This article distinguishes mathematics from science, positioning mathematics as a defined and highly compressed language and science as a discovered discipline aimed at revealing truths through observation. While mathematical theorems exhibit internal infallibility within their defined frameworks, scientific theories remain inherently falsifiable and contingent upon empirical validation. Mathematics is an indispensable analytical toolkit that enables researchers to derive "equivalent knowledge"—novel insights logically deduced from established scientific principles. However, science cannot be derived purely from mathematics because scientific laws ultimately rely on incomplete empirical observations and postulates abstracted from the natural world. Consequently, this work proposes a hierarchical taxonomy of knowledge in which epistemic reliability is highest at the lowest level of direct observation and progressively diminishes as hypotheses traverse higher layers of mathematical inference that depend entirely on the validity of their foundational empirical axioms.
Jerry Z. Liu (2026) studied this question.