theorem

theorem — noun

1. In mathematics and logic, a statement whose truth has been established by a vali

1.nounC1
Definition

In mathematics and logic, a statement whose truth has been established by a valid chain of reasoning, working step by step from agreed-upon starting points (definitions and axioms) and previously proven results.

Example

Pythagoras proved his famous theorem about right-angled triangles more than two thousand years ago.

proved + famous theorem + about [topic]

Every student in the advanced maths class had to prove the theorem using basic algebra.

prove a theorem using [method]

Common collocations
  • prove a theorem
  • Pythagoras's theorem
  • mathematical theorem
  • fundamental theorem
  • state a theorem
  • apply a theorem
  • the theorem states that
Synonyms
  • proposition

    A broader term for any statement put forward for consideration; a proposition becomes a theorem only after it has been proved.

  • lemma

    A minor theorem proved as a stepping stone toward a larger, more important theorem.

  • corollary

    A statement that follows directly from a theorem with little or no additional proof needed.

Antonyms
  • conjecture

    A statement believed to be true but not yet proved; the opposite status of a theorem.

Usage Note

Used primarily in formal mathematics and logic contexts. A theorem is distinct from a hypothesis (an unproven guess) and an axiom (a starting assumption that is accepted without proof). Common verbs used with this sense include prove, state, apply, and demonstrate.

Common Mistakes

Einstein's theory of relativity is just a theorem.
Einstein's theory of relativity is a scientific theory, not a theorem.
A scientific theory is a broad explanatory framework supported by evidence, while a theorem is a specific statement proved within a formal system such as mathematics.
The scientist made a theorem about the weather.
The scientist proposed a hypothesis about the weather.
Theorem is almost never used for natural or social sciences; use hypothesis or theory instead.

Word family

Synonym discussion

Sense: noun/1

In mathematics, these terms describe statements at different stages of proof. A hypothesis is an unproven idea that a researcher tests through experiments or further reasoning — it may turn out to be wrong. An axiom (or postulate) is a starting assumption accepted as true without proof, forming the foundation of a logical system. A lemma is a smaller result proved on the way to a larger theorem, often used as a building block. A corollary is a statement that follows so directly from a theorem that little or no extra proof is required — think of it as a bonus result. A conjecture is a statement that mathematicians believe is true but have not yet proved; once proved, it becomes a theorem. For example: the Riemann hypothesis remains a conjecture after more than a century, while Pythagoras's theorem has been proved and used for over two thousand years.

hypothesis ↗axiom ↗lemma ↗corollaryconjecture

Common collocations

Nouns

  • mathematical theorem
  • fundamental theorem
  • Pythagoras's theorem
  • geometry theorem

Phrases

  • prove a theorem
  • state a theorem
  • apply a theorem
  • the theorem states that
  • proof of the theorem

Etymology

GreekLatinEnglish

From Latin 'theorema', borrowed from Greek 'theōrēma', meaning 'that which is looked at, a spectacle, a speculation, a proposition'. The Greek root 'theōrein' means 'to look at, to observe, to consider', which is also the source of the word 'theory'. The shift from 'something looked at' to 'a statement proved by reasoning' reflects how ancient Greek mathematicians thought of proof as a form of intellectual vision — seeing why something must be true.