Weaseling and the content of science WEB

I defend Joseph Melia’s nominalist account of mathematics from an objection raised by Mark Colyvan. The indispensability argument for the existence of mathematical entities such as numbers and sets has two premisses. The first is that mathematics is indispensable to science; that is, our best scientific theories imply the existence of numbers and other mathematical entities. The second premiss is a conditional: if mathematics is indispensable to science, then there are mathematical entities. These premisses entail that there are mathematical entities. Let nominalism be the view that there are no mathematical entities. Nominalists offer different responses to the indispensability argument. One approach has been extensively discussed: that of Hartry Field (1980, 1989, 1991, 1992). Field rejects the first premiss of the indispensability argument: he claims that our best scientific theories can be re-written so as to avoid implying that there are numbers. To substantiate this claim, Field embarks on a formidable technical programme. The consensus is that it has not succeeded (see MacBride 1999 for an excellent survey); even Field (1998, p. 400) sounds rather sceptical about its prospects. Recently, several philosophers have started to explore a different style of reply to the indispensability argument (see e.g. Melia 1995, 2000; Balaguer 1998, Chs 5 and 7; Yablo 2000, 2001; Azzouni 2004). This response consists of denying the second premiss of the indispensability argument. According to these nominalists, even if mathematics is indispensable to science, it does not follow that there are mathematical entities. Three such philosophers are the target of Mark Colyvan’s ‘There is No Easy Road to Nominalism’ (2010). In this paper, Colyvan argues that the views of Azzouni, Melia, and Yablo 2 have no advantage over Fieldian nominalism: in particular, he argues that they run into more or less the same technical difficulties as Field’s view. In this note, I argue that Colyvan’s criticisms of Melia fail. For all Colyvan has said, Melia has suggested a form of nominalism which does not fall prey to the technical difficulties of Field’s programme. I first set out Melia’s view and Colyvan’s criticism (Sect. 1); then I defend Melia from Colyvan (Sect. 2). In section 3 I rebut some possible objections. 1. Melia (2000, p. 467) considers utterances of the following form: (1) Everyone who Fs also Gs. Except Harry—he’s the one exception. These utterances can be interpreted as self-contradictory: (2) Everyone who Fs also Gs. And there is an F, Harry, who does not G. But it is more charitable to interpret them as consistent: (3) Everyone apart from Harry who Fs also Gs. This is what Melia calls weaseling: conveying our view by taking back something implied by the sentence we have uttered. When articulating their theories, scientists often say things which appear to imply the existence of mathematical entities. For instance, they quantify over numbers and functions. Melia (2000, p. 469) claims that ‘almost all scientists ... deny that there are such things as mathematical objects’. It appears that these scientists are contradicting themselves. According to Melia, it is more charitable to regard them as weaseling: they convey their views by taking back something implied by the