Thursday, 11 July 2013

Lemma, Proposition, Theorem and Corollary

 If you read a lot of papers in control theory, then you have probably come across the following words: (a) Lemma, (b) Theorem, (c) Corollary and (d) Proposition. These are terms that are used a lot in articles that uses mathematics a lot. When I first saw these terms, I had no idea what they meant which makes me miserable whenever a read a paper. However, once you understand these terms, it will allow you to understand a paper much more easier. So, I decided to write a post to explain what these terms means and how to read and write them in a paper.

In order to present the meaning as precisely as possible, I have check my understanding of these terms with several references as listed below (which I found it to be similar) :
 The least important term is Lemma. As a rule of thumb should be a statement that will used in proving a Theorem (the main result(s) of a paper) or another Lemma. Sometimes, mathematician refers it as a stepping stone. The purpose of presenting a lemma is to make the proof of a theorem more readable. By proposing a Lemma and its proof separately, you essentially make the proof theorem shorter and easier to digest. So, when you are writing a long proof for a theorem, consider partition part of the theorem into Lemma(s). In particular, making statements that will be used often in other proofs into Lemmas would improve readability of a paper significantly. When skimming through a paper, you can usually safely neglect the proof of Lemmas. But make sure you read the statements of the Lemmas, as they will be used in the proofs later on.
The well-known
Phythagorean Theorem

Next, comes almighty term, Theorem. Any statements that are labelled as Theorem(s) are the main result(s) of an article or a thesis. Before you read a mathematical paper, you always want to look for the Theorems to take a look at the paper's main results. This will give you an idea of what the final results are and does it relate to what you want to read. Often to write a theorem is easy - it is not that hard to figure out which of your results are important. However, to write a good theorem is hard. Keeping in mind that a lot of people will first read the theorem, it is important to write a theorem statement as self-contained as possible. For example, all (important) assumptions and equations related to the theorem should be mentioned in the theorem.

Sometimes, Theorems are followed by Corollaries. Corollaries are statements that are easily derived from the Theorems. The key point is EASILY DERIVED. Typically, corollaries would be useful statements that one can use for computational purposes or it is a statement when you "specialized" your theorem into certain situations or problems. While skimming through a paper, after you read the Theorems, try to read the Corollaries. It will usually provide you with additional insights or it will make abstract results into something more "concrete" (something that you can compute). As for tips on writing corollaries, I would say just make sure you are stating a useful corollary. There is no point in describing something useless.

Finally, you have statements called Propositions. From reading the comments on Math Overflow, there seems to be a little bit of disagreement to how Proposition should be used. But, in current trend, it is general safe to say that Propositions are statements that somewhat important but not as important as the main results. Usage of propositions can get pretty confusing. As a writer, I trying to avoid avoid using the label Proposition. I feel that labeling proposition it is rather subjective because the proposition maybe presented as a Theorem too. It is hard to distinguish the importance based on individual view. However, when you are reading a paper, you should try to look for Propositions too and pay some attention to them. Sometimes, they just might be important results for you. Also, I have read somewhere that there are some people who use Proposition rather differently. For example, some people might used as a conjecture. So, when you come across the word "Proposition", becareful...

This basically wraps up this long post of mine. My advice is try to get familiar with these terms so that you can read a mathematical paper more easily. Also, when possible, avoid the usage of proposition. Often, I just find it confusing how important is a particular proposition in a paper. It seems that it creates more questions than answering it. I hope you guys find it useful.

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