# Theorem vs. Theory — What's the Difference?

By Tayyaba Rehman — Updated on October 23, 2023

A theorem is a proven statement in mathematics or logic, while a theory is a well-substantiated explanation in science based on evidence and facts.

## Difference Between Theorem and Theory

### Table of Contents

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## Key Differences

Often, a theorem will rely on certain axioms or postulates that are assumed to be true without direct evidence. For example, many theorems in geometry are based on Euclid's postulates. A theory, however, doesn't take its foundational facts for granted; it emerges after extensive empirical research and stands until evidence suggests otherwise.

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Oct 23, 2023

Delving deeper, theorems arise from a rigorous process of deductive reasoning. Once a theorem is proven, its truth is universally accepted within its context. Conversely, a theory, despite being well-supported, remains open to further testing, modifications, or even refutation based on new evidence or experiments.

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In mathematics, the path to proving a theorem includes using previously established theorems and axioms. This logical progression ensures the truth of the statement. Theories, in the realm of science, are formulated based on hypotheses, observations, experiments, and evidence. They offer a framework for understanding a wide range of phenomena.

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In summation, while both theorems and theories are essential to advancing understanding in their respective fields, the former deals with absolute proofs in a structured system, and the latter provides comprehensive explanations in science that withstand scrutiny but can evolve with new findings.

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A theorem and a theory, both fundamental in their respective fields, serve different purposes. The theorem is a statement in fields like mathematics and logic that has been proven to be true using a series of logical steps. On the other hand, a theory in scientific domains is a well-substantiated explanation for observed phenomena, built on a collection of facts and evidence.

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Oct 23, 2023

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## Comparison Chart

### Flexibility

Once proven, universally accepted.

Can evolve or be refuted with new evidence.

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## Definitions

#### Theorem

An established principle in mathematics or logic.

The binomial theorem explains the expansion of powers of a binomial.

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#### Theory

A well-substantiated explanation based on evidence and facts.

The theory of relativity altered our understanding of space and time.

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#### Theorem

A statement proven to be true using logical reasoning.

Pythagoras's theorem defines the relationship between the sides of a right triangle.

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#### Theory

A system of ideas intended to explain something.

Darwin's theory of evolution provides a framework for understanding the diversity of life.

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#### Theorem

A proposition that is proven based on previously accepted statements.

Fermat's Last Theorem remained unsolved for centuries until Andrew Wiles provided a proof.

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#### Theory

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

Music theory offers insight into the structures and patterns in compositions.

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#### Theorem

A deduction from axioms.

The fundamental theorem of calculus connects differentiation and integration.

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#### Theory

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.

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#### Theorem

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.

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#### Theory

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.

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#### Theorem

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

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#### Theory

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

A fine musician who had never studied theory.

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#### Theorem

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

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#### Theory

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

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#### Theorem

(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.

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#### Theory

Abstract reasoning; speculation

A decision based on experience rather than theory.

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#### Theorem

A mathematical statement that is expected to be true

Fermat's Last Theorem'' was known thus long before it was proved in the 1990s.

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#### Theory

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

Staked out the house on the theory that criminals usually return to the scene of the crime.

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#### Theorem

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

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#### Theory

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

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#### Theory

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

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#### Theorem

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

Not theories, but theorems ( ), the intelligible products of contemplation, intellectual objects in the mind, and of and for the mind exclusively.

By the theorems,Which your polite and terser gallants practice,I re-refine the court, and civilizeTheir barbarous natures.

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#### Theory

(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.

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#### Theory

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

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#### Theory

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

Knot theory classifies the mappings of a circle into 3-space.

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#### Theory

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 theory is consistent if it has a model.

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#### Theorem

A conclusion reached through rigorous proof.

The mean value theorem has applications in both geometry and analysis.

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#### Theory

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

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#### Theory

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

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#### Theory

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

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#### Theory

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

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#### Theory

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;

Theories can incorporate facts and laws and tested hypotheses

True in fact and theory

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#### Theory

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 scientific hypothesis that survives experimental testing becomes a scientific theory

He proposed a fresh theory of alkalis that later was accepted in chemical practices

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#### Theory

A belief that can guide behavior;

The architect has a theory that more is less

They killed him on the theory that dead men tell no tales

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#### Theory

A hypothesis that has withstood extensive testing.

The germ theory of disease revolutionized medicine.

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#### Theory

An idea used to account for a situation or phenomenon.

The Big Bang theory describes the origin of the universe.

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## FAQs

#### Is a theory just a guess?

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

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#### Do theorems rely on assumptions?

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

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#### Are "theorem" and "theory" interchangeable terms?

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

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#### How does a hypothesis differ from a theorem and a theory?

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

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#### Can a theorem be disproven?

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

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#### What does it mean to "prove" a theorem?

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

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#### Can a theory ever become a theorem?

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

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#### Is a theory always true?

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

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#### How does science use theories?

Theories in science offer frameworks to understand and predict phenomena.

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#### Why are some theories controversial?

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

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#### Do all theories originate from hypotheses?

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

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#### Do all mathematical statements become theorems?

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

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#### Can theorems be applied outside of mathematics?

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

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#### In which fields are theorems prominent?

Theorems are central in mathematics and logic.

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#### Is it essential to distinguish between a theorem and a theory?

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

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Oct 23, 2023

### Author Spotlight

Written by

Tayyaba RehmanTayyaba Rehman is a distinguished writer, currently serving as a primary contributor to askdifference.com. As a researcher in semantics and etymology, Tayyaba's passion for the complexity of languages and their distinctions has found a perfect home on the platform. Tayyaba delves into the intricacies of language, distinguishing between commonly confused words and phrases, thereby providing clarity for readers worldwide.