Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...) that it is only by conceiving the knowing subject(s) as embodied, fallible, and embedded in a specific context (along the lines of what has been done within social and feminist epistemology) that we can pursue an epistemology of mathematics sensitive to actual mathematical practice. I further suggest that this reconception of the knowing subject(s) does not force us to abandon the traditional framework of epistemology in which knowledge requires justified true belief. It does, however, lead to a fallible conception of mathematical justification that, among other things, makes Gettier cases possible. This shows that topics considered to be far removed from the interests of philosophers of mathematical practice might reveal to be relevant to them. (shrink)

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of (...) this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections. (shrink)

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...) on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)