Science vs. Mathematics

Mathematics is a language. By analogy with natural language, one may regard axioms as vocabulary, operational rules as grammar, and theorems as condensed idioms. Science aims to reveal truths about the world, along with the methodologies for acquiring such knowledge. Although scientific knowledge can be expressed in natural languages, mathematics offers a more precise and concise mode of expression—and often a more powerful instrument for extending that knowledge. In essence, mathematics is not a science. Mathematical theorems are infallible, whereas scientific theories must be falsifiable. Nevertheless, mathematics serves both as a precise language for articulating scientific understanding and as a powerful instrument for uncovering equivalent sets of knowledge through abstraction and extension.