How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.

Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing (...) what we may call the arbitrary reference strategy. The main claims of such strategy are 2. First, we do not know which objects are the referents of proposition and numerical terms since their reference is fixed arbitrarily. Second, our ignorance of which object is picked out as the referent does not entail that no object is referred to by the relevant expression. Different articulations of the strategy are assessed, and a new one is defended. (shrink)

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...) arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism. (shrink)

The Aristotelian Society’s Virtual Issue is a free, online publication, made publically available on the Aristotelian Society website. Each volume is theme-based, collecting together papers from the archives of the Proceedings of the Aristotelian Society and the Proceedings of the Aristotelian Society Supplementary Volume that address the chosen theme. This year's Virtual Issue includes a selection of papers from across the Society’s fourteen decades, each accompanied by a specially commissioned present-day response. The aim of the volume is to aid reflection (...) on how to go about doing moral and political philosophy, by bringing together papers that bear on their epistemology and methodology. In some cases this involves pieces whose declared focus is on one of those areas, as is the case with Margaret MacDonald’s 1941 essay ‘The Language of Political Theory’ and Mary Midgley’s 1974 article ‘The Neutrality of the Moral Philosopher’. In other cases the editor, Ben Colburn, selected papers whose primary focus is elsewhere, but which include some interesting epistemological or methodological reflections or implications along the way. The contributions from our present-day commentators vary similarly in how far their explicit attention is methodological. Nevertheless, there is throughout a concern on how to ‘be an honest reasoner’, to borrow a phrase from the late Dudley Knowles. (shrink)

I discuss Steinhart’s argument against Benacerraf’s famous multiple-reductions argument to the effect that numbers cannot be sets. Steinhart offers a mathematical argument according to which there is only one series of sets to which the natural numbers can be reduced, and thus attacks Benacerraf’s assumption that there are multiple reductions of numbers to sets. I will argue that Steinhart’s argument is problematic and should not be accepted.