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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.

In this paper I propose to reflect on contemporary philosophy of mathematics. i argue that the impartiality to methodology that is present in the current practice of philosophy of mathematics is an intellectual vice of our research community because it leaves our philosophical methods to engage with mathematics implicit. This hides difficulties our research community faces, difficulties our students can reasonably expect to learn about and which we need to engage with to overcome them. to be virtuously open-minded about methodology is not to be impartial about methodology but rather to critically assess the proposals on offer in a public debate.

In this paper I ask whether mathematicians should swear an oath similar to the Hippocratic oath sworn by some medical professionals as a means to foster morally praiseworthy engagement with the ethical dimensions of mathematics. I individuate four dimensions in which mathematics is ethically charged: (1) applying mathematical knowledge to the world can cause harm, (2) participation of mathematicians in morally contentious practices is an ethical issue, (3) mathematics as a social activity faces relevant ethical concerns, (4) mathematical knowledge itself may be ethically charged. I present a conceptual framework for oaths and discuss some empirical results on the effectiveness of oath-taking to foster morally praiseworthy behaviour. My findings suggest that enforcing oath-taking practices in mathematics is not currently a suitable means to foster ethically praiseworthy behaviour amongst mathematicians. Oaths can, however, provide needed edifying narratives about how to act ethically when acting as a mathematician.