Lemma, theorem, corollary... which one is a suitable term for an observation?

Question

I have an observation from say theorem of "the uniquness of interpolating polynomial". And I tried to prove that observation. However, I donot know whether anybody else has made that observation. And I want neither say it is completely observed by any matemathician (as I have not seen that) nore say it is completely done by myself. So I doubt about whether anybody has observed it.

Also I'm using it in computer security and I know nobody in the same field has made that observation.

Question: What shall I call my observation? lemma, theorem or corollary, etc ...?

Answer 1

I agree with @user7530. I use “proposition” as a base level for all statements and use the other names to give it additional flavor – I use “theorem” if the proposition is relatively important – e.g. the main result of the paper or a particular section; “lemma” if the proposition is kind of technical and/or used just to prove another proposition; “observation” if the proof is direct or short or can even be omitted; “corollary” if the proposition is directly implied by previous proposition(s).

Answer 2

It sounds like the context in which you present the theorem is a lot more important than the label on the theorem itself. If this is a key observation that underpins your entire algorithm, sure, go ahead and call it "theorem," and introduce it as an observation that to your knowledge is novel (assuming you have done due diligence searching the literature.)

Answer 3

Theorem, lemma,corollary seem to be a matter of taste.Some very important results are known as lemmas, e.g. Fodor's Lemma (set theory), Urysohn' Lemma (topology).And Konig's Theorem (set theory) is a fairly minor result, but Konig's Lemma is very important. When presenting a result, a lemma is an intermediate step to get to what you consider a main result, and a corollary follows briefly or readily from a lemma or theorem , often as a special case.What others call them later doesn't matter.If they call it anything at all ,you've succeeded. If you think your result may be original ,you could e-mail a mathematician, at a university, whose specialty is, or seems to be, in the matter of your result.They would very likely know, or tell you how to find out, whether it's new.

Answer 4

The words "theorem", "proposition", "corollary", etc, do not carry any connotation as to whether the statement is new or was previously proved.

In an academic paper, it is assumed that if you are aware that the statement was previously known, you will cite the paper where the original proof appeared (or, if it is a "well-known" result, you may instead cite a textbook or other source that includes a proof that you like). If you do not give a citation, readers will assume you are claiming that the result is new; that you have diligently searched for any previous appearance of the result in the literature, and not found any. If it turns out that it was already known, and you didn't cite it, you may get an icy note from the paper's referee or editor (and possibly a rejection notice). If this comes to light after publication, you can expect some indignant letters from the original author(s) and/or their friends or fans, and you will probably need to publish a correction to your paper giving the appropriate credit. If the result was the main focus of your paper, you may even have to retract it.

If you are working outside your usual area of expertise, then before publishing, you would be well advised to consult an expert in that area, who would be more familiar with the state of the art, and would have a better idea of whether the result is already known, or at least where you should look. In some cases, it could be that it is a consequence or special case of a more general result, which may be stated in terms that you might not recognize. Being unfamiliar with an area is not really an excuse for claiming that already-known results are new; as the author of the paper, it is your responsibility to give appropriate credit, full stop.

Although "new result" is in some sense the default if not otherwise stated, if you wish you can emphasize its novelty with an introductory sentence saying something like "This result is new" or "This theorem appears here for the first time", but you had better be pretty sure that it really is new. If you are not quite as sure, or wish to sound a little less bold, you can write "This result appears to be new" or something similar. This is mostly useful if you are also quoting a number of similar statements which are not new (and giving appropriate citations), and you are afraid the reader will lose track of what is old and what is new.

Answer 5

In the end, it is up to you. I have seen texts where everything that is proved is called a "proposition", and some trivial to prove facts are named "observation"; others call almost everything "theorem"; yet others call an auxiliary fact proved a "lemma" (but sometimes isn't even singled out particularly), a central, important fact, specially if hard to prove, to them is a "theorem", and if from a "theorem" something important follows directly by a simple application it is called "corollary".

Just make sure you are relatively consistent.

Also consider that most mathematician's wet dream isn't to have a theorem named for them, but get their name attached to a lemma. E.g. Gauss' lemma, Burnside lemma, and the list goes on.

Answer 6

Instead of answering your question directly, since I hear this concern about terminology so often, I'd like to offer how I think about this:

Theorem - a non-trivial observation that may or may not be easy to prove. That it's non-trivial means the proof is non-vacuous. In mathematical works, one finds theorems as provable claims of importance and that're taken in relation to the current line of reasoning, whatever it may be.

Lemma - a buttressing result proven in the course of attempting to prove a theorem. Eg., it may be necessary to prove claims A, B and C before being able to prove D, or say, it may be more helpful to prove the first three claims in order that the fourth be proven.

Corollary - a result that follows from a proven theorem. Usually it's a direct result, but sometimes such may require an additional discussion, in which case it may be indirect. Notwithstanding, in math, it's taken as a follow-along result.

Proposition - a claim that may or may not be related to a current line of reasoning and may or may not be proven. Typically, a proposition is found at either (a) the beginning of a line reasoning where it's used as a starting point or (b) out-of-context with respect to the current line of reasoning, which makes it part of a "hey, look at this!" type expression or (c) as an unproven axiom, in which case it might also be called a postulate.

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I italicized the particular word proposition because I think it fulfills the 'Cicero criterion' of that which is most befitting; in other words, I think it applies to your situation. You don't seem to indicate that you're original goal was to author a scholastic paper, but rather that you stumbled on some finding in the course of your day-to-day work. For this reason, I'd assign it the label of proposition.

And now, assuming that your finding is non-trivial (ie., eg., it's not an identity proof), that there's an absence of evidence is no indication that there's an evidence of absence; indeed, another may have shown this to be false or true somewhere you've'nt come to experience. To that end, I'd suggest contacting an expert (or several) for verification because false proofs are very common. If your finding corresponds to something similar to polynomial uniqueness per spline interpolation, perhaps a numerical analyst, an optimization gal or an applied statistician could lend a hand. There's a host of specialties to appeal to here.

If indeed it's both non-trivial and provably true, then you might switch the labeling to theorem in the course of publishing your work, assuming that's what you'd do.

Good luck.

Answer 7

I think you should start at the right end first. Unless you (or somebody else) have proven your claim then it's not fit to call a theorem/proposition or anything like that. Before you have proven it it's a hypothesis/conjecture. That distinction is crystal clear.

The distinction between theorem/lemma/corollary/proposition is more a matter of taste. One could use a rather consequent policy and always call it theorem or always call it proposition (but you can't hardly call everything a lemma or proposition).