 Welcome to our lecture series Math 3130, Modern Geometries for Students at Southern Utah University. Let me quickly introduce myself. My name is Dr. Andrew Missildine, and I will be your instructor for this lecture series. So for a class like Math 3130, like I said, the title is Modern Geometries. That's a plural thing. We are going to study geometry from a theoretical point of view, the axiomatic point of view that is to say. So this class is going to be proofs, the whole proofs, and basically nothing but the proofs. But the reason we pluralize the word geometry here is because in the modern sense, there's not just one geometry. Prior to this lecture series, your native geometric language is probably what's called Euclidean geometry, which is something we're going to build up towards later on in this lecture series. But we also want to talk about other geometries as possibilities, things like hyperbolic geometry, elliptic geometry, just to name a few other things. In our very first unit, we're going to be building up to examples of finite geometries, which is what we'll get to in about lecture three of this lecture series. Now, to give you some background of what is a student of this lecture series have as their background, let's say it that way. So as we go through this series, there is the expectation that students have an equivalent of a semester worth of mathematical proof writing prior to this. With the Southern Utah University numbering system, this would be math 3120, which is why our class is called 3130. In some essence, it's the sequel class to a first semester of proof writing. Now, the topics of this lecture series, I should say, do focus on geometry. It really should be thought of as, again, as this second course in proof writing. One of the very first topics we'll cover in this lecture series is the so-called axiomatic system. What are axioms? So from a class like math 3120, students would have developed the rudimentary skills in first-order logic. So we know things like truth values, true and false. We can do some basic implications, maybe some more advanced proof techniques like proof by contradiction, mathematical induction. We should have some basic understanding of set theory in a mathematical sense. So we should know about subsets and cardinality, unions, intersections. We're gonna use all of those terms. We're gonna use them as if the audience already knows these things. And a little bit of basic combinatorics will be helpful in our finite geometry unit, but I won't even assume much about that. So we've learned the ideas of proof. So we can write some basic implications, maybe like a biconditional statement. But in this very first lecture, lecture one, I want to introduce to the ideas of axiomatic geometry. And so why is geometry the right setting to study logic from an axiomatic point of view? Well, arguably, geometry is the oldest branch of mathematics. And applications to geometry were studied thousands and thousands of years ago by the ancient Egyptians, Babylonians, the ancient Chinese culture also had many applications, great knowledge of geometric things. And the theory of geometry was studied very extensively by the Greeks. Many of us are familiar with the School of Pythagoras. There's, of course, Euclid, just to name a few. I'm not gonna go through all the history of geometry in this lecture right now. The study of modern geometry often begins with the axiomatic method. One of the big, one thing we'll talk about a lot in this lecture series is, of course, Euclid and his book, The Elements, for which that's really where the birth of axiomatic geometry began. And really it was, to modern history's perspective, the birth of axiomatic logic in general. It's the birthplace of axioms, for which, honestly, Euclid's axioms weren't very good. There were some problems there, but as things developed over time, we were much better. And that's the sense of modern geometry that we're taking on right here. Modern geometry begins with this axiomatic method as this is the logical perspective that separates the modern study of geometry from the ancients that I mentioned a moment ago. In fact, the axiomatic method is the bedrock for all pure mathematics. Algebra, analysis, topology, you name it, they're all built upon axioms. And so from the historical perspective, it does make sense to use geometry as a vessel to learn about the axiomatic method. So the axiomatic method you can actually see on the screen right now. And so there's basically four parts to the so-called axiomatic method. And each of them has a bolded part to simplify what are those pieces. So the first step of the axiomatic method is that any axiomatic system must contain a set of technical terms that are deliberately chosen as undefined terms and are subject to interpretation by the reader. So every axiomatic system has some collection of terms that are intentionally undefined. In some degree, we don't actually know what these things are, but they're allowed to be interpreted. A lot of this will make sense when we look at some concrete examples of axioms and undefined terms that we'll of course see in the next video. Now all other technical terms of the system are ultimately defined using these undefined terms. These are called the definitions of the system. So we can define new terms using the undefined terms. The undefined terms never receive a proper definition, but everything else we define, and this is what we call a definition. And so with mathematical writing, you often see list and list and list upon definitions. Mathematicians often feel like dictionaries because they have all these list of definitions and we must know all of them if we really are to understand the theory. All right, so some terms will be undefined, most of the terms, nearly all of the terms will be defined using, of course, those undefined terms are perhaps we can make new definitions using old definitions and such. So it just compounds on top of each other. So next, the axiomatic system contains a set of statements. Now remember from your previous proof writing situation, a statement is a sentence that could be true or could be false. And so these statements, true or false statements will be dealing with undefined terms or definitions we've created. And so these statements though are unproven. That is to say these statements which are called the axioms of the system and thus are the titular topic of the axiomatic method. Axioms are gonna be statements that are taken to be true without proof. There's no proof to an axiom. It's true because it's an axiom. And then the last step is all other statements in the system are logical consequences of the axioms. These are so called theorems. For which you can give other names to theorems. A theorem is just meant to be something we've proven from the axioms. It's something we've proven inside of our logical system here. But other names you might have is like a corollary. A corollary is typically a statement that's been proven almost immediately from a theorem. So its proof is super, super short in comparison to maybe the theorem that provided to it. You might have the idea of a lemma. Lemmas are typically helper theorems. They're not necessarily super useful in their own right but they help prove a very helpful theorem. We have propositions. The typically proposition means that it's a theorem but it's not grand enough to be labeled a theorem. Theorem sort of suggests it's important. Like a fundamental theorem is one of the biggest theorems you could prove there of course. Proposition is sort of like a weaker result. Important but it's not gonna get a name or whatever. Aporism, that's always a fun one there. Aporism is a theorem you proved while trying to prove something else almost by accident. It's a statement that can be derived from the proof of another theorem. And I can go on and on of all these terminologies and such. But the axiomatic method has these four parts. We have these undefined terms that never receive a definition. We receive axioms which are statements that never receive a proof. There are other terms that receive definitions. That is they're defined using the undefined terms. And then theorems to approve it from the axioms. Now definitions precede axioms here because the axioms actually could be statements involving defined terms, not just undefined terms. And that's why they're preceded there but essentially we're gonna have definitions and theorems but some of those theorems are not proven which is why we call axioms. And some of those definitions are actually not defined believe it or not. So we have these undefined terms. And so when one first sees this description of the axiomatic method, particularly these undefined terms and these axioms might seem a little unsettling because like it needs to be proven, right? It needs a proof or it needs a definition. How can we talk about things if no one actually knows what they are? Right, again, it feels unsettling at first but this really is the proper framework to develop not just mathematics but logic in general. So axioms are considered to be the fundamental true statements of any theory, okay? Axioms are true because they're true. And this is coming from a philosophical point of view, right? Axioms don't get proofs. Now why is that? Essentially something cannot be proven from nothing. That is in order to prove a statement to be true one must already have some known true statement to derive the new statement for. You know, like if you have some theorem, theorem one, you know, and the proof of theorem one is basically, oh, it's proven from theorem two, okay? And you know, plus some other things but where did theorem two come from? It's like, okay, well, the proof of theorem two, well, this was given by some type of other, you know, some other thing but basically theorem two was proven by some other theorem, theorem three, right? And so then where did theorem three come from, right? Continuing on in this process, we must have some infinite line of logical dominoes that have already been set up just to be able to prove theorem one, right? You know, the proof of theorem two preceded theorem one but then the proof of theorem three preceded theorem two and they have all these infinite line of dominoes here. The only way to avoid this logical paradox is to accept that some statements are fundamentally true. That is, it's a statement known to be true without a proof. And that's why axioms enter the system. We have to accept something logically speaking as true. Otherwise we can't get anything else. Thus the truth of these axioms is not just an issue. All right, let me say that again, you know, the truth of the axioms is not, it's not the concern here. It's really the reader's, you know, willingness to accept them as true or not, right? And so if I might divulge this for one more moment here, you know, interesting enough, this issue about the truthfulness of an axiom or not, this is something that theist and atheist alike have used and been concerned about. Philosophers have played around with these technicalities for centuries. And there have been many theists who try to use this logical paradox I'm describing to prove the existence of deity, to prove the existence of a God. Atheists have also, you know, done the same thing to try to prove the non-existence of deity of any kind. And I'm not gonna entertain all of those arguments right here, but let me just offer one of them really quickly here and be aware, I'm not advocating any religious statement whatsoever, whether the belief in God or not. You know, I'm not advocating either one of these in the videos. I just wanna provide it as a logical example here. And so basically from a theist point of view, there's sometimes been the argument, well, you know, we're here in the present, how did we get here? There has to be something that caused it. So there's a predecessor, but then that has to have a predecessor and that has to have a predecessor. And so basically you get this infinite descent of cause and effect, right? A implies B, but A was implied by something which was implied by something which was implied by something. And so then they come to the argument that there must have been some original, some original event that caused the, you know, the eternal cause and effect of the universe. And theist will make the argument that that's where God comes into play. There must have been some original cause and effect. There must have some original cause, in which case then they take essentially axiomatically the existence of deity. A scientist might argue that this original event was the big bane or what have you, right? But I want you to be aware that all of these, you know, some initial event, some first moment in time is fundamentally an axiom, right? That's a belief. Can you prove that there was a first moment in time? Well, again, I'm not gonna get into the debate too deep here, but basically the differences that people take in religion, the differences that people take in politics, this all comes down to axioms. It really does. In the end, there's some truths that must be taken on faith. And this can actually be, this is a provable statement from a mathematical standpoint, but I won't delve into that too much more. Again, our goal here is not politics, it's not philosophy, it's not religion, it's gonna be mathematics. And so the axiomatic system, I should mention that in mathematics, it's ubiquitous, particularly in pure mathematics. The axioms are used in algebra, of course. I teach algebra a lot, algebra is my background. This leads to the axioms of groups, which groups have three axioms, the associativity axiom, the identity axiom inverses. We can talk about abelian groups, where one extra axiom is placed on top of it, commutivity. We can talk about ring theory, where you take the axioms of commutative group, abelian group and stack on top of that, more axioms about associativity of the multiplication, distributive laws, maybe you want that to be commutative, maybe you want to be a multiplicative identity. There's a lot of things you could add on to it. You get the field theory, we basically have all of the axioms of rings total. But you can also go the other way, right? What if we start pulling away axioms, right? We maybe don't want all the axioms of groups, maybe we only want associativity and identities. We talked about a monoid. Maybe we just want associativity, we get a semi-group, but maybe that's the wrong direction. Maybe we want inverses and an identity, so we get what's called a loop. But maybe we just want inverses, don't need associativity or identity, we get the idea of a quasi-group. And we can go in lots of different directions as well, linear algebra, you have the idea of a vector space, which could stack on another axiom and get a normed vector space. You could stack on another axiom and get an inner product space, or a Lie algebra. Or again, there's just a few of the examples in algebra. There's tons and tons and tons you could do. Topology is another great example where we talk about axioms all the time. So in topology, axioms are introduced to determine what's an open set, what's a closed set, how can points be separated? You know, like what's a house dwarf set, a house dwarf topology? That's some topology that has some axioms about separation. Is it connected? You have connected axioms. Is it path connected or simply connected? There's these mean different things and there's different axiomatic things going on there. Analysis, right? The theoretical study of calculus. You have real analysis, complex analysis, just to name a few things. What's a natural number, right? There's axioms that decide what is a natural number. This is the so-called piano axioms. There's axioms that determine what a real number is. There's axioms that even determine what a set is. The ZF set theory axioms, particularly what we'll be using in this lecture series. And so I could go on and on and on and on and about this, but the axiomatic method is ubiquitous in mathematics. And so a mathematician does need to learn these to become, well, really a mathematician. And so in this lecture series, we'll use the vehicle of modern geometry to help us understand the axiomatic method. And so to give you a little bit of foreshadowing what's going to do here, in this lecture series, we're going to start by taking geometric axioms and start adding on top of them, stacking them bigger, bigger, bigger, bigger, bigger until we get to something called euclidean geometry. Then at that point, we're going to pull out some of the axioms and interchange them with new axioms to create different types of geometries. Now, before we get into these geometric axioms in the next video and in the next lecture, we're going to play around with the axiomatic method, but we're not going to use geometry. We're just going to use basically made up words, truly undefined things that you have no context to what they are to truly develop what is the axiomatic method. And then in lecture three, we'll transition into geometric terms.