of girls’, with the students including some of the girls. What happens in this case, I suggest, is the use of a certain operation that is available to facilitate the application of arithmetical operations to number tropes. This is an operation of copying which creates hypothetical tropes that are distinct from, though quantitatively equivalent to a given actual trope. That is, in a case in which tropes with overlapping bearers are added, addition actually applies to hypothetical copies of the tropes. I will return to this operation of copying again later.

  19. ‘Property’ in (47a) must be understood in a sufficiently restricted, intensional sense: if there is a corresponding property for every set of sets, (47a) would be trivial. It would be fulfilled by any property of number tropes, since there would always a plural property that holds just of the bearers of a trope of which a number trope property is true.

  20. In fact, it is not clear that predicates such as natural in natural number have a meaning that is independent of the content of number, since such predicates can occur only as noun modifiers and not as predicates:

    (i):

    * This number is natural.

    This means that natural with the relevant meaning could not apply to entities other than numbers anyway.

  21. There is in principle a different way in which multiplication could be defined on the basis of particular pluralities, namely by making use of quantification over higher-order pluralities. Mayberry (2000), who also pursues a reduction of arithmetical operations to operations on pluralities, proposes hat ‘2 × 3 = 6’ be analysed as ‘two (non-overlapping) pluralities of three (distinct) entities is six entities’. On the present account, quantification over pluralities is avoided because two in two times the number of N is taken to range over number tropes that ‘measure’ two copies of the number trope ‘the number of N’. Measurement of tropes thus replaces higher-order plural quantification. Natural language appears to give evidence for that way of doing multiplication rather than the way Mayberry suggests.

    More generally, even though the trope-copying operation appears rather ‘theoretical’, it gains significant plausibility from the fact that it appears to correspond to a natural language construction. There should in fact be a general constraint on what one might posit as ‘theoretical’ operations on number tropes. Certainly, in the context of ‘natural language ontology’, only those operations should be posited for which some form of evidence can be obtained from natural language itself.

  22. This is despite Frege’s own claim to the contrary (Frege 1884). The example is equally unacceptable in German. In fact, also Frege’s other German example below, in which the numeral occurs with a definite determiner, is unacceptable to my ears:

    1. (i)

      ?? Die Anzahl der Planeten ist die Acht.

      ‘The number of planets is the eight.’

  23. There is an alternative view pursued in the literature, using higher-order equations (Jacobson 1994; Sharvit 1999). On that view, specificational sentences express an identity of semantic values, possibly of higher type if nonreferential expressions are involved.

  24. See Higgins (1973, p. 199). Higgins is worth citing in the present context: ‘Many philosophers have tended to treat sentences of the specificational variety as if they were identity sentences, and have then proceeded to build theories that rest on shaky foundations. The most impressive of such a misconstrual, which has spawned an enormous literature, is the following sentences: (1) The number of planets is nine’ (Higgins 1973, p. 199). Higgins then remarks ‘[…] it is doubtful whether the number of planets has any […] referential use at all—it seems rather to be akin to nouns such as defect and to have at most a kind of obscure referentiality associated with indirect questions’ (Higgins 1973, p. 200). Higgins then goes on citing substitution issues as a crucial argument against the view of (1) as an identity statements, such as the non-equivalence of sentences such ‘I counted up to nine’ with (the unacceptable) ‘I counted up to the number nine’, or ‘nine is the square root of eighty-one’ with (the unacceptable) ‘the number of planets is the square root of the number of eighty-one’.

  25. If specificational sentences are not analysed as question–answer structures, but as involving higher-order equations (footnote 23), the numeral in postcopula position would come out as nonreferential only as long the subject is taken to be non-referential. The latter is hard to maintain for the case of (59a), though; that is, it is hard to maintain that the number of planets is of the semantic type of a numeral.

  26. Brogaard (2007) argues against treating the sentence ‘The number of moons of Jupiter is four’ as a specificational sentence. She argues that analysing such sentences as specificational sentences by deriving them from sentences like ‘Jupiter has four moons’ or by considering them as expressing question–answer pairs is problematic in general since such analyses could not in fact account for what they originally aimed to account for, namely connectivity, a characteristic feature of specificational sentences (Higgins 1973). Connectivity consists, for example, in the unusual binding of the anaphor himself by John in The person John admires most is himself, where himself is not c-commanded by John. Brogaard argues that the failure of a syntactic treatment of connectivity (and certain other wrong predictions that such treatments would make) requires viewing The number of moons of Jupiter is four ‘as is’, that is, as an identity statement involving two referential terms, and thus treating numbers as objects. The problem with Brogaard’s argument is that she disregards the various other criteria for specificational sentences besides connectivity that have been established in the literature since Higgins (1973). What is crucial about specificational sentences is that neither the subject nor the postcopula NP need to be referential, whatever the right syntactic or semantic treatment of such sentences may turn out to be.

  27. See also Brogaard (2007) for further criticism of Hofweber’s view.

  28. Hale’s (1987) criterion involving quantifiers actually is considerably more complex.

  29. Hofweber (2005a) takes numerals to have a quantifier meaning in all contexts. This would not account for ‘adjectival’ occurrences though, as in the eight planets.

  30. The common formalization of (70a) within approaches using the Adjectival Strategy is by making use of quantification over concepts (Hodes 1984, 1990):

    1. (i)

      ∀F ∀G (∃2x Fx & ∃2x Gx & ¬ ∃x(Fx & Gx) → ∃4x (Gx v F x))

    This presupposes the view that counting means counting objects that fall under a concept. See Bigelow (1988) for a critique of that view.

  31. There is a potential problem for the Adjectival Strategy, namely quantifiers that can replace numerals in referential position, such as something or the same thing:

    1. (i)

      a. John added something to something else, namely he added ten to twenty.

      b. John added something to the number of children: he added two.

    This is not evidence against the Adjectival Strategy, however. Rather quantifiers like something are special quantifiers. Such quantifiers are well-known to be able to replace various kinds of nonreferential complements, for example predicative and clausal complements (Moltmann 2003a). In fact, it has been argued that quantifiers like something have a substitutional or quasi-substitutional status, their role just being that of enabling inferences (Hofweber 2005b), or, on an alternative analysis, that they are nominalising quantifier, introducing new entities into the semantic structure of the sentence that would not have been present in the absence of the quantifier (Moltmann 2003a). Either analysis is compatible with the Adjectival Strategy concerning simple numerals.

  32. For some reason, the difference is less apparent in main clauses:

    1. (i)

      a. Twelve interests me more than eleven.

      b. Twelve is very interesting; five is not.

    There are two possible explanations. First, the numerals here are actually focused, which may mean that the sentences, with respect to their subject position, presuppose a domain of pre-individuated objects and the numeral just serves to pick out one of those objects. Second, there may be an implicit sortal number in subject position, which for some reason may not be present in object position. In that case, the numeral in (ia, b) may in fact be in topic position and linked to a silent NP in subject position containing number as a sortal, as more explicitly in (ii):

    1. (ii)

      Twelve, that number is very interesting.

  33. One might suggest that simple numerals might be unacceptable with non-mathematical predicates simply because simple numerals primarily act as determiners and non-mathematical (but not mathematical) predicates give rise to the expectation of the numeral being followed by a noun. This is problematic, however, because the unacceptability of non-mathematical predicates remains in a conjunction in which the preceding conjunct is a mathematical predicate, as in (79a). Here the mathematical predicate should suffice to licence the simple numeral, which it does not. The suggestion is problematic for another reason. In some languages, simple numerals bear a different morphology from numerals occurring as determiners. For example, the German simple numeral for ‘one’ is eins, whereas the determiner is einer, eine, or ein. In German, eins is just as unacceptable with mathematical predicates as numerals with the morphology of a determiner:

    1. (i)

      a. ?? eins, was den Mathematiker Hans sehr interessiert.

         ‘one which interests the mathematician John a lot

      b. eins, was eine Primzahl ist

         ‘one which is a prime number’

      c. die Zahl eins, die den Mathematiker Hans sehr interessiert

         ‘the number one which interests the mathematician John a lot’

  34. Note that plus and times can easily act as functors with number tropes and other quantitative tropes:

    (i):

    a. The number of women plus the number of men is greater than the number of children.

    b. The number of women times the number of dresses is over a hundred.

    (ii):

    a. Sue’s weight plus Mary’s weight is more than that of Joe.

    b. John’s height is greater than three times the height of Mary.

    Plus obviously has a special meaning when applying to two quantitative tropes t and t’, forming a trope of the same kind as t and t’ whose bearer consists in the plurality of the bearer(s) of t and the bearer(s) of t’. Times in (iib) involves the formation of hypothetical copies of tropes.

  35. I would like to thank Per Martin-Loef for pointing out the connection to statements of calculation to me.

  36. One might think of a syntactic explanation why numerals and predicative complements require w-pronouns in German. Arguably, predicative complements and numerals are not gender-marked and w-pronouns are selected whenever the expression modified by the relative clause lacks gender-marking. However, when anaphoric pronouns are chosen that need to agree with numerals in gender, the pronouns must be neutral, which indicates the neutral gender of numerals:

    1. (i)

      Zehn ist nicht groesser als die Summe seines Vorgaengers mit eins.

      ‘Ten is not greater than the sum of its (neut.) predecessor with one.’

  37. Again a number sortal in the predicate has the same effect as a number sortal in an explicit number-referring term, rendering a plural anaphor acceptable:

    1. (i)

      a. Drei und fuenf sind beides Primzahlen. Sie sind nicht durch zwei teilbar.

         ‘Three and five are both prime numbers. They are not divisible by two.’

      b. Drei und fuenf sind nicht durch zwei teilbar. ?? Sie sind beide(s) Primzahlen.

         ‘Three and five are not divisible by two. They are both prime numbers.’

  38. It remains, of course, to be explained why the generalization seems much less strong in English than in German.

  39. The distinction between nuclear and extranuclear predicates is generally considered problematic as a distinction between types of predicates: the relevant distinction does not so much reside in a difference between types of predicates, but in a difference between a predicate predicated of an entity internally (within the story) and a predicate predicated of it externally. Some predicates, for example influential, can be predicated both internally and externally of an entity.

  40. This account faces important challenges that a proper development needs to address, such as clarifying and justifying the strategy by which a context, fictional or mathematical, individuates objects of the relevant sort.

  41. See Dummett (1973, pp. 72–73) for a discussion of parallels between expression for colors and for numbers with their adjectival and substantival uses.

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Acknowledgments

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).

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    Friederike Moltmann

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Moltmann, F. Reference to numbers in natural language. Philos Stud 162, 499–536 (2013). https://doi.org/10.1007/s11098-011-9779-1

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