Why is Pythagoras Theorem a "theory" but not a "law"? I mean we use it many times in school and to build stairs etc. and it has been proven, however it is still called a theory.

What are the conventions for calling something a "theorem" or a "theory" rather than a "law" in mathematics?

  • 5
    $\begingroup$ It’s called a “theorem”, not a “theory” – just as you wrote yourself. I never saw anyone use the term “The Pythagoras Theory”. What do you mean by your question anyway? $\endgroup$
    – k.stm
    Jan 16 '14 at 19:27
  • $\begingroup$ a theorem is a law $\endgroup$
    – janmarqz
    Jan 16 '14 at 19:28
  • $\begingroup$ we may ask Pythagoras about it? :D joke $\endgroup$ Jan 16 '14 at 19:29
  • $\begingroup$ Ok... well someone explain the difference between theorem and theory? $\endgroup$
    – itshanks
    Jan 16 '14 at 19:31
  • 4
    $\begingroup$ You are making the typical "theory"/"law" dichotomy mistake. The word "theory" is not used the way you are thinking in any of the sciences. $\endgroup$ Jan 16 '14 at 20:35

It sounds like you are using "theory" and "law" for things like "rules" as physicists sometimes do.

Traditionally mathematicians will call any significant result worth remembering a theorem. The Pythagorean Theorem is a archetypical example.

We use "law" sometimes too, but it's really the exception to the rule. (Like "Law of quadratic reciprocity.") There really isn't a standardized use of "law" in mathematics.

In mathematics, a theory is a large coherent group of results in the same field of study. It's used in the literal sense as "body of knowledge," not like "rule." So you can talk about "the theory of groups."

At any rate, since there is no formal definition of "law" or "theory" in mathematics, this is not so much a question about math as it is about language. You might try on english.SE.

English dictionary, I choose you!

Theory : 3.Mathematics . a body of principles, theorems, or the like, belonging to one subject: number theory.

Theorem: 1.Mathematics . a theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas.

  • $\begingroup$ @datodatuashvili You flatterer, you :) $\endgroup$
    – rschwieb
    Jan 16 '14 at 19:38
  • $\begingroup$ +1 but there is a formal definition of "theory" in mathematics, that is a set of propositions in some formal language, which is closed under deduction (from the rule of some logic): Set theory, the thory of real-closed fields, Peano's theory etc. $\endgroup$
    – rewritten
    Jan 16 '14 at 20:19
  • $\begingroup$ @rewritten OK, I'm sure the mathematical-logic definition of "theory" can be applied, but I guess "formal" wasn't the right word. I guess I just meant that we are thinking of the regular English definition of theory, usually. $\endgroup$
    – rschwieb
    Jan 16 '14 at 20:25
  • 1
    $\begingroup$ @rewritten And also I give exactly examples like that ("theory of groups") $\endgroup$
    – rschwieb
    Jan 16 '14 at 20:26
  • $\begingroup$ +1 great answer, but would mentioning lemmas just confuse things? I just think it might round out the proof terminology. $\endgroup$ Jan 16 '14 at 20:40

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