In particular, we can replace the old-fashioned notion of subtraction by a more modern idea: adding the opposite. Instead of “eight take away five,” we can (if we wish) view this activity as “eight plus negative five.” The advantage here is that we have only one operation to deal with: adding. We have transferred the subtraction idea away from the world of operations and over to the numbers themselves. So instead of taking off my shoe, I can think of it as putting on my “anti-shoe.” And of course my anti-anti-shoe would just be my shoe. Do you see the charm in this viewpoint?
Similarly, if multiplication is something you are interested in (that is, making repeated copies of piles of rocks), you might also notice an unpleasant lack of symmetry. What number triples to make six? Why, two of course. But what triples to make seven? There isn’t any pile of rocks like that. How annoying!
Of course we’re not really talking about piles of rocks (or anti-rocks). We’re talking about an abstract imaginary structure inspired by rocks. So if we want there to be a number which when tripled makes seven, then we can simply build one. We don’t even have to go out to the garage and get tools—we just “bring it into being” linguistically. We can even give it a name like ‘7/3’ (a modified Egyptian shorthand for “that which when multiplied by three makes seven.”) And so on. All of the usual “rules” of arithmetic are simply the consequences of these aesthetic choices. What are so often presented to students as a cold, sterile set of facts and formulas are actually the exciting and dynamic results of these new creatures interacting with each other—the patterns they play out as a result of their inborn linguistic “nature.”
In this way we play and create and try to get closer to ideal beauty. A famous example from the early seventeenth century is the invention of projective geometry. Here the idea is to “improve” Euclidean geometry by removing parallelism. Putting aside the historical motivations behind this decision (which have to do with the mathematics of perspective), we can at least appreciate the fact that in general two straight lines intersect at a single point, and parallel lines break this pattern. To put it another way, two points always determine a line, but two lines don’t always determine a point.
The bold idea was to add new points to the classical Euclidean plane. Specifically, we create one new point “at infinity” for each direction in the plane. All the parallel lines in that direction will now “meet” at this new point. We can imagine the new point to be infinitely far away in that direction. Of course, since every line goes off in two opposite directions, the new point must lie infinitely far away in both directions! In other words, our lines are now infinite loops. Is that a far out idea, or what?
Notice that we do get what we wanted: every pair of lines now meets at exactly one point. If they intersected before, then they still do; if they were parallel, they now intersect “at infinity.” (To be complete, we should also add one more line, namely the one consisting of all the infinite points.) Now any two points determine a unique line, and any two lines determine a unique point. What a nice environment!
Does this sound to you like the ravings of a lunatic? I admit it takes some getting used to. Perhaps you object to these new points on the grounds that they’re not really “there.” But was the Euclidean plane there to begin with?
The point is that there is no reality to any of this, so there are no rules or restrictions other than the ones we care to impose. And the aesthetic here is very clear, both historically and philosophically: if a pattern is interesting and attractive, then it’s good. (And if it means having to work hard to bend your mind around a new idea, so much the better.) Make up anything you want, so long as it isn’t boring. Of course this is a matter of taste, and tastes change and evolve. Welcome to art history! Being a mathematician is not so much about being clever (although lord knows that helps); it’s about being aesthetically sensitive and having refined and exquisite taste.
In particular, contradiction is usually regarded as rather boring. So at the very least we want our mathematical creations to be logically consistent. This is especially an issue when making extensions or improvements to existing structures. We are of course free to do as we wish, but usually we want to extend a system in such a way that the new patterns do not conflict with the old ones. (Such is the case with the arithmetic of negative numbers and fractions, for instance.) Occasionally, this compels us to make decisions we might otherwise not want to make, such as forbidding division by zero (if a number such as ‘1/0’ were to exist, it would conflict with the nice pattern that multiplication by zero always makes zero). Anyway, as long as you are consistent, you can pretty much have whatever you want.
So the mathematical landscape is filled with these interesting and delightful structures that we have built (or accidentally discovered) for our own amusement. We observe them, notice interesting patterns, and try to craft elegant and compelling narratives to explain their behavior.
At least that’s what I do. There certainly are people out there whose approach is quite different—practical-minded people who seek mathematical models of reality to help them make predictions or to improve some aspect of the human condition (or at least improve the balance sheet of their corporate sponsors). Well, I’m not one of those people. The only thing I am interested in using mathematics for is to have a good time and to help others do the same. And for the life of me I can’t imagine a more worthwhile goal. We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have fun.
Let’s go a little deeper into the jungle, shall we? Now, you have to appreciate that people have been doing mathematics for quite some time (and rather intensely for the last three thousand years or so) and we have made a lot of amazing discoveries. Here is one I’ve always loved: What happens when you add up the first few odd numbers?
1+3=4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
To the novice this may seem like a random jumble of numbers, but the sequence:4, 9, 16, 25, . . .
is far from random. In fact, these are precisely the square numbers. That is, these are just the numbers of rocks you need to make a perfect square design:
So the square numbers stand out from the rest as having this particularly attractive property, which is why they get a special name. The list goes on indefinitely of course, since you could make a square design of any size. (These are imaginary rocks and we therefore have an inexhaustible supply.)
But this is remarkable! Why should adding up consecutive odd numbers always make a square? Let’s investigate further:1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
(which is 7 × 7)
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
(which is 10 × 10)
It seems to keep happening! And it’s utterly beyond our control. Either this is a true (and surprising and beautiful) feature of odd numbers or it isn’t, and we simply have no say in the matter. We may have brought these creatures into existence (and that is a serious philosophical question in itself) but now they are running amok and doing things we never intended. This is the Frankenstein aspect of mathematics—we have the authority to define our creations, to instill in them whatever features or properties we choose, but we have no say in what behaviors may then ensue as a consequence of our choices.
Now I can’t make you be curious about this discovery; you either are or you aren’t. But at least I can tell you why I am. For one thing, adding up odd numbers seems like a very different sort of activity than making a square (i.e., multiplying a number by itself). These two ideas just don’t seem to have much to do with each other. There’s something a bit counterintuitive about this. I am drawn in by the possibility of a connection—a new, unforeseen relationship that will improve my intuition and perhaps permanently change the way I think about these objects. I suppose that’s really a key part of it for me: I
want to be changed. I want to be affected in a fundamental way. That’s maybe the biggest reason why I do mathematics. Nothing I have ever seen or done comes close to having the transformative power of math. My mind gets blown pretty much every day.
Another thing to notice is that the collection of odd numbers is infinite. This always makes for awe and fascination. If in fact our pattern doesn’t continue, how will we ever know? Checking the first million cases doesn’t prove anything—it might conceivably fail for the very next number. And in fact there are thousands of simple questions about whole numbers that remain unsolved to this day—we simply don’t know if the pattern continues or not.
So I wonder how you feel about this question of ours. Perhaps it’s simply not your cup of tea. Still, I hope you can appreciate why I like it. Mostly I love the abstraction of it all, the sheer simplicity. This isn’t some complicated congressional redistricting issue, or even a question about colliding electrons. It’s about odd numbers, for god’s sake. It’s the ethereal purity, the “more universal than the universe” quality that is so attractive to me. These aren’t hairy, smelly hamsters with bloodstreams and intestines; they’re happy, free, lighter-than-air constructs of my imagination. And they are absolutely terrifying.
Do you get what I mean here? So simple they’re scary? These aren’t science-fiction aliens, these are aliens. And they’re up to something, apparently. They seem to always add up to squares. But why? At this point what we have is a conjecture about odd numbers. We have discovered a pattern, and we think it continues. We could even verify that it works for the first trillion cases if we wanted. We could then say that it’s true for all practical purposes, and be done with it. But that’s not what mathematics is about. Math is not about a collection of “truths” (however useful or interesting they may be). Math is about reason and understanding. We want to know why. And not for any practical purpose.
Here’s where the art has to happen. Observation and discovery are one thing, but explanation is quite another. What we need is a proof, a narrative of some kind that helps us to understand why this pattern is occurring. And the standards for proof in mathematics are pretty damn high. A mathematical proof should be an absolutely clear logical deduction, which, as I said before, needs not only to satisfy, but to satisfy beautifully. That is the goal of the mathematician: to explain in the simplest, most elegant and logically satisfying way possible. To make the mystery melt away and to reveal a simple, crystalline truth.
Now if you were my apprentice and we had more time together, I would send you off at this point to think and struggle and see what kind of explanation you could cobble together. (And of course if you want to stop reading right now and get to work on it, that would be fantastic.) Since my goal here is to give you a taste of mathematical beauty, I will instead simply show you a nice proof and see what you think of it.
So how does one go about proving something like this? It’s not like being a lawyer, where the goal is to persuade other people; nor is it like a scientist testing a theory. This is a unique art form within the world of rational science. We are trying to craft a “poem of reason” that explains fully and clearly and satisfies the pickiest demands of logic, while at the same time giving us goosebumps.
Sometimes I like to imagine a Two-Headed Monster of mathematical criticism. The first head demands a logically airtight explanation, one with absolutely no gaps in the reasoning or any fuzzy “hand-waving.” This head is a stickler, and is utterly merciless. We all hate its constant nagging, but in our hearts we know it is right. The second head wants to see simple beauty and elegance, to be charmed and delighted, to attain not just verification but a deeper level of understanding. Usually this is the more difficult head to satisfy. Anyone can be logical (and in fact, the validity of a deduction can even be checked mechanically) but to produce a real proof requires inspiration and epiphany of the highest order. Similarly, it’s not that hard to draw an accurate portrait. One can develop an eye and master the technique. But to draw a portrait that means something, that conveys emotion and speaks to us—that’s something else entirely. In short, our goal is to appease the Monster.
Not that it’s so easy to get any proof off the ground. Most of us are so frustrated with our problems that we would gladly settle for the ugliest and clunkiest of arguments (assuming it is logically valid). At least we would then be sure that our conjecture is right and there won’t be any counterexamples. But it is an unsatisfactory state of affairs, and it cannot last. As Hardy says, “there is no permanent place in the world for ugly mathematics.” History shows that eventually (maybe centuries later) someone will surely uncover the real proof, the one that conveys not just a message, but a revelation.
But how do we do it? Nobody really knows. You just try and fail and get frustrated and hope for inspiration. For me it’s an adventure, a journey. I usually know more or less where I want to go, I just don’t know how to get there. The only thing I do know is that I’m not going to get there without a lot of pain and frustration and crumpled-up paper.
So let’s imagine that you’ve been playing with this problem for a while, and then at some point you have this realization: what the pattern is saying is that any square design can be broken into pieces which are just the odd numbers. So you try out some chopping ideas. Your first few attempts are successful, but have no real unity to them; they are random-seeming and do not generalize:
Then, all of a sudden, in one breathless heart-stopping moment, the clouds part and you can finally see:
A square is a collection of nested L-shapes, and these L-shapes contain precisely the odd numbers. Eureka! Do you see why mathematicians jump out of bathtubs and run naked through the streets? Do you see why this useless, childish activity is so compelling?
The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was “out there” all along; I just couldn’t see it. And now I can! This is really what keeps me in the math game—the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.
To me, this kind of mathematical experience goes to the heart of what it means to be human. And I’ll go even further and say that mathematics, this art of abstract pattern-making—even more than story-telling, painting, or music—is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.
Before we get too carried away, is it clear that these L-shapes do in fact follow the pattern? Is it so obvious that each successive L-shape contains exactly the next odd number , and that this pattern will continue forever? (This is the kind of skepticism typical of Head #1.) We know what we think these L-shapes are doing, and what we want them to do, but who says they will follow our desires?
This is something that happens in mathematics all the time. If proofs are stories, then they have parts, or episodes, like scenes in a novel. What our explanatory arguments do is break the problem down into subproblems. This is a big part of mathematical criticism. It’s not that our proof is wrong or bad, we’re just examining it more carefully, putting sections of it under the rational microscope.
So why do L-shapes make odd numbers? Of course the corner will always contain just one rock, and the next piece will have three, no matter how big the square is. Actually, I suppose we could entertain the possibility that our “square” consists of only one rock. It is up to you to decide if you want to include this sort of “trivial” case. The typical thing to do would be to include it, since it doesn’t break the pattern: the sum of the first odd number, namely 1, is in fact the first square, 1 × 1. (If your taste goes further, and you want to include zero—being the sum of the first none odd numbers, and also 0 × 0—then you might want to seriously consider becoming a professional mathematician.) In any case, the first few L-shapes clearly comply with our wishes.
But is it cl
ear that the pattern will keep going beyond our ability to draw pictures or to count? Let’s imagine a hypothetical L-shape way down the line:
It is important to understand that I am not committing myself to any particular size here, but keeping my mind open and arguing generally—this is any size L-shape; the nth one if you will; the generic one. Hopefully, we would then experience our next moment of clarity:
Any L-shape can be broken up into two “arms” and a “joint.” The two arms are equal, so they contain the same number, and the joint adds one more. That’s why the total is always odd! And what’s more, when we go from one L-shape to the next, we see that each arm gets larger by exactly one:
This means that each successive L-shape is exactly two more than the previous. And that’s why the pattern keeps going!
So there’s an example of what it’s like to do mathematics. Playing with patterns, noticing things, making conjectures, searching for examples and counterexamples, being inspired to invent and explore, crafting arguments and analyzing them, and raising new questions. That’s what it’s all about. I’m not saying it’s vitally important; it isn’t. I’m not saying it will cure cancer; it won’t. I’m saying it’s fun and it makes me feel good. Plus, it’s perfectly harmless. And how many human activities can you say that about?
A Mathematician's Lament Page 6