# Theorem versus Proposition

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?

• 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
• 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
• 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
• Coffee -> Theorem, Tea -> Proposition. In all seriousness, though, I agree that "Proposition" is a useful label, lying somewhere between Lemma and (Main) Theorem. – Peter Dukes Apr 28 '14 at 12:00
• A great mathematician once told me that theorems should be called 'theorems' and calling them 'propositions' is a kind of snobbery. – bof Jan 5 '15 at 12:51

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.

• 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
• 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
• 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
• @Mark: Yoneda once had that dream... – Todd Trimble Feb 5 '12 at 4:44
• 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.

• Is there a reason why an author would want or need to tell his readers which results he is proud of and which he is not proud of? Or why a reader would want to know that? – bof Jan 5 '15 at 13:35
• @bof The author would want to tell the reader about the result precisely because of being proud of it... Have you met humans before? – Najib Idrissi Feb 20 at 8:57

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.

• 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
• When I read "Proposition", I smell the "proof is left as an exercice" coming up. – Pertinax Apr 23 '15 at 10:05

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.

• 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

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.

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.

• 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....

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.

• 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
• 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

Besides the points mentioned in other answers, a theorem or proposition is usually something whose main import is reasonably clear from the statement. A lemma, by contrast, is often a statement whose interest is less obvious until one sees it used.

So “if $X$ has diameter $< 1/2$, then the ring $St(X)$ is commutative” would be a theorem or proposition, depending on how important/difficult it is, since it’s clear to the reader what the statement means and why you might want to know it. But “if $X$ has diameter $< 1/2$ and the ambient braiding is sylleptic, then there is some $k$ for which all primary ideals of $St(X)$ are $k$-dense” is more likely to be a lemma.

Edmund Landau had theorems only, and otherwise nothing (when it comes to naming special statements which required a proof; there should be also definitions and axioms on the top of the theorems).

Irving Kaplansky liked the Landau's style, thus he did the same.

(I am missing many of my books; in the case of Landau, there was a 2-volume differential and integral calculus textbook CLASSIC; in the case of Kaplansky, I think it was a monograph on ring theory).

You may always emphasize important results in several ways, without putting down the other results. One may put a name of a theorem in parenthesis, just after Theorem nn or one may create a subsection which identify the theorem in the section title, e.g. Section 2.4 The fundamental Theorem of Algebra (while, inside the section, this theorem could be Theorem 72, with a sentence of an explanation preceding that theorem).

A Theorem is an assertion that can be proved from the Axioms.

Axiom, Lemma, and Corollary are subspecies of Theorem. Axioms are distinguished by having an extremely short proof; a Corollary has a relatively short proof on the basis of a previously established Theorem; a Lemma is intended for use in proving another Theorem.

Proposition is more general than Theorem; a proposition is an assertion which may be proved or unproved, true or false.

Some mathematicians inappropriately use the word 'Proposition' as if it meant 'Trivial Theorem'.

• If a proposition is an assertion which may be proved or unproved, true or false, why is it inappropriate if a proposition happens to be a trivial theorem, i.e., a trivial assertion that can be proved from the axioms? – Yuichiro Fujiwara Jan 5 '15 at 13:14
• Labeling some results as 'propositions' meaning 'trivial theorems' is inappropriate because it smacks of snobbery and false humility. The reader can judge for himself whether the result is trivial, and his opinion may differ from the author's. – bof Jan 5 '15 at 13:30
• I said "happens to be." Do the "some mathematicians" in your mind write or explicitly say, in their papers and books, that "proposition" in their dictionary is "trivial theorem"? If they do, what's wrong with expressing one's opinion that something is trivial in their own work? – Yuichiro Fujiwara Jan 5 '15 at 13:34
• It is a matter of taste. – bof Jan 5 '15 at 13:59
• I see. You're saying a style that isn't your cup of tea is inappropriate, then. – Yuichiro Fujiwara Jan 5 '15 at 14:01