Eileen S. Nutting

Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part of Field’s challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist.

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology.

Kit Fine and Gideon Rosen propose to define constitutive essence in terms of ground-theoretic notions and some form of consequential essence. But we think that the Fine–Rosen proposal is a mistake. On the Fine–Rosen proposal, constitutive essence ends up including properties that, on the central notion of essence, are necessary but not essential. This is because consequential essence is closed under logical consequence, and the ability of logical consequence to add properties to an object’s consequential essence outstrips the ability of ground-theoretic notions, as used in the Fine–Rosen proposal, to take those properties out. The necessary-but-not-essential properties that, on the Fine–Rosen proposal, end up in constitutive essence include the sorts of necessary-but-not-essential properties that, others have noted, end up in consequential essence.

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist.

In “Mathematical Truth,” Paul Benacerraf raises an epistemic challenge for mathematical platonists. In this paper, I examine the assumptions that motivate Benacerraf’s original challenge, and use them to construct a new causal challenge for the epistemology of mathematics. This new challenge, which I call ‘Gödel’s Gap’, appeals to intuitive insights into mathematical knowledge. Though it is a causal challenge, it does not rely on any obviously objectionable constraints on knowledge. As a result, it is more compelling than the original challenge. It is also more general; the challenge applies equally to platonistic and non-platonistic accounts of mathematical truth. And it can be generalized beyond the case of mathematical knowledge to pose a challenge for, e.g., moral knowledge.