Below are some common terminologies when reading academic research papers in mathematics and related fields.
Definition
- An explanation of the mathematical meaning of a word
Theorem
- A very important true statement that is provable in terms of definitions and axioms
- Some conjectures were called “Theorems” but didn’t have proofs (e.g. Fermat’s Last Theorem didn’t have a proof from 1630 to the 1990s)
Proposition
- A statement of fact that is true and interesting in a given context
Lemma
- A true statement used to prove other true statements
Corollary
- A true statement that is a simple deduction from a theorem or proposition
Proof
- The explanation of why a statement is true
Conjecture
- A statement believed to be true, but for which we have no proof
Axiom
- A basic assumption about a mathematical situation (model) which requires no proof