Nuances of the word "proposition" (versus "theorem") in mathematical writing

Question

In mathematical writing, the word "Proposition" is often used to label lesser theorems. However, I tend to feel that there's a further difference in the way the words "Proposition" and "Theorem" are used. Namely that when a writer says: "Proposition. foo," I tend to feel that the writer is meaning "I propose that foo" or "I claim that foo holds." Whereas when a writer says "Theorem. foo," I tend to feel that this merely means "foo" or "foo holds."

Question (addressed especially to people who have written mathematical papers using these terms.) Would it be fair to say that you tend to use the words "Proposition" and "Theorem" differently, that this difference goes beyond merely a difference in the size or importance of the result being stated, and that you use the word "Proposition" to emphasize that you're actually claiming something, as opposed to merely stating its truth?

Answer 1

"Proposition. foo", in a mathematical paper, doesn't mean "The author proposes that foo". It exactly means "foo holds". The difference between Proposition and Theorem in a mathematical paper is just the importance of the result obtained. The main result is always tagged as Theorem. Results tagged as Propositions are results of a minor importance in comparison with the main result(s). They also serve to clarify the exposition of a paper. If the proof of a Theorem is too long it can be divided, by using Propositions.

Of course, there is no a universal notion of Theorem and Proposition. This can change from paper to paper in the following sense. Main theorems in some papers of lower quality can be worse than some Propositions in high level papers.

Answer 2

Lemma fits into the hierarchy in much the same way (ever noticed how the most often cited results are Lemmas?)

It's mostly a contextualized hierarchy to convey role and importance.

A lemma is usually a technical result whose intended purpose is a stepping stone to bigger results. They get cited a lot because they are specifically things that make bigger things work. They are machinery to produce results, not finished products.

Propositions are either more substantive or of expected independent interest/worth, but not a major goal of the paper. They're not quite a building block alone, not quite a final product.

And Theorem goes up from there: principle results of the paper, or substantially interesting results. They are the house that was built up from the bricks.

But this is all still relative to the paper. What's major to a mediocre researcher may be minor to a leader in his field. Even personal preferences of the author can come into play. I've seen works use only theorem and corollary, and others that are dozens of lemmas and one theorem. In the hands of other authors you would have seen some propositions!

Honestly, one of the most common questions that comes up when writing a paper is "Should this be a proposition/theorem/lemma?" There's no absolute rules, and none of us are always immediately certain which thing we should call a result.