 In this final video for lecture three, I just want to establish some definitions of things we're going to talk about going forward, things like theorem and proof and things like that. A theorem is a statement which can be proven to be true. So it's a statement that's necessarily true, but it also has a proof behind it. Well, what's a proof? A proof is a mathematical argument which logically, which is logically valid in sound which explains without any ambiguity whatsoever why the statement is true. So it's an argument that provides the truthfulness of a theorem and we know it's true. There's no doubt amongst it. That's what a proof is. Now mathematical research, what do professional mathematicians do? They do research and this mathematical research stands upon three essential pillars. So the first pillar would be like introducing new or novel concepts or methods. So you come up with new stuff, new ideas. That's one thing that researchers do. Typically new ideas to solve old problems. Also research often is bridging together existing mathematical topics in ways that have not already been seen before. So maybe I clarify this. We have things like new ideas. We have things like new methods or I should say like new connections. Let's say it that way, new connections. And then thirdly, when it comes to mathematical research, the last part basically comes down to proofs, right? We prove things. We prove new white, we prove theorems thus give truth to things we didn't know before. This last pillar is one of the most, is one of the most that we do like as in for mathematical writing often revolves around this type of thing. And that's why a class like math 3120 transition to advanced mathematics focuses so much on proof writing. This is a mathematical communication class, but we focus so much on proof writing because that's actually the type of communication that mathematicians do the most. And what's sometimes called peer mathematics, which is basically exclusively focused on proofs. I don't want you to think that if you're in other branches of mathematics like statistics, applied math, education, data science, they do proof writing too. Now their proofs might take on a different style than perhaps in peer mathematics. But proof writing is ever present in all of the mathematical sciences. There's a lot of emphasis that's put onto it, which is why we talk about theorems and proofs all the stinkin' time. Now you certainly have encountered many theorems prior to this moment, and maybe you've seen some of their proofs as well, although that's usually not as much common. Many of these proofs might have big fancy names like the fundamental theorem of algebra or the intermediate value theorem. You're given such names so that mathematicians and students, like yourselves, can refer back to them by name as justification for other mathematical results or calculations. That is, we might be like, oh, this proof follows by the mean value theorem. This happens all the time. We have seen a few theorems already in this course. We typically didn't call them theorems because we didn't actually define what theorem is yet, but they really were theorems. They were mathematical statements, which we proved to be true. They were theorems, all right? We've seen them already. Now oftentimes, like in a mathematical course or a textbook or even in a paper, many theorems receive no name at all, but they're given a number. So you can still cite them if you need to, like theorem. You might get something like theorem 1.3.5, just as an arbitrary number there. This happens so that readers can reference back to them or they themselves can justify their calculations or their proofs using this theorem. So you might say something like, oh, in the paper by Missildein 2014, you are referencing theorem 1.3.5 and your result didn't follow, something like that. I'm citing myself here just for the sake of example here. So we have to reference theorems all the time. So we often give them a label so we can refer back to them. So an example of this might look like something like the following for which now we know that this set of vectors is linearly independent by theorem 2.3.4. This sentence is ripped right out of my linear algebra textbook called linear algebra done openly. And so this is an actual theorem in the book and we've established the linear independence of vectors by that theorem. So this is how we write mathematics. We reference a theorem that just has a number to justify a later result. So it's important to number our theorems and to give them proofs. Now the term theorem is often reserved for major results, where here major is describing either the difficulty of the proof or the importance of the statement and its consequences. So there's a little bit of leeway on what is a major result or not. A major result is typically called a theorem. Now other statements, which are true by proof, which are technically theorems, will typically be given a different name. So the major results are theorems. The fundamental theorems, of course, are the biggest, most important of the entire discipline, right? Conversely, a proposition is a proven statement, which might be of less significance or has an easier proof. So it's a lesser statement. The theorems are the big ones. The propositions are the less ones. There's also something called a corollary. Sometimes when you have a proof, there's another result that's useful that the proof is so short or so immediate from the theorem that we call it a corollary. So it's like an immediate consequence of the theorem that's worth mentioning, but still it's pretty easy to prove. And sometimes corollaries don't get proofs. That is they don't write them down because they're so straightforward. That's not always the best practice, but sometimes it is, it depends. And then there's one other type of statement we should talk about. That is a lemma. We talked about Euclid's lemma in a previous video for this lecture here. A lemma is a helper theorem. That is, the theorem, the statement by itself might not be super useful, but you use it to prove other theorems with just high frequency that it's dubbed a lemma. It helps you prove other theorems. And that's because proofs of theorems can start to get longer and longer and longer and longer that to help parse it and make it easier to read, you break it up into lemmas. Much like Euclid's lemma we used to prove a previous theorem as well. There is another term that sometimes is used. It's called a porism. A porism is a statement you prove in a proof that you aren't trying to prove. You're trying to prove something else, but it turns out to be so useful. You adopt it as a standalone theorem itself. It's kind of like you accidentally found an interesting mathematical result. And for that reason, you can kind of see why people avoid it. But if it really was that useful, you could probably rewrite your proof to first prove the porism as a lemma and then you use that lemma to prove the theorem. Honestly, a lot of people might use porisms because I think they suggest a little bit of incompetence. It's like, you didn't see that one coming? Whoops-a-daisy. So I guess it's like the emperor's new clothes or something. No one wants to say they have a porism. So when they ever is one, they usually rewrite it as a lemma. So I mentioned it, but you rarely, rarely ever see them because if you rewrite your proofs in such a way that you have good limas and propositions, you'll never ever need a porism. All right. And so that brings us to the end of lecture three. Thanks for watching. If you learned anything about theorems or subsets in this lecture, please like these videos. 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