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As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(s) 'Theorem', any auxiliary result leading to this Theorem a 'Lemma' (and, sometimes, small observations that are necessary to prove a Lemma are labeled as 'Claim'). I avoid using the term 'Proposition'.

However, sometimes a paper consists of a number important results (which by all means earn to be called 'Theorem') that are combined to obtain a certain main result. Hence, another term such as 'Proposition' might come in handy, yet I don't know whether it suits either the main or the intermediate results.

So, my question is: When to use 'Theorem' and when to use 'Proposition' in a paper?

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as with most soft-questions, I'd recommend this for "community wiki", which you can get by clicking edit on your question, and then checking a box. –  David Jordan Mar 16 '10 at 12:56
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Just yesterday I was asked this question by a native speaker, and it's not the first time. Few people are native speakers of Math Paper. As expected, nobody at the table had anything especially definitive. –  Allen Knutson Mar 16 '10 at 13:56
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I've whapped this with the community-wiki hammer. Per Allen's comment, I think everything here is going to be pretty subjective, so I hope no one minds. –  Scott Morrison Mar 16 '10 at 17:05
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Whichever way you decide to name your results, please use just one numbering scheme for all of them. If "Proposition 1" appears two-thirds of the way through the paper, after "Theorem 3", "Corollary 4", and "Lemma 4", a poor reader trying to refer to it quickly will never find it. –  Mark Meckes Mar 16 '10 at 18:49
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Actually Theorem is from Greek and Proposition is from Latin, so going to the etymology may help a proper use (which is not necessarily the most popular one). Here's a nice list of mathematical terms etymonline.com/… –  Pietro Majer Sep 5 '11 at 10:01

10 Answers 10

up vote 32 down vote accepted

The way I do it is this: main results are theorems, smaller results are called propositions. A Lemma is a technical intermediate step which has no standing as an independent result. Lemmas are only used to chop big proofs into handy pieces.

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A Lemma is a technical intermediate step which has no standing as an independent result. But sometimes they escape, as Zorn's or Fatou's lemmas did. –  Gerald Edgar Mar 16 '10 at 13:56
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In fact I've heard it claimed that many mathematicians' fondest dream is to prove not a great theorem but one great lemma. –  Mark Meckes Mar 16 '10 at 14:45
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See also Zeilberger's 82nd opinion "A Good Lemma is Worth a Thousand Theorems": math.rutgers.edu/~zeilberg/Opinion82.html –  Philipp Lampe Mar 17 '10 at 10:25
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@Mark: Yoneda once had that dream... –  Todd Trimble Feb 5 '12 at 4:44
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If I remember correctly, in Atiyah's paper on division of distributions (ams.org/mathscinet-getitem?mr=256156), Hironaka's Theorem is stated as a Lemma! –  ACL Feb 25 '13 at 2:13

Here is a good rule of thumb:

If you are proud of a result, call it a Theorem. If not, it is a Proposition.

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I broadly agree with Anton Deitmar, except that I think "Lemma" is difficult to classify (and I tend to just avoid using them). For example, minor results of more generality than the larger theorem they're being used for are frequently lemmas, are they not?

Typically, one doesn't use "Claim" in the same way as "Lemma," "Proposition" or "Theorem" -- I would use it as a sub-heading within the proof of some bigger result, but not as a freestanding result. I use "Proposition" as my default, and "Theorem" for the most important results (e.g., the culmination of some long line of reasoning, the main result in a paper, etc.).

My only complaint with David Jordan's answer is that there are many results that don't have "genuine content" and yet which cannot be easily proven from a definition.

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Fair enough! =] –  David Jordan Mar 16 '10 at 14:48
    
I would try that statements of Theorems, Propositions and Lemmas are self contained (so that they can be re-used safely), while the notation and the properties of objects used in a Claim could be given by the context. –  ACL Feb 25 '13 at 2:16

Of course, this is a very subjective question, but I would tend to use "Theorem" only for a statement which has genuine content (whether my own, or one I am citing) and which I wouldn't expect the reader to be able to prove themselves fairly easily. Usually a paper shouldn't have many of these, probably no more than one per section.

"Proposition" I would use after having given a definition, when showing that some fairly straightforward (but not completely obvious) consequence holds; for instance showing that some linear subspace of functions is actually a subalgebra. This is probably close to how you said you use "claim", although I suppose the difference is that you can propose something somewhat out of the blue following a definition, while "claim" is usually directly related to some logical structure which is already moving forward, say to highlight a point midway through the proof of a theorem.

So I make the distinction that Proposition is something that the reader, if so inclined, could easily prove for themselves once they understand the definition. It highlights a result that could just as well have been stated in plain text, emphasizing that while it may be straightforward to prove, it is nevertheless worthy of note.

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It seems from your answer that you deem a Proposition as being less significant than a Lemma. To me it's the other way around. But hey, it's subjective as you said. –  Greg Martin Sep 5 '11 at 5:57

There is no stylebook in mathematics dictating which term to use in which situation, as I think the earlier comments reflect. Every proved statement (even a corollary) might be labelled "theorem", but no one wants to go that far. For me a "lemma" is a technical step in a proof of something bigger, isolated for convenience and possibly for later use. (Unless the "lemma" acquires a life of its own, graduating to "Lemma".) A "theorem" means to me a major result, perhaps the goal of an entire paper. The use of "proposition" is most subjective, but it gets tedious to read a paper containing numerous secondary results claiming to be theorems. Even "corollary" is somewhat subjective, since it might follow instantly from an earlier result or else require other inputs and/or some cleverness to derive. In German there is "Satz" but also "Theorem" to confuse translators.

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In fact, in German I would use "Satz" for a minor theorem and "Theorem" for a major, fundamental theorem. However, I think it will be regarded as hubris to call a theorem of your own a "Theorem". –  MRA Mar 17 '10 at 9:47

Not that I think I have the definitive answer on this question, but:

Suppose I'm writing a reasonably long paper, broken up into sections. I use Lemmas for technical statements and so on, as many other answerers do. I would rather use Propositions, however, when the result is more global in scale: if it's going to be used outside its section, for example, I'll call it a Proposition, but if it's only used to build up things inside its own section I'll call it a Lemma. Then the Theorems are the statements I want people to take away from the paper.

In this paradigm, a Proposition is more like a Super-Lemma than a Mini-Theorem. I doubt that's a universal sentiment - probably it's not even universal among my own papers....

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I have seen "Proposition" used for a difficult result that is cited (without proof) from the literature, but which is central to the arguments of the paper. Theorems are usually those big results which the authors of the paper prove themselves. I don't know how prevalent this practice is though.

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I use, like many others, Proposition by default and Theorem for the main results. But I always (I think) cite other people's results as Theorem.

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That's a good policy, even if the result is called "lemma", as in en.wikipedia.org/wiki/List_of_lemmas –  Igor Pak Mar 16 '10 at 17:48
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So do I, although I do like to use the author's own terms when using their results. –  Anthony Labarre Mar 16 '10 at 18:35

The trouble with non-quantifiable terms is that said terms are used in a wide variety of areas so they acquire different connotations and as such are "soft" words. One such term is the word "proposition." In Logic it is loosely synonymous with the word statement. E.g "The A-proposition is a Universal Affirmative "proposition" or statement; All humans are mortal beings. Christian Goldbach's statement about "prime #'s has been called his "Theorem." The term "proposition" does imply, indeed, a theory which is rather "tentative" than one we could rightly called "theorem."

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I'm sorry, but I don't see how this addresses the question. But the link to propositional logic is nonetheless an intriguing terminological aside. –  David Roberts Sep 5 '11 at 5:49

Every proved statement is a theorem and in mathematics there are only axioms definitions and theorems.And of course initial concepts like the concept of a set that can not have strict definition.There is ambiguity in whether a theorem is lemma or proposition or corollary. In LANDAU'S Differential and integral calculus there are only theorems.

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