Minimalism and Trivialism are two recent forms of lightweight Platonism in the philosophy of mathematics: Minimalism is the view that mathematical objects arethinin the sense that “very little is required for their existence”, whereas Trivialism is the view that mathematical statements have trivial truth‐conditions, that is, that “nothing is required of the world in order for those conditions to be satisfied”. In order to clarify the relation between the mathematical and the non‐mathematical domain that these views envisage, it has recently (...) been proposed that both Linnebo's notion of sufficiency and Rayo's “just is”‐operator can, or even should, be interpreted in terms of metaphysicalgrounding. This interpretation makes Minimalism and Trivialism akin to Aristotelianism in the philosophy of mathematics, according to which mathematical entities depend for their existence and their properties on non‐mathematical ones. In this paper we raise a general objection to this interpretation. We highlight a tension – a “Big Picture” Problem – between the metaphysical picture underlying both Minimalism and Trivialism, on the one side, and the metaphysics of Platonism and Aristotelianism on the other. We then consider various ways in which Linnebo and Rayo could respond. We finally argue that Minimalism and Trivialism are closer to Aristotelianism than to Platonism; however, even though these positions are not forms of traditional Platonism, they are not standard forms of Aristotelianism either. (shrink)

Linguistic ontologists and antilinguistic, ‘serious’ ontologists both accept the inference from ‘Fido is a dog’ to ‘Fido has the property of being a dog’ but disagree about its ontological consequences. In arguing that we are committed to properties on the basis of these transformations, linguistic ontologists employ a neo-Fregean meta-ontological principle, on which the function of singular terms is to refer. To reject this, serious ontologists must defend an alternative. This paper defends an alternative on which the function of singular (...) terms is not generally to refer and on which they are generally ontologically noncommittal. This is the best way to reject linguistic, ‘easy’ arguments for the existence of properties. The account recommends neutralism about quantification (drawing on Barcan Marcus and Meinongianism), coherently bringing together two important yet uncombined meta-ontological movements. Moreover, it employs Ramseyan insights about the transformations to provide a nonreductionist, non-error-theoretic redundancy approach to explicit talk about properties. (shrink)