How did you get on with assignment six? If you followed everything so far and managed to do fairly well on all six assignments, you should have a good idea with the kind of linguistic precision required in mathematics. Now we can start to put up precision to use, improving mathematical statements. In the natural sciences, truth is established by empirical means, involving observation, measurements, and the gold standard, experiment. In mathematics, truth is determined by constructing a proof. A logically sound argument that establishes the truth of the statement. The use of the word argument here is, of course, not the more common everyday use to mean a disagreement between two people. But there is a connection in that a good proof will preemptively counter, explicitly or implicitly, all the objections the counterarguments that a reader might put forward. When professional mathematicians read a proof, they generally do so in a manner reminiscent of a lawyer cross examining a witness. Constantly probing and looking for flaws. Learning how to proof things forms a major part of college mathematics. It's not something that can be mastered in a few weeks, it take years. What can be achieved in a short period, and what I'm going to try to help you do here, is gain some understanding of what it means to prove of mathematical statements. And why mathematicians make such a big deal about proofs. First, what is a proof and why do we use them? I'll answer the second question first, since their purpose dictates what they are. Proofs are constructed for two many purposes, to establish truth and to communicate to others. Constructing or reading a proof is how we convince ourselves that some statement is true. I might have an intuition that some mathematical statement is true. But until I've proved it or read a proof that convinces me, I can't be sure. But I may also have to convince someone else. And that's the second purpose of a proof. For both purposes, a proof of a statement must explain why that statement is true. In the first case, convincing myself, it's generally enough that my arguments is logically sound and I can follow it later. In the second case, well, I have to convince someone else, more is required, the proof must also provide that explanation in a manner the recipient can understand. Proofs written to convince others have to succeed communicatively as well as be logically sound, there's actually not as much of a distinction here as words might imply. For complicated groups the requirement that a mathematician can follow his or her own proof a few days, weeks, months or even years later can also be significant. So even proofs written purely for personal use need to succeed communicatively. The requirement that proofs must communicate explanations to intended readers consider high bar. Some proofs are so deep and complex that only a few experts in the field can understand them. For example, for many centuries most mathematicians believed, or at least held a strong suspicion that for exponents n greater than or equal to 3, the equation x to the n + y to the n = z to the n has no positive whole number solutions for x, y, and z. That was conducted by the French mathematician Pierre de Fermat in the 17th century. But, it wasn't finally proved until 1994 when the British mathematician Andrew Wiles constructed a long and extremely big proof. Over the centuries, it became popularly known as Fermat's last theorem. Since it was the last of several mathematical statements Fermat announced that remain to be proved. Most mathematicians, myself included, lack the detail domain knowledge to follow proofs ourselves, but it did convince the experts in the field, the field by the way is analytic number theory. And as a result, Fermat's ancient conjecture is now regarded as a theorem. Fermat's last theorem is an unusual example, however. Most proofs in mathematics can be read and understood by all professional mathematicians. Though it can take days, weeks, or even months to understand some proofs sufficiently to be convinced by them. I've chosen the examples in this course to be understood by a typical student in a few minutes or possibly an hour or so. Examples given to college mathematics majors can usually be understood with at most a few hours effort. Proving a mathematical which is much more than gathering evidence in its favor. To give one famous example, in the mid 18th century, the great Swiss mathematician Leonhard Euler said he believed that every number beyond two can be expressed as the sum of two primes. This property of even numbers had been suggested to him by Christian Goldbach. And became known as the Goldbach Conjecture. It's possible to in computer programs to check the statement for many specific even numbers and to date, 2012, it's been verified for all numbers up to and beyond 1.6 quintillion. Most mathematicians believe it to be true. But it's not yet been proved. All it would take to disprove the conjecture would be to find the single even number n for which it could be shown that no two primes sum to n. Incidentally, mathematicians don't regard the Goldbach Conjecture as important. It has no known applications or even any significant consequences within mathematics. It's become famous solely because it's easy to understand, was endorsed by Oiler, and has resisted all attempts at solution for over 250 years. Whatever you may have been told at school. There's no particular format that an argument has to have, in order to count as a proof. The one absolute requirement, is that it is a logically sound piece of reasoning, that establishes the truth of some statements. An important secondary requirement, is that it's expressed sufficiently well. That an intended reader can perhaps with some effort, follow the reasoning. In the case of professional mathematicians, the intended reader is usually another professional with expertise in the same area of mathematics. Proofs written for students or lay persons generally have to supply more explanations This means that in order to construct a proof, you have to be able to determine what constitutes a logically sound arguments that convinces not just yourself but also an intended reader. Doing that is not something you can reduce to a list of rules. Constructing mathematical proofs is one of the most creative arts of the human mind and relatively few are capable of true original proofs. But with some effort, any reasonably intelligent person can master the basics. And that's my goal here. Euclid's proof that there are infinitely many primes, which I gave in the first lecture, is a good example of a proof that requires an unusual insight. Let's look at it again. Here's what we did. The idea is to show that if we list the primes in increasing order, then the list can be continued forever. So we imagine we've listed the primes p1 is 2, p2 is 3, p3 is 5, etc, all the way up to some stage pn. And we're sure that we can always add another prime to the list. And to do that, we look at this number N which we obtain by multiplying together all the primes in the list so far and then adding 1. Now this number N consists of the product of all those numbers p1 through pn + 1. So it's certainly bigger than all of those numbers. So N is bigger than all the primes in the list. Well, if N is prime, then we know that there's a prime bigger than pn, namely N, in which case we can continue the list, probably not by adding N itself. N, because it's the product of all these plus 1 is going to be a lot bigger than pn. So N is almost certainly not the next prime. But that doesn't matter. If N is prime, it shows there is a prime bigger than the one at the end of the list, and that means we can continue the list. The alternative is that N is not prime, in which case there's a prime q less than N that divides it. But none of the primes in the list can divide N since if you divide N by any of those primes, you're left with that remainder 1. If you try to divide p1, or p2, or any of these primes into this number N, it gets swallowed up by this part. And then there's a remainder of 1, okay. So q has to be bigger than pn. Those are the first n primes. So if q's not equal to one of those, it must be later on in the list, in which case, we shown again that there's a prime bigger than pn, and the list can be continued. Again, this particular q that divides N is not necessarily the next prime. But as before, that doesn't matter. Showing that there is another prime is all you need to do because then you can check the next prime, whatever it is, and add it to the list. Either way, either if N is prime or if N is not prime, either way there's another prime to add to the list. It follows that there are infinitely many primes, and the theorem's proved. There were two creative ideas in this proof. The first one is here to show that if we list the primes in increasing order as p1, p2, etc, then the list can be continued forever. So the first creative idea is to think about listing the primes and showing that the list can always be continued. The second creative idea was this one. Defining this number N in such a way that it guarantees that we can always find another prime. I would say that this idea is one that most mathematicians would come up with sooner or later. It's a fairly obvious one. This one is genius, okay, this is true genius. Let me give you another example. And this time, I'm going to prove that result I promised earlier that the square root of 2 is irrational. And I'm going to write it the way mathematicians typically do when they write up results for publication in professional journals or in books. Namely, we call it out by calling it a theorem. So in mathematics, a result that's sufficiently significant or important, that it's worth mentioning as such is called a theorem. In this context, let me mention, there's another word we often use called lemma. And a lemma is a result which is worth calling out for some reason but doesn't quite merit the status of being called a theorem. It's actually, if you like, a little theorem. Okay, the next thing a mathematician typically does is indicate that we're going to begin the proof, okay? So this is just part of the way mathematicians lay things out. We specify the theorem, and then we say we're going to give the proof. You don't have to do it this way, it's just a convention. The essence of being a proof is what comes next. Proofs are about their logical structure, not the way we write them down. I'm going to begin by assuming on the contrary that square root of 2 were rational. Now if you've never seen the proof that the square root of 2 is irrational before, this first step is going to seem pretty mysterious. Why do I begin by assuming the opposite of what I'm trying to prove? Well by the time I get down here, you'll see why. The reason this is a great example is in a about six or seven lines I can make it clear while I'm doing something right in the first step, okay. Okay, in that case if square root of 2 were rational, then there are natural numbers p, q with no common factors such that the square root of 2 is p over q. Remember a rational number is one that can be expressed as the quotient of two integers. In the case of a positive number it would be two natural numbers. And we can always pick those natural numbers, or those integers, to have no common factors. In other words, when we right a rational number as a quotient, we can always cancel out any common factors and express it as a quotient where the two numbers themselves have no common factors. Again, it might seem a little bit mysterious why I'm been particular about cancelling out common factors. But as with the first step, by the time I get down here, it'll be clear why I'm doing this. Well squaring that equation gives me 2 = p squared over q squared. Rearranging, I get 2 q squared = p squared. I multiply both sides by q squared. That gives me a 2 q squared on my left. And then when I multiply the right by q squared, it cancels that q squared, and I'm left with a p squared So p squared is even. It's equal to twice something. Hence, p is even. Why? Because the square of an even number is an even number. The square of an odd number is an odd number. So the only way I could get the square of a number p to be even is if the number p itself is even. Even squared is even, odd squared is odd. So p is 2r for some r. I'm now going to take this equation p=2r and use it to substitute back in this equation. So I take this equation, I've got 2q squared = p squared and p = 2r. So I've got 2q squared = (2r) all squared, which is 4r squared. We will forget the middle term now. I've got 2q squared = 4r squared, I can cancel the 2. But if q squared is 2r squared, then q squared is even. But exactly as happened before with p squared, if q squared is even, then q is even. Ah-ha, see what's happened? I've deduced here that p is even. I've deduced here that q is even. So p and q are both even. But they can't be because we assumed p and q have no common factors. If they're both even, then they have two as a common factor. But this is impossible, since p and q have no common factors. Well the logical reasoning here, the algebra, the arithmetic is all sound. Absolutely everything's perfectly sound. How can we have arrived at an impossible conclusion by a piece of sound reasoning? Well the only thing that can possibly have gone wrong is we began by making a false assumption. Remember, we began with an assumption, the only way we could have reached a false conclusion by a valid argument and this is valid. If you don't believe me, go on and check these for yourself. There's not many steps, see if there's anything wrong in any of these steps, there isn't. If we reach a false conclusion by a logical argument, then we must have started with a false assumption. Hence, the original assumption that the square root for 2 were rational must be false. Hence, square root of 2 must be irrational. And when I was at school, teachers used to insist that we write QED at the end of a proof, [FOREIGN], which is Latin. It is actually not a bad idea to indicate when a proof ends and mathematicians have different ways of doing it. Sometimes they write a little box at the end. How you express it, how you lay it out on the page is not that critical. The idea was just to lay it out in a way that can be followed. What makes a proof a proof isn't the fact that you call it a proof, it isn't the fact that you end in a QED. It's the logical flow of the steps, it has to be logically precise and you have to be able to follow it. But there's the proof. The reason this is a good example to give is it's short, it's concise, and all of the steps are simple, arithmetical steps. It's very easy to follow every step and yet, when you follow the small number of simple steps, you've proved a significant result. In fact, this result when it was first discovered by one of the Pythagoreans in Ancient Greece was dramatic. It changed the course of Greek mathematics because until then, they had felt that quotients of integers was sufficient to measure any length. But square root of 2 is the length of the diagonal of a right angled triangle, whose sides measure 1. And when this result was proved by one of the Pythagorean mathematicians, it showed that quotients of integers were not sufficient to measure all lengths in geometric figures. And that changed the course of Greek mathematics and subsequently, the rest of mathematics. It was extremely dramatic. Incidentally, there's a story you'll read about in books and on websites that say that this was discovered by young mathematician. And the Greek mathematicians, the Pythagoreans, were so annoyed and so scared that this would kill their career and their profession that they threw him overboard. There's absolutely no evidence whatsoever that that was the case. It's a great story, but like many stories, it's probably not true. Okay, but in any case, we've now shown that the square root of 2 is irrational. And there we go, this is really a remarkable result, very short, very elegant. Incidentally, when mathematicians talk about aesthetics, when they talk about an elegant proof, a beautiful proof, this is the kind of thing they have in mind. Not that it looks beautiful the way it's written out, in fact I've just went through it and it doesn't look particularly beautiful to look at, but the logical structure is beautiful. Every step counts, the result is established, and every step can be understood fairly straightforwardly. The complexity doesn't come because there's deep results involved, deep facts, deep concepts. The proof works because of the structure, the logical structure, okay? So now you know why root 2 is irrational.
