Mathematical language is also difficult for beginners. How to make these words have more accurate meanings than everyday language also puzzles beginners. Words such as openness and domain have special meanings in mathematics. Mathematical terms also include proper nouns such as embryo and integrability. However, these special symbols and proper nouns are used for a reason: mathematics needs accuracy more than everyday language. Mathematicians call this requirement for linguistic and logical accuracy "rigor".

Stiffness is a very important and basic part of mathematical proof. Mathematicians want their theorems to be derived from axioms through systematic reasoning. This is to avoid drawing wrong "theorems" or proofs by unreliable intuition, which has happened in many examples in history. The expected rigor in mathematics changes with time: the Greeks expected careful argumentation, but in Newton's time, the methods used were not so rigorous. Newton's definition of solving problems was not properly handled by mathematicians through rigorous analysis and formal proof until19th century. Today, mathematicians have been arguing about the rigor of computer-aided proof. In the case that a large number of calculations are difficult to verify, it is hard to say that the proof is effective and rigorous.