Zhouwanyue (Nata) Yang, M.A.

PhD project

Title: Pluralism in Mathematical Understanding

Kant's philosophy of mathematics can be viewed as an application of his epistemology to the realm of mathematics, to the effect that the mathematical understanding is justified by the use of cognitive faculties. Due to the indispensable role of Sinnlichkeit (sensitivity) within Kant's framework, his philosophy of mathematical understanding is fundamentally challenged by modern advancements in mathematics, which have evolved without leaning on sensory intuition. Recognizing this, this project seeks to formulate a new systematic account of how cognitive faculties (which correspond to Kantian "cognitive faculties") justify mathematical theorizing. The account modifies the Kantian strategy of attributing the justification of mathematical understanding to our cognitive faculties while embracing the paradigm(s) of modern mathematics. To achieve this objective, the project focuses on mathematical agents, individuals equipped with cognitive faculties, who enhance their understanding through inquiry processes. Through the examination of concrete cases from mathematical practice, the understanding of a mathematical agent will be argued to be characterized by: 1) her use of cognitive faculties that justify her actions performed in her processes of inquiry, and 2) the objects on which she performs these actions to form an understanding of her subject of interest. These three concepts — cognitive faculties, actions, and objects — serve the dual purpose of identifying mathematical understanding and differentiating amongst different kinds of understanding. Rooted in this analysis, the project probes the normative principles for the constitution of mathematical understanding. Given the centrality of actions within the present process-based framework, the normativity of mathematical understanding will be defined by drawing from Sosa's normative theory of performance, as exemplified by his work on virtue epistemology. This will allow comparison of different kinds of mathematical understanding across different kinds of mathematical agents. On this basis, the project's epistemological stance will be illuminated through Carnap's pluralistic stance on mathematical languages. Ultimately, this will lead the project to advocate a pluralistic conception of the constitution of mathematical understanding.