 So let's talk about one of the most important and distinctive aspects of mathematics, but something that's also, at the same time, one of the most misunderstood aspects of mathematics, and that's the concept of proof. Now in everyday life, we're often convinced of the truth of some statement by the use of empirical evidence. And empirical evidence is just a fancy way of saying we're going to find some examples where the statement is, in fact, true. So for example, a physicist might conduct an experiment and show that the acceleration of a falling object doesn't depend on the mass of the object, or an epidemiologist might examine a smoker and determine that the smoker has lung cancer, or somebody who talks about politics on television might find a person who paid more in taxes this year than they did last year. Now a physicist, or an epidemiologist, or a political pundit, at some point the weight of empirical evidence they collect may allow the observer to draw some sort of conclusion. So a physicist might repeat this experiment thousands of times, and every time that they find that the acceleration is independent of the mass, they have a confirming instance of the truth of the statement. Likewise, an epidemiologist may examine thousands of smokers and find that, yeah, overall the smokers have a higher incidence of lung cancer. And the political pundit may examine, well, one person, and they may say that on the basis of this one person that I heard about, taxes are obviously too high. Obviously, there's a problem here in that we have to figure out when the weight of evidence is sufficient. And the problem with empirical evidence is not that it's empirical evidence. The problem is that there's no clear consensus on how much evidence is enough. So if you're in the legitimate sciences, like physics or biology, you want thousands of pieces of evidence. If you're in politics, then one is enough, then maybe that explains a lot about politics. Turning to mathematical proof, a mathematical proof is a lot more like a game of chess. When you sit down to play a game of chess, what you do is you begin with a set of rules for how you're going to play the game. We make certain agreements that if I play a game of chess, then the players alternate moves and there's a victory condition and so on. And these have their analog and mathematical proof in what are called the rules of inference. These are essentially how we're supposed to make our moves. The other thing we have in a game like chess is we have a starting configuration of the pieces. And you can think about these starting configuration of the pieces as the premises in a mathematical proof. Now in a game of chess, we have a very specific way that all the pieces start, but if you look at a chess problem in, for example, the newspaper or something like that, what they do is they say, well, suppose my pieces look like this particular arrangement. How can I go from here to a checkmate position? And very much the same thing happens in mathematical proof. We don't necessarily start at the very, very, very beginning. We often start halfway through the problem. And then finally, in a game of chess or in a mathematical proof, we have some sort of final configuration we're trying to achieve. This is the desired conclusion. So again, in a game of chess, we start with our pieces, however they're arranged, and what we want to do is we want to end up with a position where the opponent's king cannot move anywhere without being taken. And so that's our checkmate position that we're trying to get to. Mathematically, if we're trying to find a proof, we want to have some sort of final conclusion that we arrive at. And so what we're going to try and do is we're going to try and move our pieces according to the rules to achieve new configurations, and in a mathematical proof, these are the deductions. What we're going to do is we're going to start with our premises, where the pieces are all at the beginning. We're going to move them according to our rules of inference, and ideally, we'd like to get to this desired conclusion. The most important thing to remember is that the real reason you're playing a game is to play the game. It's not about winning. Well, it is if you're pathological, but it's not really about getting to the end game. It's about the actual movement of the pieces, and that's where things are interesting. When you finish the game, most people who actually like to play games want to play another game. And so, again, similarly, the reason that you want to prove a mathematical statement is not really that you care about whether the statement is true. The whole point of a mathematical proof is to provide a mathematical proof. Well, given that it's natural to ask, well, why do we bother? And part of the reason is that we bother providing a mathematical proof because there are certain benefits that are associated with it. And in particular, almost every proof that you do is going to do three things. In the course of the proof itself, you're going to have to clarify what you mean by particular concepts. You're also generally going to have to review what you already know about these concepts, and frequently you lead to the discovery of results that you didn't know before you started the proof. For example, let's take a very, very simple statement and see how we might prove it. So let's consider the statement, the square of an even number is even. So if I'm a political commentator on television, I might make the following observation. Well, 3 times 5 equals 15, so the statement is true. And if I repeat this statement loudly enough and often enough, I'll have millions of people agreeing with me. And George Orwell realized this back in 1948, but we won't go into that. Now, if I'm a legitimate researcher, I might make a couple of observations, I might try to run some experiments or collect some empirical data for the truth of the statement. And so I might say, well, let's see, 2 times 2 is 4, that's even. 4 times 4 is 16, that's even. 6 times 6 is 36, that's also even. And I repeat this, I collect a whole bunch of data. And at some point, I have enough evidence to convince me that the statement is actually true. Now, if I'm a mathematician, I'm going to go a little bit farther. And it's worth pointing out one key aspect of mathematical proof. I'm not going to bother to even try to prove a statement that I don't already believe is true. If I am going to prove this statement on some level, I have to believe that the statement is actually true. And generally speaking, what that means is I've already done the equivalent of an empirical proof. But the difference here is that with the empirical proof, I have to gauge how much evidence is enough to convince somebody else. For a mathematician, the level of evidence is enough to convince me that it's worth trying to find a mathematical proof. Well, how am I to go about this? Well, in order to find a proof for this statement, I have to do two things. First of all, I have to figure out what I mean by the square of an even number. And so I review my definitions. The square of a number is the product of a number with itself. And I also remember things like, well, a number is even if it's 2 times some other number. And so this gives me a good starting point here. I'm going to start with an even number. Well, that's a number that's 2 times something else. And then I'm going to find the square of that number. And I can use a little bit of algebra here. n squared equals 4k squared. And I want to conclude that the number that I've gotten is an even number. So again, I want to show that this number is 2 times something else. And again, I'll use a little bit of algebra. n squared is 2 times 2k squared. So that tells me that n squared is even. Now, we've proven our statement, but it's worth noticing that halfway through our statement, we actually discovered something else. What we discovered is that n squared is 4 times k squared, which is to say we've actually discovered that the square of an even number is not just even. That's our last statement. But in fact, it's a multiple of 4. Now, if we go back to our empirical evidence, now that we know to look for it, we do see that the square of an even number is in fact a multiple of 4. And if we review our empirical evidence, it also supports our conclusion. But since we were trying to provide evidence for this statement, we tended to look for confirming evidence that the number is even. And we didn't necessarily look for the fact that the number is also a multiple of 4. So here's an example where proof actually reveals something that we hadn't noticed before. And here in a nutshell is the reason why mathematical proof is important. Our very simple proof of a statement whose truth was never in doubt we had to clarify what we mean by square. We had to clarify what we meant by even. We had to review what we knew about these things. And we also discovered something brand new that we might not have seen before, that the square of an even number is not just even, but it's also a multiple of 4.