Basic Terms in Mathematics

These are the basic terms used in Mathematics

CONJECTURE

Assumed to be true, but not yet proven. most famous conjectures today are shown to be true up to large numbers: the Goldbach conjecture, the abc conjecture, the twin prime conjecture, the collatz conjecture, and so on. often the most difficult mathematical problems to prove are conjectures that have stood for years. Poincare Conjecture was proved by Grigori Perelman

HYPOTHESIS

Kind of like a conjecture but has a connotation in that it is less ’sure’. generally not as ’strong’ as a conjecture, but some hypotheses are really really important (Riemann hypothesis for example). often used more in the other sciences than in the mathematics. there aren’t as many hypotheses in mathematics as there are conjectures (famous ones anyway)

THEOREM

In mathematics, theorems are statements that are true and can be proven under a given set of axioms. these are the most numerous of the seven (i think). examples include the Pythagorean theorem, Ptolemy’s theorem, the fundamental theorem of algebra, De moivre’s theorem, the ham sandwich theorem, the BEST theorem, fundamental theorem of calculus etc.

COROLLARY

Immediate consequence of a result already proved. kind of like a specialized theorem. for example, did you know that if AB and CD are perpendicular, then AC^2 +BD^2 = AD^2 +BC^2
this is a corollary of the Pythagorean theorem.

LEMMA

Kind of like a ’sub-theorem’, an helping theorem, a not-so important theorem, but otherwise a statement that is true and can be proven under a given set of axioms. when you read the solutions of Olympiad problems, there’s bound to be lots of lemmas. examples include burnside’s lemma, Bezout’s lemma, Euclid’s lemma, Gauss’s lemma, Zorn’s lemma, and so on.

AXIOM

Something accepted true without explanation or proof. these are usually foundational, and (probably) the least of the seven. they cannot be proven, because nothing can be logically before them. examples include the field axioms, the Peano axioms for arithmetic, euclid’s five postulates (that are considered to be axioms), the axiom of choice, Zermelo-fraenkel axioms, etc.

POSTULATE

Kind of like an axiom but less foundational. accepted true without proof. Euclid's famous four + one postulates of euclidean
geometry are a famous example - one of the postulates was thought to be a theorem but it couldn’t be proved. lemmas and theorems can be derived from these.

LAW

Used a lot more in physics and the sciences than in mathematics. A mathematical statement that is always true. generally theorems. Sometimes based on empirical evidence (evidence based on observation rather than proving/deduction,
examples include the (strong) law of large numbers). the law of sines, law of cosines, benford’s law, laws of exponents, law of large numbers and so on.

PRINCIPLE

A statement that can be a postulate, a theorem, etc. examples include the pigeonhole principle (actually a theorem), the double counting principle (a postulate), the principle of mathematical induction (a postulate), the principle of inclusion-exclusion (a theorem), the well-ordering principle(a postulate), etc.

THEORY

In mathematics, kind of like a system of theorems. like for example set theory, a system of theorems, postulates, etc., about sets. also number theory, graph theory, etc. the theory of relativity is in physics, and in math it would be the equivalent of a theorem.

PROPERTY

An attribute, characteristic of something. for example, addition in the real numbers has a commutative property, one of its characteristics is that a + b = b + a. trapezoids have the property that the diagonals divide the trapezoid into four triangles, two of which have the same area and two which are similar.