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What is the difference between a theorem and a theory? The two words seem to be used to describe very similar things, but yet do not seem to be interchangeable.

For example, we have Pythagoras' theorem and Einstein's theory of relativity.

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5 Answers 5

up vote 18 down vote accepted

A theorem is a result that can be proven to be true from a set of axioms. The term is used especially in mathematics where the axioms are those of mathematical logic and the systems in question.

A theory is a set of ideas used to explain why something is true, or a set of rules on which a subject is based on.

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A theorem is not limited to mathematics –  Thursagen Aug 23 '11 at 8:35
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@Thursagen: even with your answer, I don't see this. Do you have an example of a 'theorem' which is not mathematical (relies on experimental data)? –  Mitch Aug 23 '11 at 19:24
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Yes. There is a theorem in physics, and that's not mathematical. It's called the Adiabatic theorem –  Thursagen Aug 24 '11 at 1:05
    
@Thursagen The adiabatic theorem can be stated with full rigor in the language of mathematical physics, and is just as mathematical as the Pythagorean theorem. –  senshin Apr 25 at 17:01
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From the Shorter Oxford English Dictionary.

Theorem:

A universal or general proposition or statement, not self-evident (thus dist. from axiom), but demonstrable by argument (in the strict sense, by necessary reasoning); 'a demonstrable theoretical judgement'.

Theory:

A scheme or system of ideas or statements held as an explanation or account of a group of facts or phenomena; a hypothesis that has been been confirmed or estabilished by observation or experiment; a statement of what are held to be the general laws, principles or causes of something known or observed.

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Your quote is quite right, but the examples are not. Pythagorean theorem is a theorem because it can be deductively proven from the axioms of Euclidean geometry. Testing on real triangles is empirical and inductive on the contrary. –  C.R. Aug 23 '11 at 9:57
    
You are right, I lost myself trying to summarize everything and just messed it up... :/ –  Gurzo Aug 23 '11 at 11:35
    
@Gurzo: you should modify/edit your answer to fix the problem. –  Mitch Aug 23 '11 at 13:36
    
Deleted further explanations and left only the quoted definitions. –  Gurzo Aug 23 '11 at 13:39
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I found a discussion where people were talking about the same matter, and someone put the difference between "theorem" and "theory" in a nutshell:

Theory - Verifiable Explanation.
Theorem - Demonstrable Explanation.

and explained:

Verifiable will mean that you can show that there is evidence for it. Demonstrable means that you can do it again to show people the evidence, and that they can do it too.

Wikipedia puts forth further differences:

The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical

"Empirical" means, "The word empirical denotes information gained by means of observation or experiments", whereas "deductively" means more of drawing from logic and reason, not facts.

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To qualify as a theorem, something has to have been proven (or at least believed to be provable) -- the theorem is an inescapable conclusion from some set of axioms. As implied by "provable", in some cases, the word is used to refer to something that hasn't been proven yet, but is believed to be open to logical proof (e.g., Fermat's last theorem remained unproven for centuries, but was eventually proven).

For a theory, you need supporting evidence, but not actual proof. There may easily be two or more competing theories about the same particular subject, each with some supporting evidence but none actually proven.

The difference in requirements means that theories abound in the real world, but theorems are restricted primarily to systems of logical abstraction with clear-cut axioms and rules governing how those axioms can be combined.

This may initially sound like a theorem is a much stronger conclusion than a theory -- and in a way it is. At the same time, it must be noted that while a theorem is typically absolute within its domain, it's also relatively narrow. A theory may support a conclusion much more weakly (if at all), but apply much more directly to real life.

Just for one example, many economic "theories" are really theorems that have been proven -- about drastically simplified models of a real economy. For better or worse, however, many are treated as applying to real life, even though they're based on (often drastically) simplified models.

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A theorem is a mathematical deduction.

A theory is a collection of statements or 'ways' of thinking that purport to explain a circumscribed set of experiences. A theory can be supported by experimental evidence or anecdotal observation. The term 'theory' can range in connotation from synonymous with 'conjecture' in opposition to 'fact' (e.g. "I have a theory that drinking from the opposite side of the glass stops hiccups"), all the way to a set of theorems on a given set of mathematical structures (e.g. group theory).

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