 Well, now that we know how to name numbers and how to write numbers, the next natural thing to try and do is to do something with them. And so we have what I call a set of binary operations. We take two numbers and then we find a third number that has some relationship to first two. And our initial binary operation is going to be the operation of addition. Now it turns out there's two slightly different but completely equivalent definitions of what it means to add. And we'll take a look at the first one, which is based on our piano definition of addition. This piano here, Giuseppe Piano, is a mathematician at the turn of the 20th century who said, What are we really doing when we do all these basic arithmetic operations? And Piano took them all apart and said, Well, here's what we're actually doing. And Piano's idea was based on the fact that we view numbers in two different ways. One way is we can view them as a set of ordinals. That is, numbers indicate the order that something occurs. And we refer to this when we say first, second, third, and so on. And these are the numbers used ordinarily. And what this means mathematically is that given any two distinct numbers, A and B, whatever they happen to be, we can identify which one is smaller and which one is larger, and which one comes earlier in the order and which comes later in the order. And this led to the following. I can define the set N of natural numbers in the following way. And it's a set that satisfies a number of important properties which Piano described in some detail, but the two that are important for our purposes are the following. Every number, every element of this set has a successor which we designate A asterisk, A star. And that is, we can read this as this is the number that follows A, the number after A. And the element that we designate one is not the successor to any element of the set of natural numbers. One doesn't follow anything or put another way. One is the first natural number, nothing comes before it. And based on the Piano postulates, we can now define all of the natural numbers. And again, one is unusual. It's not the successor to any element. And so again, informally this means that it is the first natural number, nothing comes before it. By the first postulate, every number has a successor. So well, one is a number. So one has a successor which will designate one star. And now, because we can apply the first postulate repeatedly, one star is a number. And so one star has a successor. And we'll designate that as one star star. And likewise, one star star is a number. So it has a successor which we'll define as one star star star. And we can do this as many times as we'd like. We can talk about the natural number, one star star star star star star. But after a couple of seconds, our eyes glaze over and we can, we go blind trying to read the parentheses and the asterisks and so on and so forth, and we would find it much easier if we actually had a new symbol that indicates each of these successors. So let's make our lives much easier and we'll come up with a name for this number that we're calling one star. Thinking, thinking, thinking. We need a name for the number that follows one. Well, how about two? So we'll name the successor of one. We'll call it two and we'll use our symbol two. And again, since two is a natural number, it has a successor which we can designate two star, the number after two. And thinking, thinking, thinking. We need to come up with a name for it. How about three? And we'll call it three and we'll write it as a symbol. And likewise, we can come up with the successor of three, the number following three, call it four and write it the symbol and so on. And this leads us directly to the piano definition of addition. And there are two parts. First of all, we're going to define what it means to write n plus one. Well, n plus one, by the piano definition, is the same as n star. In other words, n plus one is the number after n. Now, we have a somewhat more complicated second part of the definition, n plus k star. Well, that's going to be n plus k star. n plus the successor of k is the same as n plus k, the successor of whatever that number happens to be. And it might not be obvious why we have to define it in this rather strange way, but we'll take a look at that when we see an example of how we apply these definitions. And so, well, let's take a little bit of a problem here. Let's prove using the piano definition of addition that three plus two is equal to five. Now, again, be very careful. This is a mathematics course. Prove has a very specific meaning. It doesn't mean tell me that this is true. I'm not going to ask you to prove something that isn't believed to be true. We already know three plus two is equal to five, so the question is not find three plus two, but rather to prove that it is using our piano definitions. And again, it may help to have our piano definition for reference. Again, remember, part of the reason for proof in mathematics is it reminds us of things that we should know. And in this particular case, the piano definition, we have two rules. The successor rule, n plus one is the number following n, and the addition rule, n plus k star is the number following n plus k. And let's organize our prove by checking what we're trying to prove against our piano definition. So we want to prove using the piano definition. And our first rule deals with addition of n plus one. And the thing to notice here is what we're adding is not one, but it's two. So we can't use the first rule, at least not directly. The second part deals with n plus the successor of k. And this looks promising because k star is the number after k. And first of all, because there's only two parts to the piano definition, I can't use the first part for this addition. I have to be able to use the second part, so let's see what we can do with that. And what makes it promising is two is the number after one. So two is the successor of one, otherwise known as one star. And so this suggests we can approach our problem as follows. First off, three plus two is the same as three plus one star, because two is the successor of one. Two is the number following one. Now, my piano definition says n plus k star is the same as n plus k, the successor of that. So here I have n plus k star. Well, that's the same as n plus k, the whole thing star. By definition, well, three plus one, well, there's our first rule. N plus one is the number following n. So this is three star star. Well, at this point I say, well, I know what the number following three is. The number following three is four. So this three star is four. And then, well, this is the number following four. Well, I know what that is, the number following four is five. So there's three plus two equals five. And what's my proof? Well, I've completed the proof, and here's the important thing. The complete answer to a question of this type, proof, is going to include everything that's in blue. This entire statement, this entire set of statements, is the proof using the piano definition. I don't need to write down the piano definition as part of the proof. I need to use it as part of the proof, but it's not really part of the proof itself. The proof itself, all this stuff in blue.