Mathematics Cannot Be Treated as the Ultimate Inquiry into Nature: The Premise Conflict Revealed by Zeno's Paradox and Aristotle's Wheel Paradox v5.0

Employing mutually contradictory premises within a single axiomatic framework and then pretending to have “solved” two problems—this is precisely what mathematics does in the cases of Zeno’s paradox and Aristotle’s wheel paradox. Mathematics is widely regarded as a self-consistent, precise, and universally applicable symbolic system. However, through a critical analysis of the standard resolutions to these two paradoxes, this paper reveals a long-concealed fact: the same mathematical axiomatic system—the real number axioms combined with calculus—relies on two mutually exclusive premises when addressing these classic problems. In resolving Zeno’s paradox, it implicitly assumes that time possesses an indivisible minimal unit (discreteness). In resolving Aristotle’s wheel paradox, it forcibly depends on the assumption that space is infinitely divisible (continuity). Those who think they understand will say, “You don’t understand mathematics!”Truly clear-minded individuals do not pretend to understand; instead, they persist in asking: Is the generation of “0 → 1” a constructive convention, or a logical necessity? Interestingly, the Eastern philosophical notion of moving from “emptiness” to “being” faces the exact same question. In practice, underlying premises are switched at will to feign that mathematics—and the concepts of “emptiness” and “being”—can answer all questions, thereby absorbing and dissolving all subsequent detailed inquiry. Is this fair? We do not claim to be correct. We merely point out this: neither Western mathematics’ “0 → 1” nor Eastern philosophy’s “emptiness → being” constitutes an ultimate inquiry into nature. On the contrary, they should, at the appropriate moment, admit: “Here, I do not know. Here, I must stop.” Mathematics is a language. And like any language, to describe the real world, it must rest upon clearly stated assumptions. More fundamentally, this paper argues that the mathematical symbol “0” has never declared its projection layer in any application. The statement “a quantity is absent in a certain projection layer” has been illicitly equated with “absolute nothingness,” causing every physical expression involving “0” to smuggle unexamined metaphysical assumptions into its symbolic foundation. We therefore propose: Any mathematical conclusion applied to the physical world must explicitly declare its premises and projection layer. Mathematics without declared premises is not precision—it is intellectual soliloquy disguised as rigor.