Appendix B Fancy Mathematical Terms
Here are some important mathematical terms that you will encounter throughout mathematics.
- Definition—a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
- Theorem—a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.
- Proposition—a proved and often interesting result, but generally less important than a theorem. Alternatively, a proposition may refer to a sentence that is either true or false but never both (see Definition 2.16).
- Lemma—a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Occasionally lemmas can take on a life of their own (Zorn’s Lemma, Urysohn’s Lemma, Burnside’s Lemma, Sperner’s Lemma).
- Corollary—a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).
- Conjecture—a statement that is unproved, but is believed to be true (Collatz Conjecture, Goldbach Conjecture, Twin prime Conjecture).
- Claim—an assertion that is then proved. It is often used like an informal lemma.
- Counterexample—a specific example showing that a statement is false.
- Axiom/Postulate—a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Euclid’s five postulates, axioms of ZFC, Peano axioms).
- Identity—a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler’s identity).
- Paradox—a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed axiomatic theory (e.g., Russell’s Paradox). The term paradox is also used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach-Tarski Paradox, Alabama Paradox, Gabriel’s Horn).
