Theorem vs. Theory — What's the Difference?

An established principle in mathematics or logic.

A well-substantiated explanation based on evidence and facts.

A statement proven to be true using logical reasoning.

A system of ideas intended to explain something.

A proposition that is proven based on previously accepted statements.

A set of principles on which an activity or process is based.

A deduction from axioms.

A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research.

In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system.

A set of statements or principles devised to explain a group of facts or phenomena, especially one that has been repeatedly tested or is widely accepted and can be used to make predictions about natural phenomena.

An idea that has been demonstrated as true or is assumed to be so demonstrable.

The branch of a science or art consisting of its explanatory statements, accepted principles, and methods of analysis, as opposed to practice

(Mathematics) A proposition that has been or is to be proved on the basis of explicit assumptions.

A set of theorems that constitute a systematic view of a branch of mathematics.

(mathematics) A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas.

Abstract reasoning; speculation

A mathematical statement that is expected to be true

A belief or principle that guides action or assists comprehension or judgment

(logic) A syntactically correct expression that is deducible from the given axioms of a deductive system.

An assumption based on limited information or knowledge; a conjecture.

(transitive) To formulate into a theorem.

A description of an event or system that is considered to be accurate.

That which is considered and established as a principle; hence, sometimes, a rule.

(sciences) A coherent statement or set of ideas that explains observed facts or phenomena and correctly predicts new facts or phenomena not previously observed, or which sets out the laws and principles of something known or observed; a hypothesis confirmed by observation, experiment etc.

A statement of a principle to be demonstrated.

(uncountable) The underlying principles or methods of a given technical skill, art etc., as opposed to its practice.

To formulate into a theorem.

(mathematics) A field of study attempting to exhaustively describe a particular class of constructs.

A proposition deducible from basic postulates

A hypothesis or conjecture.

An idea accepted as a demonstrable truth

A set of axioms together with all statements derivable from them; or, a set of statements which are deductively closed. Equivalently, a formal language plus a set of axioms (from which can then be derived theorems). The statements may be required to all be bound (i.e., to have no free variables).

A conclusion reached through rigorous proof.

A doctrine, or scheme of things, which terminates in speculation or contemplation, without a view to practice; hypothesis; speculation.

An exposition of the general or abstract principles of any science; as, the theory of music.

The science, as distinguished from the art; as, the theory and practice of medicine.

The philosophical explanation of phenomena, either physical or moral; as, Lavoisier's theory of combustion; Adam Smith's theory of moral sentiments.

A well-substantiated explanation of some aspect of the natural world; an organized system of accepted knowledge that applies in a variety of circumstances to explain a specific set of phenomena;

A tentative theory about the natural world; a concept that is not yet verified but that if true would explain certain facts or phenomena;

A belief that can guide behavior;

A hypothesis that has withstood extensive testing.

An idea used to account for a situation or phenomenon.

No, a theory is a well-supported explanation built on evidence, not merely a guess.

Theorems often depend on axioms or postulates, which are accepted without direct evidence.

No, a theorem is a proven statement, while a theory is a substantiated explanation.

A hypothesis is an untested proposition, while a theorem is proven, and a theory is substantiated by evidence.

Once proven, a theorem is accepted as true in its system; it's not subject to disproof like a theory.

Proving a theorem means establishing its truth through logical reasoning and previously accepted statements.

No, the terms apply to different domains; theories explain phenomena, while theorems are proven statements.

While well-supported, theories can be revised or replaced based on new evidence.

Theories in science offer frameworks to understand and predict phenomena.

Theories might challenge established beliefs or lack widespread understanding, leading to controversy.

Many theories start as hypotheses, but they only become theories after thorough testing and validation.

No, only those that are rigorously proven are termed theorems.

While primarily mathematical, the logical structure of theorems can influence other disciplines.

Theorems are central in mathematics and logic.

Yes, understanding their unique roles ensures accurate communication and comprehension in their respective fields.