The Deepening and Integration of Discrete Mathematics: A Series of Foundational Questions for the Future

This paper aims to establish a research program focused on future exploration in mathematics. We observe a profound tension in the interaction between mathematics and contemporary scientific frontiers: the nature of many core areas (such as arithmetic geometry in characteristic p, quantum gravity, and complex information systems) possesses an intrinsic discreteness or combinatorial character. Yet, the most systematic and powerful theoretical tools in mathematics have long been rooted in the paradigms of continuity and analysis. This is not a flaw, but an opportunity that points the way: Can we, and how can we, develop foundational theories for discrete and combinatorial structures that rival continuous mathematics in depth, beauty, and systematicity? This paper does not advocate for replacement, but for deepening and integration. We first outline four key research dimensions (Discrete Geometry, Discrete Analysis, Discrete Models for Physics, and Discrete Arithmetic Geometry) as pillars for constructing a more comprehensive system of discrete mathematics. The core of this paper is then to propose a series of carefully crafted, profound, and visionary questions. These questions are not intended to prophesy a "millennium of discrete mathematics," but to chart a challenging map for exploration over the coming decades, inspiring mathematicians, theoretical computer scientists, and physicists to jointly explore the fundamental laws of the discrete world and build solid bridges connecting it to the continuous world.