Museum-based mathematics exhibitions are increasingly prominent but under-theorized learning environments. In this study, we analyse the curriculum of United States mathematics exhibitions developed in the early 21st century in terms of their complex suggestions about the nature of mathematics and mathematical sense-making. We apply Rancière's notions of politics and aesthetics to explore what we describe as dissensus present in the texts, images, and multi-sensory exhibits of several major mathematics exhibitions. Our analysis characterizes this dissensus as a paradoxical mix of alternative and familiar mathematical aesthetics. On the one hand, we identify an alternative aesthetic emphasizing everyday ubiquity, sensuality, and informal sense-making. At the same time, we identify a countervailing emphasis on dominant notions of mathematics as esoteric, immaterial, and formal-symbolic. Museum mathematics efforts sometimes describe themselves as expanding how the public views and defines mathematics. A close examination of the exhibitions in this study reveals a complex picture, in which dominant and alternative forms of mathematics are co-present. The analysis suggests that museum-based mathematics researchers and practitioners view their work as containing political and aesthetic dimensions that can disrupt or reify what society counts as mathematics.