Manuel Bremer - Is Number a Sortal or Basically a Functional Concept?

Numbers are typically thought to be the prime example of mathematical entities. Whether abstract entities or not they are supposed to be collectable in respective number sets and can be quantified over as individual items. Numbers can be counted of. Such considered numbers are a case of a sort. “is a number” might be considered the corresponding sortal predicate. Even arithmetical nominalist or fictionalists, who deny the existence of numbers, still agree that the concept “is a number” though actually empty is a sortal predicate.
The non-realism of Philip Hugly and Charles Sayward, however, opens up a quite different perspective on the concept of number. Hugly and Sayward deny that number talk is referential at all. According to them numerical expressions do not occur essentially (i.e. without the possibility of paraphrasing) in referential positions. The basic construction according to them is “the number of”. Thus “number” may be seen as basically a functional concept.
In the first part of this paper the theory of arithmetical non-realism is set out. Of special interest are arguments relying on language use and a theory of non-referential quantification.
In the second part some weaknesses of the arguments presented are considered. Although the theory may be right in that the functional aspect of the concept of number has been neglected too much, number has to be a sortal predicate as well.