William J. Melanson

In his Realism, Mathematics, and Modality, Hartry Field attempted to revitalize the epistemological case against mathematical platontism by challenging mathematical platonists to explain how we could be epistemically reliable with regard to the abstract objects of mathematics. Field suggested that the seeming impossibility of providing such an explanation tends to undermine belief in the existence of abstract mathematical objects regardless of whatever reason we have for believing in their existence. After more than two decades, Field’s explanatory challenge remains among the best available motivations for mathematical nominalism. This paper argues that Field’s explanatory challenge misidentifies the central epistemological problem facing mathematical platonism. Contrary to Field’s suggestion, inexplicability of epistemic reliability does not act as an epistemic defeater. The failure to explain our epistemic reliability with respect to the existence and properties of abstract mathematical objects is simply one aspect of a broader failure to establish that we are epistemically reliable with respect to abstract mathematical objects in the first place. Ultimately, it is this broader failure that is the source of mathematical platonism’s real epistemological problems.

Over the last two decades, epistemologists have become increasingly perplexed by the epistemic value problem. The problem, which first came to light in Plato's Meno, is to explain how knowledge is more epistemically valuable than mere true belief. After centuries of neglect, the value problem reemerged as part of the debate between epistemic internalists and externalists. Most notably, the value problem was wielded against forms of process reliabilism. At heart, the criticism was that being the product of a reliable belief-forming process did not add any further value to a true belief. Thus, if knowledge amounted roughly to reliably produced true belief, then knowledge would not be more valuable than mere true belief. Recently, a number of prominent virtue reliabilists, including Riggs, Greco, and Sosa, have suggested that we can account for the special value of knowledge by appealing to intellectual credit. They suggest that when an agent's belief is sensitive to the truth by being the result of a reliable disposition that is both stable and well integrated, then the agent deserves credit for this achievement. Unlike the value associated with being the product of a reliable belief-forming process, the value of intellectual credit is not swamped by the value of truth. Hence, it is claimed that intellectual credit can explain why knowledge is more valuable than mere true belief. To help motivate and explicate this solution, proponents of intellectual credit have most often relied upon metaphors and analogies. In particular, they have relied heavily on sports metaphors. This essay demonstrates the shortcomings of such metaphors and poses a more general problem for intellectual credit explanations of the value of knowledge. Overall, it is argued that the intellectual credit approach does not posit a substantial enough value to explain the extent to which we care about knowledge.