Just what is Full-Blooded Platonism?

Mark Balaguer has, in his Platonism and Anti-Platonism in Mathematics [Balaguer, 1998], given us an intriguing new brand of platonism, which he calls, plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer’s attempts to characterise full-blooded platonism fail. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it. Mark Balaguer’s Platonism and Anti-Platonism in Mathematics [Balaguer, 1998] is an exciting and original addition to the recent literature in the metaphysics of mathematics. In it, Balaguer propounds an intriguing thesis: he argues that both platonism of a very particular sort, and fictionalism, are adequate metaphysical analyses of mathematics, and furthermore, that there is no telling between platonism and fictionalism about mathematical objects. There is literally no fact of the matter deciding between platonism and fictionalism. I do not intend to discuss the major strand of Balaguer’s project. Instead, I will focus on the first part of the project — Balaguer’s exposition and defence of full-blooded platonism (henceforth ‘FBP’). I will show that Balaguer has not defined a coherent variety of platonism, and that more work must be done to explain what FBP actually is. At a first glance, FBP is relatively simple to state. It is the thesis that any mathematical object which can exist, does exist. Stating the thesis like this seems to commit one to two questionable things. First, it appears to appeal to some kind of quantification over the non-existent, for if the quantifier “any” merely quantifies over what is then we all, platonist and non-platonist together, can agree with the thesis. Any (existing) mathematical object which can exist, does. Second, the claim appears to commit one to de remodality concerning these possibilia. We ask of each x whether it possibly exists. We quantify into this modal context. Now, these are not significant difficulties if you wish to embrace Meinongian quantification and de re modality applying to possibilia. They are difficulties for Balaguer, for he wishes to embrace neither of these. For him, the modality in question is logical possibility, and for this, the bearer of possibility and necessity is the sentence, and not objects, or sentences with free first-order variables. ∗Thanks to Mark Balaguer, JC Beall, Fraser MacBride, Stewart Shapiro and Ed Zalta for comments on earlier drafts of this paper. This research is supported by the Australian Research Council, through Large Grant No. A00000348.