Lemma/Proposition/Theorem, which one should we pick?

This is something that confuses me. I have read a few mathematical texts and they often seem to use Lemma/Proposition/Theorem if they have a particular statement.

Now, which one to use? A lemma can be something you need to prove a more important theorem, but then what about Fatou's Lemma?

When to pick Proposition or Theorem?

• Other less common ones are "Scholium" and "Sublemma." I was in a lecture where Nolan Wallach used a Scholium. – Cheerful Parsnip Mar 7 '11 at 23:12
• I avoid proposition as some logicians use it to mean a well-formed statement without quantifiers which might or might not be true, and some mathematicians use it for a true statement they are not going to prove (I would use assertion for the latter). – Henry Mar 7 '11 at 23:43
• @Jim: What is a "Scholium"? Sublemma, fine, but Scholium? – Jonas Teuwen Mar 7 '11 at 23:49
• According to Wikipedia, "In modern mathematics texts, scholia are marginal notes which may amplify a line of reasoning or compare it with proofs given earlier. A famous example is Bayes' Scholium, a well-known result for interpreting observations of a Bernoulli process." – Cheerful Parsnip Mar 8 '11 at 0:56
• @jim: Rightfully so. Writing a scholium instead of a remark is like writing a prolegomenon instead of a preface. It's not quite the same thing but has a distinct highbrow smell. – t.b. Mar 8 '11 at 4:19

There seem to be two issues here. One is why certain well-known results are called Lemmas, such as Zorn's, Yoneda's, Nakayama's, and so on. I don't know the answer to this; presumably it is a mixture of what was written in some original source and the results of the transmission of that original source through the mathematical tradition. (As one interesting example of how labels can be changed in the course of transmission, there is a result in the theory of automorphic forms and Galois representations, very well known to experts, universally referred to as "Ribet's Lemma"; however, in the original paper it is labelled as a proposition!)

The second issue is how contemporary writers label the results in their papers. My experience is that typically the major results of the paper are called theorems, the lesser results are called propositions (these are typically ingredients in the proofs of the theorems which are also stand-alone statements that may be of independent interest), and the small technical results are called lemmas. This probably varies quite a bit from writer to writer (and perhaps also from field to field?).

• Yes, I suppose I was conflating these two issues in my answer. – Qiaochu Yuan Mar 8 '11 at 2:01
• Hmm, then Bill's suggestion is not that bad, it turns out to be quite subjective... Thanks! – Jonas Teuwen Mar 8 '11 at 12:04

I don't know if there are any hard and fast rules, but here is a rough start for others to nitpick:

• A Theorem is a major result that you care about (e.g. "the goal of this paper is to prove the following theorem").
• A Lemma is a useful result that needs to be invoked repeatedly to prove some Theorem or other. Note that sometimes Lemmas can become much more useful than the Theorems they were originally written down to prove.
• A Proposition is a technical result that does not need to be invoked as often as a Lemma.
• I always thought of a Proposition as an independent result, like a theorem, but not as important. – Cheerful Parsnip Mar 7 '11 at 22:48
• There's always the "technical lemmas" that are used possibly only once, for some big theorem. – Bruno Stonek Mar 7 '11 at 23:01
• And there are also the lemmas which are repeatedly used to prove many theorems, e.g. Yoneda's, Zorn's, not-Burnside's, Nakayama's, Itō's, ... I think someone once said it's far better to prove a good lemma than a big theorem. – Zhen Lin Mar 7 '11 at 23:11
• I think of Lemma and Proposition exactly in the opposite way: a lemma is a technical result that you "never" expect to use again; a proposition is a technical result that is interesting in its own right, is used more often than a Lemma, but is not as important as a Theorem. Perhaps I am wrong... – wildildildlife Mar 7 '11 at 23:59
• @Bill Dubuque: I think most people nowadays will agree that Euclid's Lemma deserves to be called a theorem. Just like Urysohn's Lemma, etc. This seems to be the first issue in Matt E's answer. – wildildildlife Mar 8 '11 at 11:09

While generally the terms are used as suggested by Qiaochu, there are some authors who are bothered by these nebulous subjective terms. For example, Kaplansky wrote in the preface of his classic textbook Commutative Rings

In the style of Landau, or Hardy and Wright, I have presented the material as an unbroken series of theorems. I prefer this to the n-place decimal system favored by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets elevated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems.

From my reading in maths, I have found the distinction useful in organization and presentation of results. I find it hard to believe that Hardy would have favored a "theorem only" approach. The distinction I find useful is essentially the following:

Theorem - main result of the paper. One to three theorems per paper, unless a long paper with many sections, where one to three theorems per section may be appropriate.

Proposition - result that is used in the proofs, but which may or may not be proved in the current presentation, and for which no originality is claimed.

Lemma - technical result used in the proof of the theorem, which is claimed as original and proved, but the main interest in which lies its use in the proof of one or more theorems.

Corollary - a specialization of a just presented theorem, in terms more likely to be useful in practice, or of intuitive interest.

For example Zorn's Lemma on partially ordered chains having maximal elements is not of much interest in itself, but it is key to have that established before proving Hahn-Banach Theorem or Tychonoff's Theorem.

I believe that this categorization is actually very useful for the following reasons:

(1) It helps the reader understand the purpose of a result in the larger scheme of the presentation, and to differentiate between results that are to be identified with the paper/section/chapter as part of its raison d'etre, versus ancillary results that may be important but are only being formally identified for use, proved or not, and not claimed.

(2) If one sticks to theorems only, particularly if one is not numbering within section, then one quickly reached double digit theorems, and there is potential cognitive interference between the section numbers and the decimal notation.

I'm surprised this type of thing doesn't have a well known codification somewhere ... it probably does, but I thought I would offer these arguments in favor of the distinction since the thread consensus seems to be trending in the other direction.

Ultimately, these are tools to make a paper more readable. I have found it useful to use the terms proposition and lemma to separate different threads in a paper. Lemmas are very much in line with the theorem (helping to display progression towards the theorem). I don't agree with Matt E's answer that a lemma should be small and technical, it may be deep and interesting in its own right, but to make it a theorem would distract from the main narrative of the paper. I agree that propositions may be less aligned, more ad hoc, and hence of independent interest. If propositions are not explicitly used later in the paper, but serve to highlight something important, then perhaps they should be called remarks.