Mathematical Foundations Vocabulary

a statement accepted as true without proof, serving as a starting point for reasoning

“Euclid's fifth axiom about parallel lines sparked centuries of mathematical inquiry.”

Origin: Greek axioma `that which is thought worthy`, from axios `worthy`

a statement that has been proven true based on axioms and other theorems

“The Pythagorean theorem relates the sides of a right triangle.”

Origin: Greek theorema `speculation, proposition to be proved`, from theorein `to look at`

a proven proposition used as a stepping stone to a larger result

“The proof relied on a clever lemma about prime factorization.”

Origin: Greek lemma `something received or taken`, from lambanein `to take`

a proposition that follows directly from a proven theorem

“As a corollary, all equilateral triangles are also equiangular.”

Origin: Latin corollarium `gratuity`, from corolla `small garland`

a proposition believed to be true but not yet proven

“The Riemann conjecture remains one of mathematics' greatest unsolved problems.”

Origin: Latin conjectura `conclusion, interpretation`, from conicere `to throw together`

a logical argument establishing the truth of a statement

“Her elegant proof reduced the problem to a simple contradiction.”

Origin: Old French preuve, from Latin probare `to test, prove`

a statement assumed true as a basis for reasoning

“Einstein's postulate that light speed is constant revolutionized physics.”

Origin: Latin postulatum `demand, request`, from postulare `to demand`

a proposed explanation or assumption to be tested

“The null hypothesis assumes no significant difference between groups.”

Origin: Greek hypothesis `base, basis of an argument`, from hypotithenai `to put under`

proving a statement by showing its negation leads to an impossibility

“The irrationality of √2 is elegantly shown by proof by contradiction.”

Origin: From Latin contradicere `to speak against`

proving a statement for all natural numbers by establishing a base case and inductive step

“Mathematical induction proves the formula holds for every positive integer.”

Origin: Latin inductio `a leading in`, from inducere `to lead in`