Theorem - Terminology

Terminology

A number of different terms for mathematical statements exist, these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary and the usage of some terms has evolved over time.

• An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject. Historically these have been regarded as "self-evident", but more recently they are considered assumptions that characterize the subject of study. In classical geometry, axioms are general statements while postulates are statements about geometrical objects. A definition is also accepted without proof since it simply gives the meaning of a word or phrase in terms of known concepts.
• A proposition is a generic term for a theorem of no particular importance. This term sometimes connotes a statement with a simple proof, while the term theorem is usually reserved for the most important results or those with long or difficult proofs. In classical geometry, a proposition may be a construction that satisfies given requirements; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.
• A lemma is a "helping theorem", a proposition with little applicability except that it forms part of the proof of a larger theorem. In some cases, as the relative importance of different theorems becomes more clear, what was once considered a lemma is now considered a theorem, though the word "lemma" remains in the name. Examples include Gauss's lemma and Zorn's lemma.
• A corollary is a proposition that follows with little or no proof from one other theorem or definition.
• A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved. For example, the isosceles triangle theorem states that if two sides of a triangle are equal then two angles are equal. In the converse, the given (that two sides are equal) and what is to be proved (that two angles are equal) are swapped, so the converse is the statement that if two angles of a triangle are equal then two sides are equal. In this example, the converse can be proven as another theorem, but this is often not the case. For example, the converse to the theorem that two right angles are equal angles is the statement that two equal angles must be right angles, and this is clearly not always the case.

There are other terms, less commonly used, which are conventionally attached to proven statements, so that certain theorems are referred to by historical or customary names. For examples:

• Identity, used for theorems which state an equality between two mathematical expressions. Examples include Euler's identity and Vandermonde's identity.
• Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.
• Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.

A few well-known theorems have even more idiosyncratic names. The division algorithm (see Euclidean division) is a theorem expressing the outcome of division in the natural numbers and more general rings. The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space.

An unproven statement that is believed to be true is called a conjecture (or sometimes a hypothesis, but with a different meaning from the one discussed above). To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. On the other hand, Fermat's last theorem has always been known by that name, even before it was proven; it was never known as "Fermat's conjecture".