Infinity and the Boundaries of Mathematical Argumentation ——A Philosophical Analysis of the Seven Millennium Prize Problems, the Planar Unit Distance Conjecture, and Fermat's Last Theorem

Mathematical argumentation, when dealing with infinity, structurally collides with boundaries that cannot be circumvented. Proceeding from cognition itself, this paper takes information collapse as the basic cognitive operation, "boundary" and "action" as the two inseparable aspects of division, "idling" as the structural symptom that emerges when argumentation encounters infinity, and "mark" as the trace left by operational failure, thereby offering a unified analysis of the boundaries of mathematical argumentation. The paper argues that mathematical argumentation is essentially an act of division, and that infinity is the label of this act; whenever argumentation attempts to extend from the finite to the infinite, self-reference necessarily occurs at some point, and self-reference is the root of idling. On this basis, the paper distinguishes two types of idling—subject-idling and object-idling—and reinterprets the three categories of mathematical problems: provable, only falsifiable, and neither provable nor falsifiable. When proof is structurally unobtainable, proof through practice becomes the structurally possible endpoint of cognition. The seven Millennium Prize Problems, the planar unit distance conjecture, and Fermat's Last Theorem are treated as concrete cases for validating this framework.