 In this video, I wanted to summarize the typical structure of mathematical discourse. So when a mathematical idea is presented, it typically comes in these four parts, definition, example, theorem, and proof. We have already seen this method used throughout the lectures. We've been doing so far in this lecture series, and we're going to see it be used over and over and over again. Typically, mathematical writing begins with definitions. This is to introduce the reader to new or novel mathematical objects that will later be explored, often described as in the sequel. You see that in mathematical writing all the time, in the sequel, which does not mean like there's a second movie that goes directly to video. It just means that later on in the book or later on the paper, we're going to use this idea. So definitions are often introduced at the beginning to remind the reader about important definitions of objects that will be discussed throughout, and it's quite reasonable that many of the readers would benefit from a reminder, even if they've seen it before, but maybe they've never seen it before either. That's also a possibility and also could be the case that they're using some type of non universally accepted definition, like is zero a natural number or not in our lecture series. We do define zero to be a natural number, but not everyone does that. So it's important to describe this at the very beginning. And so in addition to definition, sometimes notation is introduced. So the writers are saying things like, oh, we're going to use this symbol to mean shorthand for this thing in the future. And again, this might not be some universally accepted thing. So the reader needs to know that before they go any further. So even though mathematics is inherently an abstract discipline, the next part of our list is examples and examples can be very useful to make it feel more concrete. An example demonstrates how a mathematical object satisfies the definitions presented previously or how an object might not satisfy the definition. The second case is actually what we would call a non example. So if we're introducing some idea like a group, what is a group from abstract algebra? We might say that integers with respect to addition forms a group. This is an example of an example. But we might also then say that the natural numbers with respect to addition is not a group. So it doesn't satisfy the properties illustrated in the definition. And so sometimes we need examples, but also not examples so we can distinguish the difference between them. So maybe that we can start to conceptualize where is the boundary between this idea of a group and everything else? So examples are those things that lie inside of the umbrella of the definition. So the definition here is satisfied. And then over here, you don't get the definition. And so after the reader has been given a sufficient framework to understand the ideas being considered, typically through examples and such, then the reader will be presented probably with theorems. Remember, theorems are statements that we can prove to be true. And so the theorem is stated and then followed up by a proof. So these often go hand in hand, be aware of that. Now, I do also want to point out that this order definition first example, second theorem, third proof, fourth, this order is not absolute. Perhaps examples appear after a proof of a theorem to illustrate the theorem's truthfulness. Now, be aware that an example by itself is not a proof, but the illustration can help make it feel more down to earth, which is very important, even in abstract fields like mathematics. Sometimes an example can sit between the statement and the theorem, sorry, the statement of a theorem and its proof. So sometimes we put examples here between them. The idea there is that we tell a statement, we state something which we know is true. Then we provide an example of why it's true. But that's only to illustrate the further proof. That is to say that the example might generalize to the proof. So the reader might better understand the logic of the proof because they've seen a concrete example. We have we've illustrated this in our lecture series a couple times already. All right, so these four things are how mathematical discourse typically happens, typically in this order, but I've told you reasons already why we might bend some things. And another one I didn't mention is that sometimes you might put a definition after a theorem. It could be that you're studying something. There's this really cool theorem. We just proved it and then because of the theorem, you then have this new idea and you give that new idea definition. So honestly, we go back and forth between these things all the time. So so there's no hard roll on the order of these things. Now, there's actually two other steps that happen in mathematical discourse and they typically happen over here. And oftentimes they're omitted, which is why I actually omitted them on my list right here. Why are they omitted in mathematical dialogues? Particularly this happens because they happen so early in mathematical stages that the the discourse might be happening at so at some place so far down the line of the logical reasoning that it doesn't even make sense to discuss these in that discourse. Clearly, they're going to happen at some point, but not other points necessarily. All right, so what are these two missing pieces? The first one, we're going to refer to it as undefined terms. Undefined terms, as the name suggests, is kind of the opposite of a definition and I'll clarify what that means in a second. And then the next one is axioms and undefined terms and axioms then precede the other four, which usually we see all the time. All right, so what are these things? The hard of the matter is that you can't prove something from nothing. Every proof extends truth from already discovered truths towards a new truth. So if one does not have preexisting truth, then we can't really prove anything else. This is what an axiom does. An axiom is a true statement. It's a true statement, which like an assumption is it's true without proof. So an axiom is true, but we don't prove axioms. Again, it's kind of like an assumption, which we'll play around the assumptions a lot here, but an axiom is a true statement that doesn't require proof. And so sometimes that can be disconcerting for people. It's like, well, how do we know it's true? I mean, because in a mathematical class, we're emphasizing why, why, why, why is it true? What's the argument? How can you justify it? But axioms are then the epithetists of that in a manner of view because you don't prove them in proof writing assumptions are used all the time. We will see many examples of this as we prove conditionals in the future. The typical proof template of a conditional, which again, this is stuff we'll do in the future, is that we assume the hypothesis is true and then argue that the conclusion follows from the assumed hypothesis. The difference between an assumption and what we mean here by axiom is that while an assumption is conditionally true, the axiom is fundamentally true. So I'm going to insert that word in there, fundamentally true. It's fundamentally true. All truth of a theory is then derived as logical inferences from these fundamental truths as opposed to you have a lot of axioms there. Without these fundamental truths, we would be trapped in an infinite descent of logic. This is the this is this is true because something else. But that is this, you know, statement A is true because something else statement B was proven to be true. But how do we know statement true is how? How do we know statement B is true? Well, something else was proven to be true that proved then B. But then there's got to be something that implies C and then there's something that implies D and then forever on, right? There has to be some type of like omega statement, some first statement that's true and everything else was derived from it. And again, that might be disconcerting here, but this is part of the axiomatic method. And as you get more and more into advanced mathematics, this will become much more understandable and much more comfortable. And actually quite reasonable. It's like, oh, yeah, you have the three axioms of group theory. A group is associative. It has an identity and it has an inverse. Why is that true about groups? Because that's what a group is in some respect. Axioms are kind of like definitions. They are because they're true because we say they are. And again, I know that might be bothersome right now, but if you if you take some faith, which is what axioms is about, right? And proceed through your transition to advanced mathematics. The on the other side of the veil, you will see that this is a this is the right way to approach mathematics. So that discusses this idea of axioms. These fundamentally true statements, but they're true without proof. Proceeding the axioms, we have these notions of undefined terms, much like the axioms help us avoid an infinite descent of logic. Undefined terms help us avoid the same infinite descent when it comes to definitions. I mean, play around with this some time. If you define something like let's just take a regular English dictionary, like Webster's dictionary. If you define a word, you shouldn't use the word itself in the definition. Otherwise, you have a circular definition. You don't really know what the word means. But the words you use to define the word just given, they themselves have definitions you could look them up. And those words and their definitions have words that have to be defined. Doesn't there eventually have to come a point where there's a word that is understood without definition? Because every dictionary like Webster's or what have you, they have circular definitions. If you follow through eventually, you'll find a cycle in terms of the definitions. They've used words to define words that later on have definitions using the words they've already used. Like, how do you define the word, without using the word, you probably could do it. But I bet eventually you're going to get a cycle really quickly. We have to have terms which have meanings, but without definitions. We have to understand something that wasn't yet defined. We define terms using other terms that are already defined. Those terms are also defined using other terms that are already defined. This is how accepted language works. But if we're to keep going back, back, back through our definitions, there has to reach a point where we have terms we understand, but we're never defined. After all, how does a human baby learn to speak when this baby doesn't know any words? The baby will ultimately learn a word like mama, mama. And this requires something beyond a dictionary. A baby doesn't learn what mama means because mama gives the baby a dictionary and like, oh, that's what I see. You're my maternal caregiver. No, you cannot tell the baby what the word means. They have no language yet. She will have to understand the meaning for herself first, because any other language skills haven't been developed yet. They'll be developed based upon this fundamental concept, which will probably be something like mama. This is what an undefined term is for mathematics. They provide the fundamental language for a mathematical theory. For us, the word well defined when we talk about a set is actually an undefined term. I brought this to our attention at one point in our lecture series, but basically avoided the definition with just some smoke and mirrors here. It's what you call math and magics, right? You know, magicians do their tricks. They use smoke and mirrors to confuse the audience. Mathematicians have to do the same thing at times. We've seen examples of when a collection is well defined and that makes it a set. But we've also seen examples of collections of objects that were not well defined and therefore didn't make a set. Russell's paradox was such an example. We have a collection that's not a set because it's not well defined. But what is well defined actually mean? So for us, this well defined is actually an undefined term. If we were to delve deeper into ZFC set theory, we'd get a better idea of what that means, but in ZFC set theory, I want to tell you the word set is an undefined term. So I've given us a definition of set, but that really is just hiding the the dirt under the rug there. We've never actually defined what a set is, and that's OK. We have an intuitive notion of what a set is, and that's how undefined terms have to exist. The baby has to understand what mama is without you telling her what it means because she can't understand words until she knows at least one word. And that typically is going to be mama, maybe dad or something like that. But my kid's person, that's usually the first thing to say. All right. So although the idea of undefined terms and axioms might be disconcerting at first, I want to end this discussion by explaining how liberating it actually is. The reality of mathematical writing is that we are not expected to prove everything at all times. I've had students who tried to do that before and their homeworks become like portfolios in and of themselves, right? You do not have to prove everything at all times. We can place faith in pre-established truth and concepts and we can build on top of that foundation. There have already been times in this lecture series where we provided proofs based upon statements we haven't yet proven, like when we use Euclid's lima. I mentioned it, but I haven't proved it yet. We'll do that in the future. But for the moment B, we can use it with the assumption that if Euclid's lima is true, then the theorem we proved using it would also be true. Those proofs are valid proofs, but they are conditional upon the validity of those statements yet unproven to us. If those theorems are in fact true, then we find all of the we find all of the things we prove with them are likewise true. It's not our responsibility to necessarily find and prove all truth. Our goal of mathematical writing is to extend truth, that is, build upon the truth that we already know for our foundation and build build on top of it, build skyscrapers on what's already been built for us. As such, we have to start at the surface and build on top of it. But there are times where maybe we actually want to dig deeper and that digging process can be slower, difficult, but sometimes it's an appropriate thing to do. It's not typically in a class like math 3120, but that is sometimes you want to dig deeper into it. You can do that. But for our purposes, our proofs will be typical, conditional of something else. You have the right to say some things without proof. But that is because you can refer to someone else who's already proven it or in the case of axioms, which are these fundamental truths. We then develop a theory based upon the axiomatic truth. Then sometime in the future, if we or someone else might want to explore further the fundamental assumptions there, are those fundamental assumptions true or false? Let them explore it. If the statement, if the axioms turn out to be provable, which you'll understand the future isn't a nonsensical thing, but for the moment be, if the statement, if the axioms are turned out to be proven true, then the roots of the theory can be digging deeper, dug deeper, I guess. If the fundamental statements are actually false, then the whole theory dies like a weed. So clearly you need your fundamentals to be true for the theory to be valid, but with axioms you assume them to be true. So it is important our foundation beyond the fertile soil and we build from there. But again, as beginning mathematicians, we are not required to go improve everything. We can use others' work to establish upon that. So really what we're doing in this course is we're often digging deeper. We are we are starting with some fundamentals, but there are some things hidden in the background. Ignore the man behind the curtain. I will reveal that curtain eventually, but not yet. The patience will work out to be to your advantage. And that's going to bring us to the end of lecture four for for this for this video, right? And so thanks for watching. If you had any questions about any topics we discussed, please put those questions in the comments below and I'll be glad to answer them as soon as I can. If you've learned anything in these videos, please like them, subscribe to the channel to see more videos like this in the future. See you next time.