REFERENTIAL USES OF ARABIC NUMERALS

Abstract

Is the debate over the existence of numbers unsolvable? Mario Gómez-Torrente presents a novel proposal to unclog the old discussion between the realist and the anti-realist about numbers. In this paper, the strategy is outlined, highlighting its results and showing how they determine the desiderata for a satisfactory theory of the reference of Arabic numerals, which should lead to a satisfactory explanation about numbers. It is argued here that the theory almost achieves its goals, yet it does not capture the relevant association between how a number can be split up and the morphological property of Arabic numerals to be positional. This property seems to play a substantial role in providing a complete theory of Arabic numerals and numbers. Referencial Uses of Arabic Numerals 143 Manuscrito – Rev. Int. Fil. Campinas, v. 43, n. 4, pp. 142-164, Oct.-Dec. 2020. 1. THE GAME: HOW NUMERALS COULD REFER INSTEAD OF WHETHER NUMERALS REFER There is a well–known debate about the metaphysics of natural numbers. Typically, the discussion takes place in a match whose players belong to one of two predefined teams: the Realist and the Antirealist. If you choose the Realist team, as Frege (1884), Burgess and Rosen (1997), Hale and Wright (2009), and others have done, prepare yourself to commit to the existence of natural numbers as abstract, objective, and (not necessarily but most likely) mind–independent entities. The realist player holds that arithmetical sentences are true in virtue of facts about the denotations of their singular terms and predicates. Her challenge in this game is to explain by virtue of what do we gain knowledge of arithmetic sentences (since we don’t have the same type of contact with abstract entities as we do with whatever entities that are supposed to make empirical sentences true). Naturally, you might like the Antirealist team better. The spirit of this popular team is to deny the existence of entities such as numbers (see Field (1989), Yablo (2010), Bueno (2016) Once you choose to become an antirealist, your challenge is to explain in virtue of what are arithmetical sentences true. This game has spawned a diverse variety of accounts in which each team shows off their most sophisticated tactics, even reaching extreme positions with consequences such as that the only possible result is that both teams ‘win’ (for example, defending that only radical realism and radical antirealism are tenable, as Balaguer (1998) does) or that both teams ‘lose’ (as in a case of unsolvable epistemic disagreement (see Rosen (2001)). In the fourth chapter of Roads to reference, Gómez-Torrente (2019) presents an attractive and novel account where the starting point is to put aside the traditional game— which has come to seem bogged down—and starts a new one. The opening move of this game consists of taking at face value our linguistic intuitions bearing on the question of how the