Let me make a couple of important points. First of all, notice that once we know why something is true, then in particular we know that it is true. A trillion instances tells us nothing; when it comes to infinity, the only way to know what is to know why. Proof is our way of capturing an infinite amount of information in a finite way. That’s really what it means for something to have a pattern—if we can capture it with language.
Another thing I want you to appreciate is the finality of mathematical proof. There’s nothing tentative or hypothetical here. It’s not going to turn out later that we were wrong. The argument is completely self-contained; we’re not awaiting any experimental confirmation.
Finally, I want to stress again that it’s not the fact that consecutive odd numbers add up to squares that really matters here; it’s the discovery, the explanation, the analysis. Mathematical truths are merely the incidental by-products of these activities. Painting is not about what hangs in the museum, it’s about what you do—the experience you have with brushes and paint.
As I see it, art is not a collection of nouns, it’s a verb—a way of life, even (or at any rate a means of escape). To reduce the adventure that we just went through together to a mere statement of fact would be to miss the point entirely. The point was that we made something. We made something beautiful and compelling and we had fun doing it. For a brief shining moment we lifted the veil and glimpsed a timeless simple beauty. Is this not something of value? Is humankind’s most fascinating and imaginative art form not something worth exposing our children to? I think it is.
So let’s do some math right now! We just saw that adding consecutive odd numbers always makes a square (and more important, we figured out why). What happens if we add up consecutive even numbers? How about adding up all the numbers? Is there a simple pattern? Can you explain why it happens? Have fun!
Now hold on a minute, Paul. Are you telling me that mathematics is nothing more than an exercise in mental masturbation? Making up imaginary patterns and structures for the hell of it and then investigating them and trying to devise pretty explanations for their behavior , all for the sake of some sort of rarified intellectual aesthetic?
Yep. That’s what I’m saying. In particular, pure mathematics (by which I mean the fine art of mathematical proof) has absolutely no practical or economic value whatsoever. You see, practical things don’t require explanation. Either they work or they don’t. Even if you could find a way to put our odd number discovery to some sort of practical use (and of course there’s lots of math out there that is indeed extremely useful) you would have no need for our gorgeous explanation. If it works for the first trillion numbers, then it works. Issues involving infinity simply don’t come up in business or medicine.
Anyway, the point is not whether mathematics has any practical value—I don’t care if it does or not. All I’m saying is that we don’t need to justify it on that basis. We’re talking about a perfectly innocent and delightful activity of the human mind—a dialogue with one’s own mentality. Math requires no pathetic industrial or technological excuses. It transcends all of those mundane considerations. The value of mathematics is that it is fun and amazing and brings us great joy. To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits. Or is that in fact what we are saying?
Let’s quickly escape back to the jungle. Now just as hamsters occupy a certain biological niche—plants and insects they like to eat, geographic areas and terrain they inhabit—math problems are also situated within an environment—a structural environment. Let me try to illustrate this idea with another personal favorite.
Here are two points on one side of a straight line. The question is, what is the shortest path from one point to the other that touches the line? (Naturally, the part about touching the line is the interesting part—if we dropped that requirement then the answer would obviously be just the straight line connecting the two points.)
Clearly the shortest path must look something like this:
Since our path has to hit somewhere, we can’t do better than to go straight there. The question is, where is “there”? Among all the possible points on the line, which one gives us the shortest path? Or could it be that they all have the same length?
What an elegant and fascinating problem! What a delightful setting in which to exercise our creativity and ingenuity. And notice: we don’t even have a conjecture. We have no clue what the shortest path is, so we don’t even know what we are trying to prove! So here we will have to discover not only an explanation for the truth, but what the truth is in the first place.
Again, the right thing for me to do as your math teacher would be nothing. That’s a thing most teachers (and adults generally) seem to have a hard time doing. Were you my student (and assuming this problem interested you) I would simply say, “Have fun. Keep me posted.” And your relationship to the problem would develop in whatever way it would.
Instead, I will use this opportunity to show you another lovely mathematical argument, which I hope will both charm and inspire you.
So it turns out that there is in fact only one shortest path and I will tell you how to find it. For convenience, let’s give the points names, say A and B. Suppose we had a path from A to B that touches the line:
There’s a very simple way to tell if such a path is as short as possible. The idea, which is one of the most surprising and unexpected in all of geometry, is to look at the reflection of the path across the line! To be specific, let’s take one part of the path, say from where it hits the line to where it hits the point B, and reflect that part over the line:
We now have a new path that starts at A, crosses the line, and ends up at the point B’, the reflection of the original point B. In this way, any path from A to B can be transformed into a path from A to B’:
Now here’s the point: the new path has exactly the same length as the original. Do you see why? This means that the problem of finding the shortest path from A to B that hits the line is the same as finding the shortest path from A to B’. But that’s easy—it’s just a straight line! In other words, the path we’re looking for is simply the path that when reflected becomes straight!
Is that great, or what? I only wish I could see your face—to see if your eyes light up, and to make sure that you get the joke, so to speak. Mathematics is fundamentally an act of communication, and I want to know if my idea got through. (If tears aren’t streaming down your face, maybe you should read it again.)
I want you to know that when I first saw this proof I was absolutely shocked. The thing that got to me (and still does) is the perversity of it. The points were both above the line. Their shortest path is also. What the hell does this have to do with anything below the line? It was a shattering argument for me; definitely one of my formative mathematical experiences.
So I want to use this problem to make a few comments about the way modern mathematicians view their subject. What is this problem really about? What are the issues here? Well the first thing to notice is the setting—points, lines, a plane on which the action takes place, a sense of distance or length—these are the hallmarks of geometric structure. This problem fits into a larger category of problems concerned with spatial environments and notions of distance. These can range from the “elementary” geometric ideas of the classical Greeks (which were themselves inspired by earlier Egyptian practical observations about the real world) to the most abstract and bizarre imaginary structures—many having nothing whatever to do with anything even vaguely resembling reality. (Not that we know what reality is, but you get what I mean.)
Essentially, the adjective “geometric” is used by mathematicians to group together those problems and theories that concern some sort of collection of “points” (which may be quite arbitrary and abstract) and some sort of notion of “distance” between them (which also may bear no resemblance to anything famili
ar). For example, the “space” consisting of all red and blue bead strings of length five can be given a geometric structure by defining the distance between two such strings to be the number of places in the bead sequence where the colors disagree. Thus, the distance between the points ‘RBBRB’ and ‘BBBRR’ would be 2, since they differ only in the first and last places. Can you find an “equilateral triangle” (i.e., three points that all have the same distances to each other) inside of this space?
Similarly, problems can be classified as having algebraic, topological, or analytic structure, as well as many other types, and of course combinations of the above. Some areas of mathematics, such as the theory of sets or the study of order types, concern objects with almost no structure at all, whereas others (e.g., elliptic curves) involve practically every structural category under the sun. The point of this sort of framework is the same as it is in biology: to help us understand. Knowing that hamsters are mammals (and this is not an arbitrary classification, but a structural one) helps us make predictions and to know what to look out for. Classifications are a guide for our intuition. Similarly, knowing that our problem has geometric structure may give us fruitful ideas and keep us from wasting our time on approaches that are not in harmony with that structural world.
For example, any plan of attack on our shortest-path problem that involves bending or twisting is almost automatically doomed to fail, since such activities tend to distort shapes and mess up length information. We should instead think about activities and transformations that are structure preserving. In the case of our problem, which takes place in a Euclidean geometric environment, the natural activities would be those that preserve distances—namely sliding, rotating, and reflecting. From this perspective, the use of reflection maybe doesn’t seem quite so shocking anymore; it is a natural element of the structural framework of the problem.
But that’s not all. The thing about proofs is they always manage to prove more than you intended. The essence of the argument is the fact that reflection across a line preserves distances. This means that our argument applies to any setting in which there is a notion of point, line, distance and reflection. For instance, on the surface of a sphere there is a notion of reflection across an equator:
This means that equators (the curves you get when you chop a sphere in half) are the natural spherical analogs of “straight line.” And in fact it happens to be true that the shortest path between two points on the surface of a sphere is to follow an equator (which is why airplanes often take such routes).
So the corresponding problem on a sphere would be: given two points on the same side of an equator, what is the shortest path between them that touches the equator? My point is that our exact same argument still works. Again it is the path that when reflected is straight:
How about if we have two points in space on the same side of a plane?
What I’m saying is that proofs are bigger than the problems they come from. A proof tells you what really matters and what is mere fluff, or irrelevant detail; it separates the wheat from the chaff. Of course, some proofs are better than others in this regard. Often a new argument is discovered that shows that what was previously thought to be an important assumption is in fact unnecessary. I suppose what I’m really trying to say here is that mathematical structures are designed and built not so much by us, as by our proofs.
The historical development of mathematics (especially in the past couple of centuries) exhibits a consistent, undeniable pattern: first come the problems, whose sources are many and varied, often inspired by the real world. Eventually, connections are made between diverse problems, usually due to common elements that appear in various proofs. Abstract structures are then devised that can “carry” the kind of information that forms the connection (the classic example being the “group” concept, which captures abstractly the idea of a closed system of activities, e.g., algebraic operations like addition, or systems of geometric or combinatorial transformations such as rotation or permutation). New questions then arise concerning the behavior of the new abstract structures—classification problems, construction of invariants, structure of sub-objects, et cetera. And the process continues with the discovery of new connections among the abstract structures themselves, generating even more powerful abstractions. Thus mathematics moves further and further away from its “naïve” origins. Some areas of mathematics, such as logic and category theory, concern themselves with spaces (so to speak) whose “points” are themselves mathematical theories!
As a small example, the key idea in our path problem was reflection. Now reflections have the amusing property that when you do them twice it’s as if you’ve done nothing at all. Does that remind you of anything? It’s just like our self-annihilating hamster—that new version of 1 with 1 + 1 = 0. So here we have a connection between an algebraic structure and a geometric one. This raises a lot of questions concerning the extent to which number systems of various kinds can possess geometric “representations.” Can you make up a number system that behaves like the rotations of a triangle?
All I’m really trying to say here is that as modern mathematicians we are always on the lookout for structure and structure-preserving transformations. This approach not only gives us a meaningful way to group problems together and to understand what they are really “about,” but it also helps us to narrow the search for proof ideas. If a new problem comes along that lies in the same structural category as one we have already solved, we may be able to use or modify our previous methods.
Ok, grab your machete. It’s back to the jungle we go! I can’t resist giving you at least one more example of the mathematical aesthetic. This is what I like to call the “Friends at a Party” problem: Must there always be two people at a party who have the same number of friends there?
The first thing is to decide what we want our words to mean. What are people? What is friendship?
What exactly is a party? How does a mathematician address these issues? Surely we don’t want to deal with actual humans and their complicated social lives. The aesthetic of simplicity demands that we shed all such unnecessary complexity and get to the heart of the matter. This is not a question about people and friendship, it’s a question about relationships in the abstract. A party then becomes a “relationship structure” consisting of a set of objects (it doesn’t matter what they are) together with a collection of (presumably mutual) relationships between them.
If we wanted, we could visualize such a structure using a simple diagram:
Here is a party of five, including one stranger (no friends) and a rather popular fellow with three friends. And it just so happens that there are two objects with the same number of connections (namely two).
So here is a simple and beautiful class of mathematical structures (known in the math biz as combinatorial graphs) and a natural and amusing question about them: Does every graph possess a pair of objects with the same number of connections? (We’re assuming of course that our graphs involve more than one object.)
So where do math problems like these come from? Well, I’ll tell you: they come from playing. Just playing around in Mathematical Reality, often with no particular goal in mind. It’s not hard to find good problems—just go to the jungle yourself. You can’t take three steps without tripping over something interesting:YOU: So Paul, I was thinking about what you said before about arranging numbers in rows, and I noticed that some numbers are so awkward they can’t be arranged evenly in any number of rows. Like thirteen—it just doesn’t work.
ME: Well, you could always arrange it as one row of thirteen . . . or as thirteen rows of one!
YOU: Yes, but that’s boring. You can do that with any number. I’m talking about using at least two rows. So anyway, I started making a list of these weird numbers. It goes like this:1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, . . .
and it seems to keep on going, but I haven’t found any real pattern to it.
ME: Well, you’ve stumbled onto
something very mysterious. The truth is, we don’t know very much about these weird numbers of yours. One thing we do know is that they go on forever—there is an infinite supply of numbers that can’t be arranged in rows. Maybe that would be a good thing for you to try to prove.
YOU: Yes, I’d like to think about it. Anyway, the thing I noticed about my list is the spacing between the numbers. It seems like they mostly thin out as they get bigger, but then sometimes you get these little clumps like 17, 19 and 101, 103 where they only jump by two. Does that keep happening?
ME: Nobody knows! Your weird numbers are called “primes” and the ones that come in pairs are called “prime twins.” Your question about whether they keep occurring is known as the twin prime conjecture. It is actually one of the most famous unsolved problems in arithmetic. Most people who have worked on it (including myself) feel that it is probably true—prime twins should keep happening—but nobody knows for sure. I’m hoping to see a proof before I die, but I’m not terribly optimistic.
YOU: How bizarre that something so simple should turn out to be so hard! The other thing I noticed is that after 3, 5, 7 you never seem to get three primes in a row. Is that true?
ME: Prime triplets! What a terrific problem for you. Why don’t you work on that and we’ll see what you come up with . . .
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